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Further Reading

Many books on Math ematical Biology include a sect ion on single-species population dynamics, sometimes with and somet imes with out age structure. Moremathematical texts are L. Edelstein-Keshet (Mathematical mo dels in biology,

Rand om House, 1988), F. C. Hoppensteadt (Mathematical methods of popu

lation biology, Cambridge, 1982), R. M. May (Theoretical ecology, Princeton,1981), J . D. Murray (Mathematical biology, Springer, 1989, new edit ion, 2002),and M. Kot (Elements of mathematical ecology, Cambridge, 2001), and morebiological ones are J. Maynard Smith (Models in ecology, Cambridge, 1974),N. J. Gotelli (A primer of ecology, Sinauer, 1995) and M. Begon, M. Mortimerand D. J. Thompson (Population ecology, Blackwell, 1996). H. Caswell (Matrix

population models, Sinauer , 2001) discusses matrix models in detail. Evolutionary quest ions in age-st ructured populations are dealt with by B. Charlesworth(Evolution in age-structured populations , Cambridge, 1980) and S. C. Stearns( The evolution of life histories, Oxford , 1992). N. Keyfitz (Applied mathemati

cal demography, second edit ion, Springer 1985) contains applicat ions to humandemogra phy. C. W. Clark (Ma thematica l B ioeconom ics, Wiley, 1990) is thestandard text on natural resource management , and includ es economic questions th at we have not dealt with here. In this context costs as well as benefits(such as yield) may be included in the utility to be maximised. Metapopulation ecology is covered extensively in 1. Hanski (Metapopu lation ecology, Oxford ,1999) . The assumpt ion that the sites are identical and identically isolated maybe relaxed , by using the incidence function method explained there .

Readable biologically based introductions to the material in Chapter 2 areby M. Begon, J . L. Har per and C. R. Townsend (Eco logy, Blackwell, 1996)and , for the host-p ar asitoid material in par ticular , M. P. Hassell ( The spatial

and temporal dynamics of host-paras itoid interactions, Oxford , 2000). MaynardSmith (1974) and J . M. Emlen (Population biology, Macmillan, 1984) are good

253

254 Essential Mathematical Biology

more mathematical treatments. Edelstein-Keshet (1988), Murray (1989, 2002)and Kot (2001) are more mathematical still , but do not lose sight of the biology.A recent book featuring case studies, which we do not have space for here,is W. S. C. Gurney and R. M. Nisbet (Ecological dynamics, Oxford, 1998).The ecosystems section is based on this book. The original works of A. J.Lotka (Elements of physical biology, Williams and Wilkins , 1925, reprintedas Elements of mathematical biology, Dover, 1956) and V. Volterra (variousdates , depending on whether you read French, Italian, or would like an Englishtranslation; try looking in F. Oliveira-Pinto and B. W. Conolly, Applicable

mathematics of non-physical phenomena, Ellis Horwood, 1982) are worth aread . R. M. May (Stability and complexity in model ecosystems, Princeton,1976) looks at the relationship between stability and complexity. Hanski (1999)is worth reading for more on metapopulation models.

An excellent book which includes both the mathematics and the biologicalbackground on a variety of diseases is R. M. Anderson and R. M. May (Infec

tious diseases of humans, Oxford , 1991). More mathematical books are D. J .Daley and J. Gani (Epidemic modelling, Cambridge, 1999), which includes thevery important stochastic approach, F. Brauer and C. Castillo-Chavez (Mathe

matical models in population biology and epidemiology, Springer, 2001), and O.Diekmann and J . A. P. Heesterbeek (Mathematical epidemiology of infectious

diseases , Wiley, 2000), which includes extensions of the basic theory in variousdirections. The early papers of W. O. Kermack and A. G. McKendrick (seeOliveira-Pinto and Conolly, 1982) are worth reading.

Charles Darwin (The Origin of Species, John Murray, 1859, or many latereditions) is required reading for anyone with an interest in evolution. D. J .Futuyma (Evolutionary biology, Sinauer, 1986) is a good modern text . Goodreferences for mathematical population genetics are J . Roughgarden (Theoryof population genetics and evolutionary ecology, Prentice Hall, 1996) and A.W. F. Edwards (Foundations of mathematical genetics, Cambridge, 2000), theformer being more biologically and the latter more mathematically oriented. J .Maynard Smith (Evolutionary genetics, Oxford, 1998) is also good . For background genetics, D. L. Hartl and E. W. Jones (Essential genetics, Jones andBartlett, 1999) is very readable. Non-technical references for game theory areK. Sigmund (Games of life, Oxford, 1993) and , specifically for the evolutionof cooperation, R. Axelrod (The evolution of cooperation, Basic Books, 1984).A more mathematical treatment is given by J . Maynard Smith (Evolution and

the theory of games, Cambridge, 1982). For full rigour, try J . Hofbauer and K.Sigmund (Evolutionary games and population dynamics, Oxford, 1998), whoinclude proofs of results only stated here.

Much of the material in this chapter is basic to the study of almost anybiological phenomenon where spatial variation is important, and so is treated

Further Read ing 255

in most books on math ematical biology that go beyond purely kinetic phenomena . Two excellent examples are Edelst ein-Keshet (1988) and Murray (1989,2002) . Texts concent rating on molecular and cellular applications are L. A.Segel (Mathematical models in mo lecular and cellular biology, Cambridge, 1980)and S. 1. Rubinow (Introduction to mathematical biology, Wiley, 1975), whilebiological invasions are covered by N. Shigesada and K. Kawasaki (B iological in

vasions, Oxford, 1997). A. Okubo (Diffusion and ecological problems, Springer ,1980, new edit ion with S. A. Levin 2001) gives a very readable account of applications in ecology emphas ising t he math ematical modelling aspec ts . An interest ing book giving the background on random walks, emphasising the physics ofthe pro cess bu t writ ten from a biologist 's point of view, is H. C. Berg (Ra ndom

walks in biology, Princeton , 1993).Many texts tr eat biochemical kinetics from a mathematical perspect ive,

including Segel (1980), Edelstein-Keshet (1988) , Murray (1989, 2002), and J .P. Keener and J . Sneyd (Mathem atical physiology, Springer , 1998) . The lastthree of these also include neur al modelling, and all four include examples ofexcitable and oscillatory behaviour , which A. Goldb eter (Biochemical osci lla

tions and cellu lar rhythms, Cambridge , 1996) trea ts exclusively. The place togo for exte nsions of these ideas to systems physiology as well as fur th er applications to cellular physiology is Keener and Sneyd (1998) . The treatmentof immunology as a problem in popul ation dynamics is elucidated in M. A.Nowak and R. M. May (Virus dynamics, Oxford, 2000) . Oth er mathematicalapproac hes to immunology are dealt wit h in A. S. Perelson 's chapter in Segel(1980) . We have been precluded by space considerations from including manyfascinating areas of molecular and cellular biology, including all aspects of genetic inform ation utilisation, cont rol and replication , mod els of pro tein foldin g,analysis of pat tern and sequence, cont inuum mechanics, combinatorics, neuralnetworks, cont rol theory and evolution, systems physiology, etc . Some aspectsof dynami cs have also been ignored , such as burstin g, chaos, and dynamic diseases.

A. M. Turing's original paper (T he chemical basis of morphogenesis, Ph ilo

sophical transact ions of the R oyal Soci ety of Lond on B , 23 7 , 37-72 , 1952) isbeautifully written and well worth read ing. A classic from the early part of th elast century is D'Ar cy Thompson 's On Growth an d Form (Cambridge, 1917 andseveral later editions) . H. Meinh ardt (Models of biological pattern form ation,

Academic, 1982) did early work on reaction-diffusion models, and has also published a book on sea-shell pat terns (H. Meinhardt , Th e algorithmic beauty of

sea shells, Springer , 1998). The books by Edelstein-K eshet (1988) and Murray(1989, 2002) have exte nded sect ions on pat tern form ation.

Biological backgro und on cancer may be found in L. Wolper t ( The triumph

of the embryo, Oxford , 1991), who gives a short non-technica l introduction , or


B. Alberts et al (Molecular biology of the cell, Garland, fourth edition 2002),who give more detail. Mathematical models of tumours are given in J . A.Adam and N. Bellomo (A survey of models for tumor-immune system dynamics,Birkhauser, 1997), who include interactions with the immune system, and T . E.Wheldon (Mathematical models in cancer research, Hilger, 1988), who includesmodels of treatment regimes.

ASome Techniques for Difference Equations

A.I First-order Equations

We shall consider the first -order difference Equation (1.2.3),

(A.1.1)

also called a recurrence equation or map, to be solved with t he init ial conditionNo given. T his defines a sequence No, N l , N2 , .. . , called a solution of t he equation with the init ial condition. It is stable if another solution N~, N{, N~ , . . .remains close to the first solution whenever it starts close, IN t - NIl is small forall t whenever INo- N~ I is small, and asymptotically stable if also IN t - NI l -+ 0as t -+ 00. It is neutra lly stable if it is stable but not asym ptotically stable. Itis a steady state (or fixed point or equilibrium) solut ion N* if N, = N* for allt ; it is clear from Equation (A.1.1) that the condition for N* to be a steadystate is that N* = j (N * ). It is periodic of period p if Nt+p = N, for all i , butNt+q i N, for any t and any q < p, and aperiodic if it is not periodic.

A .l.l Graphical Analysis

We wish to answer qualitative quest ions abo ut the solutio n of Equation (A.1.1)with the initial cond ition No given. For example, does the solutio n tend to asteady state, does it tend to a periodic solution or is it more complex than th at?If the equation represents pop ulation growt h then clearly f( O) = 0 and there

257


is a steady state at zero; we can investigate the existence of others graphicallyby sketching Nt+l = N, and Nt+l = f(Nt) together in th e (Nt , Nt+r)-plane;any intersection of these graphs is a steady state.

We can then investigate the behaviour by the method of cobwebbing. Theidea is as follows.

- Choose a starting value No, and begin at the point (No, No) in the (Nt , Nt+r)plane.

- Draw a vertical line to th e curve Nt+l = f(Nd ; this reaches the curve at thepoint (No, f(No)) = (No, N1 ) .

- Draw a horizontal line to the diagonal NtH = Nt; this reaches the diagonalat the point (N1 , Nr) .

- Repeat the process to arrive at (N2 , N2 ) , and then indefinitely until th ebehaviour of th e equation with this starting value becomes clear.

- If necessary, do the same with other starting values.

Some examples of the process are shown in Figure A.1.

A.1.2 Linearisation

It is plausible but not quite obvious that the dividing line between th e oscillatorily stable and oscillatorily unstable behaviour shown in Figure A.l isf' (N*) = -1. Let us check this by defining n = N - N* ; then subtractingN* = f(N*) from Equation (A.1.1) gives

nt+l = f(N* + nd - f(N*) = J'(N*)nt + h.o.t.,

where h.o.t, stands for higher order terms. Let us assume that for ti; sufficientlysmall the higher order terms are negligible . Then we may infer that the solutionof Equation (A.1.1) behaves similarly to that of the approximating equation

(A.1.2)

This is known as the linearised equation. The solution is nt = nof'(N*)t , andso the trivial steady state is oscillatorily unstable, oscillatorily asymptoticallystable, monotonically asymptotically stable or monotonically unstable according to whether>. = f'(N*) satisfies>' < -1 , -1 < >. < 0, 0 < >. < 1 or 1 < >.respectively. The condition for asymptotic stability is

1>'1 = IJ'(N*)I < 1,

A. Some Techniques for Difference Equ at io ns 259

(al Cobwebbing for Ricker map , R=1.8 (b) Cobwebbing for Ricker map , R=2 .21.5 2r---~----------,

1.5

1.5

~ 1z

0.5 1.5 2

(c) Cobwebbing for Ricker map , R=2.6 (d) Cobwebbing for Ricker map , R=2 .8

// '\ /

b/IA

1/ <,1/ <, -

2.5r-----------~

2.521.5N

n

0.5

0.5

1.5'+e

Z

2

20.5oo

2

1.5

0.5

'+cz 1

Figure A .I Some examples of cobweb maps, as a par ameter of the equation increases . Th e equation is essentially irrelevant , as long as it has a hump . In(a) , the solut ion tends oscillatorily to a steady state . In (b) and (c) , it tends to aperiod-2 and a period-4 solut ion respectively. In (d) , there is chaotic behaviour .This illustrates a typ ical period-doubling cascade to chaos .

and if 1>'1 = 1 the steady state is stable but not asymptotically stable. Th euse of th e not ation >. reflects the fact th at the place of l' (N*) will be takenby the eigenvalues of a matrix for systems of equations, as we shall see, andwe shall often refer to f' (N*) itself as an eigenvalue. The neglect of th e higherorder terms can be shown to be justified sufficient ly close to the st eady stateas long as we are not on the borderline between two kinds of behaviour, i.e. aslong as f' (N *) i: 0 (when the nonlinear terms determine monotonicity) or ±l(when they determine stability) . We summarise the behaviour of the linearisedequation in the diagram below.


oscillatorilyunstable

-1

oscillatorilystable

°

monotoni callystable

1

monotonicallyunstable

Figure A.2 Behaviour of solutions of Equation (A.1.2).

A.2 Bifurcations and Chaos for First-orderEquations

Now let us assume that there is a parameter J.l in the equations, and considerequations of the form

Xt+l = f(xt , J.l) .

We shall not necessarily interpret this as an equation of population dynamics, and so will allow Xt to take positive or negative values; we have changednotation from N to x to emphasise this . Often, in applications, x will be aperturbation from a steady state.

It is clear that the behaviour of solutions of such an equation can varywith the parameter u. For example, if the eigenvalue f; increases through 1as J.l increases past some value J.le, we expect the steady state to change characte r from monotoni cally stable to monotonically unstable, according to thelinearised analysis of the last section . A diagram of the solution behaviour(showing the steady states and periodic orbits, their stability, et c) against theparameter J.l is known as a bifurcation diagram , and the points where the solution behaviour changes as bifurcation points. All the bifurcations that canoccur for first-order equations are described below.

A.2.1 Saddle-node Bifurcations

A typical example of the saddle-node bifurcation is

Xt+l = f( xt,J.l) =Xt + J.l- x~.

The equation has no steady states for J.l < 0, and two (x* = ±V/i) for J.l > 0.The positive square root is stable and the negative square root unstable. Thebifurcation point is at (x, J.l) = (xc, J.le) = (0,0) , where f(O ,O) = 0, and theeigenvalue at the bifurcation point is f x (0,0) = 1. More generally, a saddlenode bifurcation is said to occur when, near the bifurcation point, the equationXt+l = f( xt , J.l) possesses a unique curve of fixed points in the (x , J.l)-plane,

A. So me Techniques for Difference Equations 261

which passes through the bifurcation point (xc, f.1e), and lies on one side ofthe line f.1 = f.1e. Conditions for such a bifurcation to occur at (xc, f.1e) aref( xe, f.1c) = 0, f x(xe, f.1e) = 1, f l"(xe, f.1e) :j:. 0, fxx (xe, f.1e) :j:. o. Such bifurcationsoccur , for example, in models for insect pests where outbreaks may occur . Atypical bifurcation diagram for such a model is sketched below.

2

0.80.60.40.2

Two node-saddle bifurcations

O'---~-~--~-~---'

o

3r--~-~----~---,

0.5

2.5

Node-saddle bifurcation at (0,0)

0.5

x 0\.

....

-0.5

-1-0.2 0 0.2 0.4 0.6 0.8

11

Figure A.3 (a) The prototype saddle-node bifurcation. (b) Typicalsaddle-node bifurcations in models of insect pests. Here xi is a stable endemicsteady state, x2 an unstable intermediate steady state, and x3 a stable outbreak stead y state x3' The intermediat e and outbreak states appear through asaddle-node bifurcation at f.1 I . There is a hysteresis effect; the insect populationwill not reach the outbreak state x3 until the endemic and intermediate steadystat es disapp ear through a second saddle-node bifurcation point at f.12, but willthen remain at outbreak levels unless f.1 subsequent ly decreases past f.1I.

A.2 .2 Transcritical Bifurcations

The prototype equation for the transcrit ical bifurcat ion is

The equation has two steady states, the tr ivial one x* = 0 and the non-trivialone x* = u, x* = 0 being stable for f.1 < 0 and x* = f.1 for f.1 > O. The bifurcationpoint is at (x, f.1) = (xc, f.1e) = (0,0 ), and the eigenvalue at the bifurcationpoint is f x(O ,0) = 1. More generally, a tr anscritical bifurcation is said to occurwhen, near the bifurcation point , the equation Xt+! = f( xt , f.1 ) possesses twocurves of fixed points in the (x, f.1) -plane, each of which passes through thebifurcat ion point (xc, f.1e) and exists on both sides of the line f.1 = f.1e. An


exchange of stability between the two curves of steady states takes place at thebifurcation point. Without loss of generality (by redefining x if necessary) , wetake one of these curves to be the line x = O. (If we do not do this the followingconditions are more unwieldy.) Conditions for such a bifurcation to occur at(x c, Pc) are then f( xc, Pc) = 0, fx( xc, Pc) = 1, f/-'( xc, Pc) = 0, f x/-,(xc, Pc) =P 0,f xx(xc, Pc) =P O. Such bifurcations occur, for example, from the trivi al steadystate in models for population dynamics with intra-specifi c competit ion whenthe basic reproductive ratio Ro increases past 1.

Transcritical bifurcation at (0,0)

-0.5

-1 L.-_~__~ ~----I

-0.5 of1

0.5

Figure A.4 A transcritical bifurcation.

A.2 .3 Pitchfork Bifurcations

The prototype equation for the pitchfork bifurcation is

Xt+! = f( xtl p) = Xt + PXt - xr

The equation has one steady state for P < 0, the trivial one z" = 0, andthree steady states for P > 0, the trivial one z" = 0 and the non-trivial onesz" = ±..jji, the trivial one being stable for P < 0 and both non-trivial onesbeing stable for P > O. The bifurcation point is at (x, p) = (xc, Pc) = (0,0) ,and the eigenvalue at the bifurcation point is f x(O ,O) = 1. More generally,a pitchfork bifurcation is said to occur when, near the bifurcation point, theequation Xt+l = f( xtl p) possesses two curves of fixed points in the (x , p)-plane,each of which passes through the bifurcation point (xc, Pc), one of which existson one side and the other on both sides of the line P = Pc. Without loss ofgenerality (by redefining x if necessary), we take the curve that exists on bothsides of p = Pc to be the line x = 0, which again simplifies the conditions

A. Some Techniques for Difference Equations 263

for the bifurcation. The trivial st ead y state is stable on one side of J.1 = J.1e,and both non-trivial solutions are stable if x = 0 is unst abl e and unst abl e ifx = 0 is stable. Conditions for such a bifurcation to occur at (xc, J.1e) are thenf( xe, J.1e) = 0, f x(xe, J.1e) = 1, f ,,(xe, J.1e) = 0, f xx(xe, J.1e) = 0, f xlJ. (xe, J.1e) :j;0, f xxx(xe, J.1e ) :j; O. These bifurcations are most important in mathematicalbiology in their role in period-doubling bifurcations, which we shall describe

next.

Pitchfork bifurcation at (0.0)

0.80.60.40.2o

1---+ -----------

0.5

x 0

-0.5

-1-0.2

Figure A .5 A pitchfork bifurcation .

A.2.4 Period-doubling or Flip Bifurcations

The pr otot ype equation for the period-doubling or flip bifurcation is

Xt+l = f (xt, J.1 ) = - Xt - J.1Xt + x~.

One steady state is the trivial one x* = 0, which exists for all J.1 . There isalso a curve of steady states given by x2 = J.1 + 2, which ar ises through apitchfork bifurcati on from the t rivial steady state at (x, J.1 ) = (0, -2); we arenot interest ed in thi s bifurcation . The bifurcation point of int erest is at (x, J.1 ) =(x e, J.1e ) = (0, 0) , with eigenvalue f x(O,O) = - 1. Here the t rivial steady stateloses its st abili ty, moving from oscillatorily stable to oscillator ily unst abl e, bu tthere is no stable steady state for J.1 > J.1e' What happ ens to solutions of theequation here? The oscillatory nature of t he instability gives us a clue, andwe can answer this question by considering the secon d iterate j2 = f 0 f of I,which is equivalent to considering every other te rm in the sequence Xt. It is easyto see tha t as J.1 increases past 0, and the derivative f x on the t rivi al br anchdecreases past - 1, the derivative (j2) x on t hat bran ch increases past 1, and


(with a bit more work) that P undergoes a pitchfork bifurcation there. Twonew steady states xi and x; of P appear which are not steady states of f. Theonly possibility is that these correspond to period-2 solutions of f , oscillatingbetween xi and x; . A stable period-2 orbit bifurcates from the trivial steadystate at (0,0) . More generally, a period-doubling bifurcation is said to occurwhen, near the bifurcation point, the equation Xt+! = f( xt , f.l) possesses a singlecurve of fixed points in the (x, f.l) -plane , while the second iterate P undergoesa pitchfork bifurcation at (xc, f.le) . Without loss of generality (by redefiningx if necessary) , we take the curve of fixed points to be the line x = 0, whichagain leads to simpler bifurcation conditions . This curve is stable on one side off.l = f.le , and the period-2 solution is unstable if it occurs where z " = 0 is stable,stable if it occurs where z" = 0 is unstable. Conditions for such a bifurcationto occur at (xc, f.le) are then f(xe, f.le) = 0, f x(xe, f.le) = -1 , l;(xe, f.le) = 0,l;x(xe, f.le) = 0, l ;I1(Xe, f.le) =I 0, l ; xx(xe, f.le) =I O.

