GLOBAL ATTRACTOR OF COUPLED DIFFERENCE EQUATIONS AND APPLICATIONS TO LOTKA-VOLTERRA SYSTEMS C. V. doc

PAO Received 22 April 2004 This paper is concerned with a coupled system of nonlinear difference equations which is a discrete approximation of a class of nonlinear differential systems wi

EQUATIONS AND APPLICATIONS TO

LOTKA-VOLTERRA SYSTEMS

C V PAO

Received 22 April 2004

This paper is concerned with a coupled system of nonlinear difference equations which is

a discrete approximation of a class of nonlinear differential systems with time delays Theaim of the paper is to show the existence and uniqueness of a positive solution and to in-vestigate the asymptotic behavior of the positive solution Sufficient conditions are given

to ensure that a unique positive equilibrium solution exists and is a global attractor of thedifference system Applications are given to three basic types of Lotka-Volterra systemswith time delays where some easily verifiable conditions on the reaction rate constantsare obtained for ensuring the global attraction of a positive equilibrium solution

1 Introduction

Difference equations appear as discrete phenomena in nature as well as discrete analogues

of differential equations which model various phenomena in ecology, biology, physics,chemistry, economics, and engineering There are large amounts of works in the literaturethat are devoted to various qualitative properties of solutions of difference equations, such

as existence-uniqueness of positive solutions, asymptotic behavior of solutions, stabilityand attractor of equilibrium solutions, and oscillation or nonoscillation of solutions (cf.[1,4,11,13] and the references therein) In this paper, we investigate some of the abovequalitative properties of solutions for a coupled system of nonlinear difference equations

where f(1)and f(2)are, in general, nonlinear functions of their respective arguments,k is

a positive constant,s1ands2are positive integers, andI1andI2are subsets of nonpositive

Copyright©2005 Hindawi Publishing Corporation

Advances in Di fference Equations 2005:1 (2005) 57–79

DOI: 10.1155/ADE.2005.57

s1≡ τ1/k and s2≡ τ2/k are positive integers.

Our consideration of the difference system (1.1) is motivated by some Lotka-Volterramodels in population dynamics where the effect of time delays in the opposing species

is taken into consideration The equations for the difference approximations of thesemodel problems, referred to as cooperative, competition, and prey-predator, respectively,involve three distinct quasimonotone reaction functions, and are given as follows (cf.[7,11,12,15,20]):

(a) the cooperative system:

popula-b(l), andc(l)are positive constants representing the various reaction rates

There are huge amounts of works in the literature that dealt with the asymptotic havior of solutions for differential and difference systems with time delays, and much of

be-the discussions in be-the earlier work are devoted to differential systems, including variousLotka-Volterra-type equations (cf [2,3,5,7,8,12,15,19,20]) Later development leads

to various forms of difference equations, and many of them are discrete analogues of ferential equations (cf [2,3,4,5,6,8,9,10,11,19]) In recent years, attention has alsobeen given to finite-difference equations which are discrete approximations of differentialequations with the effect of diffusion (cf [14,15,16,17,18]) In this paper, we considerthe coupled difference system (1.1) for a general class of reaction functions (f(1),f(2)),and our aim is to show the existence and uniqueness of a global positive solution and theasymptotic behavior of the solution with particular emphasis on the global attraction of apositive equilibrium solution The results for the general system are then applied to each

dif-of the three Lotka-Volterra models in (1.4)–(1.6) where some easily verifiable conditions

on the rate constantsa(l),b(l), and c(l),l =1, 2, are obtained so that a unique positiveequilibrium solution exists and is a global attractor of the system

The plan of the paper is as follows InSection 2, we show the existence and uniqueness

of a positive global solution to the general system (1.1) for arbitrary Lipschitz ous functions (f(1),f(2)).Section 3is concerned with some comparison theorems amongsolutions of (1.1) for three different types of quasimonotone functions The asymptoticbehavior of the solution is treated inSection 4where sufficient conditions are obtainedfor ensuring the global attraction of a positive equilibrium solution This global attrac-tion property is then applied inSection 5 to the Lotka-Volterra models in (1.4), (1.5),and (1.6) which correspond to the three types of quasimonotone functions in the generalsystem

continu-2 Existence and uniqueness of positive solution

Before discussing the asymptotic behavior of the solution of (1.1) we show the existenceand uniqueness of a positive solution under the following basic hypothesis on the func-tion (f(1),f(2))≡(f(1)(u,v,u s,v s),f(2)(u,v,u s,v s))

