Differential and Integral Equations, Volume 4, Number 1, January 1991, pp. 117-128.

GLOBAL STABILITY IN DIFFUSIVE DELAY

LOTKA-VOLTERRA SYSTEMS

Y. KUANGt AND H.L. SMITH:j:

Department of Mathematics, Arizona State University, Tempe, Arizona 85287 USA

(Submitted by: Klaus Schmitt)

Abstract. We consider the global asymptotic stability of diffusive delay Latka-Volterra systems which may model population dynamics of closed ecological environments containing n interacting species. The first part of this paper deals with discrete delay case, where both continuous and discrete diffusion situations are considered. The second part of this paper studies unbounded continuous delay cases, where the integral kernels are assumed to satisfy linear differential equations with constant coefficients. In both parts, sufficient conditions for global asymptotic stability of the unique positive steady states are derived via some proper Lyapunov functions. To some extent, these results indicate that the diffusivity of the system may not affect the global asymptotic stability of its reaction system.

1. Introduction. In this paper, we deal with the global asymptotic stability of diffusive delay Lotka-Volterra systems which may be used to model the population dynamics of a closed ecological environment containing n interacting species.

The mathematical theory of population dynamics has largely been developed by employing first-order ordinary differential equations [4, 7, 8, 10, 20]. The popu­lation growth of a species is generally assumed to respond instantaneously to the densities of its own population and the populations of other species with which it interacts. However, it is often more realistic and appropriate to allow the rates of change of the variables to depend on the previous histories of the population densi­ties as well as the possible dispersal rates caused by the spatial heterogeneity of the environment. It is these considerations which motivate us to study the diffusive de­lay Lotka-Volterra system. When the environment is assumed to be homogeneous, this system reduces to the well-known delay Lotka-Volterra system which was ex­tensively studied by many authors (see Beretta and Solimano [1], Worz-Busekros [23], Cushing [4] and those references cited therein). In the case that population growth does not depend on previous population densities, it reduces to the well­studied reaction-diffusion systems (see Brown [3], Dunbar et al. [5], Hastings [12], Smoller [21] and those references cited therein). In this circumstance, our results reduce to the ones appearing in Dunbar, Rybakowski and Schmitt [5]. In fact, our method is a generalization of the one adopted in [5]. Since the last decade, the

Received October 31, 1989, in revised March 12, 1990. tResearch partially supported by a FGIA award and a CLAS Summer Research award at Arizona State University. :j:Research supported in part by NSF Grant DMS 8722279. AMS Subject Classifications: 34K30, 35R10, 34K15, 92A15.

An International Journal for Theory & Applications

118 Y. KUANG AND H.L. SMITH

role of dispersal in the maintenance of patchiness or spatial population variation in ecological systems has also received considerable attention by some investigators (see, for example, Beretta and Takeuchi [2], Freedman and Takeuchi [9], Levin and Segel [16] and those references cited therein).

Recently, Martin and Smith [17, 18, 19] investigated various aspects of diffusive delay systems. In particular, much of their work has been centered around mono­tonicity and convergence of the solutions, when the system itself possesses some monotonicity. The work of most relevance to this paper appears in [19], where they are able to establish sufficient conditions for the global stability of a satu­rated equilibrium (for the definition, see [26]) in some very general diffusive delay Latka-Volterra systems. Their approach combines Lyapunov function theory, strong maximum and invariance principle. Some other relevant references are Hale [11], Lenhart and Travis [14], Travis and Webb [22], and Leung [25].

The first part of this paper studies the diffusive Latka-Volterra system with dis­crete delays, where both continuous and discrete diffusion situations are considered. Our approach combines the techniques introduced in Hastings [12] and in Lenhart and Travis [15]. A proper Lyapunov function is constructed to derive sufficient conditions for the global asymptotic stability of the unique positive steady state. Our results generalize results in [5, 10, 12, 15 and 25].

The second part of this paper deals with the unbounded continuous delay sit­uations. This time, we extend those results obtained by Worz-Busekros [23] to the diffusive delay systems. The key technique is the well-known linear-chain trick discovered by Fargue [6]. Again, a Lyapunov function is constructed to establish suf­ficient conditions for the global asymptotic stability of the assumed unique positive steady state.

