Uniform persistence and global asymptotic stability in periodic single-species models of dispersal in a patchy environment

Nonlinear Analysis 36 (1999) 981–996

Uniform persistence and global asymptoticstability in periodic single-species modelsof dispersal in a patchy environment

H.I. Freedman∗;1, Qui-Liang Peng2

Applied Mathematics Institute, Department of Mathematical Sciences, University of Alberta, Edmonton,Alberta, Canada T6G 2G1

Received 14 August 1996; accepted 18 April 1997Dedicated to Professor Junji Kato on this sixtieth birthday

Keywords: Dispersal; Patches; Persistence; Periodic; Single-species

1. Introduction

Models involving populations moving through patchy environments are mainly of twotypes: (i) models of populations dispersing among discrete patches involving ordinarydi�erential equations, and (ii) models of populations di�using in continuous patchesinvolving parabolic partial di�erential equations. This paper is concerned with the �rstof these, namely dispersal among discrete patches.In the work done to date, the models were concerned with patches separated by a

barrier with constant barrier strength [2, 13, 14, 16, 18–20, 31–34]. In these papers, thesystems were autonomous, i.e. the parameters of the models representing growth rates,carrying capacities, probabilities of survival while dispersing, etc., were all deemed tobe constant. However, in the “real world”, all these parameters may vary seasonallyor even diurnally.In our paper we discuss a model of a single species dispersing among n patches,

where all parameters may vary periodically in time. This would be the next step towards

∗ Corresponding author. Tel.: 001 403 492 3530; fax: 001 403 492 6826; e-mail: [email protected] Research partly supported by the Natural Sciences and Engineering Research Council of Canada, grantnumber NSERC A4823.2 Present address: The Fields Institute, 222 College St., Toronto, Ontario, Canada M5J 3JL.

0362-546X/99/$ – see front matter ? 1999 Elsevier Science Ltd. All rights reserved.PII: S0362 -546X(97)00712 -8

982 H.I. Freedman, Q.-L. Peng / Nonlinear Analysis 36 (1999) 981–996

“reality”. We then carry this even further by considering models where the parametersare asymptotic periodic. Our model is derived by modifying a submodel discussed in[18, 19]. Models involving interactions between several species are left to future work.The paper is organized as follows. In the next section we formulate our model. In

Section 3, we discuss the uniform persistence and the existence of periodic solutionsfor our model. Section 4 deals with the question of global stability of the periodicsolution. In Section 5, we consider the asymptotic periodic case. A brief discussionfollows in the last section.Throughout this paper we assume that all functions are su�ciently smooth so that

solutions to initial value problems exist, are unique and are continuable for all positivetime.

2. The patchy model

We consider a time-dependent system in a patchy environment where a populationis able to disperse among the n di�erent habitats at some cost to the population inthe sense that the probability of survival during a change of habitat may be less thanone. We also suppose that both the barrier strengths and the survival probabilitiesvary either periodically in time with the same period as the species parameters anddispersing functions or are asymptotically periodic. The model is described by thesystem of nonautonomous ordinary di�erential equations

xi= xigi(t; xi)− �i(t)hi(t; xi) +∑j∈ Ji

pji(t)�j(t)hj(t; xj)

xi(0)≥ 0; i=1; : : : ; n; (2.1)

with∑n

j∈ Ji pij ≤ 1. where ·=d=dt, Ji= {1; : : : ; i − 1; i + 1; : : : ; n}, gi; hi :R2→R arecontinuously di�erentiable, positive, and either T -periodic or asymptotically T -periodicfor some common period T¿0 in the variable t, and �i; pij :R→R are nonnegativeand T -periodic or asymptotically T -periodic.Here, xi(t) represents the same population in the ith patch, i=1; : : : ; n, at a given

time t≥ 0; gi(t; xi) is the intrinsic growth rate of the population in the ith habitat ata given time t≥ 0; �i(t), which is not necessarily small but is positive represents theinverse barrier strength at time t in going out of the ith habitat; hi(t; xi) is the rateof dispersal out of the ith patch at time t and pji(t) is the probability of successfultransition from the jth patch to the ith patch, where i is di�erent from j.We make the following hypotheses, which are modi�ed from the standard ones in

modelling such phenomena [12, 18].(H1) All solutions of the initial value problem (2.1) exist uniquely and are continuable

for all positive time.(H2) gi(t; 0)¿0, @gi(t; xi)=@xi¡0, xigi(t; xi)→−∞ as xi →+∞, i=1; : : : ; n, for all

time t≥ 0.(H3) hi(t; 0)=0; @hi(t; xi)=@xi¿0; i=1; : : : ; n, at any time t≥ 0.

