Comparison results for systems of impulse parabolic equations with applications to population dynamics

Comparison results for systems of impulse parabolic equations with applications to population dynamics

Nonlinear Analysis, Theory, Methods & Applications, Vol. 28, No. 2, pp. 263-276, 1997 Copyright a 19% Elsevier Science Ltd Printed in Great Britai...

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Nonlinear

Analysis,

Theory,

Methods

& Applications, Vol. 28, No. 2, pp. 263-276, 1997 Copyright a 19% Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/% $15.00~0.00

0362-546X(95)00159-X

COMPARISON RESULTS FOR SYSTEMS OF IMPULSE PARABOLIC EQUATIONS WITH APPLICATIONS TO POPULATION DYNAMICS MOKHTAR

KIRANET

and YURI

V. ROGOVCHENKO$§

tAntenne de I’Universite de Picardie Jules Verne, 60000 Beauvais, France; and SDipartimento di Matematica ‘U. Dini’, Universita di Firenza, 50134 Firenze, Italy (Received 10 April 1995; received for publication

21 July 1995)

Key words and phruses: Impulse parabolic equations, upper and lower solutions, comparison principle, flow invariance. 1. INTRODUCTION

Impulse differential equations [l]-[14] often present the mathematical model for the processes subjected to short-time perturbations. The theory of impulse differential equations is rather well developed (see, for example, the recent monographs [2], [6], [14] for the fundamental results and bibliography) while the corresponding theory for partial differential equations has attracted attention recently [l], [3], [4], [8]-[13]. In this paper which combines some previous experience of the first author on reaction-diffusion equations [15]-[17] with that of the second author on abstract impulse parabolic equations [lo]-[12] we are concerned with a system of initial boundary value problems (IBVP) of the parabolic type with impulse perturbations at fixed instants. We state some comparison principles and flow invariance results (see also [ 181) obtained by a method of upper and lower solutions in Section 3. In Section 4 making use of the functional theorems on stability properties of the solutions of ordinary impulse differential equations ([2], [6], [ 141) we apply our results to a mathematical model of two competing species with abrupt harvesting. Let us note that the first attempt in mathematical modelling of population growth with an impulsive parabolic equation was made in [4] and another approach to the study of reaction-diffusion systems with impulse perturbations via abstract impulse parabolic equations is discussed in [12]. The case of harvesting at constant rate modelled with the help of additional term involved directly into differential equations was considered for the reaction-diffusion system in [19]. We also mention that significant progress in the study of ordinary impulse differential equations based on further development of the method of upper and lower solutions was made in recent papers [5], [7]. 2. STATEMENT

OF THE PROBLEM

AND BASIC

NOTIONS

Let Sz be a smooth bounded domain in Rm, QT = (0, T] x S2 for some T > 0, I, = (0, T) x 13a and u be a unit outward normal vector field on I,. For given partition 8 On leave from the Institute of Mathematics, National Academy of Sciences of the Ukraine, 252601 Kyiv, Ukraine. 263

264

M. KIRANE and Y. V. ROGOVCHENKO

0 < 71 < 7, < **a < rp <

T of the interval [0, T] we introduce the following notations:

Mi = ((Zi, X): Zi E (0, T),

E CJ]

X

and

MC

(JMi, i=l

Ni = {(ti 9X): 7i E (0, T),

E an)

X

N=

and

IfjNi. i=l

We denote by 63 the set of functions u&x), u: [0, T] x fi + RR with the following properties: (i) u(t, x) is of class C&l for (t, x) E &\(A4 U N); (ii) there exists a’u(t, x)/8x2 which is continuous for (t, x) E Qr\M; (iii) there exist the following limits: lim u(t, x) = u(rk, x) t+rk+o and lim u(t, x) = u(rk - 0, x) < 00 eTk-0

for x E !3,

where v(t, x) stands for (u(t, x), u,(t, x), u,(t, x), u&t, x)), ut=$,

u*=

(g

,..., E),

u,=

(g&

,...,

$).