We complete this paper with a section devoted to discussion.

2. Discrete delay case. In this section, we consider the following diffusive delay Lotka-Volterra system with discrete delays:

on n X (0, oo),

a avui(x,t) = 0, on an x (O,oo),

Ui(x, B)= ~i(x, B), on n X [-T, OJ,

(2.1)

(2.2)

(2.3)

wherei=1,2, ... ,n,u=(ul,Uz, ... ,un),x=(xl,··· ,xN)andnisanopen,con­nected and bounded region in ~N with an E H2+<>' 0 <a< 1 (the Holder space; see §1.3 in [25]). Furthermore, d; are measures of dispersal rate and may depend on x, t or any of the ui(x, t); for simplicity, we assume di are nonnegative constants. a;av stands for the outward normal derivative on an. bi(·), i = 1,2, ... ,n are strictly increasing functions, with b;(O) = 0. The initial functions ~i : fl x [-T, OJ --+~are assumed continuous and nonnegative, with ~;(x, 0) > 0, i = 1, 2, ... , n. Through­out the rest of this paper, C[ -T, o]n denotes the space of continuous functions from [-T, 0] into ~n and for each X E fl, u(x, ·)t = (u;(x, ·)t)I denotes the member of

DIFFUSIVE DELAY SYSTEMS 119

C[-r, o]n defined by u(x, O)t =: u(x, t + 0) for (J E [-r, 0]. The functionals G; in (2.1) map C[-r, o]n into~ and are assumed to have the form

(2.4)

for all¢= (¢;)1 E C[-r, O]n and i = 1, 2, ... , n, where ri, aij and aL are all real constants, and r1 2: 0, l = 1, 2, ... , m, r = max{ rz I l = 1, 2, ... , m }. System (2.1)-(2.3) can be used to model the population dynamics of a closed ecological system containing n interacting species (which means there is no immigration and emigration). The u;(x, t) represent the population density of the ith species at time t and location x, r = ( r 1 , r 2 , ... , r n) is the vector of intrinsic population growth rates, A= (a;j) is the instantaneous interaction matrix. Z:::7= 1 (Z:::~ 1 aL¢j(-rz)) describes the effect of the historical population densities on the ith species' present per capita growth rate. Since there are no restrictions on the signs of r; and a;j, system (2.1)-(2.3) is capable of modelling competitive, cooperative and predator­prey interactions.

In this section, we always assume that system (2.1)-(2.3) has a unique, spatially homogeneous steady state solution, which is supposed to be a positive equilibrium, namely, u* = ( ui, u:;;, ... , u~). The main objective of this section is to establish conditions under which u* is globally asymptotically stable with respect to~ = (~)1,

~ E C([O x [-r, 0], ~+), and ~;(x, 0) > 0, i = 1, 2, ... , n. Local existence, uniqueness and continuability of mild solutions of (2.1) follow

from results in [18, 22]. By a mild solution of (2.1) we mean a continuous function u: [0, t*) ~ C(f.l, Rn) which satisfies a certain integral equation obtained by applying the variation of constants formula to the abstract formulation of (2.1) as a delay differential equation in the space C(f.l, Rn) (see either of the two references above). It is shown in [18] that this mild solution u(t) is a strong solution (continuously differentiable) of the abstract delay differential equation in C(f.l, ~n) fort > r and thus it is a classical solution of (2.1) fort > r. Here, we will simply assume that our solutions are classical solutions.

If u = (u;)1 is a solution of (2.1)-(2.3) on 0 x [-r, t*] and

M; = sup{IG;(u(x, ·)tl: (x, t) En X [0, t*]},

then

N a ( au) N a ( au) """'- d-' - b·(u·)M· <au) at<"""'- d·-' + b (u )M ~ axk t axk t t t - t - ~ axk t axk t t t

k=l k=l

and it follows by comparison and the maximum principle that (see Smoller [21]): (a) ~;(x, 0) = 0 implies u;(x, t) = 0 and

(b) ~;(x, 0) > 0 implies u;(x, t) > 0 for all X E f.l, t > 0. If all d;, i = 1, 2, ... , n are positive, then (b) can be replaced by (b') ~;(xo, 0) > 0 for some Xo E 0 implies ui(x, t) > 0 for all X E f.l, t > 0.