H.I. Freedman, Q.-L. Peng / Nonlinear Analysis 36 (1999) 981–996 983

The inequalities in (H3) above state that the rate of dispersal out of the ith patchis density dependent and an increasing function of the population of the ith patch forany �xed t≥ 0.We note that the positive cone Rn

+ in Rn is positively invariant.

3. Uniform persistence and periodic solutions

In this section, we suppose that all functions contained in model (2.1) are T -periodic. Firstly, following [3, 4, 15, 20, 21] we recall brie y some terminology. Letx(t) be the population density at time t. We say that x(t); x(0)¿0, is persistent iflim inf t→∞ x(t)¿0. We say that x(t) is uniformly persistent if there exists �¿0 suchthat lim inf t→∞ x(t)≥ � independent of x(0)¿0. We say that a system exhibits (uni-form) persistence if each component (uniformly) persists. Finally, we say that x(t)exhibits extinction if lim supt→∞ x(t)= 0 and a system exhibits extinction if at leastone component becomes extinct.

Theorem 3.1. Assume∫ T

0

[gi(t; 0)− �i(t)

@hi(t; 0)@xi

]dt¿0; i=1; 2; : : : ; n; (3.1)

holds. Then system (2.1) exhibits uniform persistence.

Proof. Note that the positive cone Rn+ in Rn is positively invariant with respect to (2.1)

and hence the term∑

j∈Ji pji(t)�j(t)hj(t; xj) is positive. Letting ui(t) (i=1; 2; : : : ; n)denote the solution of the following system:

ui= uigi(t; ui)− �i(t)hi(t; ui)

ui(0)= xi(0)

then clearly, we have

xi(t)≥ ui(t) for all t≥ 0 and i=1; 2; : : : ; n:

For the purpose of �nding a persistent lower bound, we consider

y=yg(t; y)− �(t)h(t; y);

y(0)≥ 0:(3.2)

Here the functions g(t; y); �(t) and h(t; y) are a representative of gi(t; xi); �i(t) andhi(t; xi), respectively. Obviously, y=0 is an equilibrium of Eq. (3.2) and R+ is alsopositively invariant. Now we will see that y=0 is unstable provided∫ T

0

[g(t; 0)− �(t)

@h(t; 0)@y

]dt¿0: (3.3)


Indeed, the characteristic equation of Eq. (3.2) about the equilibrium y=0 is

y=(g(t; 0)− �(t)

@h(t; 0)@y

)y: (3.4)

We know that in Eq. (3.4), y=0 is unstable provided Eq. (3.3) holds. Hence undercondition (3.3) we can conclude that there exists a constant �¿0, which could besu�ciently small but positive, such that [�;+∞) is positively invariant and globallyattractive with respect to R+. Similarly, under condition (3.1), for each i there exists a�i¿0 such that [�i;+∞) has the same property. Let �=min1≤i≤n {�i}. Then we haveactually proved that

�= {(x1; : : : ; xn)∈Rn+ | xi ≥ �; i=1; : : : ; n}

is positively invariant and globally attractive with respect to Rn+. Thus, system (2.1)

is uniformly persistent.

Theorem 3.2. The system (2.1) has at least one positive T -periodic solution undercondition (3.1).