A function f E C[&, x R” x R” x R”‘, R”] is said to be quasimonotone nondecreasing in u if for some i such that 1 I i I m, u I u and Ui = vi the inequality fi(t, X, U, P, Q) 5 &(t, x, v, P, Q) holds, where (t, x) E &, P E R”‘, Q E R”*. A function f E C[Qr x R” x Rm x Rm2,R”] is said to be elliptic at the point (tI , x,) if for any U, P, Rik, Si, (i, k = 1,2, . . . , m) the quadratic form f i,k=

@ik

-

Sikbipk

5

0

1

for arbitrary vector p E R”’ implies f(tl , x1, u, P, R) I f(tl , x1, u, P, S). If this property holds for any (t, x) E Qr, then f(t, x, u, P, R) is said to be elliptic in QT and for that case the differential operator T(u) = u, - f(t, x, u, u,, u,,) is called parabolic. We are concerned with a system of initial boundary value problems (IBVP) of the parabolic type with impulse perturbations at fixed moments of time. Namely, we consider the system of equations for (t, x) E QrU4 Ukf = fk(f, x, u, ukx, ukxx) (1) subject to the initial

condition u(O, x) = uow

for x E fi,

(2)

and boundary condition

Bu = b(t, x) and impulse perturbations

for (t, x) E Pr\N,

(3)

at fixed moments of time ri, i = 1,2, . . . , p

AUk(7i 9x) f uk(7ipx) - uk(7i - 0,~) = g&(X, uk(7i - 0,~))

for x E 6.

(4)

Impulse parabolic equations

265

In what follows we assume that (i) for each k E I = (1,2, . . . , n) the function

[email protected], & u, ukx, t&x) E [email protected], x Rn x Rm x Rm2,R”] is elliptic in Qr; (ii) for each k E I, Bk: 63 + C[I,\N,

with

pk(t,

X),

qk(t,

X)

R”] is a boundary operator defined by

E (?[I,, R”] such that

pk(t,

X)

2 0,

qk(t,

X)

> 0 and

Pk(fr 4 + Qk’k4 -9 > 0 on r,; (iii) for each k E I the function gk-(x, U) E C[Q X R”, Rn]; (iv) the function $(t, x) E C[I,, R”]. The function u(l, x) E 63 satisfying (l)-(4) is called a solution of ZBVP (l)-(4). The function u(t, x) E 03 is called an upper (fower) solution ofZBVP (l)-(4) if it satisfies the following inequalities: uki 1 fk (t, x, u, ukx, u,&) for

(ukt

s [email protected],

x,

u,

ukx

3 ukxx))

E QT\M,

(t,x)

u(O,x) 2 u&e Bu 2 $(t,x)

([email protected]

9 X)

Uk(ri

3 XI

-

Uk(Ti

-

w9 -9 5 %(XN for (Bu 5 9(t, 4) uk(rj

-

-

0, XI

0, X) 5

gki

3. MAIN

2 (X3

gki(x, uk (zi

for x E G, (t,

uk(Ti -

x) E I&V, -

0, XI)

for x E fi.

03.9)

RESULTS

In this section we give some basic theorems on upper and lower solutions of IBVP (l)-(4) along with the flow invariance results. For more detailed presentation and for the comparison of our results with those from [4] we refer to 1181. THEOREM

(i)

&,

3.1. Assume that the following conditions hold: w, E 63, fk E C[& X R” X R” X Rm2, R”] iS elliptic

and

for (t, x) E Qr\M, Ukl 5 fkK x, u, Vkxt Vk*J for x E W, +bj,x) - vk(Ti - 0, X) 5 gki(X, vk(Ti - 0, X)) for (t, x) E Q&M, wkt 2 fk tt, x, w, wkx 3 wkxx) wk(rii, x) - wk((zi - 0, X) 2 gki(x, for x E Sz wk hi - 0, x)) for any k E I, i = 1, 2, . . . ,p; (ii) (a) ~(0, x) < w(0, x) for x E ai; (b) Bv(t, X) < Bw(r, X) for (t, X) E I,\N; (iii) f(t, x, u, P, Q) is quasimonotone nondecreasing in u for fixed (f, x, P, Q), where (t, x) E &, P E Rm, Q E Rm=; (iv) gki(x, u) is quasimonotone nondecreasing in u for fixed x E 0. Then if one of the inequaities in (i) is strict u(t, x) < w(t, x) for (t, x) E Qr.

