convergence of solutions of reation diffusion systems with time delays

Pao∗ Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA Received 18 July 1999; accepted 22 March 2000 Keywords: Reaction–di"usion equations; Time del

Convergence of solutions of reaction–di"usion

systems with time delays

C.V Pao∗

Department of Mathematics, North Carolina State University, Raleigh,

NC 27695-8205, USA Received 18 July 1999; accepted 22 March 2000

Keywords: Reaction–di"usion equations; Time delays; Asymptotic behavior; Global attractor; Upper and lower solutions; Volterra–Lotka models

1 Introduction

Di"erential equations with discrete or continuous time delays are traditionally for-mulated in the framework of ordinary di"erential systems and much discussions are devoted to the qualitative analysis of the systems In recent years attention has been given to parabolic systems where the e"ect of di"usion and convection is taken into consideration In this paper we investigate the asymptotic behavior of solutions for

a class of reaction–di"usion–convection systems with time delays in a bounded

consideration is given in the form

∗Tel.: 919-515-2382; fax: 919-515-3798.

E-mail address: cvpao@math.ncsu.edu (C.V Pao).

0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd All rights reserved.

PII: S0362-546X(00)00189-9

elliptic operator in the form

j;k = 1

j = 1

−ri

delays, and in the case of continuous delays it may be either Fnite or inFnite It is

the property

For Fnite continuous delays the above condition is replaced by

Reaction–di"usion systems in the form of (1.1) have been treated by many investi-gators and di"erent methods have been used for the qualitative analysis (cf [1–13,15 –19]) The discussions in earlier works are mostly in the framework of semigroup the-ory and the thethe-ory of dynamical systems (cf [4,5,8,16–18] and the references therein) More recently, the method of upper and lower solutions and its associated mono-tone iterations have been used to investigate the dynamic property of the system (cf [6,10–13]) An advantage of this method is that it can lead to various qualitative in-formation of the solution as well as computational algorithm for numerical solutions (cf [6,14]) Although the above methods are useful for obtaining invariant regions

of reaction–di"usion systems, the determination of the precise asymptotic limit of the time-dependent solution is, in general, more diGcult especially when the system pos-sesses multiple steady-state solutions One of the diGculties is the lack of explicit in-formation about the steady-state solutions of the corresponding elliptic boundary-value problem when the boundary condition is of Dirichlet or Robin type On the other hand,

if the boundary condition is of Neumann type as that in (1.1) then constant steady-state

solutions can often be found from the nonlinear reaction function in the system The purpose of this paper is to investigate the asymptotic behavior of the time-dependent solution of (1.1) in relation to constant steady-state solutions, including regions of at-traction of the stable steady solutions These results are given in Section 2 SpeciFcally,

attractor and the convergence of the time-dependent solution It turns out that these conditions are independent of the di"usion–convection coeGcients and the time delays

In Section 3 we apply the general results in Section 2 to two Volterra–Lotka mod-els in ecology for studying the global stability and instability of the various constant steady-state solutions The stability conditions for these model problems are given in terms of the rate constants of the reaction function and is independent of the time delays and the e"ect of di"usion

2 Global existence and dynamics

Q0≡ Q(1)0 × · · · × Q0(N), and let IQ(i)0 ; IQ0 be the same domains as Q0(i) and Q0 deFned

the set of scalar-valued functions that are HKolder continuous in D (with exponent

In addition to the above general assumptions for parabolic equations we impose the

Recall that by writing the vectors u; v in the split form

such that the function

fi( ˜ci; [ ˜c]ai; [ ˆc]bi; [ ˜c]ci; [ ˆc]di) ≤ 0;

Under hypothesis (H) we have the following global existence–uniqueness result

Theorem 2.1 Let ˜c; ˆc be a pair of constant vectors satisfying ˜c ≥ ˆc and condition

unique global solution u(t; x) such that

Proof It is known that if problem (1.1) has a pair of coupled upper and lower solutions

(2.3) (cf [12,13]) For the present system (1.1), coupled upper and lower solutions

where D ≡ (0; ∞) × 6 and S ≡ (0; ∞) × @6 It is easy to verify from condition (2:4) and J ∗ c = c for every constant vector c that all the inequalities in (2.6) are satisFed

existence of a unique solution u(t; x) and relation (2.5) follows from Theorem 2.2 of [13] (see also [12])