Proof. From the fact that if x=∑n

i=1 xi, then

x′ ≤n∑

i=1

xigi(t; xi)

and by (H2) if ‖x‖ is su�ciently large,x′¡0;

it follows from the fact that Rn+ is positively invariant that there exists a constant k¿0

such that

xi(t)≤ k; i=1; : : : ; n; for t large enough:

By condition (3.1) and the proof of Theorem 3.1, it is obvious that the set

S = {(x1; : : : ; xn)∈Rn+ | �≤ xi ≤ k; i=1; : : : ; n}

is not only positive invariant, but also globally attractive with respect to Rn+. We de�ne

x(t; x0) to be that solution of Eq. (2.1) such that xi(0; x0)= xi0. If we further de�neF : S →Rn by

F(r)=F(r1; : : : ; rn)= (r1 − x1(T; r); : : : ; rn − xn(T; r));

then from degree theory, deg(F; Int S; 0)=1, where Int S = {(x1; : : : ; xn)∈Rn+ | �¡xi¡k;

i=1; : : : ; n}. Actually, let �r=( �r1; : : : ; �rn) be a de�nite point in Int S. For any r=(r1; : : : ;rn)∈S, since Int S is convex, then for all �∈[0; 1], ( �r1 + �(x1(T; r) − �r1); : : : ; �rn +�(xn(T; r) − �rn))=N (r; �)∈Int S, where (x1(t; r); : : : ; xn(t; r)) is the solution of system(2.1) with initial value r. We can de�ne a homotopic mapping H : S × [0; 1]→Rn by


H (r; �)= r−N (r; �). Then H (r; 1)=F(r) and H (r; 0)= (r1− �r1; : : : ; rn − �rn)=G(r). Itis obvious that deg(G; Int S; 0)=1. By homotopy invariance of degree, it follows that

deg(F; Int S; 0)=1:

Hence, there exists at least one such r0 = (r01 ; : : : ; r0n)∈ Int S satisfying

F(r0)= 0;

that is, system (2.1) has at least one positive T -periodic solution lying in S.

Remark. The proof of Theorem 3.1 does not imply that the T -periodic solution foundis necessarily nontrivial. However; what is important in the periodic literature (i.e.in systems with periodic coe�cients) is the existence of periodic solutions; whethertrivial or not.

4. Global stability

In this section we study the global stability of the positive periodic solution of thefollowing system:

xi= xigi(t; xi)− �i(t)xi +∑j∈Ji

pji(t)�j(t)xj; i=1; 2; : : : ; n; (4.1)

where gi(t; ·); �i(t) and pji(t) are T -periodic functions about the time variable t. Ac-tually, if we assume the dispersal is proportional with the density of population, thensystem (2.1) becomes system (4.1) and (H3) is still satis�ed.We suppose system (4.1) has at least one positive T -periodic solution and denote

it by x∗(t)= (x∗1 (t); : : : ; x∗n (t)) which lies in S. Consider the variational equation of

system (4.1) about x∗(t).

X =

g1(t; x∗1 (t)) + x∗1 (t)@g1(t; x∗1 (t))

@x1− �1(t) p21(t)�2(t) · · ·

p12(t)�1(t) g2(t; x∗2 (t)) + x∗2 (t)@g2(t; x∗2 (t))

@x2− �2(t) · · ·

......

p1n(t)�1(t) p2n(t)�2(t) · · ·

· · · pn1(t)�n(t)· · · pn2(t)�n(t)

...

· · · gn(t; x∗n (t)) + x∗n (t)@gn(t; x∗n (t))

@xn− �n(t)

X: (4.2)

If we set Q(t)= diag(x∗1 (t); : : : ; x∗n (t)), then Q(0)=Q(T ). Apply the transformation

Y (t)=Q−1(t)X (t)Q−1(0). Then

Y (t)=B(t)Y (t) (4.3)


where

B(t) =

x∗1 (t)@g1(t; x∗1 (t))

@x1−∑j∈J1

pj1(t)�j(t)x∗j (t)

x∗1 (t)p21(t)�2(t)

x∗2 (t)x∗1 (t)

p12(t)�1(t)x∗1 (t)x∗2 (t)

x∗2 (t)@g2(t; x∗2 (t))

@x2−∑j∈J2

pj2(t)�j(t)x∗j (t)

x∗2 (t)...

...

p1n(t)�1(t)x∗1 (t)x∗n (t)

p2n(t)�2(t)x∗2 (t)x∗n (t)

· · · · · · pn1(t)�n(t)x∗n (t)x∗1 (t)

· · · · · · pn2(t)�n(t)x∗n (t)x∗2 (t)

. . .. . .

...