But more can be shown. Under quite general conditions, the stable steadystates xi and x; of P suffer exactly the same fate as the trivial steady stateof I, producing stable steady states of r, then r, and so on. The bifurcation of a st able stead y state to an unstable steady state and a stable orbit ofperiod two is then followed by a cascade of such period-doubling bifurcations,leading to orbits of period 4, 8 and so on. This cascade accumulates at somevalue f.loo of the bifurcation parameter. For values of f.l greater than f.loo muchmore complicated behaviour is possible, including chaos. There are many definitions of chaos, and a full discussion of them would lead us too far afield. Forour purposes, chaotic behaviour in Equation (A.I.1) is characterised by thefollowing.

- There are aperiodic solutions.

- The butterfly effect occurs. By this we mean that there is sensitive dependenceon the initial condition , so that a small error in specifying the initial conditioncan lead to large differences in the predictions of the model.

A chaotic solution is almost indistinguishable from random behaviour, despit ebeing derived from a deterministic equation.

The period-doubling route to chaos typically occurs in models for population growth with humped functions l , with bifurcation parameter the basicreproductive ratio , so that there is reason to believe that chaotic behaviour mayoccur in ecological systems. However with real data, where stochastic effectsinevitably playa part, it is even more difficult to tell whether unpredictablebehaviour arises from chaos or merely from stochasticity, and there is stillcontroversy over whether chaos is observed in ecological and other biologicalsystems.

A. Some Tech niques for Difference Eq uat io ns 265

(a) Period-doub ling bifurcation at (0,0) (b) Period-doubling cascade to chaos

0.5

- 0.5

x 0 -- -- - --- ----

:32.2 2.4 2.6 2.8bifurcation parameter

1.2

.,N

~ 0.8.2~ 0.6

&0.4

0.2

020.80.60.2 0.4

"o

- 1 '-- ~_ _J

- 0.2

Figure A .6 (a) A single period-doubling bifurcation. The trivial steadystate loses its stability for J.L > 0, and the dotted lines represent stable period2 solutions taking alternately th e upper and lower values on the fork. (b) Acascade of period-doubling bifurcations to chaos.

Example A.I

A function f : [0,00) -+ [0, 00) modelling popul ation dynamics with intraspecific competit ion exhibits exact compensation (Section 1.2.2) if it is monotonic increasing , f (O ) = 0, and f (N ) -+ Nm a x , a constant, as N -+ 00.

If f exhibits exact compensat ion, solut ions of Equ ati on (A.1.!) tend to asteady state as t -+ 00.

The result is obvious by looking at cobweb maps, although a rigorous proofis slightly more demand ing.

Example A.2

A function f : [0, 00) -+ [0, 00) modelling population dynamics with intraspecific competit ion exhibits over-comp ensation (Section 1.2.2) if f(O) = 0,f(N) -+ °as N -+ 00. Such a function is uni-modal if there exists a Nt suchth at f is monotonic increasing for N < Nt and decreasing for N > Nt .

Typically, solutions of Equation (A.1.!) with f a uni-modal funct ion exhibit a cascade of period-doubling bifurcations to chaos if an increase in thebifurcation par ameter results in st eepening of the function .

Remark A.3

We can consider such a function to be a function f : [0, N] -+ [0, N] with

f (O ) = 0, f (N ) = 0, the usual definition of a uni-modal function, by choosingN > f (N t ) and redefining f on [f(Nt) , N]; this makes no difference to the

266

dynamics from time 1 onwards .

Essential Mathematical Biology

A.3 Systems of Linear Equations: JuryConditions

In Section 1.8, on Fibonacci 's model for rabbit population growth, we derivedthe second-order linear difference Equation (1.8.22), Yn+2 = Yn+! + Yn' Lineardifference equations of the mth order may be analysed by writing them as mequations of the first order. In this case, defining Xn =Yn+l, equation (1.8.22)may be written Xn+l = Xn + Yn , Yn+l = Xn, and the initial conditions (1.8.23)become Xo = 1, Yo = 1. Generalising, we may have to consider

(A.3.3)

where Zn is an m-vector and M an m x m-matrix, with Zo given. Let us lookfor a solution in the form Zn = ,\ne , where e is an m-vector. Substituting intoEquation (A.3.3) and cancelling ,\n,

or

,\e = Me

(M - '\I)e = O.

(A.3.4)

For any ,\ this has a solution e = 0, but if ,\ is an eigenvalue of the matrix witheigenvector e then this is a non-trivial solution. For this to happen M - ,\1must be singular,

det(M - '\I) = 0 (A.3.5)

This is a polynomial of the mth degree and has m roots . If they are all distinct (the usual case in Mathematical Biology) then the general solution ofEquation (A.3.3) is

where c, is the eigenvector corresponding to the eigenvalue '\i. The Ai arearbitrary constants that are determined by the initial conditions . Note that

- if each I'\il < 1 then IZnl -+ 0 as n -+ 00, and

- if there exists i such that I'\il > 1, and if Ai =P 0, then IZnl -+ 00 as n -+ 00 .

A. Some Techniques for Difference Equations

In the case m = 2 the eigenvalue Equation (A.3.5) is given by

>.,2 + al'\ + a2 = 0,

267

where al = - tr 1\1, a2 = det M . The necessar y and sufficient condit ions forasy mptotic stability, I'\d < 1 for i = 1,2 , are the Jury conditions

(A.3.6)

If 1+ al + a2 = 0 (t r M = 1+det M) there is an eigenvalue A = 1, if al = 1+ a2(1 + tr M + det M = 0) there is an eigenvalue ,\ = -1 , and if lall < 1+a2 anda2 = 1 (Itr MI < 1+ det M and det M = 1) there is a pair of complex conjugateeigenvalues on the uni t circle.

Jury conditions may be derived for m > 2, bu t they get rapidly morecomplicated. For m = 3, with eigenvalue equation ,\ 3 + al,\2 + a2'\ + a3 = 0,they ar e

(A.3.7)

A A Systems of Nonlinear Difference Equat ions

Throughout this section we sha ll consider second-order systems of the form

(A.4.8)

alt hough the resul ts may be extended to systems of higher order. Graphicalana lysis is much more difficul t tha n for the single equation, alt hough it isoften helpful to plo t solutions in (N, P )-space, bu t lineari sation and bifurcationanalyses ar e still ava ilable, and we look at these methods in this section.

Some new kinds of behaviour occur here, and we need some definitions. Aninvariant curve is a curve r in (N , P)-space such that if (No, Po) E r , then(Nt , Pd E T for all t > O. Such a curve is st able (or orbitally stable) if a solutionremains close to it whenever it st arts close to it , and asympto tically (orbitally)

stable if the distan ce between such a solution and the curve tends to zero ast ~ 00 . A solution Nt which starts and therefore remains on a closed invari an tcurve r may either return to its start ing point after a finit e number of st eps,or not . We say it has rational or irr ational rota tion nu m ber, respectively.

268

AA.l Linearisation of Systems


Let us assume that there exists a steady state (N* , P*) of this system ; it satisfies

N* = j(N* ,Po), P* =g(N* ,P*) .

Perturbations from this steady state may be defined by (n ,p) = (N,P) (N*, Po). Linearising about the steady state, in the same way as was done forthe first-order equation, we obtain the approximate equations (the linearisedequations)

or

Pt+l

:~(N* ,P*)nt + :~(N* ,P*)Pt ,

:~(N* ,P*)nt + :~(N*,P*)Pt ,

(A.4 .9)

(A.4 .10)

(A.4 .11)

where n is the column vector (n,pf , J is the Jacobian of the transformation,viz

J(N P) - ( j N(N,P) jp(N, P) ), - gN(N, P) gp(N,P) ,

and a star denotes evaluation at the steady state . Comparing Equation (A.4 .11)with Equation (A.3.3) and using the Jury conditions (A.3.6) given in the lastsection , we infer asymptotic stability of the steady state if

ItrJ*1 < detJ* + 1, detJ* < 1.

AA.2 Bifurcation for Systems

Naimark-Sacker bifurcation at (0,0)

. .. . . . .... . . . .

0.5

(A.4.12)

-0.5

o 0.2 0.4 0.6 0.8~

~ Or----------------

-1 '----~--~--~--~---'-0.2

Figure A.7 A Naimark-Sackerbifurcation. The trivial steady stateloses its stability at the origin , Thedotted line represents the invariantclosed curve in (n ,p) -space.

A. Some Techniques for Differen ce Equations 269

T he bifurcations of first-order difference equations described in the lastsect ion also occur for systems as an eigenvalue of J* passes through ±1 , bu tthere is also a new possibility. Thi s is t he Naimark- Sacker bifurcation , oftenreferred to as a Hopf bifurcat ion (see Appendix B) for difference equations.At such a bifurcation point , the Jacobian matrix has two complex conjugateeigenvalues of modulus 1. This may occur for systems of any order greaterthan 1, and the results are similar, but we shall consider second-order systemsfor simplicity. Then, in terms of the Jacobian matrix, the condit ions at thebifurcation point are It r J*I < det J* + 1, det J* = 1. Consider the case where,as a bifurcation par ameter /-l increases past a bifurcation value /-lc, the twocomplex conjugate eigenvalues cross out of the unit disc, so that det J* increasespast 1 and the steady state therefore loses its stability for /-l > /-lc. Assume alsothat, at the bifurcation point , ).k i 1 for k = 1,2 ,3 , 4. Then t here are twopossibiliti es.

- In the sub crit ical case, there exists an unstable closed invariant curve inn-space for /-l < /-lc·

- In the supercrit ical case, there exists a stable closed invar iant curve in n-space for /-l > /-lc·

Solutions on these invariant curves may have a ration al or irr ational rotationnumb er. Th erefore, if we have a syste m such as (AA.8) and we know its solut ions are bounded, and if we also know that the steady state becomes unst ablet hrough two complex conjugate eigenvalues crossing out of the unit disc, weexpect to see solut ions of the syste m on a closed invariant curve, with rationalor irrational rotation numb er, oscillating about the stea dy state in the phaseplane. Thi s is typ ical of realist ic host-parasitoid models such as those of Chapte r 2.

BSome Techniques for Ordinary Differential

Equations

B.l First-order Ordinary Differential Equations

Consider the first-order ordinary differential equation,

N = f( N ),

to be solved with initial condit ion

N( O) = No.

(B .Ll)

(B.1.2)

Definitions of steady states and their stability are similar to the differenceequation case, Appendix A, and are omit ted. The condit ion for N * to be asteady st at e is now f (N*) = o.

B.l.l Geometric Analysis

Now note that if N currently takes a value where f( N) is positive, it willsubsequently increase (since N = f (N) > 0), and if f( N) is negative it willdecrease. A sketch of the graph of f tells us all we need to know about the steadystates (points N * where f( N *) = 0) and their st ability (st able if f'(N*) < 0,unstable if f'( N *) > 0) , and the asymptotic (long-t erm) behaviour of th esolut ion. It is clear th at as t -+ 00 , N -+ ± oo or to a steady state.

271


f(N)

'7~~' N

Figure B.l A typical function f(N) with an unstable steady state at 0and a stable steady state at K .

B.1.2 Integration

Equ ation (B.1.l) with initial condition (B.1.2) may be integra ted by separatingthe variables , to obtain the implicit solution

t rNdN

t = Jo dt = JNo f(N)"

This confirms that as t -+ 00 , N -+ ±oo or to a steady state (a zero of 1) , butalso tells us how fast it does so.

B.1.3 Linearisation

Let us assume that there is a steady stat e solution N *. Defining n = N - N *and using the fact that f(N *) = 0, we obtain

it = f(N* + n) = f'(N*)n + h.o.t ., (B.1.3)

where h.o.t. is an abbreviation for higher order terms. Let us assume that for nsufficiently small the higher order terms are negligible. Then we may infer th atth e solution of Equation (B.1.3) behaves similarly to that of th e approximatinglinearised equation

it = f'(N *)n, (B.1.4)

which has solution n(t) no exp(f'(N*)t) . The neglect of th e higher orderterms can be shown to be justified sufficiently close to the steady state aslong as l' (N*) :I O. It follows that the steady state is exponentially stable if1'(N*) < 0 and exponentially unstable if 1'(N*) > 0, while if 1'(N*) = 0 thenonlinear terms determine stability.

B. Some Techn iques for Ord inary Differential Equations

B .2 Second-order Ordinary DifferentialEquations

A second-order system of ordinary differenti al equations is given by

(; = f(U,V) , V = g(U,V) ,

to be solved with two initi al condit ions

U(O) = Uo, V(O) = Va .

273

(B.2.5)

(B.2.6)

Definitions of periodic solutions and their stability and orbital stability aresimilar to the difference equation case, and are omitted. A periodi c solutionwhich is the limit as t -+ ± oo of oth er solutions is known as a limit cycle.

B.2.! Geometric Analysis (Phase Plane)

A lot of informat ion about the solutions of such systems for general initi al conditions may be obt ained by sketching th e (U, V)-plane, known as the phase plane,together with th e solution trajectories. These solution trajectories represent solutions of the ordinary differenti al equations as curves in the (U,V)-plane, withtime as a par ameter . The procedur e is as follows.

- Find out where th e nullcline f = 0 is, where f < 0 and where f > O. Do thesame for g.

- There are steady states where f = 9 = 0, i.e. where the nullclines cross. Markthese.

- In the region(s) where f > 0 and 9 > 0, both U and V are increasing. Markthem with an arrow pointing right wards and upwards . Mark other regionswith an appropriate arrow.

- On th e nullcline f = 0, U is neither increasing or decreasing. Mark it withan upward-pointing arrow where 9 > 0, downward where 9 < O. Mark 9 = 0similarly.

- Sketch in th e solution trajectori es following the arrows.

In some cases it will be necessary or useful to do more th an this . For example,it might be necessary to analys e the behaviour near th e steady states (seeSection B.2.2 below). There might also be some traj ectori es th at move alongaxes or do something else special. These tend to be important, and should bemarked.

274

Example B.1


Sketch the phase plane for the system

(; = U(l - U), V = - V.

Phaseplane0.2 r------.-.------,-----,..--------"

o 0.5U

1.5

Figure B.2 Phase plane for(; = U(l - U) , V = -v. Note thetrajectories along the axes and theline U = 1.

B.2.2 Linearisation

Let (U*, V*) be a steady state of Equations (B.2.5) , so that !(U*, V*) =g(U*, V*) = O. Defining u = U - U* , v = V - V *, we may proceed as wedid for the first-order Equation (B.l.l) . Assuming that we may neglect higherorder terms if u and v are sufficiently small , we obtain the approximate (linearised) equations

u= !u(U*, V*)u + !v(U*, V*)v,

iJ = gu(U* ,V*)u + gv(U* ,V*)v,

or, defining the Jacobian matrix J(U,V) in the usual way,

w= J*w,

(B.2.7)

(B.2.8)

where w is the column vector (u,v)T , and a star denotes evaluation at thesteady state. The behaviour of the system near (U*, V*) depends on the eigenvalues of the matrix J* = J(U*, V*).

It can be shown that the neglect of higher order terms is valid, and thenonlinear system behaves like the linear system near the steady state, as longas neither of the eigenvalues of J* has zero real part.

Making the definitions (3 = tr J*, "( = det J* , J = disc J* , the eigenvalueequation is ,\2 - (3,\+ "( = 0, and we may determine the character of the steadystate from the signs of these.

B. Some Techn iques for Ordinary Differential Equations 275

Theorem B.2 (Steady States and Eigenvalues)

- If I < °the (trivial) steady st ate of th e second-order system (B.2.8) is asaddle poin t. Both eigenvalues are real , one positive and one negative.

- If I > 0, 8 > 0, (3 < 0, it is a stable node. Both eigenvalues are real andnegative.

- If I > 0, 8 > 0, (3 > 0, it is an unstable node. Both eigenvalues are real andposi tive .

- If I > 0, 8 < 0, (3 < 0, it is a st able focus. The eigenvalues are complexconjugates, with negative real part.

- If I > 0, 8 < 0, (3 > 0, it is an unstable focus . The eigenvalues are complexconjugates, with positive real part.

- If I > 0,8 < 0, (3 = 0, it is a cent re. The eigenvalues are complex conjugates,and purely imaginary.

The proof follows from th e formula for solutions of th e quadratic A2 - (3 A+1 = 0.

In all cases above except the last , th e same is true of the steady st ate ofthe nonlinear system (B.2 .5). In the last case , it may be a centre or a stable orunstable focus depending on th e nonlinear terms in the equat ion.

Theorem B.3 (Routh-Hurwitz Criteria for Second-order Systems)

Necessary and sufficient conditions for both roots of the quadratic

(B.2.9)

to have negative real parts are

(B.2.10)

If a 2 = °th ere is an eigenvalue A = °while if a l = °and a2 > °there is a pairof complex conjugate eigenvalues on th e real axis. It follows that necessary andsufficient conditions for asymptot ic stability of th e trivial steady state of th esecond-order lineari sed system (B.2 .8) are given by

(3 < 0, , >0, (B.2 .11)

where (3 = t r J* , , = det J * . (T his also follows from Theorem B.2.) If either ofthes e inequaliti es is strictly violated , th en it is unstable .

276

Roots of a quadratic


complex complexunstable stable

a2 two twopositive negative

one positive , 0 e negative

a,

Figure B.3 Properties of the rootsof ).2 + al). + a2 = O. The roots arestable in the positive quadrant. Fromthere, zero-eigenvalue bifurcations maytake place if the (positive) aI-axis iscrossed, and Hopf bifurcations if the (positive) a2-axis is crossed. See Section BA.

Figure B.3 summarises the information we have about the roots of thequadratic Equation (B.2.9).

Example 8.4

Sketch the phase plane for the system

(; = V, if = -U(l - U) + cV,

where c is a positive constant. The nullclines are V = 0 and V = U(l - U)jc.If V > 0, then (; > 0; if V < 0, then (; < O. If V > U(l - U)jc then if > 0;and vice versa . The steady states are at (0,0) and (1,0) . The Jacobian matrixis

J(U,V) = (-1~2U ~) .

At (0,0) we have "I = det J* = 1, j3 = tr J* = c > 0 and J = disc J* = c2 - 4.The character of the critical point depends on c; for c 2: 2 it is an unstable nodewhereas for 0 < c < 2 it is an unstable focus. At (1,0) we have "I = det J* = -1,so that the critical point is a saddle point. The phase plane is sketched below.

B.2.3 Poincare-Bendixson Theory

Theorem 8.5 (Poincare-Bendixson)

Let (U,V) satisfy Equations (B.2 .5) with initial conditions (B .2.6) . Let f and9 be Lipschitz continuous. Let (U, V) be bounded as t -t 00 .

Then either (U, V) is or tends to a critical point as t -t 00, or it is or tendsto a periodic solution. The same result holds as t -t -00 .

B. Some Techniques for Ord inary Differential Equations

Phase plane for c:1

277

0.5

> a

- 0.5

- 1- 0.5 a 0.5

u1.5

Figure BA Phase plane for thesystem (; = V , V = - U(1- U) + cV ,with c = 1. The point (0, 0) is anunstable focus and the point (1,0) asaddle point. The dashed line is thenullcline V = U(l - U)jc.

Th e result follows from the fact that solut ion trajectories cannot cross. Thenthe topology of the plane gives the result .

Theorem B.6 (Dulac Criterion)

Let fl be a simply connected region of the plane. Let the functions f and 9

be in C1 (fl) . Let B E C1(fl ) be such that the expression &ttJ l + &~~l is notidentically zero and does not change sign in fl .

Then there are no periodic orbits of Equation (B.2.5) in fl.

