Bulletin of Mathematical Biology (2010) 72: 432–443DOI 10.1007/s11538-009-9453-3

O R I G I NA L A RT I C L E

Extinction and Chaotic Patterns in Map Lattices UnderHostile Conditions

Vicenç Méndeza,∗, Daniel Camposa, Isaac Llopisb, Werner Horsthemkec

aGrup de Física Estadística, Departament de Física, Universitat Autònoma de Barcelona,08193 Bellaterra (Barcelona), Spain

bDepartament de Física Fonamental, Facultat de Física, Universitat de Barcelona,Martí i Franquès, 08028 Barcelona, Spain

cDepartment of Chemistry, Southern Methodist University, Dallas, TX 75275-0314, USA

Received: 17 April 2009 / Accepted: 14 August 2009 / Published online: 4 September 2009© Society for Mathematical Biology 2009

Abstract Population dynamics in spatially extended systems can be modeled by CoupledMap Lattices (CML). We employ such equations to study the behavior of populationsconfined to a finite patch surrounded by a completely hostile environment. By meansof the Galerkin projection and the normal solution ansatz, we are able to find analyticalexpressions for the critical patch size and show the existence of chaotic patterns. Theanalytical solutions provided are shown to fit, under the appropriate approximations, thedynamics of a logistic map. This interesting result, together with our discussion, suggeststhe existence of a universal class of spatially extended systems directly linked to the well-known characteristics of the logistic map.

Keywords Critical patch size · Extinction · Chaos · Coupled logistic map · Spatialpattern

1. Introduction

There is a long history of mathematical modeling in ecology and in particular in popu-lation dynamics. Theoretical models have been developed to study spatial and temporalbehavior such as biological invasions (Williamson, 1996), population genetics (Ewens,2004), predator-prey interactions (Crwoley, 1981), and many other ecological problems.When both space and time are considered to vary as continuous variables, the most appro-priate formulation is a system of reaction-diffusion equations (Okubo and Levin, 2002)or integro-differential equations (Cushing, 1977). For continuous space but discrete timeintervals, systems of integro-difference equations are often used (Kot and Shaffer, 1986),and when both space and time are modelled as discrete intervals, an appropriate structureis the CML (Kaneko, 1993).

∗Corresponding author.E-mail address: vicenc.mendez@uab.es (Vicenç Méndez).

Extinction and Chaotic Patterns in Map Lattices 433

Since the late 80s, CML have received considerable attention from physicists, non-linear scientists and researchers working in applied mathematics. The main reason isthat CML shed light on the dynamics of dissipative spatially extended systems. Theuse of discrete time models has a long history in ecology (Nicholson and Bailey, 1935;May, 1975) and such models have been applied to problems like two-species competition(Solé et al., 1992), predator-prey systems (Solé and Valls, 1992), chaos and oscillationsin ant colonies (Solé et al., 1993), chaos in planktonic dynamics (Bascompte et al., 1992),dispersal-driven instability in two-species models (White and White, 2005), and envi-ronmental heterogeneity and spatial structure in two-patches systems (Gyllenberg et al.,1993; Hastings, 1993; Kendall and Fox, 1998; Lloyd, 1995).

When populations grow and disperse in a finite habitat surrounded by a hostile environ-ment, a critical size for the habitat can exist below which the population cannot persist, soit becomes extinct. This problem has received much attention since the work by Kiersteadand Slobodkin (1953) and the critical patch-size has been found for different patterns ofdispersal and growth (see Méndez and Campos, 2008, and the references therein).

In this work, we study the critical patch size for populations described by CML withlogistic growth and predict the emergence of chaotic patterns. In exploring ecologicalphenomena with CML, standard problems such as invasions or spatial structures wereoften explored via numerical simulations only (Tilman and Kareiva, 1997). In this work,we provide both analytical and numerical results, where the analytical prediction is madeby using the Galerkin projection method (which has already been proved useful beforefor similar models, Méndez and Campos, 2008) and the normal solution ansatz. Thesemethods allow us to convert the CML into a single logistic discrete-time map zn+1 =μzn(1 − zn). The bifurcation parameter μ for this map will be related to the intrinsicgrowth rate and the patch size, which allows to characterize completely the populationdynamics in terms of the well known behavior for the logistic map.