(H1) (i) The function (f(1),f(2)) satisfies the local Lipschitz condition

(ii) There exist positive constants (M(1),M(2)), (δ(1),δ(2)) with (M(1),M(2))≥

(δ(1),δ(2)) such that for all (u s,v s)∈᏿,

To ensure the uniqueness of the solution, we assume that the time incrementk satisfies

the condition

k

K(1)+K(2)

whereK(1)andK(2)are the Lipschitz constants in (2.1)

Theorem 2.1 Let hypothesis ( H1) hold Then system ( 1.1 ) has at least one global solution

(u n,v n ) in ᏿ If, in addition, condition (2.4 ) is satisfied, then the solution (u n,v n ) is unique

Given anyW n ≡(w n,z n)∈᏿, relation (2.6) and conditions (2.1), (2.2) imply that

whenever (u n − s1,v n − s2)∈᏿ This leads to the relation

P(2)W1)≤(M(1),M(2)) By Brower’s fixed point theorem,ᏼ≡(P(1),P(2)) has a fixed point

U1≡(u1,v1) in᏿ This shows that (u1, v1) is a solution of (1.1) forn =1, and (u1,v1) and(u2− s1,v2− s2) are in᏿ Using this property in (2.9), (2.10) forn =2, the same argumentshows thatᏼ has a fixed point U2 ≡(u2,v2) in᏿, and (u2, v2) is a solution of (1.1) for

n =2 and (u3− s1,v3− s2)∈᏿ A continuation of the above argument shows that ᏼ has afixed pointU n ≡(u n,v n) in᏿ for every n, and (u n,v n) is a global solution of (1.1) in᏿

To show the uniqueness of the solution, we consider any two solutions (u n,v n), (u  n,v  n)

It follows again from (2.4) that| w2| = |z2| =0 The conclusion| w n | = | z n | =0 for every

n follows by an induction argument This proves (u n,v n)=(u  n,v  n), and therefore (u n,v n)

Remark 2.2 (a) Since problem (1.3) may be considered as an equivalent system of thescalar second-order differential equation

u  = f

u,u ,u τ1,u  τ2



(t > 0), u(t) = φ(t)

− τ1≤ t ≤0

, u (t) = ψ(t)

− τ2≤ t ≤0

, (2.16)the conclusion inTheorem 2.1and all the results obtained in later sections are directlyapplicable to the difference approximation of (2.16) with (u n,v n)=(u n,u  n) and (f(1),

for (u n+1,v n+1) which can be computed by a marching process for everyn =0, 1, 2, and

for any continuous function (f(1),f(2)) From a view point of differential equations, theforward approximation may lead to misleading information about the solution of thedifferential system One reason is that a global solution to the differential system may fail

to exist while the difference solution (un+1,v n+1) exists for everyn.

(c) The uniqueness result inTheorem 2.1is in the set᏿, and it does not rule out thepossibility of existence of positive solutions outside of᏿

3 Comparison theorems

To investigate the asymptotic behavior of the solution we consider a class of tone functions which depend on the monotone property of (f(1),f(2)) Specifically, wemake the following hypothesis

quasimono-(H2) (f(1),f(2)) is aC1-function in᏿× ᏿ and possesses the property ∂ f(1)/∂u s ≥0,

∂ f(2)/∂v s ≥0 and one of the following quasimonotone properties for ( u,v,u s,v s)∈

Notice that if (f(1),f(2))≡(f(1)(u,v), f(2)(u,v)) is independent of (u s,v s), then the aboveconditions are reduced to those required for the standard three types of quasimonotonefunctions (cf [15,18]).

It is easy to see from (H2) that for quasimonotone functions the conditions on (M(1),

M(2)), (δ(1),δ(2)) in (2.2) are reduced to the following

(a) For quasimonotone nondecreasing functions:

Then (w n,z n)≥ (0, 0) for every n =1, 2,

Proof Consider the case n =1 Sincew n ≥0 forn ∈ I1andz n ≥0 forn ∈ I2, the ities in (3.7) yield

inequal-γ(1)w1≥ a(1)z1, γ(2)z1≥ a(2)w1. (3.9)

The positivity ofγ(1)1 ,γ1(2)implies that

It follows again from (3.8) that (w m,z m)≥(0, 0) The conclusion of the lemma follows by

The above positivity lemma can be extended to a function (w n,z n,w n,z n) satisfying therelation

whereγ(n l),a(n l),b n(l), andc(n l),l =1, 2, are the same as that in (3.7) and ˆγ(n l), ˆa(n l), ˆb(n l), and ˆc(n l)

are nonnegative with ˆγ(n l) > 0, n =1, 2,

Lemma 3.2 Let ( w n,z n,w n,z n ) satisfy (3.13 ), and let

Then (w n,z n,w n,z n)≥ (0, 0, 0, 0) for every n.