Theorem B.7 (Bendixson Criterion)

This is a particular case of the Dulac criterion with B(U,V ) = 1 for all (U, V ) Efl .

These results follow from Green's theorem for integrals in the plane. Notethat they are negative criteria . We can never deduce the existence of a periodicsolut ion from them.

B.3 Some Results and Techniques for mthOrder Systems

Consider the systemx= f (x) ,

to be solved with initial conditions

x (O) = XQ.

(B.3.12)

(B.3.13)

278

Here f is a column vector of functions Ii.

B.3.! Linearisation


Linearisation may be applied to systems of any order. A steady state is asymptotically stable if all eigenvalues of the linearisation have negative real part.This may sometimes be analysed using the Routh-Hurwitz criteria, which maybe derived for general m, but these get rather complicated as m increases. Thecondit ions for m = 3 are as follows. Let the eigenvalue equation be given by

A3 + a1A2 + a2A+ a3 = O.

Then all the roots of this cubic have negative real part if and only if all thefollowing inequalities hold :

(B.3.14)

Another test that is sometimes useful in determining stability and related questions for higher-order systems is Descartes' rule of signs. Let the polynomial pbe given by

(B.3.15)

Let k be the number of sign changes in the sequence of coefficients ao, a1 , . . . , an,ignoring any zeroes, and let m be the number of real positive roots of the polynomial (B.3 .15). Then m ~ k, and k and m have the same parity (even orodd) . Setting J.L = -A and applying the rule again, we may obtain informationon the number of real negative roots .

B .3.2 Lyapunov Functions

Definitions: a function q, : IRm ~ IR is positive definite in Q C IRm about a pointx = x* if (a) q,(x*) = 0 , and (b) q,(x) > 0 for all x E Q\{x*} . A function IJ! isnegative definite if -IJ! is positive definite . For a system x = f(x) and a functionq, E C1 (IRm , IR) we may define a derivative, the derivative of the function alongtrajectories of the system, by

. m 8q,q,(x) == L ~(x)j;(x) .

i=1 x ,

A Lyapunov function q, : IRm ~ IR for the system x = f(x) is a continuouslydifferentiable positive definite function q, in Q whose derivative along trajectories of the system satisfies eP(x) ~ 0 in Q . If a Lyapunov function exists for

B. Some Techniques for Ordinary Different ial Equat ions 279

a system then x* is a stable steady state of the system. If also <P is negat ivedefinite in f? then x* is globally asymptotically stable in f?, that is all solutionsx of the system with initial conditions in f? satisfy x(t) -+ x* as t -+ 00 .

B .3.3 Some Miscellaneous Facts

- Let each component function f i in Equation (B.3.12) be Lipschitz continuous.Then syste m (B.3.12) with initial condit ions (B.3.13) has a solution in aneighbourhood of t = O. Moreover the solution is unique.

- Under the same conditions, the only way the solution can cease to exist isby blow-up in finite time.

- Hence a priori bounds can give existence for all time. Th ese are usuallyobt ained by finding positive ly invariant sets. (A set D c jRm is said to bepositively invariant for Equation (B.3.12) if whenever x (O) ED then x(t) ED

for all t > 0; it is negatively invariant if the same is true for all t < 0.)

BA Bifurcation Theory for OrdinaryDifferential Equations

The stability of a steady state solution of a system of ordinary differentialequations

x = f'(x .j»)

depends on the eigenvalues of the Jacobian mat rix there; it is asymptoti callystable if all the eigenvalues have negative real part , and unstable if at least oneof them has positive real part . As for difference equations, we use bifurcationtheory to study the qualitative changes in solution behaviour that may occuras the parameter f-l varies.

BA.l Bifurcations with Eigenvalue Zero

The description and analysis of bifurcation s with eigenvalue 0 is almost identicalto that of bifurcat ions with eigenvalue 1 for difference equations. It is easy tosee why this is so if we compare the difference equation Xt+l = Xt + f ( Xt , f.1-)with the differential equat ion

x = f( x , f.1-) . (B.4.16)

280 Essenti al Mathematical Biology

The steady states of each are the same , so the bifurcation diagrams for steadystate solutions are the same. Moreover the eigenvalue of the difference equationis greater by 1 than the eigenvalue of the differentia l equation. We can therefore tra nslate all the results of Sect ion A.2 on saddle-node, transcritical andpitchfork bifurcations directly to the differential equation case. The condit ionsfor each of these bifurcations of Equation (B.4.16) to occur at a bifurcationpoint (xc, J.le) are as follows. Again we have simplified the conditions by takingthe tra nscrit ical and pitchfork bifurcations to be from the trivia l solut ion.

- Saddl e-node bifurcation: f (xe, J.le) = 0, f x(xe, J.le) = 0, f p, (xe, J.le) =j; 0,

f xx(xe, /-le) =j; 0.

- Transcritical bifurcati on, with one branch of solut ions x = 0: f( xe, J.le) = 0,f x(xe, J.le) = 0, f p, (xe, J.le) = 0, f xp,(xe, J.le) =j; 0, f xx(xe, J.le) =j; 0.

- Pitchfork bifurcati on, with one branch of solutions x = 0: f( xe, /-le) =0, f x(xe, J.le) = 0, f p, (xe, J.le) = 0, f xx(xe, /-le) = 0, f xp, (xe, /-le) =j; 0,

f xxx(xe, J.le) =j; 0.

These three bifurcations of steady states are the only ones that are possiblefor first order ordinary differential equations; th ere is no counterpart of theperiod-doubling bifurcat ion in difference equations (which has eigenvalue -1).

BA.2 Hopf Bifurcations

In dimensions higher than 1, there is anot her way for a steady state to losestability, by a pair of complex conjugate eigenvalues crossing the imaginary axisinto the right half plane. The bifurcation associated with this loss of stabilityis usually called after Hopf (who analysed it for mth order syste ms in 1942),although others point to the fact that the bifurcation appeared 50 years earlierin Poin care's work , and was analysed (for second-order systems) by Andronovin 1929, and call it the Poincare-Andronov-Hopf bifurcation. The prototypefor th e Hopf bifurcati on is

(B.4.17)

where w is a constant. The bifurcation point is (x ,y , J.l) = (0, 0, 0) , and theJacobian matrix at (0, 0, J.l ) has eigenvalues /-l ± iw. Transforming to polar coordinates (R ,¢) by ta king R2 = x2 + y2 , ¢ = arctan(y / x), the equations become

(B.4.18)

T his has the trivial solut ion R =°for all values of u; and a periodic solutionR = Vii, ¢ = wt for /-l > 0. T he trivial solut ion loses stability as /-l increases

B. Some Techniques for Ordina ry Differential Equat ions 281

past 0, and the periodic solutio n is stable where it exists. More genera lly, letx* = 0 be a solut ion of a system of ordinary different ial equations for all u,and, for /l near /le, let the Jacobian matrix J* of the syste m have two complexconjugate eigenvalues >' (/l ) and ~(/l ) which are on the imaginary axis at /l = /le,

all other eigenvalues having negative real par t . For second-order systems thisoccurs if tr J* = 0 at /l = /le while det J* > O. Assume also th at th e complexconjugate eigenvalues cross the imaginary axis into the right half plane as /l

increases past /le, Re A' (/le) > O. Then there exists a periodic solution, uniqueup to phase shifts , for every /l in a one-sided neighbourhood of /le' There aretwo possibilities.

- In the subcrit ical case, an unstable periodic solut ion exists for /l < /le, whereth e trivial solution is stable.

- In the supercriti cal case, a stable periodic solut ion exists for /l > /le, wherethe trivial solution is unstable.

(a) Supercritical HopI bilurcation at (0,0) (b) Subcritical HopI bifurcation at (0,0)

0 .2o~

t----------r- - - - - - - -

1.5 ,-----~------~-___,

0.5

>: 0~

- 0.5

-1

-1 .5-0.20.80 .60 .2 0.4

~

o

f---f- - - --- ------

- 1L-_~_~__~_~_ ___'

-0.2

0.5

>: 0s.

-0.5

F igure B.5 (a) Super- and (b) subcrit ical Hopf bifurcations. The supercrit ical bifurcation is stable; the subcritical bifurcation is unstable near thebifurcation point, but frequent ly such bifurcations become stab le further alongthe branch through a saddl e-node bifurcation, as shown here. In th is case, weexpect to see the appearance of a large amplitude oscillation as /l increasesthrough zero.

The condit ion for sub- or supercrit icality is algebraically hairy. Make a tr ansformation so that the bifurcation point is at the origin (x,y , /l ) = (0, 0, 0), andso that the system wit h /l = 0 is given by

( ~ ) ( ~ f ((x , y, 0)) ) .9 x, y, O

282

Define a by


1a = 16 (Jxxx + f xyy + gxxy + gyyy)

1+ 16w (Jxy(Jxx + f yy) - gxy(gxx + gyy) - f xxgxx + f yygyy), (B A.19)

evaluated at the origin. Th e condition for sup ercriticality is a < o..Typical bifurcation diagrams are given in Figure B.5. Since periodic be

haviour is common in biology, th e Hopf bifurcation is a useful tool. It arises,for example, in models for the propagation of a train of nerve impul ses, III

oscillatory metabolic pro cesses, and in predator-prey syst ems.

cSome Techniques for Partial Differential

Equations

C.l First-order Partial Differential Equationsand Characteristics

Consider McKendrick's par tial differenti al equation

o P oP- + - = - j.lp.oa ot

(C.1.1)

This is to be solved in the posit ive quadr ant of the (a, t)-p lane, and condit ionsare given at a = 0 and t = O. Now think of travelling in this plane along one ofthe lines of the Lexis diagram (Figure 1.12), a straight line of slope 1, t = t(a) =a + c, for c constant . Wh at happ ens to P as we follow one of thes e lines? Onthe line, P may be given in terms of a only, P( a, t) = P(a, t(a)) = P (a , a + c).Taking th e total derivative with respect to a,

dP _ aP oP dt _ _ Pda - aa + at da - j.l ,

using Equation (C.l. I). The part ial differential equation for P redu ces to anordinary differential equation, which may be solved by separation of variables.A condition on t = 0 or a = 0 is required to complete the solut ion, dependingon which is hit first by the line we are following in the Lexis diagram , i.e,depending whether c > 0 or c < O.

283


Any curve along which a partial differential equation reduces to an ordinarydifferential equation is known as a characteristic curve (or just a characteristic)for the equation. More generally, consider a partial differential equation of theform

au aufax +9 ay = h.

The curve (x(s), y(s)) , given parametrically, is a characteristic curve for thispartial differential equation if the equation reduces to an ordinary differentialequation on it. But since

du au dx au dy----+--ds - ax ds ay ds '

we needdx = f dy duds 'ds = g, ds = h.

These are the equations of the characteristics.

C.2 Some Results and Techniques for theDiffusion Equation

C.2.l The Fundamental Solution

Consider the one-dimensional version of the diffusion Equation (5.2.4) with

f = 0,au = Da2Uat ax2

on the whole of JR, with initial condition

u(x,O) = 8(x).

(C.2.2)

(C.2.3)

Here 8 is the Dirac delta function which is zero everywhere except at x =°andwhose integral is unity. We claim that the solution of this initial-value problemis given by

1 (x2)u(x, t) = J41i15t exp --D '

41fDt 4 t

called the fundamental solution of the diffusion equation on JR.assertion, we need to check two things.

(C.2.4)

To verify this

C. Some Techniques for Partial DifFerential Equations 285

- The function sat isfies Equation (C .2.2). This is an exercise in partial differentiation . Some algebra can be avoided by using the fact that log u =-t log(41iDt) - x2/ 4Dt, so that

- The function satisfies the initial condit ion (C.2.3) . Since 6(x) is not a t ruefunction this needs to be interpreted as meaning that u(x , t ) -+ 6(x) ast -+ 0+ , i.e. that

- u(x ,O+) = 0 for x ::j:. 0,

- J~oo u(x, 0+ )dx = 1.

The first of these is clear because of the exponential in u ; for th e second, weneed t he standa rd result that, for (72 independ ent of x ,

I:exp ( - 2:2 ) dx = V21i(72.

T hen , for any t ,

100 1 100 ( x2) 1u(x ,t )dx = -~ exp --D dx = .j4iilli/21i(2Dt) = 1.

- 00 V 41iDt - 00 4 t 41iDt(C.2.5)

Taking the limit as t -+ 0+ , and assuming that we can interchange limitsand integrals, the result follows, and our proof is complete .

It is then easy to show that th e soluti on of

on the whole of 1R2 , with initial condit ions

u(x, y, O) = 6(x )6(y)

is given by

(C .2.6)

(C .2.7)

u(x , y , t ) = 41i~t exp ( - 4~t ) exp ( - 4~t ) , (C .2.8)

called the fund am ental solut ion of the diffusion equation in two dimensions.This may be genera lised in the obvio us way to n dimensions ,

1 ( IXI2)U x t = ex - -( ,) (41iDt)n/2 p 4Dt ' (C .2.9)


The fundamental solution in two dimensions may be written

1 (R2)u(R, t) = 4nDt exp - 4Dt '

and in three dimensions,

1 (r2

)u(r , t) = (47rDt)3 /2 exp - 4Dt .

(0.2 .10)

(0 .2.11)

(0.2.12)

The fundamental solutions may be used to solve more general initial-valueproblems.

Example C.1

Let u be the fundamental solution of the diffusion equation in one dimension.Show that the solution of the initial-value problem

au = Da2uat ax2

on JR,u(x,O) = f(x) ,

is given by

u(x, t) =i: u(x - y, t)f(y)dy,

or equivalently, by a change of variables,

u(x, t) = i: u(y, t)f(x - y)dy.

Idea of proof: we have

(Ut - Duxx)(x, t) =i:(Ut - Duxx)(y , t)f(x - y)dy = 0,

so that the given function u satisfies the diffusion equation, and

(0 .2.13)

(0.2 .14)

(0 .2.15)

(0 .2.16)

lim u(x, t) = lim Joo u(x - y, t)f(y)dy = Joo 6(x - y)f(y)dy = f(x),t--.o t--.o -00 -00

(0.2.17)so that it also satisfies the initial condition.

C. Some Techniques for Partial Differential Equations 287

Now consider equations where there is a linear source of u as well as diffu-sion,

au cP uat = au + D ax2

on JR,u( x,O) = f( x) .

Define a new function v by v (x , t) = u( x , t) e-at . Th en

av au -s cxt -r od :- = - e -aueat at '

(C.2.1S)

(C.2.19)

(C.2.20)

(C.2.21)

and

(C.2.22)

Moreover,v( x ,O) = u( x , 0) = f( x) , (C.2.23)

so we can solve for v.If f( x) = 5(x) , v is the fundamental solut ion of the diffusion equation on

JR, and

u(x , t) = eatv (x, t) = k exp (at - xD

2

) •47l'Dt 4 t

It sat isfies the integral condition

I:u(x, t)dx = eat .

C.2.2 Connection with Probabilities

(C.2.24)

(C.2.25)

(C.2.26)

On the website, we show that the right-hand side of Equ ation (C.2.4) is theprobability density function at time t of the position of a particle performinga diffusion random walk in one dimension starting at the origin .

The normal (Gaussian) distribution in one dimension with mean zero andvari ance 0'2 is given by

N(O, 0'2) ,....., _1_ exp (_ x2

) .V27l'O' 2 20'2

Hence the particle probability density function is normally distributed withmean zero, variance 2Dt . Its root mean square distance from th e origin is thestandard deviation v2Dt . We may interpret this as meaning th at th e dist ance

288 Essent ial Mathematical Biology

of a diffusing particle from its starting point after a time t is given on averageby v2Dt.

The right-hand side of Equation (C.2.9) is the probability density functionat time t of the position of a particle performing a diffusion random walk in n

dimensions starting at the origin . Each component of the position of the particleis normally distributed with mean zero and variance 2Dt. Its root mean squaredistance from the origin is V2nDt, by Pythagoras' theorem.

C.2.3 Other Coordinate Systems

In cylindrical polar coordinates R, ¢>, z,

(C.2.27)

(C.2 .28)

and2 1 a ( au) 1 02U o2u

\7 u = R oR RoR + R2 a¢>2 + OZ2 · (C.2.29)

For cylindrically symmetric flow, all ¢>- and z-derivatives are zero. The formulae for plane polar coordinates are the same without the z-components andderivatives.

In spherical polar coordinates r , f), ¢>,

\7u = (au ~ au _1_ au)or ' r of) , r sin f) a¢> ,

1 a 2 1 a . 1 oj¢!\7 . J = 2!:l(r Jr ) + -.-f) !If)(smf)Je) + - . -f) !lA-. '

r or r sm u r sin u,+,

and

\72u =~~ (r2au) + __1_~ (sinf)ou) + 1 o2U.r2or or r2sin f) of) of) r2sin2 f) a¢>2

For spherically symmetric flow, all f)- and ¢>-derivatives are zero.

(C.2.30)

(C.2.31)

(C.2.32)

C. Some Techniques for Partial Differential Equations

C.3 Some Spectral Theory for Laplace'sEquation

289

The spectrum of a differential operator is the set of it s eigenvalues. We givesome facts about the spectra for the Laplacian operator with various boundarycondit ions. Let fl E jRn be a well-behaved finite domain with boundary ofl. Anysufficiently smooth domain is well-behaved, as are such domains as rectanglesin jR2 . The equation

-\72 F = AF

in fl , with homogeneous Dirichlet boundary conditions F = 0 on ofl , hassolut ion F = 0 for any value of A, but for certain values of A it has nontrivial solut ions. These values of A are the eigenvalues of th e operator _\72

on fl with Dirichlet boundary condit ions, and the corresponding solutions arethe eigenfunctions. Th e operator has an infinite sequence Fn of eigenfunctionsformin g an orthogonal basis for the Hilbert space consist ing of square-integrablefunctions on fl , and the corresponding eigenvalues An are real and sat isfy

(C.3.33)

and An -+ 00 as n -+ 00. If Dirichlet condit ions are replaced by Neumann(zero-flux) conditions n \7 F = 0, also written ~~ = 0, where n is the outwardpointing normal on ofl , th e same conclusions hold , except now

(C.3.34)

(C.3.35)

Th ese results are useful because they allow us to writ e any function u of x satisfying the boundary conditions (and residing in the correct function space) asa linear combination of the appropriate eigenfunctions, u(x) = l:~=o anFn(x) .This is a (generalised) Fourier series. Moreover , since any function v of x and tis just a different function of x at each tim e t , we can write any such functionsatisfying the boundary condit ions (and residing in the correc t function space)as a linear combination of the appropriat e eigenfunctions, the combinationchanging with t ,

00

v(x , t) = L Gn(t)Fn(x) .n=O

We shall use this many times when analysing linearised reaction-diffusion equations . We give some examples below th at appear repeatedly.

Example C.2

Find the spectrum and th e eigenfunctions of - \72 on fl = (0, L) , with homogeneous Dirichlet boundary condit ions.


In one dimension, \72 = ~. We need to solve the problem

_ d2F = \F ( ( ) ()A in 0, L) , F 0 = F L = O.

dx2

If A = _k2 < 0, then F(x) = Aek x +Be-kx , which cannot satisfy the boundaryconditions unless A = B = O. If ,\ = 0, then F(x) = A + Bx, which alsocannot satisfy the boundary conditions unless A = B = O. If ,\ = k2 > 0,then F(x) = A sin kx + B cos kx. The boundary condition F(O) = 0 impliesB = 0, and the boundary condition F(L) = 0 then implies that A sin kL = 0,so A = 0 (which we reject as trivial) or kL = mr for some positive integer n .

I dexi h h . h 0 h k (n+l)rr \ (nH)2 rr2n exmg t ese so t at we start Wit n = ,we ave n = L ' An = L2 '

Fn(x) = sin (n+I)7rX , taking A = 1 since we recognise that any constant multipleof an eigenfunction is also an eigenfunction. Here and generally where we havesinusoidal solutions, kn is called the nth wave-number. The wave-length of E;is 2

krr = 2;

Example C.3

Find the spectrum and th e eigenfunctions of - \72 on [l = (0, L) , with homogeneous Neumann (zero-flux) boundary conditions.

We need to solve the problem

_ d2F = \F ( , '(dx2 A in 0, L) , F (0) = F L) = O.

Using a similar argument to th at of Example C.2, but now noting that ,\ = 0does zi . . I I ' h k nrr \ n2rr 2 d F nrrxoes give a non-trivia so ution, we ave n = T' An = --rr , an n = cos "T>

Example C.4

Find the spectrum and the eigenfunctions of - \72 on [l = (0, 27f) , with periodicboundary conditions .