266

M. KIRANE

and Y. V. ROGOVCHENKO

Proof. We assume that one of the inequalities in (i) is strict and define the function m by the formula m(t, x) = v(t, x) - w(t, x). If the conclusion of the theorem is not true there exist an index k and t, L 0, x1 E fi such that one of the following two cases holds. Case 1. For t, #

i = 1,2, . . . , p

ri,

mj(t,X)

< 0

1,2 ,..., ~,t, mj(t,X)

j # k

for (t, x) E [0, tr) x fi and mk(t, , xl) = 0.

mk(t, x) < 0 Cuse2.Forsomei,i=

for (t, x) E [0, tr) x n,

= riand

C 0

for (t, x) E [0, tr) x !ZJ,

j z k,

for (t, x) E [0, tr) x El and mk(tr , x1) 1 0.

mk(t, x) < 0

For the first case the proof is carried out in analogy with [20]. For the second case we have 0 5 mkb,

x1)

=

uk(7i,X,)

=

uk(7i

-

Wk(7i,Xl)

0, XJ

-

Wk(7i

-

0, XI)

+ gki(Xl

Y vk(7i

-

0, XI))

-

gki (X1 P wk (7i

-

0, XI))

According to our assumptions mk(q - 0, x1) < 0 and the summand in the brackets is nonpositive due to condition (iv) of the theorem. That leads to the contradiction 0 5 mk(ri, x1) C 0 which completes the proof of the theorem. It is possible to dispense with the strict inequality in (i) in theorem 3.1 and the conclusion of the theorem remains valid provided the functions f and gi satisfy the following conditions. (A) There exists function zk E 6%such that

z&,X) > 0 and for sufficiently

&kk

onr,\N

av

small & > 0 and for two arbitrary given

‘%kt > fk(f, x, w + ‘% 60 on Qr\M and (4

k,(t, XI -zvk>o

On QT\M,

(7i 9 X)

-

1 gki(X,

Zk (75i Wk(7i

Wkx

+

&zkx,

wkxx

+

&zkxx)

ok, wk E (8 either

fUnCtiOnS -

fk6

x,

wkx,

we

- 0, x)I -

0, X)

+

Ezk(7i

-

0, X))

-

gkj(X,

Wk(7i

=kx

, uku

-

0, X))

for x E a, k E I or

‘%kt> f&f, x, v, ukx, vk,) - fk (t, x, u - ‘% (b) on Qr\Mand (4

dzk(7i,

vkx

-

-

‘=kxx)

x) - zk(7i - 0, x)1

1 gki (X, uk (Ti - 0, X)) - gki(x, uk(Ti - 0, X) for x E fi, k E I.

wkxx)

‘%k(Ti

-

0,

X))

267

Impulse parabolic equations

THEOREM3.2. Let the assumptions

(i)-(iv) of theorem 3.1 hold and let the condition

(A) be

satisfied. Then the relations ~(0, x) 5 ~$0, x)

for x E a,

Bu(t, x) 5 Bw(t, x)

on I,\N

imply v(t, x) 5 w(t, x) on QT. Proof. Assume that the conditions (a) and (c) of (A) hold and let us define the function tik = wk + czk for k E I. Then one can show that the functions v and i? satisfy the assumptions of theorem 3.1 and hence, uk(t,x) < G&,x) = wk(t, x) + &Z&,X) on &r for k E I. Taking the limit as E + +0 we get the desired inequality ~~(f, x) 5 w,(t, x) on QT. For the case when the conditions (b) and (d) of (A) hold it is necessary to define the function fik = vk - EZ& for k E Z and to show that the functions 0 and w satisfy the assumptions of theorem 3.1 which completes the proof of the theorem. Let us assume now that solutions of the IBVP (l)-(4) exist on QT. A closed set F C R” is said to be flow invariant relative to ZBVP (l)-(4) if for every solution u(t, x) of (l)-(4) uO(x) E F on a implies u(t, x) E F on QT. The following results on flow invariance can be useful in obtaining bounds for the solutions of the IBVP (l)-(4). THEOREM3.3. Assume that the following

conditions hold: 0) (a) fdf, x, 0, 0, 0) 2 0; (b) gki(x, 0) 1 0 and conditions (a) and (c) of (A) are satisfied with u = U, where u = u(t, x) is any solution of IBVP (l)-(4) for @(t,x) 2 0; (ii) conditions (iii) and (iv) of theorem 3.1 are satisfied. Then the closed set u, where U = lu E R”, u > 0), is flow invariant relative to the system

U)-(4).