· · · · · · x∗n (t)@gn(t; x∗n (t))

@xn−∑j∈Jn

pjn(t)�j(t)x∗j (t)

x∗n (t)

: (4.4)

In order to obtain our results, we recall some concepts and known results concerningFloquet theory.For an n-dimensional linear periodic di�erential system

x=A(t)x (4.5)

where A(t) is a continuous, T -periodic n× n matrix, let X (t) be a fundamental matrixof system (4.5) which satis�es X (0)= I . Then there exists a C2 nonsingular T -periodicn× n matrix W (t) and a constant n× n matrix R such that X (t)=W (t) exp(Rt). Aneigenvalue of the matrix R is called a characteristic exponent or Floquet exponent ofsystem (4.5).Floquet theory implies that for each characteristic exponent � there corresponds a

solution x(t) of system (4.5) with the form

x(t)= �(t)e�t

where the vector function �(t) is periodic in t with period T , i.e. �(t + T )= �(t).We denote an eigenvector of the matrix R corresponding to the eigenvalue � by �,

i.e. R�= ��. Consider the solution x(t) with initial value x(0)= �. Then

x(t)=X (t)�=W (t) exp(Rt)�=W (t) e�t�=W (t)�e�t = �(t)e�t

where �(t)=W (t)� which is obviously T -periodic.The following result is well-known.

Lemma 4.1. System (4.5) is uniformly asymptotically stable if and only if all of itscharacteristic exponents have negative real parts; that is

Re �i¡0

where �i are the roots of det(R− �I)= 0.


Now let us continue discussing the stability of the positive T -periodic solution x∗(t)of system (4.1).

Theorem 4.2. Assume∫ t

0[�i(t)− gi(t; 0)] dt¡0; i=1; 2; : : : ; n:

Then system (4.1) has only one positive; T -periodic solution which is globally asymp-totically stable with respect to the �rst octant Rn

+.

Proof.Step 1: We prove system (4.3), namely, Y (t)=B(t)Y (t), where B(t) has the form

(4.4), is uniformly asymptotically stable.Let � be a characteristic exponent of system (4.3), and y(t) a nontrivial solution of

system (4.3) of the form

y(t)= �(t)e�t

where �(t + T )= �(t) and the coordinates of � never vanish at the same time sincey(t) is nontrivial. Then from

y= �e�t + ��e�t =B(t)y=B(t)�(t)e�t ;

it follows that

�=−�� + B(t)�: (4.6)

Because �(t) is T -periodic and continuously di�erentiable, there exist an integer k anda time t0 ∈ [0; T ], such that

|�k(t0)|= max1≤i≤n

maxt∈[0; T ]

|�i(t)|;

as a result of which we have

�k(t0)= 0:

Substituting into Eq. (4.6) gives

− ��k(t0) +n∑

j=1

bkj(t0)�j(t0)= 0 (4.7)

where we denote B(t)= (bij(t))n×n. From Eq. (4.4), we know that

bkk(t) = x∗k (t)@gk(t; x∗k (t))

@xk−

∑j∈Jk

pjk(t)�j(t)x∗j (t)x∗k (t)

;

bkj(t) =pjk(t)�j(t)x∗j (t)x∗k (t)

; j 6= k: (4.8)


Hence substituting Eq. (4.8) into Eq. (4.7) gives

− ��k(t0) +

x∗k (t0)

@gk(t0; x∗k (t0))@xk

−∑j∈Jk

pjk(t0)�j(t0)x∗j (t0)x∗k (t0)

�k(t0)

+∑j∈Jk


�j(t0)= 0:

Rewriting shows(−� + x∗k (t0)

@gk(t0; x∗k (t0))@xk

)�k(t0)

+∑j∈Jk


(�j(t0)− �k(t0))= 0: (4.9)

Case 1: If �k(t0)¿0; then �j(t0)≤ �k(t0) for any j 6= k.Hence from Eq. (4.9), it follows that

−� + x∗k (t0)@gk(t0; x∗k (t0))

@xk≥ 0;

that is,

�≤ x∗k (t0)@gk(t0; x∗k (t0))

@xk:

Case 2: If �k(t0)¡0; then �j(t0)≥ �k(t0), for all j 6= k.Similarly, we get

�≤ x∗k (t0)@gk(t0; x∗k (t0))