Periodic boundary conditions occur naturally in examples. Usually the dependent variable is an angular variable such as 1> in cylindrical polars, and th eline 1> = 0 is identical to the line 1> = 27f. In order that the function be continuously differentiable , it and its derivative must be th e same on th e two lines.Hence we need to solve the problem

d2F

- d¢>2 =,\F in (0, 27f) , F(O) = F(27f), F'(O) = F'(27f).


Using a similar argument to that of Example C.2, we have k« = n, An = n2, and

Fo(¢) = 1, Fn (¢) = cosn¢ or Fn (¢) = sin n¢ if n 2:: 1. We write Fn(¢) = ei n¢ ,

where it is understood that both the real and the imaginary parts of Fn areeigenfunct ions.

Example C.5

Find the spectrum and eigenfunctions of - \72 when n = IRn, and bound ary

condit ions are replaced by the requirement that the function be bounded asIxl-t 00 .

We need to find bounded solutions of the equation

- \72F = AF in IRn.

Then F given by F (x) = exp(ik ·x) is an eigenfunction for any constant vectork E IRn

, called a wave-nu mber vector, and A = k . k is the correspondingeigenvalue. Again, it is understood that the real and imaginary parts of Fareboth eigenfunctions. Hence any non-negative real number Ais an eigenvalue inthis case, rath er than just a countable set {An} . In this case the sum (C.3.35)has to be replaced by an integral ,

v(x, t) = r G(k , t ) exp(i k · x )dk ,JRna Fourier transform.

CA Separation of Variables in PartialDifferential Equations

(C.3.36)

The method of separation of variables is a method of constructing solutions ofvarious partial differential equations. It is of limited scope, being in particularonly appli cable to linear problems, but is nevertheless important in applications, including mathematic al biology. It is used, for example, in determiningwhether spat ially homogeneous solutions of reaction-diffusion equations arestable.

The diffusion operator %t - D\72 is always separable, in the sense that thetime variable is always separable from the space variables, if th e spatial domainis fixed in time and condit ions are given at t = 0 and on the boundary of thespatial domain .


In this section we demonstrate the technique of separation of variables in nspace dimensions. The result allows us to determine the asymptotic behaviourof solutions of the diffusion equation as t ~ 00.

Let u satisfy

(C.4.38)

(C.4.37)

au- = n· 'Vu = 0an

on the boundary an of n, and initial condition

au = D'V 2u

at

on a well-behaved bounded domain n c IRn , with Neumann boundary condition

u(x, 0) = uo(x) (C.4.39)

(Here as before n is the outward pointing normal to n at points of an.)The strategy is simple . First, we look for functions of the form u(x, t) =

F(x)G(t) which satisfy both the differential Equation (C.4.37) and the boundary condition (C.4.38). We shall find a whole set of them. Since the problemis linear , any linear combination of them will also satisfy Equations (C.4.37)and (C.4.38), by the principle of superposition. We then determine which linearcombination also satisfies the initial condition (C.4.39) .

Substituting u(x, t) = F(x)G(t) into Equation (C.4.37), we obtain

F(x)G'(t) = D'V 2F(x)G(t),

so that'1

2F(x) = G'(t) =-A

F(x) DG(t) ,(C.4.40)

say. The next part of the argument is crucial. Note that since A= '1 2F (x) / F (x),it is independent oft, but since A = G'(t)/(DG(t)) , it is independent ofx. HenceA is constant. The differential operator is separable. Hence

- '1 2F = AF, G' = -ADG. (C.4.41)

The boundary condition implies

(C.4.42)ofan = n . 'VF = O.

The first of Equations (C.4.41) with boundary conditions (C.4.42) is the eigenvalue problem discussed in Section C.3, and so we know about the spatial eigenvalues An and spatial eigenfunctions Fn. In particular, the eigenvalues satisfyEquation (C.3.34). In this context of separation of time and space variation,the spatial eigenfunctions are known as the spatial modes . The correspondingGn may now be found from the second of Equations (C.4.41), and are given

C. Some Techniques for Part ial Differential Equat ions 293

by Gn(t) = exp(- AnDt). This gives functions Fn(x )Gn(t ) which satisfy thediffusion Equation (CA. 37) and the boundary condit ion (C.4.38). The genera lsolut ion of these is given by

00

u(x , t ) = L anFn(x) exp(- AnDt ).n=O

This sat isfies the initi al condit ion (CA.39) as well if

00

UO( x) =L anFn(x) .n=O

(CAA3)

(CAA4 )

(CA A5)

Since the Fn may be taken to be orthonormal, InFn(x)Fm(x)dx = 0 for m in,In F; (x)dx = 1, and the coefficient s an are given by

an =1uo(x )Fn(x )dx.

From Equ ation (CA A3) using Equation (C.3.34), u(x , t ) -t aoFo(x ) = constantas t -t 00 . For Dirichlet boundary condit ions u = 0 on an, Equation (C.3.34)is replaced by Equ ation (C.3.33), and u(x , t ) -t 0 as t -t 00.

Example C.6

Give the general solut ion of th e diffusion equation in the one-dimensional domain n = (0, L) with homogeneous Neumann boundary condit ions.

The spatial eigenvalue problem is th at of example C.3, for which we can2 2

writ e down explicit ly the eigenvalues AO= 0, An = n £1 ,and the eigenfunct ionsFo(x ) = 1, Fn(x ) = cos nzx. The genera l solut ion is

u(x , t ) = ~ao +~ an cos C~X) exp (_ n~:2 Dt) .

(The facto r ~ in the first term is conventional; without it , the formula (CA A4)

for an requir es a minor change for n = 0.)

Example C.7

Give the general solut ion of the diffusion equation in the one-dimensional domain n = (0, L ) with homogeneous Dirichlet boundary condit ions.

294

By a similar argument, we obtain


Example C.8

Write down the general solution of

au 2at = au + D\l u

(C.4.46)

(C.4.47)

with homogeneous Neumann or Dirichlet boundary conditions, where a is aconstant. Discuss the asymptotic behaviour as t -+ 00 if a ~ O.

Using the same transformation as in Section C.2.1, the general solution isgiven by

00

u(x, t) = L anFn(x) exp(at - AnDt) .n=O

(C.4.48)

For Neumann boundary conditions, when Equation (C.3.34) holds, all solutionsof Equation (C.4.47) with a < 0 decay exponentially with time . If a = 0, thesolution tends to a constant. For Dirichlet boundary conditions, when Equation (C.3.33) holds, all solutions with a ~ 0 decay exponentially with time .

There are various conditions that must be satisfied for separation of variables to work, as follows.

- There must be a step like (C.4.40) where we can separate the t variation andthe x variation in the partial differential equation. We say that the lineardifferential operator must be separable.

- All initial and boundary conditions must be on lines of constant t and x .We could not, for example, solve a problem on a growing domain directly byseparation of variables.

- There are restrictions on the boundary operators that can be used. Neumann, Dirichlet and periodic boundary conditions are suitable, as are Robinboundary conditions, au + b~~ = 0, with a 2:: 0, b 2:: o.

All of these conditions are easily violated.

C. Some Techniques for Partial DifFerential Equations 295

C.5 Systems of Diffusion Equations with LinearKinetics

The method of separation of variables may be extended to systems of diffusionequations. Although the separation of variables process itself is virtually unchanged, there are some differences in writing down the solutions, so we presentthe method here.

Consider the system of m equations on a domain n given by

(C.5.49)

(C.5.50)

where J is a matrix with constant coefficients, and D is a constant matrix,usually a diagonal matrix of diffusion coefficients, with Neumann boundaryconditions au

n · \7u = - = 0an(by which we mean that n · \7ui = 0 for each i) on an and initial condition

u(x,O) = uo(x) (C.5.51)

in a.Now look for a solution of (C.5.49) with (C.5.50) in separated form,

u(x, t) = cF(x)G(t) , where c is a constant vector . Then

- F satisfies the Neumann boundary conditions, and

- u = cFG satisfies (C.5.49) ,

i.e.cF(x)G' (t) = JcF(x)G(t) + Dc\7 2 F(x)G(t) ,

or, dividing through by the scalar functions F and G,

G'(t) \72F(x)c G(t) = Jc + Dc F(x) .

The left hand side of this equation is independent of x, so the right hand side9 2 F(x)must also be independent of x, so F(x) must be constant, -A, say. The

operator is still separable. This leads to exactly the same eigenvalue problemfor F that we had in the single equation case, so we have a set of eigenvaluesAn of - \72 on n with Neumann boundary conditions and the correspondingeigenfunctions or spatial modes Fn .

The equation for the temporal behaviour corresponding to the nth spatialmode is


say, a system of linear ordinary differential equations with constant coefficients.It has exponential solutions, so look for a solution in the form Gn(t) = exp(O"nt).(More generally, if there are repeated roots , there may be a solution in the formof a product of a polynomial and an exponential.) The equation becomes, oncancelling through by exp(O"nt),

(C.5.52)

say, an algebraic eigenvalue problem , i.e. O"n is an eigenvalue of An with eigenvector Cn ' These eigenvalues are referred to as the temporal eigenvalues, todistinguish them from the spatial eigenvalues An. The temporal eigenvalues aregiven by

(C.5.53)

a polynomial of mth degree in O"n, where I is the identity matrix. Assume forsimplicity that each of these polynomials has m distinct roots O"nl, ' " ,O"nm,with corresponding eigenvectors Cnl , ' " ,Cnm' and let Gni(t) = exp(O"nit) . Wenow have a whole set of solutions given by Uni(X, t) = CniFn(X)Gni(t) , forn = 0,1, .. . , and i = 1,2 , .. . ,m . The general solution of the system is obtainedby taking linear combinations of these,

00 m

ufx, t) = L LanicniFn(x)Gni(t) .n=Oi=l

This also satisfies the initial condition (C.5.51) if00 m

uo(x) = L L anicniFn(X).n=Oi=l

Since the Fn are orthonormal, this reduces to

(C.5.54)

for each n, a set of m equations for the m unknowns ani, i = 1, ' " ,m. (NB Ifthe initial conditions involve only one Fn , then so will the solution.) In fact, wedo not usually solve initial-value problems. Our main concern is whether anyof the 0" have positive real part, i.e. whether u = 0 is stable or not .

If n = R" , then any non-negative Ais a spatial eigenvalue, and the temporaleigenvalue problem for 0" is

O"C = (J - AD)c = Ac,

say, and 0" is a root of the polynomial

det(O"[ - J + AD) = 0

for any A 2: o.

(C.5.55)

(C.5.56)


C.6 Separating the Spatial Variables from EachOther

In some cases we need not only to separate the time variables from the spacevariables, but also to separate the space variables themselves. The Laplacianoperator with Neumann or Dirichlet boundary conditions is separable on appropriate spatial domains. The bound ary conditions have to be specified onlines (or surfaces) where a dependent variable is constant . We shall assumethat we have already separated the time and space variabl es, so that we havean eigenvalue problem in the space variabl es alone.

Example e.g

In Cartesian coordinates x, y , find the solutions of the eigenvalue problem

- \72F = -Fxx - Fy y = >..F in (0, a) x (0, b),

with boundary conditions

F(O, y) = F(a, y) = 0 for y E (0, b),

of ofoy (x ,O) = oy (x , b) = 0 for x E (0, a).

We look for separated solutions in the form F( x , y) = P( x)Q(y) . The differential equation gives us

P"(x) Q"(y)- P( x) - Q(y) = >.. ,

so that P" (x) / P (x) and Q" (y) / Q(y) must each be constants. The operator isseparable. The boundary conditions give us

P(O) = P(a) = 0, Q'(O) = Q'(b) = O.

Both the P and the Q problems are familiar , and we immediately deduce thatPm(x) = sin m:x for m 2: 1, Qn(Y) = cos~ for n 2: 0, and the eigenvalues

\ _ m 2rr2 n 2 rr 2

areA- ~+b2'

Example e.10

This is a problem on the surface of a circular cylinder of radius a, which arisesfrom the tail-pattern analysis of Section 7.5. In cylindrical polar coordinates ¢,Z, solve the eigenvalue problem

2 1-\7 F = -2F4> 4> - Fz z = >..F, (C.6.57)

a

298

with boundary condit ions


Neumann (zero-flux) boundary condit ions, and

F(O, z ) = F (21r , z ), F¢(O , z ) = F¢(21r , z ),

periodic bound ary condit ions.

(C.6.58)

(C.6.59)

(C.6.60)

Look for a solut ion of the form F (¢>, z ) = P(¢»Q(z ). Then Equ ation (C.6.57)becomes, on subst ituting in and dividing through by P (¢»Q(z ),

1 P" (¢» Q"(z )- a 2 P( ¢» - Q(z) = A,

so th at P"IP and Q"IQ must both be constants . The operator is separable.The boundary condit ions give

P (O ) = P(21r ), P' (O) = P' (21r ), Q' (O ) = Q'(h) = O.

Both th e P and the Q problems are familiar , and know tha t the solut ions arePm(¢» = eim¢ , m ~ 0, Qn(z ) = cos n~z for n ~ 0, and the eigenvalues Amn aregiven by

1 P;:" (¢» Q~ (z ) m 2 n21r2

Amn = - a2 Pm(¢» - Qn(z ) = ~ + }(2 '

for m ~ 0, n ~ 0, with eigenfunctions cos m::z ein¢ .

DNon-negative Matrices

D.l Perron-Frobenius Theory

A matrix M = (mij) is positive if all its elements are positive, and non-negativeif all its elements are non-negative. Non-negative matrices occur in mathematical biology in several contexts. Population pro jection matrices, or Leslie matrices, are non-negative; these are described in Chapter 1, and give the stagedependent bir th rates and the transit ion rat es from one stage to anot her instage-struct ured populations. So are contact matrices; these are describ ed inChapter 3, give the cont act rat es between members of a structured population,and are used there to analyse the spread of an infectious disease. PerronFrobenius theory describ es the eigenvalues and eigenvectors of such matri ces.One of the most important facts about th em, used in both contexts above, isthe following.

Theorem 0.1 (Non-negative Matrices)

Let M be a non-negative matrix. Then there exists one eigenvalue '\'1 that isreal and grea te r than or equal to any of the others in magnitude, Al ~ IAil.This is called the principal or dom inant eigenv alue of M. The right and lefteigenvectors VI and WI corr esponding to Al are real and non-negative.

In some cases more can be said about the domin ant eigenvalue and itseigenvectors. To present the theory we need to define some terms . We say th at

299


there exists an arc from i to j if aji > 0; a path from i to j is a sequence of arcsstarting at i and ending at j ; a loop is a path from i to itself. A non-negativematrix is either reducible or irreducible; it is irreducible if, for each i and j, thereexists a path from i to j . In terms of stage-structured populations, a matrix isirreducible if each stage i may contribute at some future time to each other stagej . This is almost always true, unless there are some post-reproductive stagesthat cannot contribute to any younger stages. An irreducible matrix is eitherprimitive or imprimitive; it is primitive if the greatest common divisor d of thelength of its loops is 1. If d > 1, it is called the index of imprimitivity. Mostpopulation projection matrices are primitive. The only significant exception isin cases like the Pacific salmon, which has a single reproductive stage at twoyears of age. The population projection matrix with an annual census thereforehas index of imprimitivity d = 2.

Theorem D.2 (Primitive Matrices)

If M is primitive (and therefore, a fortiori, irreducible) , then (in addition tothe results of Theorem D.1), its dominant eigenvalue Al is

- positive,

- a simple root of the eigenvalue equation 1M - AIl = 0, and

- strictly greater in magnitude than any other eigenvalue.

Moreover , the right and left eigenvectors VI and WI corresponding to Al arepositive. There may be other real eigenvalues besides AI, but Al is the only onewith non-negative eigenvectors.

Theorem D.3 (Irreducible but Imprimitive Matrices)

If M is irreducible. but imprimitive, with index of imprimitivity d, then (inaddition to the results of Theorem D.1) , its dominant eigenvalue Al is

- positive, and

- a simple root of the eigenvalue equation.

- Although Al ~ IAil for all i, the spectrum of M contains d eigenvalues equalin magnitude to AI, Al itself and Al exp(2k1ri/d), k = 1,2, ·· · , d - 1.

Moreover, the associated right and left eigenvectors VI and WI are positive.

EHints for Exercises

1.1 T his is a par t icu lar case of Equation (1.2.4), discussed in example 1.1. There isa t ra nscrit ica l bifurcation point at (x' , A) = (0, 1) ; as A increases past t his pointt he re is an interchange of stab ility from t he t riv ial to t he non-trivial steady stateA - 1. The solution is

1-px n = -----,-.,...--....,-'---------,--

p + ((1 - p )/xo - p)pn X OA n + (A - 1) - Xo .

1.2 5 is a stable steady state; C is an un st able steady state, and A and B are stableperiod- 2 solut ions of t he equat ion .

1.3 a) T he linearised equat ion is nn +1 = h' (N ' )nn , and t he steady state is linearlystable for Ih' (N " )1< l.

b) No dep ensation in t he su rvivorship functi on for t he young, f , or the fertili tyfun ction for th e ad ults, g .

c) Fixed point s satisfy a' = f (y' ), y' = g(a ') , so that t hey arc t he intersect ions of y = g(a) and y = r l (a) . Note that an +2 = f (yn+l ) = f (g(an ) ) ,

so for stability we must have Ih' (a')1 < 1, where h is defined by heal =f(9( a)) . T he resu lt follows. Stability dep ends on whether y = 9(a) crossesy = f -I(a) from below to above or from above to below. Stable steadystates ar e 50, 52 or an alte rn at ion between (Y2, 0) and (0, a2).

1.4 a) Nn/ (Nn + 5) is t he pr obability that a given insect picks a fertile mate, if itpicks an insect at random from t he pop ula t ion .

b) If W f; 0, W = f (N ' ) may be solved to give 5 = 5(W) = I~oa~·' - W =( R o - l -a N ') N ' . . b N' 0 d N" ( Do 1)/ h ..l + a N ' , posIti ve etwee n = an = a u - a, w ere It ISzero.

c) The maximum of 5 on this curve is at N" = (y'R;; - 1)/ a , where 5 = 5c =(y'R;; -1 )2/ a .

d ) See Figure E.1.1.5 a) Immedia te, on substit ution of t he given fun ct ions.

b) We obtain r = aC - {3, J( = (aC - (3 )/ (a-y).c) If r < 0, i.e. C < (3 / a , t hen N(t ) -T 0 as t -T 00 .

301


(a) S small (b) S larger

02 0.4 Q6 Q8population allime n

o'""""'----~-~--~-~-----'

o

'+ 0.8c:CI>

.§ 0.6iiic:.g 0.411l:;c.[0.2

0.2 0.4 0.6 0.8population allime n

o"""--~--~-~-~-_...J

o

'+ 0.8c:CI>

,§ 0.6iiic::8 0.411l:;c.[0.2

Figure E.! Co bweb maps for th e sterile insect cont rol equat ion for two valu esof S . In (a) S < S«, too sm all to drive the population to ext inc t ion, whil e in (b)S > Se, the saddle-nod e bifurcation at (N", Se) has taken place, losing the two nontrivial steady states , and th e population goes to ext inct ion.

d ) The main advantage of t he first approach over t he second is it s simplicity.The main advantages of the second are t he insight gained and its testability,so t hat, for example, we could predict t he effect of a reduction in C on thegrowth ra te and carryi ng capac ity.

1.6 The logist ic equation IV = rN( 1 - N/ I< ) is a Bernoull i eq uation and may besolved by the subs t it ution M = I/N , to obtain tV! = _ IV/N 2 = -r(M -1/I<),whi ch has solution M (t) = 1/N (t ) = 1/I< +(1/No-1/ I< )e- rt . The resul t followsafte r som e algebra, whi ch is mad e easier if we not e that we can t ake No = I</ 2without loss of gene rality.

1.7 The fun ction f is zero at 0, U, and I< , negative between 0 and U and positivebe tween U and I< . Hence N( t) ~ 0 if 0 < No < U , N( t) ~ I< if U < No < 00,

an d there is critic al dep ensation , with a minimum viabl e population size U.1.8 a) If f (N ) = rN2, j' (N ) = 2rN, an increasing fun ction of N .

b) By separation of var iab les, N( t) = No/(l - rNot) , and N (t ) ~ 00 as t ~1/ (r No).

c) The t rivia l steady state 0 is unst abl e, and t he non- tri vial steady state I< isstabl e.

d ) The ana lysis parallels that of Exercise 1.5.1.9 Integr ating the Bernoull i equa tio n dL/dt = -(/-£1 + /-£2L )L by the substitution

M = 1/L, or ot he rwise, from n + tl t o t with initial condit ion L( n + h) = Lo,we obtain

L(t) = /-£ILo(/-£1 + /-£2Lo) exp( t - n - tIl- /-£2Lo'

Now, putting t = n + t2 and usin g the ass umptions given , the resul t follows.1.10 Y" = . f (N" ) - N" = N" (er(I-W / K ) - 1), so Y ' is a posit ive fun ction in (0, I<) ,

zero at 0 and I< .1.11 a) The model with fishing is given by IV = rN( 1 - N/I<) - qEN, which has

stead y states at 0 and N' = I«1 - qS" / r ).b) Y " = qE ' W = qE" I« 1 - qS" / r ).c) For maximum Y " , qE " [r = t, so Ymax = :trI<o

1.12 a) Immedi ate.

