discretizations of a perturbed logistic...

Discrete Dynamics in Nature and Society, Vol. 4, pp. 125-131

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Discretizations of a Perturbed Logistic Equation*MIGUEL ANGEL MORELES and FRANCISCO SOLIS

Department of Mathematics, CIMAT, Apartado Postal 402, Guanajuato Gto. 36000, Mexico

(Received 5 June 1999)

The logistic equation has been used to model a population where the intrinsic rate of growthis a linearly decreasing function of the population density. We propose some models arisingfrom discretizations of perturbed logistic equation where external factors such as harvestingand migration are included. The different discretizations exhibit diverse dynamic asymptoticbehavior when dependence on two control parameters are allowed. We also give an alter-native method to provide with new models that produce better approximating solutions.

Keywords. Logistic equation, Population dynamics

1 INTRODUCTION

A general setting for population models is therelation

dP(t)dt

Births-Deaths :t: Migration

where P(t) represents the total population at time t.

Assuming analytical laws depending on the popula-tion density for the birth B(P) and death D(P) ratesand analytical migration M(P) of the form,

B(P) Z akP: D(P) Z/3:Pkk=O k=O

M(V)k=O

we obtain the differential equation

dPdt Z(a -/3 + %)P

k=0

aoP boP2 -{- coP3 q-- mo nt- O(p4). (1)

This general model includes the well knownmodels of Malthus (with bo, Co, and mo equal to

zero) and the logistic equation of Verlhust (Co andmo equal to zero). Here ao and bo are the so-calledvital coefficients of the population; aoP is the termof pure exponential growth, ao is known as the percapita growth rate and -boP2 the term of competi-tion. This last term is justified because individualmembers of the population compete for limitedresources, e.g. land and food. We will consider onlythe cases where the right-hand side of (1) is a cubic

Partially supported by CONACYT.Corresponding author.

125

126 M.A. MORELES AND F. SOLIS

polynomial and the coefficients Co and m0 are smallenough, so that the equation might be viewed asa structural perturbation of the logistic equation.The perturbation terms can model some biologicalimportant quantities as harvesting, immigration,emigration, etc. We are considering only the caseswhere the perturbation terms are simple enough sothat the harvest (emigration, immigration, etc) rateis a constant value independent of the populationsize or it can vary proportional to the third powerof the population size. Linear and quadratic powersare already included in the model.A standard technique to use the continuous sys-

tem (1) to study the asymptotic behavior of some

species that live in isolated generations is to reduceit to a discrete map. For example, for the logisticequation the following two models have been con-sidered as its difference equation analogue (see [7]),

Pn+l (1 + a)Pn(1 Pn) and

Pn+, Pn exp(a- (1 q-a)Pn). (2)

It is known that there is an optimal discretiza-tion with zero local truncation error, unfortunatelythere is no procedure to obtain it for a general sys-tem. For this reason we use, as a benchmark, theperturbed logistic equation for which in some casesthe optimal discretization can be computed in an

explicit form. We use this optimal discretisation tocompare it with some discretizations arising fromnumerical one-step explicit methods for solving dif-ferential equations, e.g. Runge-Kutta methods. Inparticular models in (2) can be obtained by using afirst order Runge-Kutta method.A drawback of these two models is that their

asymptotic behavior coincides with the continuoussystem only up to some value of a. Moreover anyconvex combination of these two schemes will notextend the range of validity of the asymptotic solu-tion for larger values of a. To improve the rangeof validity we will introduce new schemes providedby nonlinear convex combination of Runge-Kuttamethods. Of special interest is to study the biologi-cal implications of the discrete maps obtained whenwe allow dependence on two control parameters.

2 DISCRETE MODELS

The point of departure is the perturbed logisticequation

dPaoP boP 2 + coP -+- too, (3)

to simplify our analysis, let y-bo/(ao + 1)P, thuswe are led to a equation of the form

dydt

ay- (1 + a)y2 + cy + rn. (4)

Euler’s method yields the corresponding differenceequation

Y,+l (1 + a)yn(1 y,) + cy3, + m (5)

which we shall refer as the perturbed free growthmodel equation. An alternative discretization isobtained by rewriting Eq. (4) in the form:

dlnydt

a- (1 + a)y + Cy2 q- my-1 (6)

and applying the Euler scheme to the resultingequation to obtain:

y+l yn exp a (1 + a)y + cy2 + (7)

hereafter referred as the perturbed free growth ex-

ponential model.In essence (5) and (7) are obtained by applying

a one-step numerical method to solve (4) with stepsize equal to one. Notice that (5) and (7) are pertur-bations of the systems given in (2).