To investigate the dynamics of the system we deFne two sequences of constant

Ic(m)i = Ic(m−1)i + 1

Kifi( Ic(m−1)i ; [ Ic(m−1)]ai; [c(m−1)]bi; [ Ic(m−1)]ci; [c(m−1)]di);

c(m)i = c(m−1)i +K1

ifi(c(m−1)i ; [c(m−1)]ai; [ Ic(m−1)]bi; [c(m−1)]ci; [ Ic(m−1)]di)

se-quences of constant vectors are well deFned The following lemma gives the monotone property of these sequences

possess the monotone property



ifi( Ic(0)i ; [ Ic(0)]ai; [c(0)]bi; [ Ic(0)]ci; [c(0)]di)



ifi( ˜ci; [ ˜c]ai; [ ˆc]bi; [ ˜c]ci; [ ˆc]di) ≥ 0;



ifi(c(0)i ; [c(0)]ai; [ Ic(0)]bi; [c(0)]ci; [ Ic(0)]di)



− c(0)i

quasimonotone property of f(u; v),

by (2:7) and hypothesis (H),

Ki( Ic(m)i − Ic(m+1)i )

≥ 0:

for i = 1; : : : ; N The monotone property (2.8) follows by the principle of induction

In view of the monotone property (2.8) the constant limits

lim

exist and satisfy the relation

the equations

It is clear that the constant vectors Ic; c are not necessarily steady-state solutions of (1.1) unless Ic = c In the latter case, Ic (or c) is the unique steady-state solution in time-dependent solution u(t; x) in relation to Ic and c

Theorem 2.2 Let the conditions in Theorem 2:1 hold, and let Ic; c be the limits in the relation

and

lim

Proof Consider the steady-state problem

upper and lower solutions of (2.14) if ˜u ≥ ˆu on I6 and they satisfy the inequalities in

(2.6) without the time derivative terms and the initial conditions (cf [12,13]) This implies that the constant vectors ˜c; ˆc are coupled upper and lower solutions of (2.14)

−Liu(m)i + Kiu(m)i

= Kiu(m−1)i + fi(u(m−1)i ; [ Iu(m−1)]ai; [u(m−1)]bi; [ Iu(m−1)]ci; [u(m−1)]di);

−Liu(m)i + Kiu(m)i

= Kiu(m−1)i + fi(u(m−1)i ; [u(m−1)]ai; [ Iu(m−1)]bi; [u(m−1)]ci; [ Iu(m−1)]di);

Since for each i = 1; : : : ; N; fi(u; v) is a constant whenever u; v are constant vectors we

u(m)N } governed by (2.15) coincide, respectively, with the constant sequences { Ic(m)}

lim

m→∞u(m)= c and Ic and c, called quasisolutions of (2.14), satisfy the equations in (2.11) Moreover,

of the theorem follows from Theorem 3.2 of [13] (see also [12])

of these two types) we have the following global result

Proof This is a consequence of Theorem 3.3 in [13] Details are omitted

and hypothesis (H) is reduced to the following:

As a consequence of the above theorems we have the following conclusion for system (1.1) without time delay

conclusions in Theorems 2:1 and 2:2 remain true for system (1:1) without time delays Moreover, the result in Corollary 2:1 holds if condition (2:16) is satis6ed

It is obvious that a similar conclusion in Corollary 2.2 holds if f ≡ f(J ∗ u) is

independent of u

3 Applications

It is seen from the results of the previous section that under the mixed

quasi-monotone condition on the nonlinear function f(u; J ∗ u), an invariant region and the

convergence of the time-dependent solution to a uniform steady-state solution can be obtained through the construction of a pair of constant vectors ˜c; ˆc satisfying relation (2.4) The construction of these vectors depends only on the nonlinear function f, and

a suitable construction can sometimes lead to a characterization of the stability or insta-bility of the various steady-state solutions To demonstrate this possiinsta-bility we consider two model problems arising from ecology where the nonlinear reaction function f is mixed quasimonotone Our construction of the vectors ˜c; ˆc yields some simple condition

on the reaction rate constants of f so that the global asymptotic stability (or instabil-ity) of a constant steady-state solution can be determined It is assumed for physical

3.1 A prey-predator system

The well-known Volterra–Lotka prey-predator reaction–di"usion system with discrete

or continuous time delays is given by

It is obvious that the steady-state problem of (3.1) possesses the trivial solution (0,0)

