Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol165.Issue1.May2000:

E€ect of aggregating behavior on population recovery on a set

of habitat islands

V

õctor Padr

on

*

_{, Mar}

_{õa Cristina Trevisan}

Departamento de Matematicas, Facultad de Ciencias, Universidad de Los Andes, M erida 5101, Venezuela Received 10 December 1997; received in revised form 15 December 1999; accepted 6 January 2000

Abstract

We consider a single-species model which is composed of several patches connected by linear migration rates and having logistic growth with a threshold. We show the existence of an aggregating mechanism that allows the survival of a species which is in danger of extinction due to its low population density. Numerical experiments illustrate these results. Ó _{2000 Elsevier Science Inc. All rights reserved.}

MSC:92D25; 34A34

Keywords:Recovery; Allee e€ect; Aggregation; Persistence

1. Introduction

Many animals in nature enhance their chances of survival and reproduction by increasing their local density, or aggregating (cf. [1±3]). Perhaps the simplest theoretical example of a functional purpose of aggregation occurs when the net population change from birth and death exhibits the Allee e€ect ([4]). The Allee e€ect was originally proposed to model the reduction of reproductive opportunities at low population densities. A population may exhibit the Allee e€ect for a variety of reasons, see [5] and the literature therein.

The main feature of the Allee e€ect is that below a threshold value the death rate is higher than the birth rate because, on average, the individuals cannot reproduce successfully. This, in turn, imposes a threat to the survival of the species at low densities.

We will show in this paper that a set of habitat islands, interconnected by anisotropic and asymmetric density-independent dispersal rates, contains mechanisms for the survival of a

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*_{Corresponding author. Tel.: +58-74 401 409; fax: +58-74 401 286.}

E-mail addresses:padron@ciens.ula.ve (V. PadroÂn), trevisan@ciens.ula.ve (M.C. Trevisan).

population menaced by extinction due to the Allee e€ect. In fact, we will give conditions for a population located in a neighborhood of low density, to cluster together in an attempt to raise their local population above the threshold at which the birth rate begins to exceed the death rate. We give the name ofrecovery, properly de®ned below, to this kind of behavior.

The concept of recovery, ®rst used without this name by Grindrod [6], was introduced by Lizana and Padron [7] to study this type of phenomenon and can also be seen as a measure of the capability of a model to produce aggregation.

In this paper, we study an island chain model for a single species described by a system of di€erential equations

u0_i…t† ˆX

n

jˆ1

‰dijuj…t† ÿdjiui…t†Š ‡f…ui…t††; iˆ1;. . .;n; tP0: …1†

Here ui…t† is the population in patch i at time t. The coecients dij are the rates at which the individuals migrate from patch j to patch i, and satisfy the following conditions: (i) dijP0,

i;jˆ1;. . .;n; (ii)Pn

jˆ1dij>0 and

Pn

jˆ1dji>0, iˆ1;. . .;n.

The functionf…u† that describes the net population change from birth and death satis®es the following hypothesis:

Hypotheses 1.

1. f is continuously di€erentiable.

2. There exists 0<a<b<1 such that f is negative in …0;a† [ …b;‡1† and positive in…a;b†. 3. f…0† ˆ0 and there exists a constantM >bsuch thatÿf…u†>d1u, for alluPM. Hered1 is a

positive constant such that Pn

jˆ1dij<d1 and

Pn

jˆ1dji<d1 , iˆ1;. . .;n.

The second condition above states that the population is subject to an Allee e€ect. The ®rst condition is technical and is needed together with the third condition to ensure, given initial data, the existence and uniqueness of solutions for (1) globally de®ned for tP0 (see Proposition 1 below). The third condition assumes, as should be expected, that the migration mechanisms cannot override the carrying capacity of the environment, represented by the parameterb, beyond a ®xed value M >b. That is, under this condition a population located belowb can surpass this value at a later time but it can never go beyond the level M (see Proposition 1 below). The pa-rameter M represents the relaxation of the carrying capacity b that is allowed by the migration mechanism.