In regards to truncation error, Euler’s methodsis a first order method, meaning that the errorgrowths as O(1). It is natural to do the analoguefor better numerical methods, like Runge-Kuttamethods of higher order (this has been done in [1]for the logistic equation). To illustrate this fact,we consider a second order Runge-Kutta methodknown as the improved Euler’s method. Applying

DISCRETIZATION OF LOGISTIC EQUATION 127

this scheme to model (4) we obtain the followingsystem:

z, +l Az, ( + A) + e(rn, A)

where A (a2/2) + a + 1, zn fix,, /3 (1 + 2.5a +2a2 + 0.5a3)/A, C(x, A) ((1 + a)3x(1 x/2))/, h(x, A) + (a/2) (1 + a)2x + (1 + a)2x ande(rn, c) a small parameter depending on m and c.

System (8) can be considered as a perturbed discretelogistic equation with an extra term that can beinterpreted as a regulatory mechanism of harvest-ing or migration.Our next step is to study numerically the discrete

systems (5), (7) and (8) in order to compare theasymptotic behavior of these models and the effectof discretization.

3 NUMERICAL RESULTS

For the perturbed logistic equation the behaviorof asymptotic solutions are well known but thedynamics of their discretized counterparts are verydifficult to analyze (specially for higher ordermethods). We will give in the next section somedescription of the dynamics of the discrete modelsthat we propose above.

3.1 Constant Migration

In order to study the behavior ofthe systems (5) and(7) under constant migration, we set the value of cto be equal to zero. For system (5), the case m 0corresponds to the quadratic logistic equationmodel that has been studied extensively (see [3]).This case shows a typical period doubling sequenceto chaos. The case m > 0 have a similar behaviorbut the system becomes chaotic for smaller valuesof the growth rate. The figures that follow arebifurcation diagrams of the grow rate a againstpopulation densities. In these diagrams the firstthousand points are discarded to let a stable period(or chaos) becomes established and then we plotthe next hundred points for each value of a. The

0.6

0.4

0.2

1.6 1.8 2’.2 2.4 21.6 2A.8

u;ll .[, :,4::.," .Wit’!; I%; :.::"L’;’..:.m=O .........h’,; ’,il;:,’l ’,’,’,Uh,

=o.o8 ,--o.o m=O.02 ;!:!!,; +i’,:,’ ";i:: :ii: ::-!:::i! !iii.:.":". ............ .::::.:’. .... ,:...;,’.; ’L

:,"i::" ’::".:[

"’I :"..a :;:;.i"l,!!:,.;i’;..’;:,.;""..’,;i[": "’,

,i’li:’ ]"!:l

FIGURE Bifurcation diagram for free model with con-stant migration.

numerical simulations were carried out on a Sparcstation 5. The bifurcation diagrams move up andleft with m. This is shown in Fig. 1, where we presentseveral bifurcation diagrams of the system (5), withvalues of m taken as 0.08, 0.05, 0.0287, 0.0. Form > 0 the diagrams are truncated at the first bifur-cation point, only the case m- 0 is shown in full.

In contrast with the previous model which has a

homogeneous behavior, the exponential model pre-sents three different characteristics. For values ofm < 0.04 the behavior of the system is nonchaotic,the model starts with a single period limit followedby a double period before extinction; for values ofrn (0.0287, 0.4) there is a period doubling reversalfeature again without chaos (see [9]). Finally forrn < 0.0287 the behavior of the system becomeschaotic in a small interval of the rate growth. Thechaotic behavior for very small values of rn pres-ents a similar behavior as model (5), but the chaoticregion starts for larger values of the parameter a.

