Spreading Speeds and Traveling Waves

in Cooperative Systems

Bingtuan Li

University of Louisville

Collaborators

Hans F. Weinberger

University of Minnesota

Mark A. Lewis

University of Alberta

• Hans F. Weinberger, Mark A. Lewis, andBingtuan Li, Analysis of linear determi-

nacy for spread in cooperative models,

Journal of Mathematical Biology, 45 (2002),

183-218

• Mark A. Lewis, Bingtuan Li, and Hans F.Weinberger, Spreading speed and linear

determinacy for two-species competition

models, Journal of Mathematical Biology,

45 (2002), 219-233

• Bingtuan Li, Hans F. Weinberger, andMark A. Lewis, Spreading speeds as slow-

est wave speeds for cooperative systems,

Mathematical Biosciences, 196 (2005), 82-

98

Fisher’s Model (1937)

ut = ru(1 − u) + duxx

where

u(x, t) = Mutant alleler = Intrinsic growth rated = Diffusion coefficient

Two constant equilibria:

u = 0 (unstable), u = 1 (stable)

Spread with Fisher’s Model

• Step function initial data of the form 1−H(x) converges to a rightward moving

wave with speed

c∗ = 2√

rd.

Kolmogorov et al. (1937)

• Compact initial data expands at speed c∗

C *C*

1

• c∗ is the slowest traveling wave speedTraveling wave: u(x, t) = u(x − ct), c ≥ c∗

Aronson and Weinberger (1975, 1978)

Integro-difference models

un+1(x) = Q(un)(x) =∫ +∞

−∞k(x−y)f(un(y))dy

where k is the dispersal kernel and f is the

growth function.∫ +∞−∞ k(x)dx = 1.

k (x)

f (u)

Assume f(u) ≤ f ′(0)uMoment generating function of k:

K(s) =∫ +∞

−∞{esuk(u)}du

Spreading speed c∗ = infs>0

{1sln(f ′(0)K(s))}

Traveling wave: un(x) = w(x − cn), c ≥ c∗

Weinberger (1982)

Linear determinacy

Nonlinear models:

ut = f(u) + uxx

un+1(x) =∫ +∞

−∞k(x − y)f(un(y))dy

Linearized models:

ut = f′(0)u + uxx

un+1(x) =∫ +∞

−∞k(x − y)f ′(0)un(y)dy

f(u) ≤ f ′(0)u

Linear determinacy:

Propagating speed of nonlinear system =

propagating speed of the linearized system.

110 .

-

Invasion of competing species Competition between grey and red squilTels . 111, ,

The North-American "greysquirrel (Sciurus carolinensis)was released,from various sites in Britain around the turn of this century; Althoj.lgi!;iBritain had, as.an indigenousspecies,'.the red squirrel (Sciurusvulgaris),i!1began disappearing from are.as which the introduced grey squirrel, had'invaded and where it had successfully spread its range (see Fig. 6.3)'

(a) major poinlS 01 inl,oducl,on0 from 1676 10 1929,II prc-1920 disl.ibul.ionum 1~20-1930

~ 1930-19441'15

~ 1944/45-1952131952-1955

. 19~5~1959

~ 1~5~"1971

,~,tMacK.innon,1978; Lloyd, 1983; Reynolds, 1985; Williamson and Brown,

~\;1986).The native red .squirrel is at a clea.r competitive disadvantage, with~~;pnlyhalf the. body weight ~f the grey squIrrel and a lower growth rate. In~iNorth Amenca both species are present, though they occupy separate~I;l.iches,with the red squirrel living mainly in the northern conifer forests~L~d the grey squirrel mostly in mixed hardwood forests. In Britain,~; however, the absence of the grey squirrel had allowed the indigenous red~;squirrel to adapt to mixed hardwood forests as well to occupy an extensive

I I

iii

}

',"'" ---

III[]

}1959 1944 -

1971

ft~jl'II.::"::'::~.f i~:'1

P~..t..::,~,;.;.: ,.-::.

.

,..

~...

...::...\ i: iW, : '.,:'

Fig.6.3 (a) Spread of grey squirrel in England from 1920 to 1971. (b) Distributiondecline of the red squirrel from 1944 to 1971. (After Lloyd, 1983,)

-",

:~i1-: ""

116 '. Competition for, open space Competition for largeopen spaces. 117.-

--

(f)

0 4O0IImt---;~.

21

22-16

0 4OOIImf i

~.

(c) 7.1 Geographic spread of several tree species through North America after. last ice age (after Davis, 1981):(a) spruce; (b) oak; (c) white pine; (d) hemlock;c~)beech; (0 'chestnut. The numbers refer to the radiocarbon age (in thousands of~) of the first appearance of the respective species at the site after 15,000 years1. Isopleths were drawn to connect points of similar age. The stippled areaspresent the modem range for that species.r .",'.