2. Coupled map lattice

The CML depicts a discrete patch or habitat that contains resources for populations belowa certain concentration. The habitat is finite with size L and the spacing between latticepoints is denoted by l, so the number of sites in the patch is 1 + L/l. The individuals ateach site x (note that more than one individual is allowed to be at a single site) reproduceaccording to the nonlinear map ρt+1(x) = f [ρt (x)], where x = kl with k a natural num-ber, and then they either remain at their site or move to a neighboring site. ρt (x) denotesthe density of individuals at time t at a given site and f [·] represents the growth functionfor the population.

Let us define p as the fraction of individuals that remain in their current site and 1 −p

the fraction of individuals that leave the currently occupied site to a neighboring site, thenthe balance equation for the population density ρt (x) at time t at position x is

ρt+1(x) = pf[ρt (x)

] + 1 − p

2f

[ρt (x − l)

] + 1 − p

2f

[ρt (x + l)

]. (1)

Note that Eq. (1) describes a dynamics consisting of a growth term for population followedby the dispersal of individuals to one of the nearest neighbors. If the patch is surrounded

434 Méndez et al.

by a completely hostile environment the prescribed boundary conditions are

ρt (x = 0) = ρt (x = L) = 0. (2)

The reproduction term is assumed to follow a discrete logistic growth at each site(Hastings, 1993; Lloyd, 1995; Kendall and Fox, 1998),

f[ρt (x)

] = (1 + r)ρt (x) − r

Kρt (x)2, (3)

where r is the intrinsic growth rate per generation and K is the carrying capacity. Asis widely known, the local population dynamics in (3) possess two stationary states,i.e., ρt (x) = 0 (unstable) and ρt (x) = K (stable if 0 < r < 2). In the remaining of thepresent work, the capacity K will be assumed to be 1 for simplicity. Equation (1) with theboundary conditions defined in (2) and the initial condition ρ0(x) specify the CML westudy in this paper. We have chosen for all the numerical computations performed hereρ0(x) = A sin πx

Lwith A a constant; however, we stress that our results hold for any ρ0(x)

positive in the interval 0 < x < L. The objective of this work is to analyze the stationarysolutions to the problem (1–2) and their stability. Our results will provide useful and in-teresting ecological information for persistence/extinction of populations living in finitehabitats surrounded by completely hostile environments.

2.1. Stationary patterns

The Galerkin projection method (Kantorovich and Krylov, 1958) consists of choosing anorthogonal basis of functions {φi(x)} which satisfy the boundary conditions φi(x = 0) =φi(x = L) = 0. Seeking a solution of the Fourier series form

ρt (x) =∞∑

n=1

ϕn(t)φn(x), (4)

and substituting into (1), one finds a hierarchy of discrete equations for ϕi(t) after takingthe projection

L/l∑

k=0

L[ρ]φm(kl) = 0, (5)

with L[ρ] = ρt+1(x)−pf [ρt (x)]−(1−p){f [ρt (x + l)]+f [ρt (x − l)]}/2. It is importantto point out that ϕ1(t) is expected to be the leading order term in the above expansion. Tosatisfy the boundary conditions (2), we consider φn(x) = sin(nπx/L).

Introducing (4) into (1) and taking into account the projection (5) and (3), one finds

L/l∑

k=0

∞∑

n=1

ϕn(t + 1) sin

(nπkl

L

)sin

(mπkl

L

)

= (1 + r)p

L/l∑

k=0

∞∑

n=1

ϕn(t) sin

(nπkl

L

)sin

(mπkl

L

)

Extinction and Chaotic Patterns in Map Lattices 435

− pr

L/l∑

k=0

∞∑

n=1

∞∑

n′=1

ϕn(t)ϕn′(t) sin

(nπkl

L

)sin

(n′πkl

L

)sin

(mπkl

L

)

+ (1 − p)(1 + r)

L/l∑

k=0

∞∑

n=1

ϕn(t) sin

(nπkl

L

)cos

(nπl

L

)sin

(mπkl

L

)

− (1 − p)r

L/l∑

k=0

∞∑

n=1

∞∑

n′=1

ϕn(t)ϕn′(t) sin

(nπkl

L

)sin

(n′πkl

L

)cos

(nπl

L

)

× cos

(n′πl

L

)sin

(mπkl

L

)

− (1 − p)r

L/l∑

k=0

∞∑

n=1

∞∑

n′=1

ϕn(t)ϕn′(t) cos

(nπkl

L

)cos

(n′πkl

L

)

× sin

(nπl

L

)sin

(n′πl

L

)sin

(mπkl

L

).