In order to present our main result, we need the following lemma, which estab­lishes conditions for the global stability of u* when d; = 0, i = 1, 2, ... , n.

120 Y. KUANG AND H.L. SMITH

Lemma 2.1. Consider the following discrete delay Lotka-Volterra system

dui(t) ~ = bi(ui(t))Gi(u(·)t),

ui(B)o = ~i(B) ~ 0, () E [-7, OJ, ~i(O) > 0,

(2.5)

(2.6)

where i = 1, 2, ... , n, and bi(·), Gi(·) are defined as above. If there exists a constant positive diagonal matrix C = diag(cb c2 , ••. , cn), such that CA +_Arc is negative definite, then the unique positive equilibrium u* is globally stable. Here A = ( aij), where aij = aij if i =/:- j, and

Remark 2.1. When the interactions do not depend on the historical population densities; i.e., aL = 0, for all 1 :::; i, j :::; n, 1 :::; l :::; m, then Lemma 2.1 reduces to the well-known result of Goh [10].

Remark 2.2. In the case n = 1, and b1 (u1(t)) = u1(t), our system reduces to the equation considered by Lenhart and Travis [15], and Kuang [13]. Thus Lemma 2.1 can be viewed as a partial generalization of the results contained in [13] and [15].

Proof of Lemma 2.1: Suppose there exists a constant positive diagonal matrix C = diag(cb c2, ... , en) satisfying the assumptions specified in the lemma. We define V : C ---> ~ as

n [ {'/>;(0) 8 n ( m laLijo 2 )] V(¢) = t; ci Jo Ms + ui) ds +]; ~ -2- -rz <Pi (B) d() . (2.7)

It is easy to see that V(¢) > 0 if¢ =1- 0 and ¢i +u; > 0, fori= 1, 2, ... , n, V(O) = 0. Assume Ut(-) = <Pt(·) + u* is a solution of (2.5) and (2.6). Let i[2.5J(¢t) denote

the derivative of V(<Pt) along the solution of (2.5) and (2.6). It follows that

Vr2.s] (<Pt) = t ci [t aij¢i(t)¢j(t) + t (~ aL¢i(t)¢i(t- 7!))

+ t (~ la~jl (<P;Ct)- <P;Ct- 7!))) J

:::; 1; Ci [t aij¢i(t)¢j(t) + t (~ la;jl ( ¢I(t) + ¢T(t)))]

( n ( m 1 !_I) n ( m 1 l·l)) n n = ciaii + ci]; ~ a~J +]; ~ Cj a~, ¢T(t) + ~]; ciaij¢i(t)¢j(t)

j#i j#i

DIFFUSIVE DELAY SYSTEMS 121

where </>(t) = (¢1(t), </>2(t), ... , </>n(t)), A= (aij),

~ .. -a·. l.f; ...J. J. and ~,.=a •• +~(~ la2LI) + ~(~ ccJ·· la2jil). '-'-<J - <J • I ' '-'-•• •• ~ ~ ~ ~

j=l l=l j=l l=l ' #i

Obviously, the assumption of C A + _AT C being negative definite implies that if <f>(t) =J 0, then i[2 .5J(<I>t) < 0. By virtue of the proof of Theorem 3.2 in Chapter 5 of [28], we have <f>(t) ----> 0 as t----> oo. This proves the lemma. 1

Now we are in a position to state and prove our main result.

Theorem 2.1. Consider the diffusive delay Lotka-Volterra system (2.1)-(2.3). As­sume 1s (b,(s~u:J) ?': 0, i = 1, 2, ... , n. If there exists a constant positive diagonal

matrix C = diag(c1 , c2 , ... , en) such that CA+ATC is negative definite, and u(x, t) is a bounded solution of (2.1)-(2.3) with nonnegative initial vector function~ such that ~i(x, 0) > 0, i = 1, 2, ... , n, then u(x, t) tends to u* as t----> oo, uniformly for X En. Here A is the same as defined in Lemma 2.1.