De®nition 1.We will say that problem (1) exhibits recovery, if there exists a relatively open subset

X ofQa :ˆ fn2Rn :06ni<a; iˆ1;. . .;ng with the property that for any solution u…t† of (1) withu…0† 2Xthere existt0>0 and 16i06nsuch thatui0…t0†>a. We will say that the recovery is permanent if ui0…t†>a for all tPt0.

Note that if (1) exhibits permanent recovery, then (1) exhibits persistence in a local sense, i.e., there exists a relatively open subsetKofRn‡, the set of alln-tuples with non-negative coordinates,

in persistence and stability of populations, (cf. [8±10]). Nevertheless, these results are related to the concept of persistence de®ned globally, i.e., independent of positive initial data, and do not seem to be properly suited to address the more speci®c question of recovery. For example, neither the spectral conditions nor the irreducibility of the matrix of the coecients of the system, imposed in [10] to obtain persistence, are necessary for system (1) to exhibit recovery as is shown by the example given at the end of Section 5.

In general, we are assuming that dij 6ˆdji for i6ˆj. Systems like (1) with dijˆdji have been extensively studied in the literature (see [11] and the literature therein). Under this assumption the dispersal behavior is di€usive and no aggregation takes place. In particular, we show in Propo-sition 1(ii) that in this case the population cannot exhibit recovery. When the dispersal rates are asymmetric thenP

dij 6ˆ P

djiand some patches receive more immigrants than they lose causing the clumping of some of them. Godfrey and Pacala [12] studied the in¯uence of this behavior on the stability of host±parasitoid and predator±prey interactions. That clumping enhances persis-tence of the population was already established by Adler and Nuernberger [13].

This paper explores the importance of clumping of patches in achieving recovery. We will show in Theorem 1 that a necessary and sucient condition for problem (1) to exhibit recovery is the existence of at least a patchi0 such that

Pn

jˆ1di0j>

Pn

jˆ1dji0. This condition implies that there is a greater movement of the population from other patches into patchi0 than out of patch i0.

In the simplest case, when nˆ2, we can already see some of the main features of the more complex situation described by (1). In this case, using the new variablesx…t†and y…t†, the system (1) is reduced to

x0ˆ ÿcx‡dy‡f…x†;

y0ˆcxÿdy‡f…y†: …2†

Here d21ˆc and d12ˆd. It follows from Theorem 1, that (2) exhibits recovery if and only if

c6ˆd.

This is illustrated in Fig. 1(a), in which the threshold valueafor the functionfis equal to 1. This picture shows a number of trajectories starting in di€erent locations on the lineyˆ1. We can see

a relatively small region of recovery in the upper right-hand side corner of the unit square. Nevertheless, the recovery exhibited here is not permanent.

In general, we need to impose further restrictions on the coecients dij to obtain permanent recovery. In the case of the particular system (2), this can be accomplished by choosing c small enough. Fig. 1(b) illustrates permanent recovery. It is clear from this picture that there is a region of permanent recovery in the upper right-hand corner of the unit square.

In Theorem 2, we will show that a meta-population exhibits permanent recovery as long as at least one population can go from below the thresholdato the `safe' region described by Lemma 1. In both Theorems 1 and 2 we give simple algebraic conditions that guarantee the existence of the regionXof recovery and permanent recovery. These conditions can be tested and the parameters involved can be computed (see Remark 1). The experiments in Section 4 suggest a method, that can be applied in real situations, for a numerical estimation of X.

This paper is organized as follows: In Section 2, we will show, for completeness, some results of existence and uniqueness of solutions to the initial value problem (1). In Section 3, we will show the main results of this paper: Theorems 1 and 2. These results are illustrated in Section 4 with the aid of some numerical simulations. In Section 5, we establish some results regarding the rela-tionship between recovery and persistence.

2. Existence and uniqueness of solutions

Since the functionfisC1_{, the existence and uniqueness of a locally de®ned solution to the initial} value problem for the system (1) in Rn, follows.

Before proving global existence, we describe some positive invariant regions for the solutions of (1).

A subsetS Rispositive invariantfor the problem (1) ifui…0† 2S for all iˆ1;. . .;n, implies

ui…t† 2S for all iˆ1;. . .;nand all tP0.

Proposition 1 (Positive invariant regions).