@xk:

Case 3: If �k(t0)= 0, then �(t)≡ 0. This is a contradiction. Thus

�≤ x∗k (t0)@gk(t0; x∗k (t0))

@xk¡0 (4.10)

by (H2) and hence from Lemma 4.1, we know that system (4.3) is uniformly asymp-totically stable.Step 2: Suppose X (t) is a fundamental matrix of the variational equation (4.2)

satisfying X (0)= I . We prove that there exist two constants a; b¿0 such that

|X (t)| ≤ ae−bt : (4.11)

From Step 1, we know there exist a1; b1¿0 such that

|Y (t)| ≤ a1e−b1t :

The transformation Y (t)=Q−1(t)X (t)Q(0) gives

X (t)=Q(t)Y (t)Q−1(0)


and hence

|X (t)|= |Q(t)Y (t)Q−1(0)| ≤ |Q(t)||Q−1(0)||Y (t)|:

Here Q(t)= diag(x∗1 (t); : : : ; x∗n (t)) and the periodic solution x∗(t) lies in S so |Q(t)|

|Q−1(0)| are bounded and then Eq. (4.11) holds, that is, Eq. (4.2) is uniformly asymp-totically stable.Step 3: From step 2, we see that every positive T -periodic solution of Eq. (4.1) is at

least locally asymptotically stable. From step 1, we see that all the Floquet exponentsof system (4.3) have negative real parts, which are independent of the speci�city ofa positive periodic solution x∗(t). Indeed, from Eq. (4.10), we see that all Floquetexponents of system (4.3) are negative since the considered periodic solution x∗(t)¿0and the hypothesis (H2) implies @fi(t; xi)=@xi¡0 in [0; T ]× (0;+∞). Moreover, underthe transformation Y (t)=Q−1(t)X (t)Q(0), the linearized variational system (4.2) aboutx∗(t) of system (4.1) is transformed into system (4.3) and since Q(0)=Q(T ), systems(4.2) and (4.3) have the same Floquet exponents, that is, all the Floquet exponents ofsystem (4.2) are also negative. This implies that every �xed point of the correspondingPoincar�e periodic map, if it exists, must be isolated and has index 1, where the indexof a �xed point of the corresponding Poincar�e periodic map is de�ned as usual bythe Browder degree of the Poincar�e periodic map at the �xed point [27]. Note thatin either step 1 or step 2, we required no condition for the stability of x∗(t), exceptfor the positivity of x∗(t). This is the key to show the global stability of a positiveperiodic solution and this is true only for certain systems. Recall that the compact setS is globally attractive with respect to Rn

+. So if a positive periodic solution existsfor system (4.1), it must be in S. It follows from a simple compactness argument thatthere are at most �nitely many positive periodic solutions in S. Furthermore, from theproof of Theorem 3.2, we know that deg(F; Int S; 0)=1: It follows immediately fromthe additivity of the �xed point index that the positive T -periodic solution of system(4.1) if it exists, must be unique since the sum of all indices is equal to deg(F; Int S; 0):Here @hi(txi)=@xi ≡ 1. So Eq. (3.1) becomes

∫ T0 [�i(t)− gi(t; 0)] dt¡0, under which the

existence of a positive T -periodic solution holds. Hence system (4.1) does contain onlyone positive T -periodic solution.Step 4: Note that S = {(x1; x2; : : : ; xn)∈Rn

+ | �≤ xi ≤ k; i=1; 2; : : : ; n} is globally at-tractive. So without loss of generality, we can limit out attention to these solutions ini-tiating in S. Generally, let A(t; x0) (t≥ 0) denote the semi ow of solutions of Eq. (4.1)initiating at some point x0 ∈ S, i.e. A(0; x0)= x0. Then one can see that A(t; x0) is ac-tually a monotone semi ow in the sense that A(t; x0)≥A(t; y0) if x0 ∈ S, y0 ∈ S andx0≥y0. Here a vector x0≥y0 is in the componentwise sense. So for any x0 ∈ S, if wedenote �0 = (�; �; : : : ; �)∈ S and k0