Proof. We define the function m(t, x) = u(t, x) + ez(t, x), where u(t, x) is arbitrary solution of the IBVP (l)-(4) such that U,,(X) E oaand E > 0 is sufficiently small. In view of conditions (a) and (c) of (A) we have m(t, x) > 0 on QT and Bm(t, x) > 0 on I’r\ZV. Actually, if this conclusion is not true, there exist an index k and t, 1 0, x1 E fi such that one of the following two cases holds. Case 1. For tl # ri, i = 1,2, . . .,p mj(t,X)

mk(t,x) Case2.Forsomei,i=

>

0

for (t, x) E [0, tl) x a,

j Z k

for (t, x) E [0, tl) x r;i and mk(t, , x1) = 0.

> 0

1,2 ,..., p, t, = riand mj(t,X)

mk(t, 4 > 0

>

0

for (t, x) E [0, tl) x a,

j f k,

for (t,x) E [0, tJ x fi and mk(t,,x,)

I 0.

268

M.

KIRANE

and Y. V. ROGOVCHENKO

For the first case one can show in analogy with [20] that (tr . x1) E QT and m(t, x) = u(t, x) + ez(t, x) > 0 on Qr\M. For the case 2 using the condition (c) of (A) and the assumption (i) (b) of the theorem we get 0 2

mk

(Ti,

=

Uk(tj,Xl)

=

Uk(Ti

1

uk(ri +

=

x1) +

&z&,x1)

-

0, XI)

+

gki(xl,

uk (7i

-

0, X1))

+

&zk(ri

9 XI)

-

0, XI)

+

gkj(Xl,

uk (ti

-

0, XI))

+

Ezk(7i

-

gki(x1,

mk(ri

uk(Ti -

The assumption

-

0, x1)

0, XI)

+

+

gkih,

&zk (a

mkh

-

0, XI)

+

gki(X,

9 uk(Ti

-

0, XI))

0, XI))

0, xl)).

(ii) of the theorem along with mk(ri - 0, xi) > 0 implies gki(X1,

mk(ri

and thus we arrive at the following

-

0, XI))

1

gki(X1

3 0)

contradiction:

0 1 mk(ri,xl)

1 mk(zj - 0,x1) + &(x1,

0) > 0.

Thus we can conclude that m(t, x) = u(t, x) + .sz(t, x) > 0 on Qr and hence, taking the limit as E * +0 we get flow invariance of the set ii relative to IBVP (l)-(4) which completes the proof of the theorem. COROLLARY 3.1. Suppose that the following ci)

(a)

fk(f,-%

0, 0, 0)

5

conditions

hold:

0;

(b) g&x, 0) L 0 and conditions (b) and (d) of (A) are satisfied with w = U, where u = u(t, x) is any solution of IBVP (l)-(4) for +(t, x) s 0; (ii) condition (ii) of theorem 3.3 is satisfied. Then the closed set u, where U = (u E R”, u c 0), is flow invariant relative to the IBVP (l)-(4). The proof of the corollary

is similar

to that of theorem

3.3 with the evident choice

m(t, x) = u(t, x) - az(t, x), and natural changes of the signs in the inequalities. Combining

theorem 3.3 with corollary 3.1 we get the following result.

COROLLARY 3.2. Suppose that the condition (A) is satisfied with u = w = u, where u = u(t, x) is any solution of IBVP (l)-(4). Suppose further that

fk(f, x, a, 0, 0) 5 0,

gki(Q, O) s O;

fktf,

[email protected],

&

h

0, 0)

1

0,

0)

1

0;

and the condition (ii) of theorem 3.3. holds. Then the closed set @‘, where W = (u E R”, a < u < b), is flow invariant IBVP (l)-(4).

relative to the

Impulse parabolic equations

269

Next we consider the result concerning upper and lower bounds for the solutions of IBVP (l)-(4) in terms of solutions of the corresponding impulsive ordinary differential equations. THEOREM 3.4. Suppose that the following conditions are fulfilled: (i) condition (A) holds with u = w = U, where u = u(t, x) is an arbitrary IBVP (l)-(4); (ii) there are functions f, , f2 E C[[O, T] x R”, R”] such that f2k(f, u) 5 fk(f, XI u, 030) = flkk

solution

of

for k E I

u)

and fi ,f2 are quasimonotone nondecreasing in u for fixed t E [0, T]; (iii) there are functions gli, g,i E C[R”, R”] such that g2ki

t”)

s

gkihs

u,

s

for k E I

glki(u)

and gli , gzi are quasimonotone nondecreasing in U; (iv) r(t), p(t) are the solutions of the following impulsive $ = flK

t # Ti,

r),

ordinary differential

equations

Ar(Ti) = g,,(r),

r(0) = r,

(5)