(i) IfNPM, then the interval Sˆ ‰0;N† is positive invariant. (ii)If Pn

jˆ1dij6

Pn

jˆ1dji for alliˆ1;. . .;n, thenS ˆ ‰0;aŠ is positive invariant.

Proof.Letu…t†be a solution of (1) de®ned in a maximal interval…T1;T2†containingtˆ0. Suppose that ui…0† 2S ˆ ‰0;N† for all iˆ1;. . .;n. Standard arguments show that ui…t†P0 for all

iˆ1;. . .;nand all 06t<T2.

Suppose now by contradiction that there existi0andt0 >0 such thatui0…t0† ˆN andui…t†<N for 06t<t0and for alliˆ1;. . .;n. It is obvious that in this caseu0i0…t0†P0. By Hypothesis 1(3) we obtain

u0_i

0…t0† ˆ

Xn

jˆ1

di0juj…t0† ÿ

Xn

jˆ1

dji0N‡f…N†6d1N‡f…N†<0:

To prove (ii) we ®rst notice that Pn both sides of this inequality are actually equal.

Suppose now thatui…0† 2S ˆ ‰0;aŠfor alliˆ1;. . .;n. Assume by contradiction that there exist

This is a contradiction that proves (ii).

As an immediate consequence of Proposition 1, all solutions of system (1) are de®ned for all

tP0. Indeed, letu…t†be a solution of (1) de®ned in a maximal interval…T1;T2†containing tˆ0, such that ui…0†P0 for all iˆ1;. . .;n. Then choosing NPM such that ui…0†6N for all

iˆ1;. . .;n we obtain that ui…t†6N for all iˆ1;. . .;n and all 0<t<T2. This shows that

T2 ˆ ‡1.

3. Recovery and permanent recovery

The ®rst result of this section gives a necessary and sucient condition for problem (1) to exhibit recovery.

Theorem 1. System (1)exhibits recovery if and only if there exists 06i06n such that

Pn

jˆ1di0j

Pn

jˆ1dji0

>1: …3†

Proof.It follows from Proposition 1(ii) that (3) is a necessary condition for system (1) to exhibit recovery.

From (1) and (3) it follows that on the initial data, it follows that Xis open. This ®nishes the proof.

Next, we will give some conditions to assure that system (1) exhibits permanent recovery. For this we will need the following Lemma. For any d>0 let

D…d†:ˆ fs2R:a<s<b and f…s† ÿds>0g: …4†

Theorem 2. Assume that there exists a partition of the set f1;. . .;ng in two non-empty disjoint subsets I1;I2 such that

Proof.Sincef is positive in the interval…a;b† it follows that there existsd0>0 such that for any

d <d0, D…d†, de®ned by (4), is non-empty. Moreover, the continuity of f implies that

Choose >0 such that satis®es the requirement of the de®nition of permanent recovery.

Hence, from this inequality, (10), and sincer…d†>r…d0† and g…d†<g…d0† it follows that

This is a contradiction. Hence there are two alternatives

1. There exists<1and 16i06nsuch thatvi0…s† ˆsandc<vj…t†<sfor alljˆ1;. . .;nand all

and we reach a contradiction. Therefore i0 2I1. In the second case it follows from (11) thatv0

i…t†>0 for alltP0 and alli2I1.

Hence, from both cases we obtain that there existsi0 2I1such that vi0…t†>afor allt>0. This ®nishes the proof.

Remark 1. Knowing the functionf, one can computes…d† to ®ndd>0 satisfying (8)±(10). This can certainly be done with the aid of a computer to obtain a numerical estimate ofd suitable for applications.

4. Numerical simulations

As an illustration of the results of Section 3 we will consider a habitat of 100 localities dis-tributed into a square grid of points…h;k†wherehˆ ‰…iÿ1†=10Š ‡1 and kˆ …iÿ1† mod 10‡1 withiˆ1;. . .;100. Here‰pŠdenotes the greatest integer less than or equal topandpmodqis the remainder ofpdivided byq. The population densityv…h;k;t†at the point…h;k†at timetis given by

v…h;k;t† ˆu10…hÿ1†‡k…t†;h;kˆ1;. . .;10, whereu is the solution of system (1) .