Neglecting higher order terms, like ϕ1ϕ2, ϕ1ϕ3, . . . , ϕ2ϕ3, . . . , ϕ22 , . . . and performing first

the sum over k one gets the equation for ϕm(t):

ϕm(t + 1) = (1 + r)M1(m)ϕm(t) − rM2(m)ϕ1(t)2, (6)

where

M1(m) = p + (1 − p) cos

(mπl

L

),

M2(m) = M1(m)l

L

sin2( πlL

)[1 + cos(mπlL

)][1 − (−1)m]sin(mπl

L)[1 + cos(mπl

L) − 2 cos2( πl

L)] .

(7)

The normal solution ansatz (Harris, 1971) consists in assuming that the time dependenceof all the coefficients ϕk(t) with k > 1 is through their functional dependence on ϕ1(t),that is,

ϕm>1(t) = ϕm

[ϕ1(t)

] = amϕ1(t) + bmϕ1(t)2 + · · · . (8)

The accuracy of this method requires that limϕ1→0 ϕk>1/ϕ1 = 0, a condition that we shallprove as follows. Since, as noted above, we consider small densities, the solution will bewell approximated outside the initial time layer by ϕ1(t). Inserting the expansion (8) into(6) and using (7), one finds

am>1 = 0,

bm>1 = rM2(m)

(1 + r)[M1(m) − (1 + r)M1(1)2](9)

436 Méndez et al.

by equating the coefficients of ϕ1(t), ϕ1(t)2, . . . at each side of Eq. (6). Then ϕm>1(t) =

bmϕ1(t)2 + · · · and ϕm>1/ϕ1 → 0 as ϕ1 → 0. The approximated solution to (1) is finally

ρt (x) � ϕ1(t) sin

(πx

L

)+ ϕ1(t)

2∞∑

m=2

bm sin

(mπx

L

). (10)

To find the stationary patterns, emerging from the initial conditions, we set ϕ1(t) = ϕ∗

(constant) into (6). The two different solutions are

ϕ∗ = 0,

ϕ∗ = (1 + r)M1(1) − 1

rM2(1),

(11)

for extinction and survival states, respectively. So that, from (10) and (11), the stationarypatterns are the extinction pattern ρ∞(x) = 0 and the survival pattern, which is given by

ρ∞(x) � (1 + r)M1(1) − 1

rM2(1)sin

(πx

L

)+

[(1 + r)M1(1) − 1

rM2(1)

]2 ∞∑

m=2

bm sin

(mπx

L

).

(12)

In Fig. 1, we plot the stationary pattern (12) reached by the population when conditions forsurvival are met. Our theoretical predictions are depicted with squares and are comparedto numerical computations drawn with circles, showing a very good agreement.

Fig. 1 Population survival pattern for r = 0.1, p = 0.2, and L = 11l. With circles are depicted numericalcomputations directly performed on (1) and with squares joined with dotted lines we depict theoreticalpredictions from Eq. (12). The initial condition used is ρ0(k) = 0.5 sin(πkl/L).

Extinction and Chaotic Patterns in Map Lattices 437

2.2. Critical patch size

To check the stationary states stability, we define ϕ1(t) = ϕ∗ + ε(t), with ε(t) � 1. In-serting this definition into (6) and linearizing, we get

ε(t + 1) � [(1 + r)M1(1) − 2rM2(1)ϕ∗]ε(t), (13)

hence, the stability condition reads

(1 + r)M1(1) − 2rM2(1)ϕ∗ < 1. (14)

The extinction state ϕ∗ = 0 is stable if (1 + r)M1(1) < 1 and unstable otherwise. Thenthe equation (1 + r)M1(1) = 1 determines the critical patch size Lc as

Lc

l= Int

{π

arccos[( 11+r

− p) 11−p

]}, (15)

where Int{·} denotes the integer part. With this definition, the extinction state is stable ifL < Lc and unstable otherwise.

It is not difficult to show from (15) that r is a monotonically decreasing function ofLc , i.e., it reaches its maximum value at Lc = 2l. Note that 2l is, in fact, the minimumpossible patch, i.e., a patch composed by three sites: two boundaries and one site in themiddle. In Fig. 2, we plot the diagram p versus L/l for the regions of extinction andsurvival. Let pc the maximum value of p corresponding to the critical curve (15) when

Fig. 2 Diagram of p in front of L/l for r = 0.1. With circles are depicted numerical computations directlyperformed on (1) with the initial condition ρ0(k) = 0.5 sin(πkl/L) and with squares joined with dottedlines we depict theoretical predictions from Eq. (15).