E. Hints for Exercises 303

b) The yield-effort relationship for critical depensation is as shown in Figur e 1.7(b) j for non-critical depensation it is similar except that the unstablebranch reaches the E*-axis at a valu e E, = F(O)/q which is non-zero (unlessF(O) = 0).

1.13 a) The term m(1 - p) represents colonisation of an empty patch from themainland.

b) The equation is p= I(p) , where I is a quadratic with 1(0) = m , 1(1) = -e,so there is a singl e steady state in (0,1) , which is stable.

1.14 The non-trivial steady state is N * = K . The equ ation linearised about thissteady state is n(t) = - rn(t - 7) , so the characterist ic equat ion of the linearisation is 8 = -,«:", and no instabili ty is possible.

1.15 a) Immediate.b) Immediate.c) Substituting into the equat ion and performing the integration, we obtain

th e characteristic equat ion 8 = 7/( 1 + 78), or 782 + 8 - 7 = O.

d) The charact eristic equation has one positive and one negative root, so thesteady state is unstable.

e) The result follows by differentiating the expression for P and integrating byparts .

1.16 The linearisation is again given by Equation (1.7 .21) . Substituting in n(t) =no expfsz}, and the form given for k, and performing the integration , we obtain8 = 1/(1 + 78)2 . This charact erist ic equation still has at least one positive root,and the steady state is unstable.

1.17 No stochastic effects, no death , no effects of competition , no predators , discretetime, no effects of ageing on fertility, etc.

1.18 Define Yn = 2::;:'=0 Yk,n . We sh all use the fact that YO ,n+l = Yl ,n + Y2,n + .. ..Then

Yn+2 = YO,n+2 + Yl ,n+2 + Y2,n+2 + Y3,n+2 + ...= (Yl,n+l + Y2,n+l + ...)+ YO ,n+l + (Yl,n+l + Y2,n+l + ...)= (YO ,n + Yl ,n + ...)+ (YO ,n+l + Yl,n+l + .. .) = Yn + Yn+l ·

Sinc e YO ,1 = 1, Yl = 2, and the resulting sequ ence is 1, 2, 3, 5, 8, 13, "' , theFibonacci sequ enc e shifted by one place (which it must be since no rabbits everdie) .

1.19 a) , is the potential per capita production of offspring , taking into accountall factors but over-winter survival and germination. a is the probability ofsurviving the winter, G the fraction of one-winter survivors that germinate,(3 th e fraction of two-winter survivors that germinate.

b) Censusing the population at th e flowering stage (just before the new seedsare produced), when there are Pn plants and Sn one-year-old seeds, we have

(Pn+l ) = (S n+l

, aG, a (1 - G)

(Note that this is not in the form of Equation (1.9.26), because there isreproduction and not merely survival between the plant and the seed stage.)

c) The eigenvalue equation is Q(A) = A2 -,aGA - ,a2(3(I- G) = 0, which hasone positive and one negative root, and therefore a solution A > 1 if andonly if Q(I) < O.

d) Ro = , aG+ , a 2(3 (1 - G).1.20 a) Immediate j y is the necessary constant of proportionality.

b) The charac terist ic equat ion is (1 - I - A)(-A) - ,I = 0, so the principaleigenvalue is Al = t(1- I + )(1 - IF + 4,f) .


c) Since t he pr ob lem is linear, A = 1 mu st be a root of th e charac te rist icequation , so I - "11 = 0, 1=0 or "I = 1.

d ) Although C' is stable, it is not asymptotically stable. Also, t he mod el isnot structurally stable, meaning that a small change in paramet er valu escan lead to drastic changes in solution behaviour.

1.21 A proo f by induction is straightforward.1.22 T he res ult follows on expanding the det erminant by its top row.1.23 Since AlVI = LVI, we obtain AlVI,i = 8iVI ,i-1 for each i , where the VI,i ar e the

components of VI, and t he resul t follows.1.24 a) The context shows that Ii 2: 0 for each i, so j is th e sum of monotone

decreasing terms , each of which ex hibits t he limit ing behaviour given.b ) Ro = j(O), while r l is t he real root of j(r) = 1. If Ro > 1, j(O) > 1, so

r i , t he real roo t of j(r) is positi ve, by th e properties of j in (a) . A simi larargument holds for Ro < 1, an d t he resul ts for Al follow immediately.

c) Expanding j about zero , we have 1 = jh) ~ j(O) + r l1'(O)+ h.o.t. Theresult follows, for r l small.

1.25 a) Ro = L:;~I Lim, = 5.904.b ) Using t he resul t of t he previo us exercise , bas ed on the assumption t hat rl

is small, r l = (R o - 1)/11' (0)1, where 11'(0) 1 = L: ;~I i l . rru; t he result isr l = 0.273.

c) Use Newton 's method or simi lar.1.26 a) T he resu lt follows from Eq uation (1.10.29) , on mult iplying by s" and sum -

ming.b) Im mediate.c) Immediate.d) Since b(8) is t he ratio of two po lynom ials g(8) and 1 - 1(8) wit h t he degree

of 9 less than the degr ee of 1 - 1(8), th e t heory of partial fractions tells us

t ha t b(8) = L:~= I ( I !~ , s) , where each B, = limH si (1 - Ai8)g(8)/ (1 - 1(8»is a constant . Since 10g(1 - 1(8» = L:log(1 - Ai8), so 1' (8)/ (1 - 1(8» =L: Ai/(I- Ai8), we may mu ltiply t hrough by (1 - Ai8)g(8) and take limits as8 -+ s . , to obtain B , = Aig(8;)/ I' (8;), and the result follows on expanding(1 - Ai8) - I .

1.27 a) Since aRo = - j' (0) , t his is just a rest atement of Exercise 1.24(c) .b) From part (a) , Ro ~ 1/(1 - ri a ) , so log Ro ~ r ia, and t he resul t follows.

1.28 The first claim follows since t he integrand is decreasing for each a, and the cla imson limi ts follow on dividing t he ran ge of t he integral into (0, €) and (e, 00) , for €sufficiently small.

1.29 The first claim follows from t he defin it ion Ro = foco I (a)da, and t he second from

the monotoni city and limit pr operties of t he previous exerc ise.1.30 a) Multiply Equation (1.10.36) by e- s t and int egrate, and t he result follows

from t he Laplace t ransform convolut ion theorem.b) The first part is the pr evious exercise re-s ta ted , and t he second from the

inequ ali ty 1](8)1$ foco I/ (a)e- salda = foco I (a)e- (Res)ada.

c) This follows from pa rts (a) and (b) and t he form ula for t he inverse Laplacet ra nsform .

1.31 a) l(b)/l(a) is t he probability of survi ving from age a to age b, and exp( - r (b a» the amount a birth to an indi vidu al at age b mu st be discounted if t heindiv idual is now at age a.

b) The repr oduct ive value will increase from birth, reach a maxim um somet ime between first and maxim al reprodu ct ion , and t hen decline, becomi ngzero for individua ls of post-reprod uct ive age.


c) Heart diseas e ca used the loss of 250 units of reproductive valu e, mal aria967, almost four times as much .

1.32 a) We may think of 1 as th e fraction of a cohort surviving to age a.b) Integrate Equation (1.11.40), with 1 = u ind ependent of t and 1(0) = 1.

1.33 a) The ra te at whi ch members of a cohort die when the cohort is aged a isd(a)l(a) , so the average age of death is Jo

co ad(a)l(a)da j Joco d(a)l(a)da . The

result follows sinc e Joco d(a)l(a)da = 1 is the probability of eventual death .

b) A similar argument gives the required result , once the number of age a alivenow is discounted by e- r a to account for th e smaller population in the past .

c) If d is constant, I(a) = e- da, life expectancy is 1jd and th e mean age of

those dying simultaneousl y is 1j(r+d) . The second expression takes accountof the smaller number of older individuals, arising from the fact that thepopulation was smaller when they were born.

2.1 If the effects of host intra-sp ecific compet it ion ar e negligible during the searchperiod , th en we may simply replace Ro by Ro(1 + aHn)-b in the H-equation .The solutions are now bounded , and may tend to the steady state (H' , P') or toa closed invariant curve, as we would expect from a Naimark-Sacker bifurcation ,in (H , P)-space.

2.2 The searching efficiency is aop- m, a decreasing function of P. This stabilises

the steady state.2.3 Assume that hosts always find an available refug e. Then Sn = min(H, Hn) hosts

find a refuge, U'; = Hn - Sn do not . The equat ions become

Hn+1 = Ro(Sn + Un exp(-aPn)) , Pn+1 = cUn( l - exp(- aPn)).

The solutions are now bounded. If there are too many refug es the parasitoid maygo extinct , but otherwise the solutions may tend to the st eady state (lr ,P')or to a closed invariant curve in (H , P)-space.

2.4 a) The eigenvalue equation for the linearised equations is A2 +w ' A+au' v" = 0,and the result follows .

b) 8 j8u(Bf) + 8j8v(Bg) = -EjV < 0, which does not change sign .c) It is easy to see that <P (or, strictly, <P - <P(u' ,v')) , is positive definite about

(u' , v' ) in th e positive quadrant. Now note that 1 - EU - V = -E(U - u') (v - v') . It follows t hat 1>(u , v) = - aE(u - u·)2 :5 0, as required .

d) All solutions in the positive quadrant spiral in to the steady state.e) <P is decreasing, bounded below , so 1> -t 0, u -t u" . The result follows from

the Poincare-Bendixson theorem, Section 8 .2.3 of the appendix.2.5 a) We have to assume that u starts above k ; then it never drops below it , so

u - k > 0. Only these u - k prey are available to th e pr edators.b) The steady state (u' , v' ) = (k + 1, k + 1) is st able (node or focus).

2.6 a) Predators switch from handling to searching mode at constant rate (3, andfrom searching to handling on encounte r with a prey at rate aU.

b) Immediate.c) U equat ion immediate , V equ ati on assumes births proportional to rate of

consumption of prey, deaths at constant rate.2.7 a) On g, u has carrying capacity K . On p, predation reduces population of pr ey

(unless there is non e) . On q; for the predators, the more prey the better.b) Equations (2.4 .13) satisfy Gause's condit ions. Ifthere is a coexistence steady

state, the nu llclines ar e as in Figure 2.9; there is no coexisten ce steady state(in t he positive quadrant) if th e vertical nullcline v = 0, or q(u) = d, is toth e right of the st eady state at (K ,O).


c) Trajectories ente r (0,0) along t he v-axis, leave it along th e u-axis; t heseare t he stable and unstabl e manifolds of the saddle point. det J (K, 0) =Kg ' (K) (-d + q(K )) < 0 (implying a saddle point) if q(K ) > d, when a coexistence state exists , det J(K,0) > 0 (implying a stabl e node) if q(K) < d.(Thinking of d as a bifurcation paramet er , t here is a transcritical bifurcation as d decreases past d = de = q(K) when the coexiste nce state ente rsth e positive quadrant through (K,O) and takes over stability from it .)

d) detJ(u' , v') = p(u')v' q' (u' ) > 0, so the condition for instability istrJ(u' , v' ) = u' g' (u') + g(u' ) - v' p'(u') > O.

e) The ph ase plane is similar to t hat of Figur e 2.9(d ).2.8 a) Prey-predator relationship.

u can survive at low v populations, v cannot survive at low u populations.

There is a pr ey-on ly ste ady state.

For small u , eit her u has a depensatory growth rate or there is a nonlinearsaturating functional response; v is purely compensatory.

Dep ensation occurs at the steady sta te.b) The phase plan e is very sim ilar to that of Figure 2.9(d) , except that 9 = 0

has a positive rat her than an infinite slope .c) Boundedness follows by const ructing an invariant set . Then use the Poincare

Bendixson t heorem (Section B.2.3 of the appendix).2.9 a) No relationship between predation term in u equa tion and any growth te rm

in v equation; probl ems with sin gulari ty at u = 0 in v equat ion.b) See Figure E.2.

(a) Leslie phase plane (b) Leslie time series1.2

/1.2

'CQ> \

/ s:/ '" \

.... /eu

.... :3. \

'" 0.8 / e \.... / S!g eu \

~ 0.6 ¥0.8

••l!? c.c.

~0.4

/ .... ~0. 60.2 / .... >-

/ -, l!?/ .... C. -

0 0.40 0.2 0.4 0.6 0.8 0 2 4 6 8 10

prey time

Figure E.2 The phas e plane and a typical time senes for the Leslie pr ey-pr edator equat ions.

c) All solut ions in the positive quadrant tend to the co-existence stead y state ,which is a stable nod e or focus .

2.10 The slope of t = 0 is - t~/t: , etc.2.11 a) Immediate.

b) u r21'2 / v r

1'Y1 = C exp(TIT 2 (-y2 - ')'r)t ), where C is a constant of integrati on .Compet it ive exclusion follows since u and v ar e bounded , with u winning if')' 2 > ')' 1.

c) u is less affected by th e fact that th e food is being consumed than v .


d ) n species cannot survive on less than n food resources , mod elled in t his way.2.12 a) Take B (u , v ) = (uv) -I .

b) <P(u , v ) = - c((u - u*)2 + (a + b)(u - u*)( v - v*) + (v - v*)2) ~ 0 since , ift he coexiste nce state is stable, a < 1 and b < 1, and so (a + b)2 ~ 4.

2.13 The m te rms represen t t he experimental rem oval. The coexiste nce results followdirectly from th e usu al stability condit ions .

2.14 a)du dvdi= u(l- u+ av ), dt =cv( l + bu - v) .

b) There is a coexist ence steady st ate (u" , u") = (( 1 +a) /( l - ab), (1 + b)/(l ab)) if and only if ab < 1. No solut ions in t he positive quadrant te nd to (1,0)or (0, 1) , since th ese are saddle points with th e axes as st abl e manifolds . Anyset D = { (u , v )IO ~ u ~ M u' ,O ~ v ~ M v*} with M 2': 1 is invariant , soall solut ions are bounded . Dulac's crite rion with B(u, v ) = (uV)-1 showsth ere are no periodi c solut ions , so the Poin care-Bendixson theorem givesthe result .

2.15 a) cP is the photosynthesis term.b) So, (P,H) = (cP/a,O); S I, (P,H) = (P' ,W) = (c/(eb), (ebcP - ac)/(bc)) .

The eigenvalues at So are -a and ebo] « - e, whil e the Jacobian matrix atS I has t r r < 0, det r > 0, stable, if H* > O.

c) There is a t ranscrit ical bifurcation where stability switches from So to S Ias cP increases t hrough cPc = (ae) /( eb), and S I ente rs th e positive quadrant.

d ) See Figure E.3.

(b) High primary production(a) Low primary product ion

2 2

sn 1.5 Ul 1.5~ ~0 0> >:e 1 :e 1Q) Q)

.J::: .J:::

0 .5 -, 0.5

S0 0

0 0.5 1 1.5 2 0plants

0.5

I

II

III

IS,,1

I ,

I 'I

1plants

.... -c S

1.5 2

F ig u re E .3 The ph ase plan e with (a) low and (b) high primary production cP .Aft er the bifurcation to S I, steady state plant density is const ant .

2.16 A mod el for this is given by

dN dPdt = aP + bZ - eN P, dt = eN P - dP Z - aP,

dZ dG- = dP Z - bZ - eZ G - = eZ G - f G& ' & '

with N + P + Z + G = A . The st eady states are So, S I and S2 in t he text , with


C := 0, and S3 := (N" , P" , Z" , CO), where

P" := _ e_ (A _ ~ + ~ _ (c + d)f) , Z " := Ld + e c e ce e '

C" := .A: (A _9:. _ ~ _ (c + d)!) .d+ e c d ce

As A increases , we hav e a sequ ence of t ra nscritica l bifur cations as SI takes overstability from So, S2 from SI , and finally S3 from S2.

2.17 T he model is

dP := -I. _ P _ bP H dH := ebP H _ Hdt 'I' a 1 + kP , dt 1 + kP c .

T he steady states are (<p /a ,O) and (r ,H") , where P" := c/(c - ek) , H"(<p-aP")( I+kr) /(br) , in t he positi ve quadrant as long as c > ek, <p -aP" > 0.T he J acobian matrix at t he coexistence steady state has tr P := -a - bH "/ (1 +kP ")2 < 0, det P := eb2P: H "/ (1 + kr)3 > 0, so is always stable.

2.18 a)dPI dP2b := (O'IN - (3J)PI, b := (0'2N - (32 )P2,

where N := A - 'Y IPI - 'Y2P2.b) The resu lt follows on defining u := PI/KI, V := PdK2, t := (O'IA - (3I) T,

c := (0'2A- (32)/ (0'IA- (3I) , where K I := (O'IA- (3I)hl, K 2 := (0'2A - (32)h2 .c) T he slopes are l/a and b.

2.19 a) So:= (0,0) , SI := (pi , O) := (1 - et/cI , O), S2 := (0, 1 - edc2 ), S" := (pi , 1 pi - ed c2 - cIPi / C2) .

b) Immediate; sp ecies 2 t hen cannot survive even in the absence of sp ecies 1.c) See Figure E.4. Species 2 survives whenever S" is in the positive quadrant,

(a) No coexistence steady state (b) Coexistence steady state

0.8

IIIII

I0.2 \ I

\.-I- 5. 1

o,r . ---.Jo 0.5

compet itor 1

C\I

B0.6~a.§ 0.4o

0.2 \\

\

0.5competitor 1

Figure EA The phas e plan e for t he metapopulat ion competitio n system wit hand without a coexistence steady state.

i.e. P2 := et/ci - ez f c: - cI/c2(1 - et/cI) > 0.d) In particular, it su rv ives if C2 is sufficiently la rge.

2.20 Steady states ar e (0,0) , (1 - et/cI, O), and (pi , pi3), where pi := eI3/ c3, Pi3 :=(CI(1 - pi) - eJ) / (cl + C3 ).Condition for survival is Pi3 > 0, or e13/c3 < 1 - et/c i .tr P := - clPi < 0, det P := c3pi3(CI + c3)pi > 0, so st able.


2.21 The Pl3-nullcline remains fixed , while the PI nullcline moves leftwards. Thereis a transcritical bifurcation when the coexistence state passes through theprey-only steady state, at D = Dc = 1 - ei/ci - eI3/c3, and the predators goextinct.

2.22 We are given coexistence for D = 0, (C2 - e2)/(c2 + CI) > 1 - eiic«. It followsthat (c2(1- D) -CJ)/(C2 -l-cr ) > 1- D-el/CI for all D, and this is the conditionfor the coexistence state S; to be in the positive quadrant. S· does not passthrough SI. The only possibilities for bifurcation from the coexistence phaseplane is through either SI or S2 passing transcritically through So, and thetrade-off condition shows that SI does so first. If there is such a trade-off, thebetter competitor is always the one that goes extinct first .

3.1 a) Let the required probability be p(7). Then p(7 + 67) = p(7) - "(p(7)67 +0(672). Subtracting p(7) from both sides, dividing by 67 and taking thelimit as 67 -t 0, P= -"(P, and since p(O) = 1 the result follows.

b) Let T be the time spent in the T class . Then !P' {7 :$ T < 7 + 67} = p(7) p(7 + 67) :::::: -p(7)67 = "( exp(-"(7)67, so

T =100

7"(exp(-"(7)d7 = 11"1,

as required.3.2 a) An individual is infective at time 7 if he or she was infected at some time

7 - a and remained infective at least for a time a . Thus 1(7) = Jooo i(7a)f(a)da, and

i(7) = (3IS:::::: (31 N = (3N100

i(7 - a)f(a)da,

as required.b) Trying a solution i(7) = ioexp(r7), we obtain

1 =(3N roo exp(-(r+"()a)da= (3N ,Jo r+"(

so r = (3N - "(.c) Hence r = "((Ro - 1), familiar as Equation (3.2.3) .

3.3 The incidence function is as for the SI disease, but now results in entry to theexposed class . The time in the exposed class is exponentially distributed withmean 1/6, and is followed by entry to the infective class .