:= (k; k; : : : ; k)∈ S; then

A(t; �0)≤A(t; x0)≤A(t; k0); (4.12)

since �≤ xi ≤ k. Furthermore, one can verify that

�0 =A(0; �0)≤A(t; �0)≤A(t + T; �0)≤A(t + T; k0)≤A(T; k0)≤A(0; k0)= k0


for all t≥ 0. Now for all t ∈ [0; T ], the monotone sequences {A(t + nT; �0)}∞n=0 and{A(t+nT; k0)}∞n=0 are uniformly convergent to two positive T -periodic solutions. Basedon the uniqueness of positive T -periodic solutions proved in Step 3, the global attrac-tivity of x∗(t) follows from Eq. (4.12). In Step 2, we also proved local stability, andtherefore x∗(t) is globally asymptotically stable.

5. The asymptotic periodic case

In this section, we study the global properties of solutions of system (4.1), when allfunctions are asymptotically periodic.

De�nition 5.1. (Freedman et al : [17]). Let ’; : [0;+∞)→ R, ’ is said to approach asymptotically, in notation ’∼ , if

limt→∞ |’(t)− (t)|=0:

De�nition 5.2. Let col(’1; : : : ; ’n); col( 1; : : : ; n) : [0;+∞)→Rn. Then col(’1; : : : ;’n) is said to approach col( 1; : : : ; n) asymptotically; in notation col(’1; : : : ; ’n)∼col( 1; : : : ; n); if ’i ∼ i for all i=1; 2; : : : ; n.

It is easy to verify that “∼” is an equivalence relationship. The main result of thissection is the following theorem.

Theorem 5.1. For all i; j=1; : : : ; n; let �i(t); �i(t); pij(t); pij(t) : [0;+∞)→R+ be suchthat �i(t)∼ �i(t); pij(t)∼ pij(t) and let gi(t; xi); gi(t; xi) : [0;+∞)×R+→R+ be suchthat gi ∼ gi for each �xed second variable. Here �i ; pij and gi are T -periodic in t. If;in addition; the inequalities

lim inft→∞

∫ t+T

tgi(s; 0) ds− lim sup

t→∞

∫ t+T

t�i(s) ds¿0; i=1; 2; : : : ; n (5.1)

hold; then for any positive solution col(x1; : : : ; xn) of system (4.1), we have

col(x1; : : : ; xn)∼ col(x1; : : : ; xn)where col(x1; : : : ; xn) is the globally asymptotically stable positive T -periodic solutionof

xi= xigi(t; xi)− �i(t)xi +∑j∈Ji

pji(t)�j(t)xj; i=1; : : : ; n: (5.2)

We defer the proof to a later part of this section.

Corollary 5.2. If Eq. (5.1) holds then system (4.1) is uniformly persistent.

Firstly, we prove a general result concerning di�erential systems of the form

xi=fi(x1; : : : ; xn; t); i=1; 2; : : : ; n; (5.3)

xi=fi(x1; : : : ; xn; �; t); i=1; 2; : : : ; n: (5.4)


Theorem 5.3. Assume that both systems (5:3) and (5:4) are T -periodic systems. Sup-pose

lim�→0

fi(x1; : : : ; xn; �; t)=fi(x1; : : : ; xn; t) for all i=1; 2; : : : ; n

and there exist unique positive T -periodic solutions col(u1; : : : ; un); col(u�1; : : : ; u

�n) of

systems (5:3) and (5:4) respectively. Denote their trajectories by � and �� respectively.If in addition; {�′

� | 0¡�¡�0} is a bounded subset of Rn; then for any �¿0; no matterhow small; there exists an �¿0 such that �� ⊂O(�; �) (the �-neighbourhood of � inRn) for all 0¡�¡�. In notation

col(u�1; : : : ; u

�n)

�∼ col(u1; : : : ; un):

Proof. Because of the continuity of solutions of di�erential equations with respect toinitial values and parameters, to prove col(u�

1; : : : ; u�n)

�∼ col(u1; : : : ; un), it su�ces toshow that for any �¿0, there exists an �¿0 such that

d(�)=n∑

i=1

|u�i (0)− ui(0)|¡� (5.5)

for all 0¡�¡�. Suppose Eq. (5.5) does not hold. Then

lim sup�→0

n∑i=1

|u�i (0)− ui(0)|= lim sup

�→0d(�)=d¿0:

Since {�� | 0¡�¡�0} is bounded in Rn and the Euclidean space Rn is complete, thereexists a sequence {�j}∞1 ⊂ (0; �0) such that

lim�j→0

d(�j)=d¿0

and lim�j→0 col(u�j1 (0); : : : ; u

�jn (0))= col( �u1(0); : : : ; �un(0)).