438 Méndez et al.

Fig. 3 Forward bifurcation for the density ρ∞(L/2) versus L/l for r = 0.1. With circles are depictednumerical computations directly performed on (1) with the initial condition ρ0(k) = 0.5 sin(πkl/L) andwith squares joined with dotted lines we depict theoretical predictions from Eq. (12).

Lc → 2l (see Fig. 2). Then if

1

1 + r= pc < p < 1 (16)

the population survives, regardless of the patch size. The practical implication of thisresult is clear: if the sedentary population at each generation is higher than the fractionof newborn individuals in this period, then the population survival is guaranteed for anypatch size. This result establishes a strategy for a population to prevent extinction whenit lives surrounded by a hostile environment. In addition, one can conclude that, beingL∗

c = limp→0 Lc, then if

L > L∗c = l Int

{π

arccos( 11+r

)

}, (17)

the population survives, regardless of the patch size and of the fraction of sedentary indi-viduals. In Fig. 3, we plot a bifurcation diagram for the density at the central site of thepatch as function of the patch size, where a forward bifurcation is found both analyticallyfrom (12) and numerically.

2.3. Chaotic patterns

As we have seen above, the temporal behavior of ρt (x) is governed essentially by ϕ1(t).As Eq. (6) is a nonlinear discrete time map, it may present chaotic behavior. In fact,defining zt+1 = r

1+r

M2(1)

M1(1)ϕ1(t), Eq. (6) turn into the logistic map zt+1 = μzt(1 − zt )

and, in consequence, the dynamic properties of (6) are linked to the logistic map with

Extinction and Chaotic Patterns in Map Lattices 439

bifurcation parameter μ = (1 + r)M1(1). Since chaos for the logistic map begins whenμ = μc = 3.569945 (Ott, 2002) the CML with the boundary condition (2) predicts chaoticspatial patterns, i.e., chaotic persistence, when

p ≥ 1

1 + r

μc − (1 + r) cos( πlL

)

1 − cos( πlL

). (18)

As our model is spatially extended, one can compute in fact a set of Lyapunov expo-nents, one for each lattice node. If i = 0,1, . . . ,L/l, one can compute up to L/l Lyapunovexponents from

λi = 1

tmax

tmax∑

t=1

ln|ρt+1(i) − ρ̃t+1(i)|

|ρt (i) − ρ̃t (i)| , (19)

where the tilde indicates that the corresponding density evolves from a slightly differentinitial condition.

The minimum value in (18) is plotted in Fig. 4. In agreement with (18), the pointsabove the critical curve correspond to chaotic persistence patterns while the points fallingbelow the critical curve correspond to deterministic persistence patterns. In order to facil-itate understanding on the appearance of these chaotic patterns reported, we also show inFig. 5 the time evolution of the density at the center of the patch ρt (L/2) (left) and theprofile ρt (x) for two-consecutive time steps from our numerical simulations (right).

We have thus found two critical curves in the plane p versus L/l separating extinc-tion/persistence (15) and deterministic persistence/chaotic persistence (18). Taking into

Fig. 4 Critical curve for a chaotic/deterministic persistence spatial pattern for r = 3. Squares joined bydotted lines represent theoretical predictions computed from (18) while circles represent numerical simu-lations of (1) for which the Lyapunov exponent changes its sign. In our computations, t = 3 × 105 gener-ations, ρ0(k) = 0.5 sin(πkl/L) and ρ̃0(k) = (0.5 + 10−7) sin(πkl/L).

440 Méndez et al.

Fig. 5 Representation of the chaotic dynamics obtained numerically from our CML with the initial con-dition ρ0(k) = 0.5 sin(πkl/L). Left panel: Evolution of the population density at the center of the patch(x = L/2) as a function of time for L/l = 50, p = 0.2 and r = 0.1 (circles) or r = 3 (squares). Rightpanel: Density profiles obtained at two consecutive time steps after the asymptotic regime has been reached(L/l = 50, p = 0.2 and r = 3).

account (15) and (18), we see that, if r < μc −1 = 2.569945, the p-L/l diagram has onlyone critical curve (15) and population can extinct or persist deterministically as shown inFig. 2. If r > μc −1 = 2.569945, the p-L/l diagram has also only one critical curve (18),population always persists, but the persistence can be deterministic or chaotic as shown inFig. 4. In addition, when r > μc − 1 we can also establish the following general results:if p > μc/(1 + r) population persists chaotically regardless of the patch size and if L > l