3.4 a) At time 7 , the I class consists of those who have been infected between7 -71 and 7.

b) r = 17" (3/* S' do = (3/* S· 71, and the result follows.T-TI

c) Ro = (3N7I .3.5 At W = 0, the derivative of exp( -Row) equals -Ro , and the qualitative features

follow. UI satisfies RoUIexp(-RouJ) = Roexp(-Ro), i.e. Ro and RoUI are twopositive values of x where xe-'" is equal. Since xe-'" increases in (0,1) anddecreases in (1,00), the result follows.

3.6 a) If Row is small, then exp( - Row) :::::: 1 - Row + ~ Rl,w2. The result follows.b) The approximation above holds if RoWI is small, and then w -t WI .c) The incidence of death is pili, where p is the probability of death given

infection and w satisfies the logistic equation with r = Ro - 1 and K = WI.This is a sech 2 curve.

3.7 There is a one-parameter family of solution trajectories, given by Ro(u + v) log U = A, where A is a constant of integration.


3.8 If Ro > 1 th ere are N PWI deaths, where WI is as in th e text .

3.9 a) Since , = 1/25 per year , Ro = (3N h = 4.b) For (i) , since Raul < 1, WI > 1-I/Ro, and NWI > 750. In fact, numerical

calculations show that this is a conservative estimate. For (ii) , R!J = qRo =(0.3)(4) = 1.2, so (R!J - 1)/R!J = 1/6 is small, and the approximation ofExercise 3.6 holds ; the final size of the epidemic is given approximately byqNWI = 2H300 = 83.

3.10 a) dS = 8R - (3I S dI = (3IS - 'VI dR = 'VI- 8R.dr ' d-r J ' d T /

b) R o = (3Nh , since the time spent in I is unchanged.c) The endemic steady state is

( * * *) (1 lId 1)u ,v,W = R o'l+ d Ro'l+dRa '

so th e condition is Ro > 1.

3.11 a) Susceptibles enter exposed state on infection , and leave it after an exponentially distributed time of mean 1/8.

b) Ra = (3Nh·c) Since we still have dw/du = -1/(Rou) , the final size is unchanged.

3.12 Parts a) to d) are immediate.e)

f()1

u s -o s -l'(u -s)da = poe e s .o

f)

R o = p8(3'N 100

(e-OU _ e-l'U)dcr = (3N .,-8 0 ,

g) Ro depends on how long is spent in I , not on whether there is a delay inentering it.

3.13 Immediate.

3.14 a) Expresses probability of not dying from disease-unrelated causes before disease age cr .

b) Immediate.c)

R - (3N100

-h+d)Ud _ (3N0- e cr- .o ,+d

d)

f( ) -1u s -(o+d)s -h+d)(u-S)da - poe e s ,o

Ra = (3N100

f(cr)dcr .

3.15 a) Straightforward.b) Working with the (N,S, I)-system, the Jacobian at the endemic steady state

(N*, S* ,1*) is given by

(

-dJ*= ~

o-(31* - d

(31*


whose eigenvalues are -d and the roots of the quadratic .\2 + ((31* + d).\ +(321*S * = O. Now ((31* + d)2 = R6d 2 « 4(Ro - l)dh + c + d) = (321*S*, sothe roots of this quadratic are .\ :::::::±i~, as required.

c) T::::::: 271h/(365/12)(1/70)12 -+ 271"/}(365/12)(1/70)11 = 2.75 -+ 2.87. Thetime series from Providence, Rhode Island (Figure 3.6) has 9 peaks in 24years, a period of 2.67.

3.16 If an ende mic steady state exist s, S* = h + b)/(3, (31* + b = qbRo , so (31* =b(qRo - 1).

3.17 a) Immediate.b) Ro = (3 N */h + c + d(W)) .c) Usin g the (N, S , I)-system , we look at the stability of the disease-free st eady

st ate (N* , N* , 0) . We have

(

r'( N*) Wr = r'(N~)N *

o-d(W)

o- c )- (3 N * ,

(3 N * - ,- c-d(W)

where r (N ) = b(N)-d(N) , whos e eigenvalues are r'(N*)N* < 0, - d(N *) <0, and (3W r rt > c - d(N*) .

d) p '2. pc = 1 - 1/Ro .3.18 First note that .\(i)(a ,t) = (3 1(t ), independent of a. Then the integration is

straightforward.3.19 a) Let subscript 1 represent females, 2 males. This is an SIS disease with

criss-cross infection; (312 is the infectious contac t rat e for infectious femalesinfecting sus ceptible males, and (321 the contrary. (312 may be higher becausemany infectious females do not know they are infectious; ,2may be higherbecau se males know they have t he disease and seek treatment; there mayalso be behavioural differences.

b) Sl + I, = N 1, S 2 + 12 = N 2, N 1 and N 2 constant , V1 = Ii/N1, V2 = h/N2,R0 1 = (321N2h1, R02 = (312Nd'2.

c) The eigenvalue equat ion for the wholly sus ceptible steady state is .\2+ (,1 +' 2).\ - (Ro 1R02 - 1),1/2 = 0, which is unstable if and on ly if Ro1Ro2 > 1.

3.20 a) Delete th e , ;Ii t erms from the susce pt ible equat ions , and add removed classequa t ions dRi/dT = , Ji .

b) T he eigenvalue equat ion for the wholly suscept ible ste ady state is simplyA2 times t he eigenvalue equat ion for the corresponding SIS disease, foundin the las t exercise , so the condit ion for an epide mic is again R01R02 > 1.

c) Separate vari ab les to obtain dwddu2 = - ,dh2Ro2U2), and vice versa,interchanging 1 and 2. Integrating from th e disease-free st eady state, U2 =exp( - h 2R o2/ , d w 1, and vice versa . In the limit as t -+ 00 , 1 - W2 =ex p(- h 2Ro2hd w 1, and vice versa .

3.21 a) W is t he m ean worm burden , so the W equa t ion is equivalent to Equation (3.8.34) divided through by N for humans. 1 is the total number ofinfected snails, so the 1 equat ion is not divided t hrough by N for snails.

b) Always have (0,0) , which is st abl e; two non -trivial st eady states, the smallerone unstable and the larger one stable, exist when N > N; = (8/c)(d/b +

2..jd[b).c) As c decreases , the two non-trivial steady st ates disappear by a saddle-node

bifurcation when c passes through c, = (8/N)(d/b+2..jd[b) , and the levelof disease in a population at the endemic st ab le steady state drops sudden lyto zero.

312

d)


dW bcNW2-~ - <5W.dt bW 2+ W + 1

3.22 Pl otting Ro = (/3N)/ (,o c- o. +c+d) against c, Ro has an intermediate max imum,and the virulenc e t ends to the value of c that achi eves it.

3.23 a ) The J acobi an matrix a t t he ende mic ste ady st ate (S ·, Ii , 0) is given by

(

- /3di.- dJ = /3d l

o- /32 S · )

o ./32S· - , 2 - C2 - d

Two of its eigenvalues are stable, and the other is given by A = /32 S · -,2 C2 - d = (,2 + C2 + d)(RodRol - 1), positive if R02 > ROI . There is nocoexistence stat e since i. = 0 and i, = 0 cannot hold simult aneously withneither II nor h zero .

4.1 If, as the information given strongly suggest s, the gene is dominant, and Elisabe t h Horstmann 's daughter was heterozygous for it , then the probability ofpolydactyly in her children is ~ , so the expec ted number of her children to showthe trait is four .

4.2 Let the pure-bred rounded yellow and wrinkled green ph enotypes have geno typesRRYY and WWGG resp ectively. Then all those in the H generation havegenotype RWYG, and phenotyp e rounded yellow.

Female gametesRY RG WY WG

Male RY RY RY RY RYgametes RG RY RG RY RG

WY RY RY WY WYWG RY RG WY WG

The F2 generation is produced by rando m mating, summarised in the diagram.This shows the phenotype that results from each union of gametic genotypes.Each of t hese possibilities is equally likely, so that the ratio of RY : RG : WY :WG is 9 : 3 : 3 : 1.

4.3 Consider a population in Hardy-Weinberg equilibrium; then x = p2, Y = 2pq,z = l , so that y2 = 4xz . Conversely, consider a population with y2 = 4x z .Then

2 1 2 2 1 2 2P = (x + 2'y) = x + x y + 4'y = x + x (1 - x - z) + x z = x .

Similar calculations show t hat y = 2pq, z = q2, so the population is in HardyWeinberg equilibrium.

4.4 The ph enotypes resulting from the various unions of gametic genotypes areshown in the diagram below.

Female gametesA B 0

Mal e A A AB Agametes B AB B B

0 A B 0Let the frequ encies of the alleles A, B and 0 be p, q and r, and the frequ enci es ofthe blood groups A , AB, Band 0 be A , AB, etc . If Hardy-Weinberg proportionshold, then 0 = r2, B+O = (q+r)2 , so r = JO= 62.0%, q = VB + 0 - JO=78.0% - 62.0% = 16.0%, and so p = 22.0%. Then A = p2 + 2pr = 32.2%,AB = 7.1%, and we are very close to Hardy-Weinberg equilibri um .

E. Hints for Exercises

4.5 a) The offspring will have genotypes AA , AB, BB in th e ratiousual.

b) We have

313

2 1, as

Hence1 1

pn+l = X n+l + iYn+1 = X n + '2Yn = pn ,

so P» = p, qn = q, Yn = Yo(~r -+ 0 as n -+ 00 , X n = p - Yo(~r+l -+ P asn -+ 00 , Zn = q - Yo(~r+l -+ q as n -+ 00.

c) Heterozygotes disappear, and the population splits into two non-interbreedingsubpopulations. This may be the basis for some speciation events.

4.6 1 in 400.4.7 Assume random mating and no selection , and other Hardy-Weinberg assump

tions. For males, there are two genotypes A and B , with frequencies m and n,say, and , ,

m =p, n = q,

where p and q are femal e allele frequ encies. For females , with the usual notationfor genotypes,

Hence

x' = mp, y' = mq + rip,,

Z = nq .

, ,1, ( 1) 1p = x + '2Y = m p + '2q + '2np,

" '(' 1 ') 1" (' I ') 1 , 1, 1p = m p + '2q + '2n p = p p + i q + "i qp = "i P + "i p.

Solving this,2 1 1 1 n

pn = 3po + 3mo + 3(po - mo) (- "i ) .As n -+ 00, p.; tends to a constant equal to th e initial frequen cy of allele A inthe who le population.

4.8 a) Let W y < W z . Since w > 0 th en Equation (4.3.8) implies that op < 0 ifw '" < W y , and op < 0 as long as p < p' = :; -w~ if w '" > W y • In either

W z Wy W x

case Sp < 0 for p small, and (Pn) is a positive decreasing sequence, so thesteady state P = 0 is stable. Biologically, the condition wy < W z says thatthe heterozygote is less fit than the homozygote BB .

b) The steady state p = 1 is stable if the heterozygote is less fit than thehomozygote AA , W y < w"' .

c) An interior steady state p' exists if and only if

• wy - W zP = E(O,I) ,

(wy - wz ) + (wy - w"')

i.e. either

i) W y > W z and w y > W "" the het erozygote is fitter t han both homozygotes, or

ii) W y < W z and w y < w"' , th e heterozygote is less fit than both homozygotes .

It is stable if neither p = 0 nor p = 1 is, i.e. in case i) above.

314 Essential Mathem atica l Biology

4.9 a) From the FHW Eq uation (4.3.2) with W x = 1, w y = 1 + s, W z = 1 - t ,

I - sp+ (s+ t )(l- p) s(1 -2p) +t(1 -p)p = p+ pq = p+ pq .

p2 + 2(1 + s )pq + (1 - t )q2 1 + 2spq - tq2

b) There are steady states at p = 0, p = 1 and p = p" , where

( *) (*) * s + ts l -2p + tl - p =0, p =--.2s + t

The cobweb map is as shown in Figure E.5, an d it is clear that P» -T p" asn -T 00 for any po E (0,1) .

c) Give n t = 0.8 (fitness of BB homozygotes 0.2) , p* = 0.8, t hen

1 - p* 0.2s = t - - - = 0.8- ~ 0.27,

2p* - 1 0.6

so that the fitnesses of AA and A B are in t he ratio 1 : 1.27, and the cha nceof dyi ng from malaria before mat urity is approx imate ly 0.21.

(a) Cobweb map (b) Wright adapt ive topography1.2,.---------------,

0.80.2Ol.--~-~~-~-~----'

o

0.2

~ 0.8Ql

E~ 0.6C1lQl

E 0.4

0.2 0.4 0.6 0.8frequen cy pn

0.8

Figure E .5 (a) Cobweb map and (b) Wright 's adaptive topo graphy for t hesickle cell an aemia gene . Wright 's adaptive topography is discussed in Sect ion 4.7an d , for t his example, in exercise 4.15.

4.10 Routine. Use t he gene ratio v = q/p for t he depa rture from p = O.4.11 Routine .4.12 a) From Equa t ion (4.5.15) with h = 1, k = 1,

1 r 9 dps = 35 l o.oJ p(1 _ p)2 ~ 0.45.

b) Nume rical. The weak selecti on ap pro ximation is exce llent , desp it e the relati vely large value of s .

4.13 In selection-mutation balance,

op = a~p _ u w~P + v w~q .w w w


a) For a recessive deleterious gene W x = 1 - s, w y = W z = 1, so that w =(1 - s )p2 + 2pq + q2 = 1 - Sp2, W p = (1 - s) p + q = 1 - sp , W q = 1,Q p = wp - W = - spq, and

8p = _ Sp2q - up + vq + li.o.i,

assuming u, v « s « 1. At steady state 8p = 0 and p = p' , small, so

_ sp· 2 + V ~ 0, p" ~ v;;r;.b) If t he deleterious gene is dominant W y = 1 - s, w = 1 - sp, W p = 1 - s ,

W q = 1 - sq, Q p = - sq, and

8p = - spq - up + vq + hoL.,

assuming u , v « s « 1. At steady state 8p = 0 and p is small, so

- sp'+ v~O , p· ~v/s .

c) Here A is recessive, so that (a) holds, and leth al , so that s = 1. Sincev = 4 X 10- 4

, p" ~ 2 X 10- 2.

4.14 Differenti ating Eq uation (4.3.5) with respect to p , and noting that q = 1- p, wehave

dii: aw awdp = ap - aq = 2wxp + 2wyq - 2wyp - 2wzq = 2(w p - wq ) ,

and Eq uat ion (4.7.18) follows. Also,

8w = wx(2p8p + 8p2) + 2wy(q8q + p8q + 8p8q) + wz(2q8q + 8l)

= 2(W p - W q )8p + (w x - 2wy + W z )8p2,

since 8p + 8q = O. P ut t ing W p - W q = !!!f:-, t he resul t (4.7.17) follows.

4.15 From w = 1 + 2spq - tq2, we can sket ch the Wright ad ap ti ve topogra phy, shownin Figure E.5. T here is a maxi mum at p" , so P» -t p" as n -t 00, as expec ted .

4.16 a) Since p is an int erior point , r = (1+ E) p - eq E Sn-l for E sufficiently small.b) Since p is an ESS W(r,p) - W (p,p) ::; 0, and W (q, p) - W (p,p) ::; 0, and

since q is an ESS W (p,q ) - W (q,q ) :=:; O. But , since W is linear in its firstvariable, W (r,p) - W (p,p) = E(W( p,p) - W(q , p)), with E > O. T his is onlypossible if

W (r ,p) = W (p, p).It also follows t hat W(p,p) = W(q,p) . Now, since W is linear in its firstand second vari ables, it follows after some algebra th at W (r ,r) - W (p,r ) =E2(W (q, q) - W (p,q) ), so that

W(r , r ) ~ W (p, r ).

These two resul ts cont radict t he alt ernativ es (4.9.23) and (4.9.24) requiredfor p t o be an ESS .

4.17 Let p be a mixed ESS, wit h i in t he sup port of p. We need to pr ove t hatW( ei ,p) = W (p,p) . We know t hat for all q =I p,

W (q,p) :=:; W (p,p).

Assume for cont radict ion that W (ei, p) < W (p, p). Conside r q = (1 + E)p - ee. .As long as E is su fficient ly small, q E Sn- l , since i is in the suppo rt of p. Now

W(q,p) - W (p, p) = E(W(p,p) - W (ei , p)) > 0,

giving us our cont radiction.

316 Essential Mathematica l Biology

4.18 a ) If your par tn er were to play D , you would be be tter off playin g D . If yourpa rtner were to play G, you would be better off playin g D. Hence you playD; your partn er , followin g the same logic, also plays D. You both obtainpay-off 2, missing out on t he benefit of mutual co-op eration , with pay-off 3.This is the prisoner 's dilemma.

b) The pay-off matrix if th e game is played te n t imes is given below.

on encounteringthis strategy

D TFTPay-off to D 20 22this st rategy T FT 19 30

Both D and T FT are ESSs.c) Co-operat ion can persist , but how does it start?

4.19 Let x be th e frequ ency of the hawk strat egy. Then (4.10.31) with U as in Section 4.9 gives

x= ~x(G - Gx )(1 - x ),

which is to be solved subject to x(O) = Xo, 0::; Xo ::; 1. Both x = 0 and x = 1 aresteady states, so we need on ly consider 0 < Xo < 1. If G < G, then x (t ) -+ GIGas t -+ 00 , whereas if G ~ G, th en x (t ) -+ 1 as t -+ 00 . The solution agr ees withthat in Section 4.9, with x ' , th e evolut ionarily stable state of the system , takingthe place of p", the evolut iona rily stable strategy.

4.20 The pay-off matrix becom es

(

- E

U = ~1

1- E

-1~1 )- E

a) Let p be the symmetric st rategy (t,t ,t), and q = (ql ' qz, q3) anotherstrategy, q E 52. Then W(p,p) = - tE, W(q,p) = - h W(p,q) = -tE,W(q ,q) = - E(qi + q~ + qj) . It is easy to show that W(q ,q) < W(p ,q) forq =j:. p , so (4.9.24) holds and p is an ESS .

b) The rep licator equations ar e

with similar equat ions for if and z , Under these equat ions,

d ( (x if i) (2 2 2 )-d x yz ) = x yz - + - + - = EX Y Z 3(x + Y + Z ) - 1 ,t x Y Z

so that xyZ increases with t. It tends to (~, ~ , ~) , the maximum of x yZ on52, where the RHS of the equat ion above is zero , as shown in F igure 4.7.In fact , x yz is a Lyapunov function for the system (see Chapter B of theappendix).

c) If th e penalty is replaced by a reward , then p is no longer an ESS; in fact ,it is invadabl e by any other strategy. The function xyz now decreases witht , and any non-constant solution of the repli cator equat ions with initialconditions in th e interior of 52 approaches a trajectory whi ch visits eachapex of the triangle 52 in turn .

4.21 See the website for one possibility.


Phase plane, epsilon negat ivez=1

5.1

y=1

317

Figure E.6 Evolutionary dynamics for thelizard (rock -scissors-paper) game with a reward.

.!!:..- r udV = r au dV = r -\7 . JdV = - r J. ndS = O.dt Jv Jv at Jv Js

5.2 The flow is spherically symmetric, so J = J(r, t)er . Conservation of mass for Vgives

u(r, t + ot)4 7lT2or = u(r, t)47rr2or + J(r , t)47rr2ot

- J(r + or, t)47r(r + or)2ot + f(r, t)47rr2orot + h.o.t.

(The spheres have surface ar eas 47rr2 and 471'( r + or)2 respectively.) Dividingthrough by 47rr2orot and taking limits as Or -+ 0, Ot -+ 0, we obtain

AU 1 a 2

7ft = - r 2 Or (r J) + f.

5.3 The flow is cylindrically symmetric, so J = J(R , t)eR . Conservation of mass forV gives

u(R, t + ot)27rRoRh = u(R, t)27rRoRh + J(R, t)27rRhOt

- J(R + oR, t)27r(R + oR)hOt + f(R , t)27rRoRhOt + h.o.t.

(The cylindrical surfaces have areas 27rRh and 27r(R + oR)h respect ively.) Dividing through by 27r RoRhOt and taking limits as oR -+ 0, ot -+ 0, we obtain

au 1 a7ft = -R oR (RJ) + f.

5.4 In two dimensions, some concepts are interpreted slightly differ ently: for example, u is the amount of substance per unit area, and J is the rate at whichsubstance crosses a curve per unit length in the direction perpendicular to theflow. Otherwise, the derivation essent ially follows from the previous exercise ondeleting h .

5.5

318

5.6


dN l baudt(t) = a [jt(X , t)dx

lb(au aZu)= a -v ax + D axz (x,t)dx = [

au ]b-vu(x, t) + D ax (x, t) a'

For conservation this must be zero , so vu - Dux = 0 (zero-flux boundary conditions) at x = a and at x = b.