AH

Pm = P 0

(6)

and $ = fi(t, Ph

t # ti,

= gzi(P),

which exist on [0, T] and such that p,, I u0 I r, on fi andp(t, xlp(t) 5 Bu(t, x) I p(t, x)r(t) r,\ N. Then p(t) 5 u(t, x) 5 r(t) on QT. Proof. Let us define the function following system: m kt = Fk(fr

m,(t, x) = ~(f,x) &

m,

mkx,

m(0, x) = uo(x) - r(0) Bkmk(t9

x)

Amk(ti)

=

Bkuk(t,

x)

=

Gki(x,

mkb

mkxx)

- rk(t),

pk(f, -

x)rk(f)

satisfies the

QT\M

on

on a,

= uo(x) - r, -

then mk(t,x)

on

s

0

on I’,\N,

ukxx)

-

(7)

O)),

where Fk(f,

X9 111, mkx,

mkxx)

= fk(f?

Gki(Xv m) = gki(x,

x3 m

+

r9 ukw,

m + r) - glki(x,

fik(fy rh

r)-

One can show that (7) satisfies the assumptions of theorem 3.3 and hence, u(t, x) I r(t) on QT. Making use of corollary 3.1 one can show that the inequality p(t) 5 u(t, x) holds on QT. Now the proof of the theorem is complete. If we know a priori that the solution following result may be useful.

u of the IBVP

(l)-(4)

is such that a 5 u < b the

M.

270

KIRANE

and Y. V. ROGOVCHENKO

COROLLARY 3.3. If the closed set I?‘, where W = (U E RR:a < u < b, a, b E R”], is flow invariant relative to the IBVP (l)-(4) then there exist functions fi , fi , gli, gzi satisfying the assumptions of theorem 3.4 provided fk is elliptic for each k E I.

Proof. The functions fi , fi , gii, g,, are constructed in the following way: fik(f, U) =

SUP[fk(t,X,

?I,

f2k(f, u) = inf(fk(t,x, glki(U

U,

U),

aI

X E Sz,

U I

2.4, Ui

=

IBVP (l)-(4).

3.5. Suppose that the following conditions (i) the functions uk, W, E CBare such that

are satisfied:

THEOREM

fk (6

for all Q such that

&

u(t,

0, X)

ukx

9 ukxx)

I CJI

w(t,

and X)

Wkr= fk (t, x, c, w,, w,,,) for all Q such that v(t, (3 (a)

X)

I 0I

~(t,

=

Ui],

Uij*

result related,to

U&t s

Ui

a 5 0 I U, Ui = Uj),

U), X E iI,

= inf(g,i(X,

Finally, we present a comparison

aI u5

E b,

u, O,O), x E a, a 5 u 5 U, ui = ui),

= SUp(gki(X,

g,ki(li)

O,O), X

and ~71= Ui(t, and

X)

uk (zi 3 XI

wk(zj

-

uk (ri

-

0, x)

5

gki(x,

0)

x); 3 XI

and Gi = Wi(t,

-

wk(zi

-

0, x)

1

gki(x,

0)

x);

whenever ek 2 ok, where ak E c[Q, x Rn x R”’ X RmZ,R”] and ykj E C[fi X R”, R”] are nondecreasing in u and there exists a fUnCtiOII zk E a, zk > 0 on Q&W, azk(t, x)/& 1 8k > 0 on Ir\N for k E Z such that for sufficiently small & > 0 qUaSimOnOtOUe

(C) czkt > ak (6 x, &Z, czkx , &zkn) On QAM

and

(d) &[?&(?j, X) - Z&j - 0, X)] 2 \yki(X, EZ) for X E fi, k E 1; (iii) u(t, x) is any solution of the IBVP (l)-(4) such that ~(0, x) I u0 I ~(0, x) for x E fi and & uk 5 Bk wk On I-,\ N. Then u(t, x) I u(t, x) I w(t, x) on QT.

Proof. To begin with we prove the conclusion of the theorem under the assumption that u, w satisfy strict inequalities in (i), (iii). Let us define the functions m and n by m = u - w and n = u - u on QT. If the conclusion is not true there exist an index k E Z and the point (tl , xi) E &r such that one of the following two cases holds.