Int{ π

arccos( μc1+r

)} population persists chaotically regardless of the sedentary population frac-

tion. The bifurcation diagram depicted in Fig. 6 shows the density at the center of thepatch x = L/2 for different values of the intrinsic growth rate r . It displays the differentpossible stable states available for the population, starting by a single state for small r .After that, the period doubles in cycles of 2, 4, and 8. However, at r > 2.6, the trajectorybecomes chaotic, although for certain values of r, there are deterministic states where thesystem oscillates in odd cycles.

3. Conclusions

We have investigated the dynamics of reactive and dispersive populations in discrete en-vironments using CML surrounded by a hostile environment, so that the survival of theindividuals is not possible beyond a certain point of the lattice. Similar to the continuouscase, which has been extensively studied before (Kot, 2001), there exists a critical size ofthe lattice above which the survival of the population is possible while, below that critical

Extinction and Chaotic Patterns in Map Lattices 441

Fig. 6 Bifurcation diagram obtained numerically for the chaotic behavior at the center of the patch(x = L/2) as a function of the growth rate r in the asymptotic regime (t → ∞) (L/l = 12, p = 0.2,

and ρ0(k) = 0.5 sin(πkl/L)).

size, the effects of the hostile environment leads the global population to extinction. Us-ing the Galerkin method and the normal solution ansatz, the critical size of the lattice andthe concentration profile for the case of persistence have been determined analytically, sothat the regions of survival/extinction are fully characterized. These results, though basedon approximate methods, provide an excellent approach to the numerical solutions, asshown in Figs. 1 and 2. These results are of special interest since models based on CMLhave been often explored only from numerical approaches (computer simulations), so wehope that the possibility of introducing such analytical techniques for the study of thesesystems can be positively appreciated by the researchers in this area.

In the last part of the article, we have focused on the chaotic properties of the system.While the chaotic patterns of CML have been broadly studied before (Lemaitre and Chaté,1999), we have shown here that for the case of hostile boundary conditions the model canbe reinterpreted as a logistic map zn+1 = μzn(1 − zn) with a bifurcation parameter μ

which is obtained explicitly from our approach. As a consequence, the threshold betweenthe region of deterministic persistence and chaotic persistence can be easily identified,showing again a good agreement with simulations.

In general, our work provides a new insight into the old field of population dynam-ics within spatially extended heterogeneous environments, and more specifically to theproblem of persistence as a function of the patch size. From the point of view of islandbiogeography and habitat fragmentation, models based on hostile conditions represent aquite useful approach in order to understand persistence (Cantrell and Cosner, 2001). Inagreement with similar previous works and intuitive arguments, the critical patch size inour model is reduced as the local population growth rate (r) becomes larger, and it isincreased as a consequence of increased dispersal probability (p). However, the fact thatthe evolution dynamics of CMLs can be linked to the well-known logistic map is a resultwhich has not been reported before, to the best of our knowledge. This result is of greatecological interest, as it could even suggest the existence of a universal class of spatiallyextended models directly linked to the logistic map, and so the dynamics of many differentpopulations could be analyzed from a common perspective.

442 Méndez et al.

It remains to be seen if all these ideas can be extended to the case where more complexdispersal kernels are considered for the dispersal of individuals (here we have restrictedourselves for simplicity to the case of interactions between nearest neighbors). Also, westress that our method is not restricted to the logistic growth model we have used herefor the sake of simplicity. For example, the local population dynamics could also be rep-resented by any polynomial growth term, as bistable terms usual in the analysis of theAllee effect (Lewis and Kareiva, 1993). Nonpolynomial terms, like those appearing in theRicker model (Ricker, 1954) or the Hassell and Beverton–Holt models (Brännström andSumpter, 2005) could also be considered, provided that one performs first a polynomialexpansion of the corresponding equations in order to obtain a hierarchy of polinomialequations for the functions ϕi(t), as defined in Section 2.1. Finally, systems of differenceequations could be taken into account, also; this would just involve writing the hierarchyfor ϕi(t) in a matrix form, and then solve again the problem neglecting those terms withϕi>1(t). This shows the wide range of applicability of the tools presented here.

Acknowledgement

This research has been partially supported by the Grants Nos. FIS 2006-12296-C02-01,SGR 2005-00087 (VM).

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