" lmine this problem using the following set of equations, which isntical to eqns (6.1):

,',

ant a2nt

at' =D1 ax2 + (81 - JLunl :...JLI2n2)nl'an2 . a2n2

-at=D2 ax2 + (&2- JL21nl - JL22n2)n2'rhere we denote by nl(x, t) the population density of the earlier invading~cies 1, and by n2(x, t) that of the later arriving species 2. We arefterested in the case when, compared with species 1, species 2 is 'weaker"'.mpetitively but has a higher diffusive capability' (i.e., it is a fugitive

occupied by'a-resident species;'the former must be cOplpetitivelystronger~ " cies). Thus, the competitive relationship satisfies (Hi) of (6.4), and thethan the latter in order to :;\J~essfuflyestablish' fts.elf. However, if two ~sion coefficient of species 2 is higher than that of species 1:species consecutivelyinvade an opep.area and the earlier species is still in ~- 81 82 82 81the process of eWIUJ,(iingits range"it ro.ilYbe possible for a competitively;, - > -,. ,.,..,,--< -'-, and D1 < D2.weaker species arriving later to successfullyspread its range by reaching '.: JLu JL21 JL22 JL12residual open spacesbefore the earlier invadingspeciesoccupiesthe whole 'lie assume here that before species 2 is introduced, species 1 is the sole.area. How large must the d~ive capabilityof the competitivelyweaker~vader. Thus the population dynamicsof species 1 can be expressed bysoecies be in' order for it to successfully.reach the open space? We-~uing n2= 0 in the first equation of (7.1).Because this is the same as the

(7.1)

0 4OCiIun. >--:-:--<

~. 0 . 4OOIImt---;

(7.2)

0

Competition for large open space

u

,-...I \I II II II I

v , II II II II II II \

I \, ...

u

v".--

/~ "/. "

I C ~/ 2 \

, ' \

/'~ \, ,~

106 . lnvasion of competing speciesn2 (a) n2 (b)

c:/,u22~~2~"'" !:L

c:J.U 121\.'\ ,....

~ ""'"

~4:JE1 '.

c:l,ull c:2/,u2!

!:L

~ A:~...,;;:......~

0 Eo Eo Elnl n{

n2 (c) (d)n2

E2 r.~.~ ""\...

\

... EJ

(

!:L

E2 r..".

"'" ~".

". ,". ~

~~

!:L

.-,.

Eo Eo El nlEl nl

Fig. 6.1 Phase plane diagrams of Lotka-'-Volterra competition model. Solid lines,null clines of species 1; dotted lines, null clines. of species 2; solid arrows, typicaltrajectories. Four possible cases are: (a) species 2 always wins; (b) both speciescoexist; (c) species 1 always wins; (d) either species 1 or species 2 wins dependingon initial conditions. Circles, unstable equilibria; solid dots, stable equilibria.

The null clines divide the (nl' n2) plane into regions where dnl/dt ordn2/dt is either positive or negative; within each region the direction inwhiChnl and n2 change with time is indicated by a short arrow. Thus, thephase plane trajectory can be roughly depicted by following the arrow'sdirection. Typical trajectories are drawn by the solid arrows in Fig. '6.1: anytrajectory approaches equilibrium point E2 for (a), EJ for (b), and El forcase (c), while in case (d) trajectories reach either El or E2' depending onthe initial point. These results are reinterpreted in terms of competition:

81 82 82 810) when - < -, - > -, only species 2 wins;

JLll JL2l JL22 1L12 '

81 8, B, BlOi) when - < ~, ~ < -,

. ILIl 1L21 1L22 1L12, Bl B2 8,. 81

Wi) when - > -, ~ < -,ILIl fJ-21 1L22 1L12 '

. BI 82 82 BI(IV) when ->-, ->-

ILIl 1L21 1L22 1L12

depending on the initial condit

Since we are interested in whether2 can become established in an ar,determine whether n2 will increas(equilibrium point El:(8lIJLll'0) atinspection of Fig. 6.1 reveals that n2case 0), the invading species outcondisplacing it, while in case 00, the scpoint given by E3' where the campspecies coexist.

~,c~,

6.3 Retreat of resident sped

Returning to eqn (6.1), we now exinvading and resident species are be

First, we assume that the entire a(species 1) so that its population deni(x,O) = 8111L11at all points. lnt

, species 2 invade the vicinity of thethe invasion to be successful the (number. Mathematically this ffiI(nlx), n2(x» = CBll JLll'0) of eqn fsome simple analysis, it can be shovthe same as in the case where nGO of (6.4).