We will assume that individuals move only to the four adjacent cells (three or two when they are located in the boundary), with dispersal rates dij ˆdj depending only on their present locationj. That is, the decision about moving is based only on the local conditions of the habitat (described here by the vectordi;iˆ1;. . .;100) and it is the same towards any of the adjacent locations. For instance, if …h;k† is an interior point of the grid, then for jˆ10…hÿ1† ‡k we have dijˆdj for

iˆjÿ1;j‡1;jÿ10;j‡10 and dijˆ0 otherwise.

Under these conditions, the matrixAˆ …aij† of system (1), with

aijˆ

Coming back to the square grid representation, since every locus is associated with a grid point

Applying Theorem 1 to this particular case, we obtain that (1) exhibits recovery at a grid point

…h;k†if and only ifD…h;k†is smaller than the average of its adjacent neighbors. This suggests the

The entries of the matrix M give a measure of the total migration (i.e., the di€erence between immigration and emigration) at a given location. In our particular example, the matrixMis given by 3(a) and (b), respectively. According to Theorem 2, we should expect permanent recovery to occur among the grid points of positive entries of the matrixM.

In Fig. 4, we show a sequence of graphs of the solutions v…h;k;t† of (1), with initial data

v…h;k;0† ˆ4:55; h;k ˆ1;. . .;10, below the threshold level aˆ5 at di€erent times. We observe permanent recovery as the population aggregates at the points of minimum values of D. If we think ofD…h;k† as inversely proportional to the amount of resources available to the population in the location …h;k†, then this is telling us that the aggregation behavior that yields permanent recovery occurs at the points of higher resources (the smaller values of D…h;k†). The numerical values of the population density v…h;k;t† attˆ100 are given in Table 1.

In the next example, we show that in general this is not the case. Comparing matricesDandM we notice that the points of more resources are not necessarily the ones with higher total mi-gration rates. As a consequence, the aggregating behavior that yields permanent recovery may occur at points that are not necessarily the points of higher resources. This is illustrated in Fig. 5 where we show a sequence of graphs of the solutions v…h;k;t† of (1), with initial data

v…h;k;0† ˆ4:25; h;k ˆ1;. . .;10, at di€erent times. Table 2 shows the numerical values for

v…h;k;t† attˆ100. Here, the individuals aggregate in a spatial pattern that surrounds, but does not include the points of higher resources available to the population.

In both examples, permanent recovery also takes place in a region ofQa, whereQa is given in De®nition 1, bounded below by the given initial data. This suggests a method for a numerical estimation of the recovery region X.

5. Recovery vs persistence

It is clear from De®nition 1 that if (1) exhibits permanent recovery, then (1) exhibits persistence. The natural question of whether persistence implies permanent recovery arises. We already know, Fig. 4. Population density v…h;k;t† at times tˆ0:005;tˆ10;tˆ25 and tˆ100, respectively, with initial data v…h;k;0† ˆ4:55,h;kˆ1;. . .;10.

Table 1

Numerical values of population densityv…h;k;t†attˆ100 forv…h;k;0† ˆ4:55,k;hˆ1;. . .;10

hnk 1 2 3 4 5 6 7 8 9 10

by Theorem 1, that we will have to assume that (1) satis®es the inequality (3) for somei0. We will show that, under this condition and with the added assumption that the matrixAˆ …aij†de®ned in (13) is irreducible, persistence implies that system (1) exhibits permanent recovery.

A matrix is said to be irreducible if it cannot be put into the form

B C

0 D

Fig. 5. Population density v…h;k;t† at times tˆ0:005; tˆ5; tˆ10 and tˆ100, respectively, with initial data v…h;k;0† ˆ4:25,h;kˆ1;. . .;10.