271

Impulse parabolic equations

Case 1. For tr # ti, i = 1,2, . . .,p tTZj(t,X)

0

<

<

for (t, x) E [0, tr) x Q

tlj(t*X)

j#k

and mku,x) and either m&r,

x,) = 0 or n,(t,,

Case2.Forsomei,i=

1,2 ,..., ?Tlj(t,

X)

<

for (t, x) E [0, tr) x !3

c 0 c n/c(t,xl

0

<

x,) = 0. p,

nj(t,

=tiand

tl

for (t, x) E [0, tl) x si,

X)

j#k

and mk(t,x)

for (t, x) E [0, tl) x fi

< 0 < n&,x)

and either mk(t, ,x1) 1 0 or n,(t,,x,)

s 0.

The first case is handled in the same way as in [20]. For the second case we have either mk(t;, x1) 1 0 or nk(ri, x1) 5 0. Let us suppose that the last inequality holds, then we easily arrive at the following contradiction:

=

Uk(Ti,

2

uk(ri

x,) -

0,

uk(T;, XI)

-

x1) uk(ri

-

0, XI)

+

gki(Xl

9 Uk(Ti

-

0, XI))

-

gki(xl)

0)

since nk(t; - 0,x1) > 0 and u(tl,x,) I u(t,,x,) 5 w(t,, x1). Thus the conclusion of the theorem is true for the strict inequalities in (i) and (iii). To finish the proof for the general case let us define the functions iLk and 0, setting iirk= wk+&zk, tik = vk - &zk on QT\M. Evidently we have 6(0, X) < u,,(X) < G(O, X) on fi and B,i& < Bkuk < Bki& on r,\N. Define the function Sk by sk(t,

x, a) =

max(vk(t,

x), min(ak,

wk(t,

x))).

Then for a, 6 I 0 I ii, and 6jk = tik it follows that the function o = p(t, x, (3) satisfies the inequality b 5 o 5 W and ok = wk. Making use of the assumptions (i), (ii) of the theorem we get by quasimonotonicity of @‘k wkt

=

Wkt

2 fk (t,

+

=kt X,

0,

wkx

, wkxx)

+

‘=k,

since (fik - okI 5 &zk for all k E 1.

272

M. KIRANE

and Y. V. ROGOVCHENKO

Moreover, ii)k(Ti,X)

= Wk(Ti,X) +

EZk(Ti,X)

1 wk(Tj - OSX) + gki(X, 0) +

EZk(Ti

9X)

= wk(Ti - 0,X) + gki(X,a)

- yki(X, Ia.1 - 611,

1 W,(Ti -

0,X)

+ &Z&(Ti -

0,X)

= iG,(Ti -

0,X)

+

g&--(X,

+

***,EZk,

*em*IBpj - O,l) +

EZk(Tii,X)

a)

g,i(X,

a).

Thus the functions fik and i& satisfy all the conditions required for the strict inequality proved above and we get w,

result

x) - EZk(f, xl < Uk (6 xl < WkK 4 + EZk(6 xl

on Qr for arbitrary sufficiently small E > 0 and k E I. Taking the limit as E + +0 we get u(t, x) I u(t, x) 5 w(t, x) on QT. Now the proof of the theorem is complete. 4. MODEL

OF

TWO-SPECIES

COMPETITION

WITH

ABRUPT

HARVESTING

In this section we consider a mathematical model for the growth of populations of two competing species under the presence of abrupt harvesting which is defined by the system of reaction-diffusion equations 24, - D,V2u = fl(U, u) = u(u* - b, 24- Cl u), (8) tr, - D2V2v = f2(u, v) = v(a2 - b,u - c, u) on Qr\M

along with the boundary conditions

at4 i$- - 0, on rr\N,

av 0 ar= ’

(9)

and the initial conditions

w 4 = ho,

(10)

NO,4 = %(X)

in fi and equations of impulse perturbations ANTis X) = gli(u, v) E U(Ti - 0,

X)(-d,

+ e, U(Ti - 0,

X))

Av(ri 9X) = gzi(U, V) E U(ri - 0,

x)(-d2

+ e2 U(Ti

0,

X))

in a, where u and u are population distributed throughout a bounded constants and u,(x), r,+,(x) L 0 on a. and condition (9) implies that there is a nonempty set of real numbers