5.7 We have x = z + vs , t = s , so we may think of x and t as functions of z and s.Then, from U(z, s) = u(x, t) by the chain rule,

au au ax au at----+--as - ax as at as'

or Uz = vUx + iu . Similarly Uz z = Uxx, and the result follows.5.8 a) The total flux is given by J = Jk, where J = -D~~ -agu. The conservation

equation is dJ = _D d2u

_ ag du = O.dz "dz2 dz

b) The boundary condition at the surface is the zero-flux condition J =-D~~ - o qu. = 0; the boundary condition at infinity is u bounded.

c) Integrating, J is constant , and applying the boundary condition, the constant must be zero . Thus D~~ + ccqu. = 0, which we can integrate to obtainu = uoexp(-1§'z), where Uo is the (undetermined) plankton concentrationat the surface.

5.9 a) We need to solve the problem

du dZu0= -v- + D- in (0, L), u(O) = Uo, u(L) = O.

dx dx?

The general solution of the equation is u = A + Bexp(vxID), and theboundary conditions lead to

u(x) = Uo exp(vLID) - exp(vxID) .exp(vLID) -1

The flux J (equivalent to the diffusion current I) is given by

du exp(vLID)J = vu - D-d = vuo (LID) .x exp v -1

b) Since u is a line concentration then the amount of matter N in (0, L) isgiven by

N = rLu(x)dx = Uo Lexp(vLID) - (Dlv) (exp(vLID) -1) .

Jo exp(vLID)-l

c) Hence the transit time T is given by

T= N =!:._ D (l_exp (_VL)).I v v Z D


d ) To find t he limi ts as v -t 0, we may eit he r expand t he exponent ials as Taylorser ies or use L'H opi tal 's rule. Using Taylor series, we have u -t uo(l - x / L ),J -t uo D / L , and

L D ( v L 1 v2L

2) 1 L

2

T = ;- - v 2 1 - (1 - D + 2152 + h.o.t. ) = 2D + h.o.t ,

as required .5.10 a ) Integr a ting 0 = *-;J; (r 2D ~~ ) , with D constant , we obtain the general

so lution u(r ) = - 1- + B . The boundar y condit ions u(a) = 0, u(b) = Uodetermine A and B , and

b(r - a)u(r) = Uo r( b _ a) ' J (r ) = _D du (r ) = _ Du oab .

dr r2(b - a)

b ) The diffusion current , i.e , the rate at whi ch subs tance flows out of theregion throu gh the sur face r = a , whose a rea is A = 41ra 2, is given byI = - AJ(a) = 411"~_u~ a b . Integrating over t he spherica l she ll a < r < b,o< 0 < zr, 0 < rjJ < 2rr in spherica l pola r co-ordinates,

1211" 1 11" J b 2 . 4rrbuo [r3

3_ a

2r

2]ba

.N = u( r)r sin Bdrd BdrjJ = - _-4> = 0 8=0 r=a b a

c) Hence the aver age ti me it t akes a particle to diffu se from a poin t on r = bto a poin t on r = a is given by

T = 2- (b3- a

3_ b2

- a2

) .

D 3a 2

d ) If a « b, t h is is given app rox imately by

b3

T :::::: - - .3aD

5.11 a) Integr ating 0 = -k d~ (R D ~~ ) , with D const ant, we obtain the generalsolut ion u( R ) = A log R + B . The bo undary condit ions u(a ) = 0, u( b) = uodet ermine A and B , and

(R ) _ log(R / a)u - u o log(b/a ) '

du DuoJ (R ) = - D dR( R) = R log(b/a)

b) The diffu sion current, i.e. th e rate at whi ch subs tance flows out of the regionthrough the surface R = a, whose a rea is L = 2rra, is given by 1= - LJ (a) =l:;~;~) ' In tegr at ing over the annulus a < R < b, 0 < rjJ < 2rr in plane polarcoordinates ,

12 11" l b 2rruo [R 2 R R 2] b

N = u(R) RdRdo = I (b/) -2 log( - ) - - .4> = 0 R = a og a a 4 a

c) Hen ce t he average t ime it takes a particle to d iffuse from a poin t on R = bto a po int on R = a is given by

N 1 (b2( b) 1 2 2 )

T = T = D "2 log -;; . - :t (b - a) .


d ) If a « b, t his is given ap proximately by

b2 (b )T :::::: -log - .2D a

5.12 Immedia te, from Section C.2 .1 of t he appendix.5.13 a)

100 1211'N(t) = u( R, t )RdRd<jJ

R = R 2(t ) 4> = 0

2rrM 100

( R2

) (R~ (t))= -D R exp a t - -D dR = M exp at - -D '4rr t R 2(t ) 4 t 4 t

b) Defining R2(t) by taking N(t) = m , and taking logs,

(m) R~(t)log M = at - 4Dt '

for t large, as before.c) If a = 0, R?(t) '" t log t but m (t ) '" t .

5.14 a)au 2at = au + D'V u.

b) We measure t in generations . The RM S disp ersal dist an ce for a par ticl ediffusin g in two dimensions over one generation is J4i5, so assumption (iv)gives J4i5 ::; 50. Using the resul t of Exercise 5.12, t he number of t reesprodu ced in a generation is eO< , so e" :S 9 x 106

, a :S log(9 x 106) = 16.

c) The dist an ce R 1 moved in 20000 years is t hus R , ::; .j4aD t = 200t =200 x 20000160 metres , or ab ou t 67 km . Skellam concluded t hat diffusionwas insufficien t , and t hat ot her disp ersal mechan isms such as transport byanimals and birds mu st be importan t .

d) The estimates sound high for D , out rageous ly high for a, and low for generat ion time, bu t since we are looking for an upper bound to the distan cetravelled this is not a problem .

5.15 Referring to Figure 5.5, we need to show th at T is below t he upper boundaryv = mf(u) of D"n ear (1, 0). But T is below the nullcline v = (1/e)f(u) since itis pointing sou th east , and hence is below v = mf(u) if m > lie, e > 11m. Butthis holds sinc e e > 11m + mK.

5.16 It is clear that no trajec to ry may leave D through the lower part of the boundary,where v = 0, Ul < u < 1, as s decreases, since v'is negative there. On theupper part of th e boundary where p (u, v ) := v - 2(1 - u - w(u)) = 0, v' =e(u + w(u) - 1 + v) = ~ev , and

'( ) I I '( ) I 1 1 1 1P u,v = v +2u +2w u u = - cv-2-uv+ 2Ro

- uv2 e u e

( 2 4 ) v ( 2 4 ) v= e - - (Rou - 1) - > e - - (Ro - 1) - > 0,Ro 2c Ro 2e

since e2 > 4(Ro - 1)1Ro , an d t he result follows.


6.1 The equat ion is

321

1 «; + S 1 tc; 1-= =-+--V VmS Vm Vm S '

so l/Vm is the intercept and Km/Vm the intercept of the best straight line fit.6.2 For the substrate , the inn er solution and the common part ar e both identically

equal to 1, so the result follows. For the complex , the common part is 1/(1+Km ) ,

and the result follows from co,unif(t) = co(t) + CO(t/E) - 1/(1 + K m) .6.3 a) Immediate.

b) The leading order equations are

ds o tc,--;It = K

mCo - So, 0 = So - Co,

so dso/dt = (Kd/Km -l)so = -Kso , and the result follows.c) The inner equations are

1 dS tc,-- = -C-S+wSC,EdT k ;

dC- =S-wSC-CdT '

so So = 1, dCo/dT = So - Co = 1 - Co , and the result follows.d) Matching gives A = 1, and we obtain So unif(t) = «!", Co unif(t) = e- K t

_

-tie "e .6.4 a) The term k_ 2PE should be added to the C equation and subtracted from

the E and the P equations.b) Immediate , using E + C = Eo .c) Dire ctly from the P equation, using E = Eo - C and the expression for C .d) Immediate from d.F[tlr = O.

6.5 a) Immediate .b) Since

Y(S) = 2R2 + RI + TI + 2T22(R2 + RI + Ro + To + T I + T2) '

th e result follows from the quasi-steady-state relationships between thedimer and its complexes.

c) All states of the dimer are R states, which are non-cooperative, and wereduce again to simple Michaelis-Menten kinetics .

6.6 a) Immediate.b) The quasi-steady-state hypothesis gives ES:; = KeXI, SIXI = K mX2 , and

the result follows.c) Immediate.

6.7 a) The Jacobian is immediate. For stability we require tr J* < 0, det J* > O.b) This condition violates tr J* < 0, which with these conditions on the pa

rameters is the more stringent of the two .c) The humps are at x = ±l.d) The existence of the steady state is immediate, and tr J* = 0 there since

x' = "Y.e) We have to show that tr J* increases as I increases past L.: But since I =

_~x*3 + z " + (a - x ' )/ b, dI/dx* = _X*2 + 1 - l/b = b/c2 - l/b > 0 at thebifurcation point, z " decreases, and tr J* increases .

6.8 a) The drug essentially switches (3 to zero , so we have

dV- =aY -bYdr '

dX- =c-dXd.r ' ~~ = -fY.


b) Then Y = Yoe- f t, V = Vo(be- f t - f e-b t) / (b - I) . The beh aviou r of Vfollows from the assumption on half-lives , so that f « b.

6.9 a) Ware th e uninfectious virus particles, which start to be produced fromthe infected cells Y aft er therapy starts. Infectious virus particles ar e stillpr esent, and die as before, but are no longer produced.

b) With X = X' = (bf)/(a f3) , the equations become a linear system that maybe integrated to obtain

V = Voexp(-br) ,f e-b-r - be-fT

Y=Yo f-b '

W - v, b ( b (-fT -bT) f -bT)- 0-- -- e - e - r eb-f b-f

The result follows, using f « b.6.10 a) Y1 is the productive infected class , Y2 the latent infected class ; the proba

bilities of entering these on infection are ql and q2 respectively. On ly the Ylcells produce virions, and Y2 cells leave for Y1 at a per capita rate 0.

b) Y2 cells produce Y1 cells at a rate 0 for a t ime 1/ (0 + iz) . Hence , adding

contributions from Y1 and Y2 cells, Ro = ~ (ql + q2 0;h) 7J '7.1 a) Define x = 7fx/L , t = 7f2t/L2, and u by u( x ,t) = u(x,t) , We obtain the

required equ ation with , = L/7f.b) Immediate from the Tayl or expansion of f , since f(u*) = o.c) By Fourier analysis , the solution must contain a term in sin nx for n =

1 only. Because the equat ion is linear with constant coefficients, we tryv( x, t) = sin x exp(at) , to obtain

v( x ,t) = sinx exp ((,2 J'(u*) - D)t) .

d) From the expression for v, with , = L/7f , it may be seen that increasingD te nds to stabilise the solution (making it more likely to decay to zero) ,whereas increasing L tends to de-stabilise it .

7.2 On ly the ¢ derivatives in V'2 are non-zero. Hence the spatial eigenvalue problemis given by

_ V' 2 p = _2.d2p =)..P

a2 d¢2 '

wit h periodic boundary conditions P(O) = P(27f) , P'(O) = P'(27f) . From Section C.4 of the appendix, the solutions of this are P( ¢) = exp (in¢) , for anynon -n egativ e integer n , with corresponding eigenvalue )..n = n 2

/ a2. The resu lt

follows from the general theory.7.3 a) Since tr J = 0, det J = 4, then the eigenvalues of u , = Ju are ±2i, so any

solution is of th e form u (t ) = Re {A exp(2it )}, and the trivial steady stat eis stable.

b) The spatial eigenvalue problem is -P"(x) = )"P(x) on (0,7f) with boundary condit ions P'(O) = P'(7f) = 0, one of the standard problems in Section C.4 of the appendix, which has eigenvalues n2

, n a non-negative integer, and corresponding eigenfunctions (sp atial modes) cos nx . QuotingEquation (7.4 .33) , the mode n is unstable for any n with ~ < )..n < >:, i.e.~ < n 2 < >:, where ~ and>: are positive real roo ts of a2()..) = 0, and

a2()..) = D1D2)..2 - (Dz!~ + Dlg~) .. + det J* = 9)..2 - 24)"+ 4.

But this has two positive real roots with 0 < ~ < 1, 2 < >: < 3, and theonly unstable mode is n = 1.


7.4 a) It is simple algebra to show that A = g~2 > 0, B = 2f~g~ -4(f~g~ - I; g:) <0, and C = f~2 > 0, so that

B 2_ 4AC = 4f~2g~2 - 16f~g~f:g~ + 16f: 2g~2 - 4f~2g~2

=-16f:g~detJ· > 0.

b) Hence the quadratic AD? + BD1D2 + CD~ =°has two real positive roots(for D 2 / Dt) , and the result follows.

c) Immediate.d) Immediate from the fact that the Turing bifurcation cur ve is the envelope of

the hyp erbolae a2(.x) = 0, (whi ch in turn is immediate from its constructionby eliminating .x between a2(.x) = °and a~(.x) = 0) .

e) On ly the second mode is unstable, so we would expect this to grow expon entially until nonlinear terms becam e important . We would exp ect thefinal solution to be close to a multiple of th e second mode.

7.5 a) If a2 is a perfect square, with repeated root .xc , t hen a2(.x) = D, D2(.x - .xc)2,so D1D2.x~ = 0

2det J., from Equation (7.4 .22) , and the resu lt follows.b) Immediate .

7.6 a) The pos itive spatially uniform steady state is given by (u · , v · ) = (l/b , l/b2).The Jacobian matrix her e is given by

J. = ( 2u· Iv: - b2u ) (

b _ b2)

= 2/b - 1 .

b) For asymptotic stability of (u· , v") to spatially uniform perturbations, werequire ts J" = b-l < 0, det J" = b > 0, i.e. 0< b < 1.

c) For spatially non-uniform perturbations , we look at the roots of a2(.x), whichis given by

a2(.x) = d.x2+ (1 - bd).x+ b.

The cur ve of marginal stability is the part of (bd - 1)2 = 4bd where bd > 1.

Solving the quadratic (bd - 1) 2 = 4bd, bd = 3 ± 2V2, and applying thecondition bd> 1, the plus sign must be taken .

d) See Figure E.7 .

Instability in the (b,d)-piane50 r-r-----.,..---~-__,

40

30

20

10

Turing

unstable

st ble

0.5 1b

unstable

unstable

1.5 2

Fi gure E .7 The region of Turinginstabi lity in (b,d)-space. Right of the lineb = 1, (u· , v·) is unstable to spatiallyuniform per turbations, but in t he regionmarked "Turing unstable" it is only unstable to spatially nonuniform perturbations .

e) From Equation (7.4 .27) .xc = Jbid = .jbd/d2 = (1 + V2)/d, or fromEquation (7.4 .35) , .xc = t(b - l/d) = t(bd - 1)/d = (1 + V2)/d.

324

Mimura-Murray phase plane2.--------,---------.,


1.5

0.5

0.2 0.4 0.6prey

0.8

Figure E.8 A possibl e ph ase plan efor the Mimura-Murray prey-predatormodel. The steady state here is stableto spatially uniform perturbat ions , bu tcan become unstabl e to spat ially nonuniform perturbati ons and exhibit ecological pat chin ess.

7.7 a) See Figure E.8.b) The Jacobian at t he spatially uniform stead y st ate (u' , v') is given by

r = ( u· f) u') -u' )• I ( . ) ,- v 9 v

which is a cross-act ivato r-inhibitor syst em as long as J'(u' ) > 0, so thatthe steady state is to the left of the hump on the nullcline v = f (u ).

7.8 The spatially uniform steady state is given by

( • • ) = (a + l (a+ l)c)u , v b' b .

The Jacobian here is given by

J • _ ( -b + cu· c-

1Iv'- * c-!cu

•ci . 2 )- u v

-1 '

which gives a pure activato r-inhibito r system near (u', v') as long as cu· c-

1 [v" >b, i.e. c > bu· = a + 1.

7.9 a) (u·,v· ) =(a + b,bl(a + b?) .b) The Jacobian matrix here is given by

J . = ( -1+2u' v'-2u· v·

which gives a cross- ac t ivator-inhibitor syst em near (u' , v·) as long as2u' v' > 1, 2b/(a + b) > 1, b> a.

c)

T 2 =

d)

2D1(b+ a)Ic ~ 27r

b- a

7.10 a) The nth mode is unstabl e if a2(>"n) = a2(n2) < 0, where

a2(>") = 10>..2 - 14a >" + 4a2 = 2(5)'' - 2a )(>.. - a) .

For instability, ~a < n 2 < a.


b) T he bifurcation diagram is as in Figure 7.6(b), with successive bifurcat ionpoints 1, 5/2, 4, 9, 10, 16 (not shown) and 25 (m od e 5 bifurcation notshown).

7.11 a) Immedi ate.b ) Must have Arn ,n = Ae, and t he result follows.

7.12 a) The steady state is given by (n *, c*) = (,J'i, Bo,J'i/(3), as long as

Bo ';::;/(3 > co·

If this condit ion does not hold, the conce nt rat ion c decays to zero.b) We have Q(a) = a 2+al(A)a+a2(A) = 0, where al(A) = (D n+De)A+2,+(3 ,

a2(A) = DnD eA2+ ((3Dn + 2,De - BoX';::;)A + 2, (3.

A sketch of a2(A) = 0 in the (A, x)-plan e is again as in Figure 7.9, and theresult follows, as long as inequality in (a) also holds.

c) The crit ical wave-length i, = 21f/A, where Ae = J r="2-, (3-;:'/-;-(:-:D::-"n'""'D=-c....,.).7.13 a) Here al (A) = (Dn + Dc)A + (3,

a2(A) = DnDcA2 + ((3Dn - a xn*)A .

Since al is always positive, and a2 = 0 always has a root A = 0, potentialinstabi lity occurs if the other root of a2 = 0 is positive, a x n* > (3 Dn.

b) As X increases beyond Xc = (3 Dn/ (an* ), long wave-length patterns areform ed ; their wave-length decreases as X increases further .

c) In this case A = n 21f 2 / £2 , and

For X < Xc, there are no bifurca tions, and the spatially uniform steady staterem ain s stabl e. For X > Xc, the first mode to becom e un st ab le is mode 1,a t J DnDc

L = £1 = tt (3 D .axn * - n

Other modes becom e unstabl e in turn as L increases further , but this remains the most un stable one .

8.1 Ifu= log( N /K) ,

du = ~ dN = -budt N dt '

so that u = uoe- bt = 10g(No/ K) e- bt = _ A e- bt , and the result follows.8.2 a) Immediate, on defining K = (a / (3 )I/V.

b) Withu=( N /K )-V ,dudt = - (311(1 - u),

so u = 1 - (1 - uo)e-{3vt ,

1u

and the result follows.

1/uo(l /uo)(l - e-{3vt ) - e-{3vt'

326 Essentia l Mathematical Bio logy

c) With u = 10g(N /K) ,

du = ~ dN = f3V u _ 1).dt N dt

Separating variables,

f du f e-vudu 1 ( e-vu

- 1 )t - - -- - - - -10f3 - eVU _ 1 - e- VU - 1 - l/ g e- VUO - 1 '

and the result follows.8.3 a) Immediate, from

1 d ( dC) kR dR R dR = D '

b) Ifthere is no necrotic core, the boundary conditions :~(O) = 0, c(r2) = C2

give c(R ) = -~~(R~ - R 2) + C2, whi ch is val id as long as c(O) 2: c i ,

R~ ~ R~ = 4(C2 - cd¥.e) For R2 > ti; we have

1 k 2C1 = 4D R 1 + A log R 1 + B ,

1 k 2C2 = 4D R 2+ A log R2 + B ,

1 k A0= - -R1 + - .

2 D R 1

d) Subtracting the first of these from the second and substituting for A fromth e third,

1 k 2 ( Ri Ri R2 )C2 - C1 = - - R2 1 - 2 - 22 log - .

4 D R2 R 2 R 1

As R 2 -+ 00, the quantity in parentheses must t end to zero, so R 2/ R 1 -+ l.But then, if oR = R2 - R 1,

k ( 2 2 2 st: )C2 - C1 = 4D R2 - (R2 - oR) + 2(R2 - oR) log(1 - R

2)k 2

::::; 2D OR ,

and so oR 2 -+ h2 = 2(; (C2 - C1) as R2 -+ 00.8.4 The model for the necrotic layer 0 < r < r1 is unchanged . In the qui escent

layer r1 < r < r2, 0 = -k1 + D'i72c, in the proliferative layer r2 < r < r3,0= -k2+ D'i72c, with k2 > k1. Continuity of concentration and flux should beapplied at r = r1 and r = r2.