Table 2

Numerical values of population densityv…h;k;t†attˆ100 forv…h;k;0† ˆ4:25,k;hˆ1;. . .;10

hnk 1 2 3 4 5 6 7 8 9 10

(where B and D are square matrices) by reordering the standard basis vectors. This condition implies that anyith patch can in¯uence anyjth patch. In fact, there is a simple test to see if ann

nmatrixAis irreducible. WritenpointsP1;P2;. . .;Pn in the plane. Ifaij6ˆ0, then draw a directed line segmentPiPj connectingPi toPj. The resulting graph is said to be strongly connected if, for each pair…Pi;Pj†, there is a directed path PiPk1;Pk1Pk2;. . .;PklPj. A square matrix is irreducible if

and only if its directed graph is strongly connected, cf. [14, p. 529].

In the rest of this section, we will assume that the matrix A is irreducible. The following Proposition gives a useful description about the distribution of the equilibria of system (1).

Proposition 2. Letv2_Rn

‡ be an equilibrium of system (1).Ifvjˆ0for some j, then viˆ0for all

iˆ1;. . .;n. Moreover, ifvj>0 for some j, then either vˆ …a;. . .;a† orvi >afor some i.

Proof.Suppose thatvjˆ0 for somej and leti6ˆj. We will show that viˆ0.

Since Ais irreducible there exist distinctk1;. . .;kl such that djk1 6ˆ0;dk1k2 6ˆ0;. . .;dkli6ˆ0.

Now, sincevj ˆ0, by (1) we have

dj1v1‡ ‡djjÿ1vjÿ1‡djj‡1vj‡1‡ ‡djnvnˆ0:

Thus, sincedjk1 6ˆ0,vk1 ˆ0 and by (1) we have that

dk11v1‡ ‡dk1k1ÿ1vk1ÿ1‡dk1k1‡1vk1‡1‡ ‡dk1nvnˆ0:

Therefore, sincedk1k2 6ˆ0 it follows that vk2 ˆ0. We can continue this argument to show that viˆ0.

The second part of the Proposition follows from the ®rst part and the fact that

f…v1† ‡ ‡f…vn† ˆ0 and f…s†<0 for 0<s<a.

Next, we will recall some results of the theory of irreducible cooperative systems that will be used later. Consider the autonomous system of ordinary di€erential equations

x0ˆf…x†; …16†

wheref is continuously di€erentiable on an open and convex subset ERn. The system (16) is said to be a cooperative system if

o_fi

o_xj…x†P0; i6ˆj; x2E:

The system (16) is said to be cooperative and irreducible if it is a cooperative system and if the Jacobian matrix…o_f=o_{x†…x† is irreducible for any x2E.}

SincedijP0 and the matrixAis irreducible, (1) is a cooperative and irreducible system inRn_‡. Moreover, the set C:ˆ fn2Rn‡ :limt!1u…t;n† existsg of convergent points of (1) contains an

open and dense subset of_Rn

‡, cf. [15, Theorem 4.1.2, p. 57]. Here,u…t;n†denotes the solution of (1)

Proposition 3.Suppose that the inequality(3)holds for somei0. Assume that there exists a relatively open subset K of Qa :ˆ fn2Rn:06ni<a;iˆ1;. . .;ng such that if n2K, then lim inftP0ui…t;n†>0for some i. Hence, ifn2K\C, thenlim inftP0ui…t;n†>0for alliˆ1;. . .;n. Moreover, (1) exhibits permanent recovery.

Proof.If the inequality (3) holds, then the point …a;. . .;a† 2Rn‡ cannot be an equilibrium for (1).

Therefore, if n2C, by Proposition 2 if limt!1uj…t;n†>0 for somej, then limt!1ui…t;n†>0 for all iˆ1;. . .;n. Moreover, by the second part of Proposition 2 and since …a;. . .;a† is not an equilibrium, limt!1uj…t;n†>a for some j.

The proposition follows by choosing Xˆint…K\C† and noticing that, since C contains an open and dense subset of_Rn_‡,Xis an open and non-empty subset ofQasatisfying the requirement of the de®nition of permanent recovery.

The local concept of persistence that we have used in this paper seems to be better suited, in the light of the previous results, to the problem of recovery than the global concept, independent of positive initial data, considered by other authors in the literature. We would like to stress this point by showing, with the aid of an example, that the conditions imposed in [10] to obtain persistence for a class of discrete di€usion systems are too restrictive for recovery.