Let col( �u1; : : : ; �un) denote the solution of system (5.3) with the initial value col( �u1(0);: : : ; �un(0)). Continuous dependence of initial values and parameters for system (5.4)leads to

lim�j→0

col(u�j1 (t); : : : ; u

�jn (t))= col( �u1(t); : : : ; �un(t))

for all t ∈ [0; T ]. Then col( �u1(t); : : : ; �un(t)) is positive and T -periodic since col(u�j1 (t);

: : : ; u�jn (t)) is T -periodic and positive. Because of the uniqueness of positive T -periodic

solutions of system (5.3), when we take col( �u1; : : : ; �un) as the unique T -periodic so-lution col(u1; : : : ; un) of system (5.3) as given in the hypotheses of this theorem, wehave

0=n∑

i=1

| �ui(0)− ui(0)|=d¿0:


This contradiction shows that Eq. (5.5) holds. Thus

col(u�1; : : : ; u

�n)

�∼ col(u1; : : : ; un):

Assume that �i ∼ �i, pij ∼ pij and gi(t; ·)∼ gi(t; ·), i; j=1; 2; : : : ; n; i 6= j. Then for any�∈ (0; �0], there exists a T�¿0 such that

|�i(t)− �i(t)|¡�;

|pij(t)− pij|¡�;

|gi(t; xi)− gi(t; xi)|¡� when x=(x1; : : : ; xn)∈ S

(5.6)

for all t≥T�. We construct the following two auxiliary systems:

xi = xi(gi(t; xi)− �)− (�i(t) + �)xi +∑j∈Ji

(pji(t)− �)(�j(t)− �)xj

i=1; 2; : : : ; n (5.7)

and

xi = xi(gi(t; xi) + �)− (�i(t)− �)xi +∑j∈Ji

(pji(t) + �)(�j(t) + �)xj

i=1; 2; : : : ; n; (5.8)

where we choose � so small as to guarantee that

gi(t; xi)− �¿0; pji(t)− �¿0; �i(t)− �¿0

i; j=1; 2; : : : ; n; i 6= j for all t ∈ [0; T ]; 0≤ xi ≤ k;

and ∫ T

0[(gi(t; 0)− �)− (�i(t) + �)] dt¿0 (5.9)

under the assumption (5.1).

Theorem 5.4. Assume inequalities (5:9) hold. Then there exist unique positive T -periodic solutions col(u−�

1 ; : : : ; u−�n ); col(u�

1 ; : : : ; u�n) and col(x1; : : : ; xn) of systems (5:7);

(5:8) and (5:2); respectively; which are globally asymptotically stable with respectto Rn

+.

Proof. The proof is analogous to the proofs of Theorem 3.2 and 4.2, and weomit it.

Finally, in order to prove Theorem 5.1, we give the following comparison relationshipof solutions among systems (4.1), (5.7) and (5.8).


Lemma 5.5. Suppose that col(x−�1 ; : : : ; x−�

n ) and col(x�1 ; : : : ; x

�n ) are solutions of sys-

tems (5:7) and (5:8) respectively; satisfying

col(x−�1 (T�); : : : ; x−�

n (T�))= col(x�1 (T�); : : : ; x�

n (T�)), r� ∈Rn+:

Then

x−�i (t)¡xi(t)¡x�i (t); i=1; : : : ; n (5.10)

for all t¿T�; where col(x1(t); : : : ; xn(t)) is the solution of Eq. (4.1) with initial valuecol(x1(T�); : : : ; xn(T�))=V�.