8.5 While r2 < r- , there is no necrotic core, and v = kpr . Thus ~ = kPr2 , givingexponent ial growth .

8.6 The model for the nu trient concentration is as above. For the velocity field ,'i7. v = 0 in the qui escent layer r1 < r < r2, with cont inuity of the velocity fieldat r1 and rs .

8.7 a ) We have two relations betwe en r~ and r2 ' From Equation (8.4 .20) ,


Now, substituting into Equa tio n (8.3. 14) ,

( ) ( )

2k P 1/ 3 P 1/3 *2

C2 - CI = - 1 + 2 (-) 1 - (-) r26D P + L P+ L '

whi ch gives r2'b) In t he lim it as L/ P -+ 0, t his becomes

C2 - CI ~ :~ (1- (1 + ~) - 1/3f r ; 2 ~ 2~9~2r;2 ,

and t he result follows.8.8 See Figure E.9 .

Inhibitor bifurcat ion diagram

Figure E .9 Using the inequa lity (8.5.29) , t he bifur cation curve is givenby CI = (A/j..t )(1 -1 /f(aR*)) . T he curvehas pos iti ve slope at t he or igin , and tendsasymptot ically to t he line CI = A(3 /( j..t (1 +(3)). Here we have t aken a = (3 = A = j..t =1.

10

8

6

a:4

2

00 0.1 0.2 0.3 0.4 0.5c,

8.9 a) The in hibit or is produced at constant rat e A within the t umour , decays atspec ific rate j..t everywhere, and diffuses wit h diffusion coefficient D everywhere.

b) Continuity of concentration and flux.c) T he concent rat ion inside t he t umour is given by c(x ) = A/ j..t+ A cosh( J j..t/ Dx) ,

using t he symmetry condit ion c' (0) = 0. Outside t he t umour , it is givenby' c(x ) = B exp( -JJ.! / D lx l), since C is bou nded as Ixl -+ 00 . T he conti nu ity condit ions now give two equat ions for the constants of int egrat ion A and B , which may be solved to give A = - (A/ j..t ) exp(- Jj..t/ DL),B = (>" / j..t)sinh(Jj..t /DL) .

d) The concent rat ion of inhibitor in t he tumour is leas t at t he surface, so thetumour stops grow ing at L = L * when t he concent rat ion at t he surfaceincreases to C1, where CI = (>,, / j..t)sinh(Jj..t /DL*) exp(-Jj..t /DL*) . T his is

eas ily solved to give 2Jj..t/DL* = log (>.. / (>.. - 2j..tcd) , and the bifur ca t iondiagram is simi lar to t hat in Figure E.9 above with asy m ptote ~ >" / j..t.


8.10 a) Immediate.b) See Figur e E.I0 .

Immune response bifurcation diagram2,"","-~--~--~-~---,

Figure E.I0 Bifur cat ion diagramfor tumour imm une response. The equat ion gives a parabola wit h nose pointingin th e positive ,B-direct ion , whose slopedd"'r:; is negative or positive at x' = 0according to whet her () is less than orgreater than 1. T here is a transcritical bifurcation in each case, and also asaddle-node bifurcation (at the nose ofthe parabola) if () > 1.

2.521.5

8=0.8

0.5O'---~--'--~-~--"""<""---'

o

1.5

0.5

x

c) It is clear from th e figur e that for () < 1, x' rises continuously from 0 as,B drops below (), but for () > 1 it jumps from 0 to a positive value; theproduct ion of tumour cells is suddenly turned on.

advection equation , 150advection-diffusion equation, 150age structure, 27-44- disease, 103-106- McKendrick approac h, 41-44, 103-106- st abl e, 31, 34Allee effect , see dep ensationasy mptotic expansions, matched ,

176-1 83, 194-196

basi c reproducti ve ratiocont inuous time, 12

- cont inuous time age-st ruct uredpopulation, 39

- discrete time age-s t ruc t ured popula-tion ,36

- empirical disease valu es , 102- esti mat ing for diseases , 106- for a metapopulation , 22- in population dynamics, 3- in populations with non- overlapping

generations, 3- macroparasitic disease, 111- sexually transmitted disease, 109- SIR disease, 91- - thresho ld for epide mic, 92- SIS disease, 88- - threshold for epide mic, 88- vector-borne diseases , 108- virus within an individual , 199- - effect of immun e resp onse, 201Bernoulli , Dani el , 83Beverton-Holt stock-rec ru it me nt, 19, 20bifurcation- difference equations , 260

Index

- Hopf, 62, 63, 197, 276, 280- Naimark-Sacker , 268, 305- ordinary differential equat ions, 279

parti al differenti al equat ions, 211- period-doubling, 6, 8, 263- pitchfork , 262, 279- saddle-node, 260, 279, 301, 311, 328- tran scritical , 7, 22, 72, 261, 279, 301,

306- 309, 328- Turing, see under patt ern- with de lay, 25- 26- zero-eigenvalu e, 213-233, 276, 279biochemic al kinet ics , 176-190- acti vation and inhibition , 184- allostery and isoster y, 184- competit ive inhibition , 184- cooperat ive ph enom ena, 186-190- glycolysis , 189- Hill equation, 188- metabolic pathways , 183- Michaelis-Menten , 176-183- Monod-Wyman-Changeux model,

188- quasi-steady-state hypothesis, 178biogeography, island, 24blood groups, 122blowflies, Nicholson 's- adult resources limited , 24- larval scra mb le compet it ion , 9butterfly effect, 264

cancer- angiogenesis, 237, 247-248- avascular, 237, 240-245- chemotaxis, 247-248

329

330

- Gompertz model, 239- growth inhibitors , 245- 247- immunology, 249-2 51- logistic model, 238- met astasis, 248-249- necrosis, 240-245- nutrient limitation , 240-245- prevascular , 237, 240-245- quiescent layer , 242- t umour progression , 236- vascularisation, 237, 247-248- von Bertalanffy model, 238carr ying capacity, environme ntal, 13ca tchability, 18, 55chaos- difference equat ions, 260, 263-266- in ecology, 9- in insect population dynamics, 6- period-doubling route, 263-266characteristi cs- partial differential equat ions, 283chemotaxis, 148- bacteria , 154- cancer, 247-248- pattern formation , 229-233cobweb map , 257Colorado beetl e population dynamics , 8colour-b lindness, 123compensation- cont inuous t ime, 14- dis crete time, 5compe t it ion- Beverton-Holt model, 20- competitive exclusion, 66-70- contest , 4-9- disturban ce-m ediated coexiste nce, 70- fugitive coexistence, 76- interspecific, 66-70- intrasp ecific, 4- metapopulation models , 74-76, 78-80- Ricker model , 20- scramble, 5-9compe t it ion coefficient, 67conservation- of enzyme, 177- of mass, 242- of matter, 148- - equat ion, 150- of substrate, 177control theory- harvesting and fishing , 18- sterile insects, 11cottony cushion scale insect , 57


Darwin, Ch arl es, 117death rate, 12delay , 24-27densi ty -dependence, 4- population regulation by, 4dep ensation- and patchin ess, 222- consequ ence for natural resource

management, 20- continuous time, 14- critic al , 14, 302- discret e time, 5- promotes limit cycles, 63difference equations- bifurcation, 260- chaos, 260, 263-266- cobweb map, 257- invariant cur ve, 267- Jury condi t ions, 266- linearisat ion , 258- Naimark -Sacker bifurcation , 268- period-doubling bifurcation, 263- period-doubling route to chaos, 263- periodic solutions, 257- pitchfork bifurcation , 262- saddle-node bifurcation , 260- stability , 257- steady state, 257

transcritical bifurcation, 261diffusion equat ion, 150- Cau chy problem, 284- connect ion wit h probabilities, 287- fundamental solution, 284- initial-valu e problem , 284- random walk , 287- separation of variables, 291disease status, 84diseases- age at infection , 106- age structure, 103-106- basi c reproductive ratio- - estimating, 106-- macroparasitic disease, 111- - sexually transmitted disease, 109

SIR disease, 91SIS disease, 88values, 102vector-borne diseases, 108

- black death, 168- cancer , 235-252- congenital rubella syndrome, 106- criss-cross infection , 107-109- cystic fibrosis , 121, 133

Index

- disease age , 89- ende mic, 85- epidemic, 85- eradication and cont rol, 100-103- evolution of virulence, 113-115- force of infection , 86

est imat ing, 106- - macroparasiti c disease, 110- - with age structure, 104

fun ctional response, 87gonorrhoea , 108herd immunity, 102

- HIV / AIDS, 197-202- inc idence, 85- infectious contact rate, pairwise, 86- - with age structure, 104- infectivity, 95

infiuenza, 198Kermack-McK~ndrick model, 90-96macroparasiti c, 85, 109-113malaria , 84, 107-108

- - DDT resistance, 128- - sickle-cell anaemia , 126- mean infective time, 88- meas les, oscillations, 84, 100- microparasitic , 85-109- oscillations in SIR endemics, 99- pr eval ence, 85- rabies, 168- 172- rate of recovery, 87- roundworm , 111- rubella , 106- schistosomiasis, 112- SEIR disease, 95- sexually transmitted , 108- 109- sickle-cell an aem ia , 126- simple epidem ic, 86-87- SIR disease , 90-100- - threshold for epidemic, 92- SIR ende mic, 96-100- SIR epide mic, 90-96- SIRS disease, 94- SIS disease, 87, 90- - threshold for epide mic, 88- size of SIR epide mic, 92-93- smallpox

eradica t ion , 102first vaccine, 198in Aztecs, 94vario lation , 83

- spatial spread , 168-172- vaccination , 100-103- - with age structur e, 103-106

331

- vector-borne , 107-108- vertica l transmission , 96dispersion relation, 212

ecosyste ms models, 70-74- resource-based approach to logist ic, 15enzymes, see biochemical kineticsEuler renewal equat ion- continuous t ime, 38-40- discrete time, 34-38- generating function method, 37- Lap lace transform method, 40Eu ler , Leonhard, 34Euler- Lotka equation- cont inuous time, 38-40- discrete t ime, 34-38evolut ionary ecology- evolut ion of virulence, 113-115

int erspecific compet it ion, 67- r - and I<-selection , 17- r-K trade-off, 16excitability , see neural modelling

Fibonacci, 27fisheries, 17- 21- Peruvian an chovy, 20fitn ess- absolute, 123- mean , 125- mean excess , 125- overall mean , 125- relative, 123flip bifur cation, see period-doubling

bifurcationforce of infection , 86, see under diseasesfree boundary problem , 240-245fun ctional resp ons e, see under pr ey-

predator syste ms, etc

gam e theory, 136-145adaptive dynamics, 140-141Bishop-Cannings theorem , 141convergence-st ability, 141evolutionar ily stable strategy, 137, 138

- hawk-dove game, 137- 140invasibility plots, 140lizard , side-b lotched , 143Nash equilibr ium, 138Nash equilibrium , strict, 139normal form games, 138pay-off matrix, 137prisoner 's dilemma, 142rep licator dynamics , 142-145rock-scissors -p ap er gam e, 144

332

- strategy set , 137gene, 119genetics, 119-136- allele , 119- assortative mating, 122- blood groups, 122- central dogma of evolutionary biology,

131- chromosome, 119- cystic fibrosis , 121, 133- diploid, 119- dominant, 118, 119- evolution of dominance, 135- evolution of the genetic system,

134-136- filial generations, 120- Fisher-Haldane-Wright equation, 124- frequency, 119- gamete, 119- genetic drift, 134- genotype, 119- germ line , 131- haploid, 119- Hardy-Weinberg law, 120, 145- heterozygote, 119- homozygote, 119- industrial melanism, 131- locus, 119- modifier theory, 135- parental generation, 120- phenotype, 119- polydactyly, 118- polymorphism, 125- recessive, 118, 119- selection

against an allele, 129for a dominant allele, 127for a recessive allele, 127the additive case , 129weak ; analytical approach, 130-131

- selection-mutation balance, 131-133- soma, 131- spatial spread , 164-167- with selection, 123- 136- without selection, 119-123- Wright's adaptive topography,

133-134- X-linked genes, 123- zygote, 119.geotaxis, see under motionglycolysis, 189grey lar ch bud moth population

dynamics, 9

Essent ial Mathematical Biology

growth rate, net , 12growth ratio, 3guillemot, 14

habitat destruction, 23- effect on competitors, 79-80Hamilton, William, 136harvesting , 17-21heteroclinic orbit, see under travelling

wavesHodgkin , Alan , 176, 191Hopf bifurcation, 62, 63, 197, 276, 280host-parasitoid systems , 48-53- fun ctional response, 52- interference, 53- Nicholson-Bailey model , 48-51Huxley, Andrew, 176, 191

immunology- cancer, 249-251- HIY/ AIDS , 197-202invariant curve- difference equat ions, 267- Naimark-Sacker bifurcation, 268

Jury conditions, 266

Laplace's equation- eigenvalues and eigenfunctions, 289- separation of variables, 297- spect ra l theory, 289law of mass action

in chemistry, 176- in disease modelling, 86- in population interactions , 55Leslie matrices, 30-34Lexis diagram, 41life expectancy, 44limit cycle, 59, 273linearisation, 258logistic model

cancer, 238definition, 13ecosystems or resource-basedapproach, 15

- empirical approach, 12- generalisation to int erspecific

competition , 66- simple epidemic, 87- SIS disease, 89- use in forecasting, 13Lotka's intrinsic rate of natural increase- continuous time, 39- discrete t ime, 36Lotka, Alfred , 34, 54

Index

Ma lt hus, Thom as, 3Malthusian model- cont inuous t ime, 12- discret e time, 3map, see differen ce equat ionmaternity functi on- cont inuous t ime, 38- discrete ti me, 31maternity fun ction , net- continuous t ime, 38- discrete t ime, 35mat rices- irred ucible matrices, 31, 299- Leslie , 30- 34- non-negativ e matri ces, 31, 299- Perron-Frob eniu s th eory, 31, 299- pop ulation projection matrices , 31- primitive matrices , 31, 299- principal eigenvalue, 31, 299Maupertuis, Pierre-Lou is Moreau de,

118maximum susta inab le yield , 18May nard Sm it h , John , 137Mendel, Gr egor, 117- first law, 119- second law, 121metapopula t ions- competit ion, 74-76, 78- 80- predator-m ediated coexiste nce, 78- 79- prey-predator sys te ms, 74, 77- sing le species, 21-24metered mod el, 2Michacl is-Ment en kin eti cs , 176-1 83Morgenstern , Oskar , 136mortality , 3, see also survivorshipmor t al ity rate, 12motion , see also travelling waves- advect ion , 147- adve ct ion equa t ion, 150- advection-di ffus ion equation , 150- chemotaxis, 154-1 55- diffusion , 147- diffusion equation , 150- flux, 149- geo taxis, 155, 157- macroscop ic t heory, 148- taxis, 154-1 56- t ransit ti mes, 156-1 60moving boun dar y prob lem , 242-245mutation- germ-line , 132- somatic, 131

Naimark-Sacker bifurcation . 268, 305

333

Nas h, John, 139natural select ion , see geneticsneural mod elling, 191-197- excitability, 191- FitzHugh-Nagumo model, 193- 197- Hod gkin-Huxley mod el, 191-193niche, 66

Occam 's razor, 13ordinary differenti al equat ions- Bendi xson crite rion , 277

bifurcat ion , 279charac te r of ste ady states , 275Descar tes' rul e of signs, 278Dul ac crite rion, 277geomet ric an alysis , 271, 273globa l existence, 279Hopf bifurcation , 276, 280int egration , 272invariant set s , 279limi t cycle, 273lineari sa t ion , 272, 274, 278local existence and uni qu eness, 279Lyapunov functi ons , 278phase plan e, 273pitchfor k bifur ca t ion, 279Poincar e-Bendixson theory, 276Rou th -Hurwit z crite ria , 275, 278saddle-node bifurcation , 279solut ion traj ectories , 273tr anscritical bi furcation , 279zero-eigenvalue bifur cation, 276, 279

par ad ox of enrichme nt, 63paradox of the plankton , 74parti al different ial equat ions- charac te ristics, 283- coordina te syst ems , 288- diffusion equat ion, 284- Laplace's eq ua t ion , 289- separa t ion of var iab les, 291-298pattern- act ivato r-inhibito r syste ms, 214- 228- act ivator-inhibitor-immobiliser

syste ms, 223- ba ct eria , 229, 231- chemotax is, 229-233- crit ica l wave-length , 221- cross -diffusion , 231- curve of margin al stability, 213- diffusion-driven inst ability, 206- 228- domain geomet ry, 224- 228- dom ain size, 224- 228- mammalian coa ts, 224, 227

334

- mechanochemical models, 233- pure and cross-ac t ivat ion-inhibition,

220- shor t-range activat ion , long-range

inhibi t ion , 219- 222- slime mould, 229, 232- spat ial eigenvalues and modes, 208- ten t acles in hydra , 211- te rm ite nests, 229- Turing bifurcation , 211, 233- Tu rin g inst ab ilit y, 206-233- wave-number and wave-length , 210period-doubling bifurcati on , 263- cascade to chaos, 6, 8periodi c solutions , 257, 273- Fi tzHugh-Nagumo eq ua t ions, 196- pr ey-predator systems, 56- 66Perron-Frobenius th eory, see matricesPetty, Sir Will iam, 1pitchfork bifurca t ion , 262, 279Poin care-Andronov-Hopf bifurcation ,

see Hopf bifurca tionpopulation genet ics, see genet icspopula tion structure, see also age

structure- by diseas e st at us, 84- 115- by sex , 108- by stage, 27- 34pr ey-pred ator syst ems , 54-66- ecosyste ms models, 70-73- fun ctional resp onse, 55, 60-64, 66- - the three archetypes, 61- Ga use's mod el, 64- har e-lyn x data, 58- interference, 55- Kolmogorov 's condit ions for a limit

cycle , 65- Leslie's model, 65- limit cycles, 62-66- Lotka-Volterra mod el, 54- 60- metapopulation models , 74, 77- Mimura-Murray mod el for patchin ess ,

222- periodi c solut ions , 56-66- plankton models , 70-73- Rosenzweig-MacArthur mod el, 60-64- Volterra's principle, 57principal eigenvalue, see matricesproducti on , 3- density-dep enden t , 4production ra te, 11Punnet t square, 120

quasi-steady-state hypoth esis

Essentia I MathematicaI Biology

- biochem ical kineti cs , 178- cancer-immune syste m , 250- nu t rient in cancer, 243

r - J( t rade-off, 16R a, see basic reproductive ra t iorandom mating, 120reaction- diffusion equat ions- pat tern format ion , 206- 228- separa t ion of variables, 295- t rav elling waves, 160-172- - syst ems, 168-172recurrence equat ion, see difference

equationrelax ation oscilla t ion , 196reproducti on , 3reproducti on rate, 11reproductive valu e, 43rotation number- difference equat ions, 267Routh-Hurwitz criteria , 275, 278

saddle-nod e bifurca tion , 260, 279, 301,311, 328

separation of variables, 291- 298- diffusion equat ion, 291- Lapl ace 's equa t ion, 297- pattern format ion , 208- reaction- diffusion equat ions, 295- sinu soida l spatial modes, 210Sigmund , Karl , 117singu lar perturbat ion theory, see

asy mptotic expansions, matchedstability , 257, 271, 273stage struct ure, see under po pulat ion

structurestationary process, 2st ead y sta te, 257, 271structural instability, 59survival fraction , 4, 31survival fun ction- cont inuous time, 38- discrete t ime, 35survivorshi p- Gompertz , 43- th e three archetypes, 42

t axis, see under motiontrade-off- r-K; 17t ranscritical bifur ca tion , 7, 22, 72, 261,

279, 301, 306-309, 328travelling waves- bistable equa t ion, 167

Index

- black death , 168- epidemics, 168-1 72- genes , 148, 164-167- heteroclinic orbit , 165- invasion , 148, 160-1 72- - muskrats, 160- 163- monostable equat ion, 166- nerv e impulses , 164-1 68, 193- node-saddle orbits , 166- oa ks , 163- rabi es, 168-172- reaction-diffusion equat ions, 160-172- - syste ms, 168-172- saddle-saddle orbits , 167Turing bifurcation , see und er patternTuring inst ability, see under pattern

Turing, Alan , 205

ut ility, 18

Verhulst mod el, see logisti c mod elVolterra 's pr inciple, 57Volt erra , Vito, 54von Neuma nn, Joh n, 136

weevil population dyn amics, 9World populat ion, human , 1

yield-effort relationship , 19

zero-eigenvalu e bifurcation, 213-233,276, 279

335

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