In model (5) chaos appears approximately at 2.57whereas in model (7) it appears well after 2.6. InFig. 2 we present several bifurcation diagrams ofthe exponential model, with the values of m takenas 0.08, 0.06, 0.04, 0.03. In this case all diagramsare shown completely.

In Fig. 3 we show how chaos is concentrated ina small region of parameter space, in this case we


m=O.06

m=O.03

m=O.04

1.6 1.8 2.2 214 216 2’.8

FIGURE 2 Bifurcation diagram for exponential model withconstant migration.

FIGURE 4 Bifurcation diagram for exponential model with-out migration.

0.5 ..................... ................ "’"’::;film........................% 118 212" 4 216 2.8

FIGURE 3 Bifurcation diagram for exponential model withconstant migration m 0.0287.

0.8

0.6

o.41

(?.21

---2---- ,.--".’’" --""" 0-07 .....’/-......

,’:-" ..---’% oo "..,.,..... ,,, ).....;., ;...’,.:,?1

/.. ....... ......:..:::" ./ "..{=. ;"4

,.....Z’, .

’/0.5 1,5 2.5

FIGURE 5 Bifurcation diagram for free model with cubicmigration.

took the value of m--0.0287. As m gets smaller thechaotic regime increases and the feature of perioddoubling reversal disappears, these facts are shownin Figs. 4 and 5 where we set the value of m equalto zero.

3.2 Nonconstant Migration

In order to study the behavior of the systems (5)and (7) under variable (cubic) migration, we set

the value of m equal to zero. Consider now thesystem (5) with m 0 and c > 0. We obtain a similarbehavior as in system (5) with c 0 and m > 0 ofa period doubling sequence to chaos. The appear-ance of chaos is controlled by the value of c. Forlarger values of c larger values of a are required forchaos to appear. This effect is shown in Fig. 5 wherethere are three diagrams corresponding to valuesof c equal to 0.69, 0.7 and 0.8. Again we only showthe case c--0.69 completely and the other two are


truncated at the first bifurcation point. In this figurewe notice how the value of c delays the appearanceof chaos. Another feature to note is the curvatureof the one period branches due to the nonlinearperturbation. Compare this effect with those givenpreviously with c 0 and m > 0.

For the exponential model with variable migra-tion, chaos is always present, in contrast with thecase of constant migration. Even though chaos ispresent in this model, its appearance starts farbeyond than its counterpart (free growth modelwith variable migration). Thus the value of c playsthe role of a delay of the appearance of chaos inboth models, but it has a stronger effect whenconsidering the model (7). In Fig. 6 we show threediagrams for the same values of c as in Fig. 5. Herewe note that the chaotic behavior starts for valuesof a close to four whereas in its counterpart startsapproximately at a- 2.5.

3.3 Migration with Control

The bifurcation diagrams shown in Fig. 7 corre-spond to the system (8) for different values ofe(rn, c). We show the complete diagram when e 0,and truncated diagrams at the first bifurcationpoint when e > 0. The dynamics of this system fore 0 are similar to those when e > 0.The system exhibits the same characteristic of

becoming chaotic for smaller values of the param-eter a when larger values of e are chosen. Recallthat the same effect was present in the case of freegrowth. In this case, we observe three importantnew features.

(1) The presence of a tangent bifurcation for allvalues of m.

(2) There is a parameter region where perioddoubling reversal is present, a feature that hasnot been documented for polynomial mapsbefore.

(3) Finally and more importantly is the fact thatchaos appears for larger values of a than thecases when Euler’s method was used.

3.5

2.5

..:::....:;.:" o:o.h’:

:;: c=O.7 il:,,:,,,"" ::::"::::: :; ;.7.,,..2

":.. "-.

FIGURE 6 Bifurcation diagram for exponential model withcubic migration.

-0.10.5 1.5 2.5 3.5

FIGURE 7 Bifurcation diagram for several values of e.

ALTERNATIVE METHOD OFDISCRETIZATION

From our previous numerical experiments we real-ize that the discretizations obtained are only validup to some value of the natural parameter. Thesame feature is present when we use higher orderRunge-Kutta methods as shown by the methodsabove.As we remarked before, a basic principle to ob-

tain a discrete model of population dynamics from


a system of differential equations is to apply a nu-merical method.The discrete map need not come from a single

numerical method. It is natural to pose the follow-ing problem.