Proof: Assume u(x, t) = <f>(x, t) + u* is the solution of (2.1)-(2.3), satisfying

ui(x,B) = ~i(x,B), on n x [-r,O] and ~i(x,O) > 0.

Then <f>(x, t) satisfies

a av </>i(x, t) = 0, on af! X (0, oo) (2.9)

and (2.10)

where

Qi(</>(x, ·)t) = t aij</>j(x, t) + t (~ aL</>j(x, t- rz)). (2.11)

Since Ui(x,t) > 0 on n X (O,oo), we can define w: C(n X [-r,O],IR) ____, lR as

W(<f>t) = k V(<f>t(x, ·)) dx, (2.12)

where

n [ {'/>;(x,t) 8 n ( m lal. ·I JO )] V(<f>t(x, ·)) = ~ ci Jo bi(s + u;) ds + ~ t; ~1 -r, </>](x, t +B) d(} .

(2.13)

122 Yo KUANG AND HoLo SMITH

We see W 2: 0, W = 0 if and only if c/Jt(o, B)= Oo The derivative of W with respect to timet along solutions of (208)-(2010) is

0 ~ [ (Pi(x, t) (~ {) ( ac/Jt(x, t)) V[2 0 8J(¢t(X, o)) = 6 Ci b·(lj>·( t) + *) 6 ~ di {)

i=l t t X, Ui k=l UXk Xk

+ bi(¢i(x, t) + u;)Qi(¢(x, o)t))]

+ tci [~(~ la~jl (¢J(x,t)- ¢](x,t- rz)))]

=h+h

The above expression is obtained via integration by parts and the boundary condi­tion (209)0 By the assumption of fs (b;(s~u;)) 2:0, we see that

In hdx:::; Oo (2014)

Similar to the argument as made in the proof of Lemma 201, we have

where ¢(x, t) = (¢1 (x, t), 0 0 0 , ¢n(x, t))o By the assumption that cA +Arc is negative definite, we see J0 12 dx < 0 if ¢(x, t) =/= 0° Hence W = f0 (h + h)dx < 0, if ¢(x, t) =/= Oo Since ¢(x, t) is assumed to be bounded, by a similar proof as the Lemma 201 in [27], we see 1/>t is bounded in C 1

o A similar argument as given in

DIFFUSIVE DELAY SYSTEMS 123

[30] indicates that (Pt ----> 0 uniformly, since {0} = { <f> E C 1 (0 x [0, oo )); W(<f>t) = 0}. This proves our theorem. I

In addition to the continuous diffusion model (2.1)-(2.3), we can also establish a similar result for discrete diffusion systems. We assume the closed environment consists of K discrete patches. Then the discrete diffusion version of (2.1)-(2.3) takes the form

(2.15)

1:::; i:::; n, 1:::; k, l:::; K, where Gi(·) is defined as in (2.4). Here u~ is the population density of species i in patch k, uk = (u~, ... , u~), and the Df1s are dispersal coeffi­cients which may depend on tor on U(t), where U(t) = (ul(t) ... , u;(t), ui(t), ... , u~(t)). Systems similar to (2.15) have been studied in Beretta and Takeuchi [2], Levin and Segel [16] and Hastings [12].

A similar argument as made in Lemma 2.1 proves the following theorem, which is an analogue to Theorem 2.1 for the discrete patch case.

Theorem 2.2. Consider the system (2.15) where it is assumed that

Dkl = Dlk > o t t - . (2.16)

If the assumptions of Lemma 2.1 are satisfied, then U* = ( ui, ... , u:V ui, ... , u~, ... , u~) E 1R+K is globally asymptotically stable for all nonnegative initial vector func­tions ~(0) E C([-T, 0), JR+.K), such that ~(0) > 0.

Proof: We note that the boundedness of solutions follows if we can show V ( <f>t) :::; 0. Since the argument for proving this is very similar to Lemma 2.1, we choose to omit the details in order to avoid repetition.