We numerically solved eqn (6.mentioned above and found that fl

. proceeds as shown in Fig. 6.2(a) ~spread of the invading species (spewave of constant speed whose fronthe Fisher model. The rear of thetwo cases: in (a), the invading' spone, whereas in (b), while the resi(tion between the two species evenas given by (6.3), allowing them

.r.,"

.i,

..~.1

~1

----- ---

Spread of invading species u in a region

pre-occupied by species v

v

v

21a < 1, a 1, a

Observations

• Spreading speed of the linearized system

c̄ = 2√

r1d1(1 − a1)

• Numerical experiments (Hosono, 1998)

c̄ is the spreading speed of the full sys-

tem when d2/d1 is small and a1a2 < 1,

but not when d2/d1 is large or d2/d1 is

small and a1a2 is large

• Allee effect introduced by the invadingspecies

General recursions

un+1 = Q[un]

i. Q is cooperative: if u ≥ v, then Q[u] ≥Q[v].

ii. Q[0] = 0 and Q[β] = β (β >> 0) which is

minimal. The number of constant equi-

libria ν with 0 ≤ ν ≤ β is finite. β is aglobally stable.

iii. Q is translation invariant: Q[Ty[v]] = Ty[Q[v]]

for all y.

iv. If vn(x) converges to v(x) uniformly on

every bounded set, then Q[vn] converges

to Q[v], uniformly on every bounded set.

v. Every sequence vn(x) has a subsequence

vnℓ such that Q[vnℓ] converges uniformly

on every bounded set.

Definitions of spreading speeds

Define

an+1(c; x) = max{a0(x), T−c[Q[an(c; ·)]](x)}.

Initial function a0(x): each component is non-

increasing in x and vanish for x ≥ 0; 0

Spreading speeds in cooperative

reaction-diffusion systems

Let Qτ [u0] be the value u(x, τ) of the solu-

tion at time τ . Then un(x) = u(x, nτ) satis-

fies un+1(x) = Qτ [un](x). Suppose that for

each τ the recursion has a spreading speed

c∗τ . Then c∗τ = τc∗1.

Theory on spreading speeds

Theorem 1 (i) No component spreads at a

speed less than c∗ and at least one componentspreads no faster than c∗.(ii) No component spreads at a speed more

quickly than c∗f and at least one componentspreads no more slowly than c∗f .

Remark 1. c∗ ≤ c∗f ≤ c∗+

Remark 2. If there are only two equilibria

0 and β, then c∗ = c∗f = c∗+

Conjecture 1. In general c∗f = c∗+

Remarks on spreading speeds

i. If there are only two equilibria 0 and β, all

species spread at the same speed

ii. If there is an extra equilibrium, species may

move at different speeds.

iii. The two-species Lotka-Volterra competi-

tion model can be converted into a co-

operative system that has two or three

equilibria in the relevant region, depend-

ing on parameter values.

Linear determinacy for recursions

Assume that the moment generating matrix B(µ) isa block lower triangular (Frobenius form) in which allthe diagonal block are irreducible. Let λ1(0) be theprinciple eigenvalue and ζ(0) a principle eigenvector ofB(0). Assume λ1(0) > 0 and ζ(0) >> 0. Let

c̄ := infµ>0

[µ−1 lnλ1(µ)] = µ̄−1 lnλ1(µ̄).

Theorem 2 Assume that either

a. µ̄ is finite,

λ1(µ̄) > λσ(µ̄) for all σ > 1,

and

Q[e−µ̄xζ(µ̄)] ≤ M [e−µ̄xζ(µ̄)];or

b. there is a sequence µm ր µ̄ such that for each m,the above inequalities with µ̄ replaced by µm hold.

Then c∗ = c∗f = c̄

Linearization of Reaction-diffusion

systems

The matrix B(µ) for the time 1 map is given

by

B(µ) = exp [C(µ)] ,

where

C(µ) = diag(

diµ2)

+ diag (eiµ) + f′(0).

Assume that γ1(0) is the principle eigenvalue

and ζ(0) is a principle eigenvector of C(0).

PDE version of linear determinacy

Let

c̄ := infµ>0

[γ1(µ)/µ] = [γ1(µ̄)/µ̄].

Theorem 3 Assume that either

(a) µ̄ is finite,

γ1(µ̄) > γσ(µ̄) for all σ > 1,

and

f(ρζ(µ̄)) ≤ ρf ′(0)ζ(µ̄)

for all positive ρ;

or

(b) There is a sequence µν ր µ̄ such thatfor each ν the above inequalities with µ̄

replaced by µν are valid.

Then c∗f = c∗ = c̄.

Characterization of c∗ as the slowestspeed of a class of traveling waves

Theorem 4 If c ≥ c∗, there is a nonincreas-ing traveling wave solution W of speed c with

W(−∞) = β and W(∞) an equilibrium otherthan β. If c < c∗, there is no traveling waveW with W(−∞) = β and W(∞) 6= β.

Spreading speed of the Lotka-Volterra

competition model

ut = d1uxx + r1u(1 − u − a1v)

vt = d2vxx + r2v(1 − v − a2u).

Suppose that a1 < 1, and that

r1(2d1 − d2)(1− a1) ≥ max{d1r2(a1a2 − 1),0}

is satisfied. Then c∗+ = c∗f = c̄1 = 2

√

d1r1(1 − a1)

I

Traveling waves in the Lotka-Volterracompetition model

"

If 0 < al

Discrete-time competition model

un+1(x) =∫

R1(1 + ρ1)un(x − y)

1 + ρ1(un(x − y) + α1vn(x − y))

k1(x − y)dy,

vn+1(x) =∫

R1(1 + ρ2)vn(x − y)

1 + ρ2(vn(x − y) + α2un(x − y))

k2(x − y)dy.