Given a finite collection of numerical methodsfind a discrete map, better in the sense it extends therange of validity of asymptotic solutions.We propose a solution as follows.Let C be the family of continuous functions from

R into (0, 1) and let {..}/N__ be a finite collection ofmaps obtained by discretizations of one given dif-ferential equation. Choose /3; E C for i= 1,...,nwith the property that }-/3;(A)= 1. Associate thediscrete map given by

(9)

The choice ofthe weights (/3’s) may depend on theparticular differential equation. For instance if weuse the weights as constant functions the range ofvalidity improves at least in one hundred percent. Inparticular with the two Runge-Kutta methods thatwe use in the previous section and choosing/31 =/3(the weight for the improved Euler’s method) as thevalue that optimizes the function E(/3)= At., whereAt. is the value of the parameter where the discre-tization has its first bifurcation, a 160% improve-ment is achieved. We obtain in this case/3 =0.255and A.= 5.2 for m =0 and c=0. Compare thesevalues with the ones given in Fig. 7.

5 CONCLUSIONS

We have consid.ered the perturbed logistic equa-tion (4) to study population evolution with the aidof some discrete models associated with it. Ecolog-ical external factors, such as migration, were pre-sent as perturbation of the logistic equation. Suchperturbations terms resulted by considering a

higher order polynomial for the birth and deathrates as in (1) or by adding to the logistic equationitself the corresponding terms. We clarified that two

of the most popular discrete models associated with(2) are nothing but different discretizations of thelogistic equation. Thus we propose that if externalfactors are to be considered, they should be addedto the continuous equation and then discretizethe resulting equation with appropriate methods.Compare with [7] where the migration term isadded to (2).The dynamics exhibited by the discrete models

were very diverse. Phenomena like nonchaoticperiod doubling and period doubling sequence to

chaos, chaotic and nonchaotic period doublingreversal appeared. When chaos is present, we haveshown at least numerically that its appearancedepends on the method of discretization. Moreprecisely, for better methods, the interval of stabil-ity is larger and the appearance of chaos is delayed.We noticed that in some cases, chaos in the logisticequation can be controlled by appropriate per-turbations; in ecological terms, populations withchaotic behavior are stabilized if specific externalfactors are included.As observed in our study, the dynamics described

by the different discrete versions is similar withinan interval of confidence, but special care has to betaken for larger values of the parameters. Hence an

important conclusion of our study is that a singlediscrete model is not enough to make assertionsabout a population. Thus we suggest that to checkthe re-ability of a continuous model is convenientto check several versions of discretizations of themodel and base conclusions in the interval of confi-dence. Moreover, we may include all discretizationsin the discrete map proposed in (9).For further insight on this subject the interested

reader is refered to [2,4,5,6,8].

References

[1] D.F. Griffiths, P.K. Sweby and H.C. Yee, On spuriousasymptotic numerical solutions of explicit Runge-Kuttamethods. IMA J. Numer. Anal., 12, 319-338, 1992.

[2] R. Devaney, An Introduction to Chaotic Dynamical Systems,Redwood City, California, Addison-Wesley, 1985.

[3] F. Hoppensteadt, Mathematical Methods of PopulationBiology, Cambridge University Press, Cambridge, pp. 247-254, 1982.


[4] R. May, Biological populations with non-overlapping gen-erations: Stable points, stable cycles and chaos, Science, 186,645-647, 1974.

[5] R. May, Biological populations obeying difference equa-tions: stable points, stable cycles and chaos, J. Theor. Biol.,51, 511-524, 1975.

[6] R. May and G. Oster, Bifurcations and dynamic complexityin simple ecological models, Am. Nat., 111), 573-599, 1976.

[7] S. Sinha and S. Parthasaranthy, Behavior of simple popula-tion models under ecological processes, J. Biosci., 19, 247-254.

[8] S. Sinha and S. Parthasaranthy, Unusual dynamics ofextinction in a simple ecological model, Proc. Natl. Acad.Sci., 93, 1504-1508, 1996.

[9] L. Stone, Period-doubling reversals and chaos in simpleecological models, Nature, 365, 617-620, 1993.

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