3. Continuous delay case. In this section, we still consider the diffusive delay Lotka-Volterra system (2.1)-(2.2). Compared to the previous section, the only difference appears in the definition of Gi(u(x, ·)t)· Throughout this section, we assume

i = 1, 2, ... , n, where ri, aij, rij, i, j = 1, ... , n are real constants and Fij : [0, oo) ---->

lR are continuous nonnegative functions normalized by

1= Fij(s) ds = 1, i,j = 1, ... , n. (3.2)

We suppose that the population densities in the past are known, nonnegative bounded functions

ui(x, t) = <f>i(x, t), -oo < t :::; 0, <f>i(x, 0) > 0, x E 0, i = 1, ... , n. (3.3)

124 Y. KUANG AND H.L. SMITH

Given the initial conditions (3.3), the existence, uniqueness and positivity of local solutions of (2.1)-(2.2) follow from the standard technique of integra-differential equations. In [6], Fargue observed that the integro-differential equation

u(t) = H(u, t) + j_too F(t- s)G(u(s)) ds

with initial condition u(t) = u0 (t), -oo < t < 0, is equivalent to a system of ordinary differential equations with proper initial conditions if and only if the kernel F is a linear combination of functions

Motivated by this observation, in the rest of this section, we assume that every kernel Fij appearing in (3.1) is a normalized convex combination of functions

(3.4)

The derivative of Fm is given by

with F0 (t) = 0. We define

vi(x, t) = ui(x, t), i = 1, 2, ... , n,

and introduce further distinct functions Vn+l, ... , Vp of the form

j_too F,_.(t- s)vj(x, s) ds, JL = 1, ... , m, (3.5)

where F,_. runs through all density functions (3.4) appearing in Fij, ... , Fnj (omit­ting those which have been already defined) for j = 1, ... , n.

The new functions Vn+I, ... , Vp satisfy a system of linear constant coefficient differential equations in VI, ... , vp. Together with the differential equations for VI, ... , Vn, we obtain a system of ordinary reaction-diffusion equations of the form

avz(x,t) -f-. ot = L...J azjvj(x, t), j=I

l=n+1, ... ,p, (3.7)

with initial conditions

i = 1, 2, ... , n, X E f2, (3.8)

vz(x,O) = j_0

00F,_.1(-s)vj1(x,s)ds, x En, for some 1 ~ J-Ll ~ m, 1 ~ jl ~ n.

(3.9)

DIFFUSIVE DELAY SYSTEMS 125

The coefficients appearing in (3.6) and (3.7) are determined by those appearing in (2.1), (2.2) and (3.1). To simplify the presentation, in the following, we do not specify these coefficients; thus, theoretically speaking, system (3.6)-(3.9) is more general than (2.1), (2.2), (3.1) and (3.3). It is easy to see that every solution of system (3.6)-(3.9) with initial values in C(O, JR+.) x C(O, JRP-n) has values in C(O, JR+.) x C(O, JRP-n).

The following theorem generalizes Theorem 2 in Worz-Busekros [23] to the reaction-diffusion system (3.6)-(3.9).

Theorem 3.1. Suppose (3.6)-(3. 7) possess an isolated equilibrium v* E C(O, JR+.) x

C(O,JRP-n), and fs(b,(s~v;))?:: 0, i = 1,2, ... ,n. A sufficient condition for v*

to be globally asymptotically stable with respect to initial values in C(O, JR+.) x C(O, JRP-n) is the existence of positive real numbers c1, c2, ... , Cn and a positive definite (p- n) x (p- n)-matrix C4 such that CA+ATC is negative definite, where A= (aii), 1 ::; i,j ::; p, C1 = diag(c1, c2, ... , en) and C = diag(C1, C4 ).

Proof: The proof is very similar to the one for Theorem 2 in Worz-Busekros [23]. We define V : C(O, JR+.) x C(O, JRP-n) _.. lR as

{ [ n 1v,(x,t) 8 _ v* P ]

V(v(·, t)) = Jr: 2 2::>i . bi(s)' ds+ .. L (vi -v;)Cij(vi -vj) dx, (3.10) 0 •=1 v, •,J=n+l

where C4 = (Cii), i,j = n+ 1, ... ,p. It is easy to see that V?:: 0, and V = 0 if and only if v(·, t) = v*. The derivative of V along a solution of (3.6)-(3.9) is given by

p

+ L (vicii(vi- vj) +(vi- v;)ciivi)] dx i,j=n+l

where

1 n * N a a Vi- V· Vi I1 = _ 2LCi-b·( ·)· (2: -a (di-a )) dx.