Proof By (3.13) withn =1, we have

γ(1)1 w1≥ a(1)1 z , γ(2)1 z1≥ a(2)1 w1, ˆγ(1)1 w ≥ ˆa(1)1 z1, ˆγ(2)1 z ≥ ˆa(2)1 w (3.15)

This implies that

whereK(l),l =1, 2, are the Lipschitz constants in (2.1) Since σ1(1)≤ K(1),σ2(2)≤ K(2),

it follows that kσ1(1)< 1 and kσ2(2)< 1 Notice that σ2(1) andσ1(2) are nonnegative while

σ1(1)andσ2(2)are not necessarily nonnegative The following comparison theorem is forquasimonotone nondecreasing functions

Theorem 3.3 Let hypotheses ( H1), (H2)(a), and condition ( 3.20 ) be satisfied Denote by

(u n,v n ), ( u n,v n ), and ( u n,v n ) the solutions of (1.1 ) with (φ n,ψ n)=(M(1),M(2)), ( φ n,ψ n)=

(δ(1),δ(2)), and arbitrary ( φ n,ψ n)∈ ᏿, respectively Then

we conclude from Lemma 3.1that (w n,z n)≥(0, 0) This leads to (u n,v n)≥(u n,v n) Asimilar argument using the property (φ n,ψ n)≤(M(1),M(2)) yields (u n,v n)≥(u n,v n) This

For quasimonotone nonincreasing functions, we have the following analogous rem

theo-Theorem 3.4 Let hypotheses ( H1), (H2)(b) and condition ( 3.20 ) be satisfied Denote by

(u n,v n ), ( u n,v n ), and ( u n,v n ) the solutions of (1.1 ) with (φ n,ψ n)=(M(1),δ(2)), ( φ n,ψ n)=

(δ(1),M(2)), and arbitrary ( φ n,ψ n ) in ᏿, respectively Then

Theorem 3.3 that (w n,z n)≥(0, 0) This leads tou n ≥ u n,v n ≥ v n A similar argumentusingφ n ≥ δ(1),ψ n ≤ M(2)givesu n ≥ u n,v n ≥ v n This proves the theorem

For mixed quasimonotone functions, we consider the solution ((u n,v n), (u n,v n)) of thecoupled system

The existence and uniqueness of a solution to (3.28) can be treated by the same argument

as that for (1.1) The following theorem gives an analogous result as that inTheorem 3.3

Theorem 3.5 Let hypotheses ( H1), (H2)(c) and condition ( 3.20 ) be satisfied Let also ((u n,

v n), (u n,v n )) be the solution of (3.28 ) and (u n,v n ) the solution of (1.1 ) with arbitrary (φ n,ψ n)

n ) which are nonnegative in view of the mixed quasimonotone property (3.3) Since

by hypothesis (H2)(c), all other coefficients (a (l)

n,b(n l),c(n l)) and ( ˆa(n l), ˆb(n l), ˆc(n l)) are tive, and by conditions (3.19), (3.20), and (3.23),

we see from (3.31) and (3.32) that condition (3.14) holds if

be-to a common limit as n → ∞ In this section, we show the monotone convergence of

(u n,v n) and (u n,v n) to equilibrium solutions (or quasiequilibrium solutions) of (1.1) foreach of the three types of quasimonotone functions (f(1),f(2)) Here by an equilibriumsolution (or simply equilibrium), we mean a constant (u,v) ∈᏿ such that

Our first result is for quasimonotone nondecreasing functions

Theorem 4.1 Let the conditions in Theorem 3.3 be satisfied Then the solution (u n,v n )

con-verges monotonically to a maximal equilibrium (u,v), the solution (u n,v n ) converges

mono-tonically to a minimal equilibrium (u,v), and

δ(1),δ(2) 

≤(u,v) ≤(u,v) ≤
M(1),M(2) 

If, in addition, (u,v) =(u,v)( ≡(u ∗,v ∗ )), then ( u ∗,v ∗ ) is the unique equilibrium in ᏿,

and for arbitrary (φ n,ψ n ) in ᏿ the corresponding solution (u n,v n ) converges to ( u ∗,v ∗ ) as

n ,b n(l), andc(n l)are nonnegative It follows from the proof

ofTheorem 3.3 that (w n,z n)≥(0, 0) forn =1, 2, This proves the relation (u n,v n)≤

(u n+1,v n+1) A similar argument gives (u n+1,v n+1)≤(u n,v n) and (u n,v n)≤(u n,v n) for eryn =1, 2,

ev-This leads to the relation

u n ≤ u n+1 ≤ u n+1 ≤ u n, v n ≤ v n+1 ≤ v n+1 ≤ v n, n =1, 2, (4.4)The above monotone property ensures that the limits

exist and satisfy relation (4.2) Letting n → ∞in (1.1) shows that (u,v) and (u,v) are

solutions of (4.1) Now if (u,v) is another solution of (4.1) in ᏿, then by considering(u,v) as a solution of (1.1) with (φ n,ψ n)=(u,v), Theorem 3.3ensures that (u n,v n)≤