1; c∗

1= (a1c2− a2c1)=(b1c2+ b2c1); c∗

see from Theorem 2.1 that a unique global solution (u; v) to (3.1) exists and satisFes

the relation (0; 0) ≤ (u; v) ≤ (M1; M2) The maximal principle for standard parabolic

we Frst consider the case

This ensures that a unique positive constant steady-state solution exists and is given

1; c∗

{c(m)1 ; c(m)2 }, governed by (2.7) with ( Ic(0)1 ; Ic(0)2 ), (c(0)1 ; c(0)2 ) given by (3.6) and (f1; f2) given by (3.2) possess the monotone property (2.8) and converge to some limits

Subtraction of the corresponding equations (3.7) gives

1; c∗

2)

1; c∗

1; c∗

Moreover, the standard comparison theorem for parabolic boundary-value problems

implies that u(t; x) ≤ U(t; x) in D, where U is the positive solution of the scalar

boundary-value problem

D, where V is the solution of the scalar boundary-value problem

with 2= a2+a1(b2=b1) Since V (t; x) → 2=c2 as t → ∞ we see that there exists t2¿0

an application of Corollary 2.1 we conclude that if the time delay is Fnite then the

1; c∗

results we have the following

be satis6ed Then a unique global positive solution (u; v) to problem (3:1) exists and lim

1; c∗

inequalities in (3.4) if ( is suGciently small and

c(m)1 = c(m−1)1 +K1

1f1(c(m−1)1 ; Ic(m−1)2 )

= c(m−1)1 +K1

1c(m−1)1 (a1− b1c(m−1)1 − c1Ic(m−1)2 );

or

Theorem 3.2 Let (1; 2) ≥ (0; 0) with 2(0; x) = 0; and let a2=a1≥ c2=c1 Then a unique global positive solution (u; v) to (3:1) exists and

lim

Remark 3.1 Theorems 3.1 and 3.2 imply that under condition (3.5) the positive

1; c∗

Lyapunov stability) Moreover, for Fnite continuous or discrete time delays, including

1; c∗

stable This implies that there can exist no nonuniform positive steady-state solution

On the other hand, if the reversed inequality in (3.5) is satisFed then the

continuous delays

3.2 A competition model

As a second application we consider the Volterra–Lotka competition model with time delays:

the same as that in (3.1) It is obvious that this system possesses the same trivial and semitrivial steady-state solutions as that in (3.1) If the rate constants in (3.10) satisfy the relation

1; c∗

1= (a1c2− a2c1)=(b1c2− b2c1); c∗

conditions in (3.1)) has a unique global solution (u; v), and (u; v) is positive in (0; ∞)×

To investigate the asymptotic behavior of the solution (u; v) we Frst consider the case where the condition (3.11) holds Our aim is to show that (u; v) converges to the

positive constant (c∗

1; c∗

This is possible because of condition (3.11) It is easy to verify from (1.4) and

leads to

1; c∗

2)

1; c∗

stan-dard comparison theorem for scalar parabolic boundary-value problems, there exists

Corollary 2.1 we have the following conclusion

a unique global positive solution (u; v) to (3:10) exists and

lim

1; c∗

Theorem 3.3 implies that under condition (3.11) the positive steady-state solution

1; c∗

Fnite continuous or discrete time delays, including the case without time delays, the

1; c∗

perturbations In this situation, there exists no nonuniform steady-state solution despite

We next show that if condition (3.11) is replaced by either

then there exists no positive steady-state solution, and the time dependent solution (u; v)

constants satisfying

that satisfy the equations in (3.15) Since by (2.7) and the monotone property of the

c(m)2 = c(m−1)2 +K1

2f2(c(m−1)2 ; Ic(m−1)1 )

= c(m−1)2 +K1

2c(m−1)2 (a2− b2Ic(m−1)1 − c2c(m−1)2 );

and using either one of the conditions in (3.17) we obtain

non-trivial nonnegative initial perturbations) while the non-trivial and seminon-trivial solutions (0; 0)

On the other hand, if either

≡ 0 the solution (u; v) of (3.10) converges to (0; a2=c2) as t → ∞ This implies that

summarize the above conclusions we have the following results

Theorem 3.4 Let (u; v) be the solution of (3:10) under the boundary–initial conditions

lim

if one of the conditions in (3:17) holds; and

lim

if one of the conditions in (3:19) holds

Remark 3.2 The results in Theorems 3.3 and 3.4 for the competition model (3:10) have been obtained in [16] using the approach of order-preserving semiRows (see also [18]) In this work the time delays are of continuous type and the initial function

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