0 i=l ' v, k=l Xk Xk

The last equation used the fact that v* is an equilibrium of (3.6)-(3.9). In matrix

126 Y. KUANG AND H.L. SMITH

notation, we have

-I +1( *)(e1A1+Afe1 A§e4+e1A2)( *)Td - 1 v - v e A ATe e A ATe v - v x n 4 3+ 2 1 4 4+ 4 4

= h +In (v- v*)(eA + ATe)(v- v*f dx

= h +12,

where

According to our assumption, h ::; 0, 12 = 0 if and only if v = v*. From the proof of Theorem 2.1, we have h ::; 0. Hence, dV/dt ::; 0 along every solution of (3.6)-(3.9), and dVjdt = 0 if and only if v = v*. Therefore, by Theorem 2 of [30], the steady state v* is globally asymptotically stable with initial values in e(n, JR+.) x e(n, JRP-n). This proves the theorem. 1

Examples to illustrate the application of this theorem are readily available by modifying those presented in Worz-Busekros [23].

As in the case of the previous section, a result analogous to Theorem 3.1 can be established for the situation when the closed environment consists of K discrete patches. This approach can easily lead to a generalization of Theorem 3.1 in Beretta and Takeuchi [2] where K = 2 is considered. To avoid repetition, we leave the details to interested readers.

In [20], Post and Travis investigated a system similar to (2.1) and (3.1) where di = 0, i = 1, 2, ... , n. By combining a similar method as employed by Worz­Busekros [23] and the well-known M-Matrix theory, they obtained a very practical global stability result. A similar argument as made in the proof of Theorem 3.1 indicates that their result is valid even with diffusion terms added to their system.

4. Discussion. In this paper, we have investigated the diffusive delay Lotka­Volterra system in various contexts. In all these situations, the sufficient conditions for global asymptotic stability of the unique positive steady state, are derived via some proper Lyapunov functions. To some extent, our results indicate that the diffusivity of the system may not affect the global asymptotic stability of its reaction system. A well-known result established by Hale in [11] asserts that if the diffusivity is large enough, then the solutions of the diffusive delay system are asymptotic to the solutions of its reaction system.

In both §2 and §3, we have assumed that ,fs(b,(s~u;)) ::::-:0, i = 1,2, ... ,n. We would like to point out here that this is not a strong restriction. For instance, if we assume the standard expression of bi(ui) = f3iui, where f3i are positive constants, then Js ( b,(s~u;J) ::::-: 0 automatically for Ui ::::-: 0.

By employing a very different method, Martin and Smith [19] obtained suffi­cient conditions for the general diffusive delay Lotka-Volterra system with bounded

DIFFUSIVE DELAY SYSTEMS 127

distributed delays. However, their sufficient conditions require that the coefficient matrix of the system be negative diagonal dominant, which is rather restrictive. The example presented in Post and Travis [20] indicates that for a particular predator­prey interaction, the region of global stability found by our criterion is at least eight times larger than the one found via Martin and Smith's criterion. However, it should be mentioned here, our method cannot be generalized to deal with the general distributed delay situation.

In the discrete diffusion case, we have assumed that each patch has identical dy­namics and the diffusion rates satisfy Df1 = D~k. However, this is not essential. The global stability of the unique positive steady state can tolerate small perturba­tions of the coefficients and coefficient functions. Such an argument can be found in Kuang, Martin and Smith [24].

It should also be pointed out that the results in Dunbar, Rybakowski and Schmitt [5] have considerable overlap with our theorems. In fact, when there are no delays in (2.1) and bi(ui) = ui, our results coincide with theirs.

Our method can be generalized to deal with some of the more general diffusive delay Kolmogorov interaction systems, where the per capita growth rates may be nonlinear. This generalization will be similar to the approach adopted by Freedman and So [8] in dealing with the global stability problem of a general food chain model.

Finally, we would like to mention that if system (2.1)-(2.3) satisfies the so-called food pymmid condition (cf. [31]), then the assumption for u(x, t) to be bounded can be removed from Theorem 2.1. This can be seen from the proof of the Theorem 4.1 in Alikakos [31].

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