(u,v) ≤(u n,v n) for everyn Letting n → ∞yields (u,v) ≤(u,v) ≤(u,v) This proves the

maximal and minimal property of (u,v) and (u,v), respectively It is clear that if (u,v) =

(u,v)( ≡(u ∗,v ∗)), then the above maximal and minimal property implies that (u ∗,v ∗) isthe unique equilibrium solution in᏿ Moreover, byTheorem 3.3, (u n,v n)→(u ∗,v ∗) as

If f(1)≡ f(1)(u) is independent of (v,u s,v s) and there exist constants M ≥ δ > 0

such that

f(1)(M) ≤0≤ f(1)(δ), (4.6)then all the conditions in (H1) and (H2)(a) for the scalar problem

u n = u n −1+k f(1)

u n

, u0= φ (n =1, 2, ) (4.7)are satisfied In this situation,Theorem 4.1ensures that the solutionsu n,u nof (4.7) with

φ = M and φ = δ, respectively, converge to some constants u and u as n → ∞ Moreover,

u and u satisfy the equation f(1)(u) = f(1)(u) =0 and the relation δ ≤ u ≤ u ≤ M If

u = u( ≡ u ∗), then for anyφ ∈[δ,M], the corresponding solution u nof (4.7) converges

tou ∗ asn → ∞ In particular, if f(1)(u) = αu(1 − βu) for some positive constants α, β,

then condition (4.6) is fulfilled by any constantsM, δ satisfying 0 < δ ≤ β −1≤ M, and the

limitsu, u are both equal to β −1 Sinceδ can be chosen arbitrarily small and M arbitrarily

large, we have the following result which will be needed in later applications

Corollary 4.2 If f(1)≡ f(1)(u) is a C1-function such that kσ1(1)< 1 and condition ( 4.6 ) holds for some constants M ≥ δ > 0, then the solutions u n , u n of ( 4.7 ) with φ = M and φ = δ, respectively, converge to some equilibrium solutions u, u such that f(1)(u) = f(1)(u) = 0 In

particular, if f(1)(u) = αu(1 − βu) for some positive constants α, β, then for any φ > 0, the corresponding solution u n of ( 4.7 ) converges to β −1as n → ∞

For quasimonotone nonincreasing functions, we have the following analogous result

Theorem 4.3 Let the conditions in Theorem 3.4 be satisfied Then the solution (u n,v n )

con-verges monotonically to an equilibrium (u,v), the solution (u n,v n ) converges monotonically

to an equilibrium (u,v), and

δ(1)≤ u ≤ u ≤ M(1), δ(2)≤ v ≤ v ≤ M(2). (4.8)

If, in addition, (u,v) =(u,v)( ≡(u ∗,v ∗ )), then ( u ∗,v ∗ ) is the unique equilibrium in ᏿ and

for any (φ n,ψ n)∈ ᏿, the corresponding solution (u n,v n ) converges to ( u ∗,v ∗ ) as n → ∞ Proof Consider the solution (u n,v n), and let (w n,z n)=(u n − u n+1,v n+1 − v n) Then,

exist and are equilibrium solutions in᏿ If (u,v) =(u,v)( ≡(u ∗,v ∗)), then the uniqueness

of the equilibrium (u ∗,v ∗) and the convergence of (u n,v n) to (u ∗,v ∗) follow from (3.26)

For mixed quasimonotone functions, we say that the constants (u,v), (u,v) are a pair

of quasiequilibrium solutions of (1.1) if (u,v) and (u,v) are in ᏿ and if

f(1)(u,v,u,v) =0, f(1)=(u,v,u,v) =0,

It is obvious that quasiequilibrium solutions are not necessarily true equilibrium tions unlessu = u or v = v In the following theorem, we show the convergence of the

solu-solution of (3.28) to quasiequilibrium solutions

Theorem 4.4 Let the conditions in Theorem 3.5 be satisfied Then the solution ((u n,v n),(u n,v n )) of (3.28 ) converges monotonically to a pair of quasiequilibrium solutions ((u,v),

(u,v)) that satisfy ( 4.11 ) If (u,v) =(u,v)( ≡(u ∗,v ∗ )), then ( u ∗,v ∗ ) is the unique

equilib-rium in ᏿ and for any (φ n,ψ n)∈ ᏿, the corresponding solution (u n,v n ) of (1.1 ) converges to

(u ∗,v ∗ ) as n → ∞