Proof. Inequalities (5.6) lead to

�i(t)− �¡�i(t)¡�i(t) + �;

pij(t)− �¡pij(t)¡pij(t) + �;

gi(t; xi)− �¡gi(t; xi)¡gi(t; xi) + �

for all t¿T�; i; j=1; 2; : : : ; n. Then it follows that

xi(T�)− x−�i (T�)¿0; xi(T�)− x�i (T�)¡0

and hence inequalities (5.10), namely,

x−�i (t)¡xi(t)¡x�i (t)

will hold for t− T�¿0 and su�ciently small. If inequalities (5.10) do not hold for allt¿T�, there exist T1; T2¿T� such that

x−�i (t)¡xi(t) for all t ∈ (T�; T1); i=1; : : : ; n;

xi(t)¡x�i (t) for all t ∈ (T�; T2); i=1; : : : ; n:

and there exists atleast one component, which we denote by i0, such that

x−�i0 (T1)= xi0 (T1); x−�

i (T1) ≤ xi(T1) (5.11)

and

xi0 (T2)= x�i0 (T2); xi(t2) ≤ x�i (T2) (5.12)

for all i 6= i0. Suppose Eq. (5.11) holds. The fact that

x−�i0 (t)¡xi0 (t) for all t¡T1 and x−�

i0 (T1)= xi0 (T1)

gives that

x−�i0 (T1)− xi0 (T1) ≥ 0:


However, from Eqs. (5.7) and (4.1), we have that

x−�i0 (T1)− xi0 (T1) = x−�

i0 (T1)( gi0 (T1; x−�i0 (T1))− �)− ( �i0 (T1) + �)x−�

i0 (T1)

+∑j∈ Ji0

(pji0 (T1)− �)( �j(T1)− �)x−�j (T1)

−xi0 (T1)gi0 (T1; xi0 (T1)) + �i0 (T1)xi0 (T1)

−∑j∈ Ji0

pji0 (T1)�j(T1)xj(T1)

¡ 0:

This contradiction shows that

x−�i (t)¡xi(t); i=1; 2; : : : ; n

holds for all t¿T�. Similarly, Eq. (5.12) is also impossible. Thus we have provedEq. (5.10) is true.

Now let us show the main result of this section.

Proof of Theorem 5.1. Let col(x1; : : : ; xn) be a given positive solution of system (4.1).Since system (4.1) approaches system (5.2) asymptotically, given any �¿0, thereexists a T�¿0 such that Eq. (5.6) holds. From condition (5.1), we can pick up asu�ciently small �¿0 such that Eq. (5.9) holds. For the constructed systems (5.7) and(5.8) and from Theorem 5.4, we know that there exist positive T -periodic solutionscol(us�

1 ; : : : ; us�n ); s=−1 or +1 and col( x1; : : : ; xn) of systems (5.7), (5.8) and (5.2)

respectively, each of which is globally asymptotically stable with respect to RN+.

From Theorem 5.3, we have

col(us�1 ; : : : ; us�

n )�∼ col( x1; : : : ; xn) (5.13)

where s=±1. Take col(xs�1 ; : : : ; xs�

n ); s=−1 or +1, as the solutions of systems (5.7)and (5.8) respectively, satisfying

col(xs�1 (T�); : : : ; xs�

n (T�))= col(x1(T�); : : : ; xn(T�)):

Then from Lemma 5.5, it follows that

x−�i (t)¡xi(t)¡x�i (t); i=1; : : : ; n; (5.14)

for all t¿T�. Since col(xs�1 ; : : : ; xs�

n ) ∼ col(us�1 ; : : : ; us�

n ); Eqs. (5.13) and (5.14) implythat col(x1; : : : ; xn) ∼ col( x1; : : : ; xn) and the proof if completed.

6. Discussion

In this paper we considered a model of a population dispersing among discretepatches in an environment whose carrying capacities, barrier strengths, etc. uctuateperiodically or in an asymptotically periodic manner.


Our main focus was persistence, i.e. the survival in all patches of the population, andthe existence of a periodic solution. We have shown that under very general conditions,the populations will settle down to a stable periodic uctuation.Of course, in nature, most often, two or more populations will interact with each

other in a given environment. Modelling several interacting populations can be verycomplicated, especially when periodicity and patchy environments are involved. Forone thing, what may be a barrier to one of them, may not be to the other [19]. Thiswill be the next project in these models.

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Documents

Uniform persistence and global asymptotic stability in periodic single-species models of dispersal in a patchy environment