The spectrum of a matrixA, written asd…A†, is the set of eigenvalues ofA. De®ne the stability modulus ofA, s…A†, as

s…A†:ˆmaxfRek:k2d…A†g:

Suppose that f…u† ˆur…u†, where r…u† is the net rate of population supply. De®ne the matrix

Ar:ˆA‡r…0†I.

It is shown in [10, Theorem 1] that if A is irreducible and s…Ar†>0, then

lim inftP0ui…t†Pd>0, for alliˆ1;. . .;n and all positive initial data. The following system does not satisfy these conditions. Let

u0_i…t† ˆdiÿ1uiÿ1…t† ÿ2diui…t† ‡di‡1ui‡1…t† ‡f…ui…t††; iˆ1;. . .;n; tP0;

where d0ˆd1;u0ˆu1;dn ˆdn‡1, and un ˆun‡1. Then clearly, the matrix Ais not necessarily ir-reducible, and since s…A† ˆ0, it follows that s…Ar†<0. Nevertheless, under the hypothesis of

Theorem 2, this system exhibits permanent recovery.

If we assume that f0_{0†<0, then the equilibrium …0;. . .;0† is locally asymptotically stable.}

Therefore, this example also shows that the set Xin the de®nition of recovery is not necessarily dense inQa, i.e., the concept of recovery is a local property.

6. Discussion

used by the applied scientist to induce a population that is in danger of extinction to ®nd its natural way of survival.

The results presented here can be generalized to systems of the form

u0_i…t† ˆX

j2I

‰dijuj…t† ÿdjiui…t†Š ‡f…ui…t††; i2I; tP0

for any subsetIof natural numbers, not necessarily ®nite. We only need to assume, in addition to

dijP0, that there is a positive constantd1 such that 0<

P

j2Idij<d1 and 0<

P

j2Idji<d1. The solutions to this equation, with initial values inl1_{I†, are globally de®ned for alltP0. We obtain}

Theorem 1 for this case, and replacing (5) and (6) by

P

respectively, for some >0, we also obtain Theorem 2.

Acknowledgements

This research has been supported by the Consejo de Desarrollo Cientõ®co, Humanõstico y Tecnico (CDCHT) de la Universidad de los Andes through project C-1002-00-05-B. We are deeply indebted to anonymous referees for detailed criticisms that helped us to substantially improve this paper.

References

[1] W.C. Allee, Animal Aggregations, University of Chicago, Chicago, 1931.

[2] D. Grunbaum, A. Okubo, Modeling social animal aggregation, in: S. Levin (Ed.), Lecture Notes in Biomath. 100, Springer, Berlin, 1994, p. 296.

[3] P. Turchin, Models for aggregating populations, in: T.G. Hallam, L.J. Gross, S.A. Levin (Eds.), Mathematical Ecology, World Scienti®c, Singapore, 1988, p. 101.

[4] W.C. Allee, The Social Life of Animals, Norton, New York, 1938.

[5] M.A. Lewis, P. Kareiva, Allee dynamics and the spread of invading organisms, Theret. Populat. Biol. 43 (1993) 141.

[6] P. Grindrod, Models of individuals aggregation in single and multispecies communities, J. Math. Biol. 26 (1998) 651.

[7] M. Lizana, V. Padron, A spatially discrete model for aggregating populations, J. Math. Biol. 38 (1999) 79. [8] L.J.S. Allen, Persistence, extinction, and critical patch number for island population, J. Math. Biol. 24 (1987) 617. [9] V. Hutson, K. Schmitt, Permanence and the dynamics of biological systems, Math. Biosci. 111 (1992) 1. [10] Z. Lu, Y. Takeuchi, Global asymptotic behavior in single-species discrete di€usion systems, J. Math. Biol. 32

(1993) 67.

[12] H.C.J. Godfray, S.W. Pacala, Aggregation and the population dynamics of parasitoids and predators, Am. Natural. 140 (1) (1992) 31.

[13] F.R. Adler, B. Nuernberger, Persistence in patchy irregular landscapes, Theret. Populat. Biol. 45 (1994) 41. [14] P. Lancaster, M. Tismenetsky, The Theory of Matrices, Academic Press, Orlando, FL, 1985.