(11) -

densities of two competing species which are continuously habitat 51 in R2, ci, bi, ci, di, ei, Di (i = 1,2) are positive We denote by a/@ the outward normal derivative on I’r\N is no migration across the boundary of CL Further, J = (ti] such that

card([a, a + l] fl J) < 00

for all a E R,

(12)

Impulse parabolic equations

213

where cardM denotes the number of elements of the set M. We also suppose that ti < tj for i < j and limi,, tj = +oo. Equations (8) mean that in the absence of competition (cr = c2 = 0) each population grows according to a Malthusian law while under competition the growth rate of each population is reduced at a rate proportional to its size and to the size of its competitor. Equations (11) mean that each population is harvested at a rate proportional to its size and to the size of its competitor and duration of harvesting is small in comparison with duration of the process of the population growth. It is natural that two competing species can coexist, both extinguish, or one of them wipes out the other and we will be especially interested in existence and stability of nonnegative steady-state solutions to the problem (8)-(11). We observe that fr(0, V) 2 0,f2(u, 0) 2 0, gn(O, u) L 0, g,i(U, 0) 2 0, and according to [20] the functions fr , f2, gri, g2i satisfy a local Lipschitz condition with the constants

Ml=

I-,,-.,$

I$=

l-,,_.,g,

Ll = l-4 + e,l,

L2

= I-d2 + e21,

respectively. Hence, we can conclude that u(t, x) L 0 on (Zr. Moreover, following estimate for the solutions of the system (8)-(11) is valid: 0 5 u(t, x) I where r(t) is the solution of the following

due to theorem 3.4 the

r(t)

on !A-, system of impulsive ordinary differential

du - = u(a1 - b,u - Cl u), dt

t # Ti,

du - = v(a2 - b2u - czu),

t # Tj,

dt

Au(rJ = u(-d,

(13)

+ eru), (14)

Au(tj) = u(-d, + e2u), u(O) = uo,

equations

u(0) = uo.

(15)

Now we are interested in the behaviour of the solutions of the system of impulse differential equations (13), (14), and steady-state solutions are of special interest to us. It follows from (13), (14) that there are two stationary solutions to the system of impulse differential equations (13), (14)-the trivial solution u = 0, u = 0 and the nontrivial one as2 - a2cl ‘* = blc2 - b2c, ’

a24 - 02 ‘* = blc2 - b2cl

provided d, = e, ad1 - alb2 b,c2 - b2c, ’

d, = e,

w2 - a2c1 b,c2 - b,c, ’

(b,cz - b2cl # 0).

(16)

Naturally, we are interested in stability properties of nontrivial solution, so in what follows we suppose that condition (16) is fulfilled. Taking into account that (11) implies that the coefficients e, , e2 from the biological point of view must be rather small, e, , e, 4 1, we conclude from (16) that the same concerns also the coefficients dl , d2, so the harvesting rate cannot be very high.

274

M.

KIRANE

and Y. V. ROGOVCHENKO

Linearizing the system of impulse differential stationary point u, , v, , we get dx --AX, dt

equation (13). (14) in the neighbourhood

of the

# Zi

X

Ax(q) = Bx, where

In accordance with [14, theorem 16.31 the stationary solution U, , v, of the system (13), (14) is asymptotically stable if the following condition holds: cr+plnjI
(17)

where

a = ReL,,(A),

P2 = &,,,(V + B)=U + BN,

and p = lim card Jv, t+ T)

?‘a3

T



J(t,r+~) = Jn It, t + Tl.

It turns out that the eigenvalues of the matrix A are real and they are given by the formula A 1,2 = +[-blU, Straightforward A 1,2

-

C,V,

+ d(blU,

-

C2V*)2

+

‘&C,U,V,].

calculations give us also two real eigenvalues for the matrix (1 + B)‘(I + B):

- 1 + iKel wd2 + (e2ud2 + dl((e,w+J2- (e2vd2j2 + 4(elu* + e2vd21,

-

so we get a2 = 1 + R, where

R = i[(e, u*)? + (e2v*)~ + d((e, u*)~ - (e2v*)2)2 + 4(e, u, + e2v,)~]. Thus condition (17) which guarantees the asymptotic for our case can be written in the following form:

stability of steady-state solution u, , v*

b,u,+~~v~-Q-pln(1+R)>O,

(18)

where R is as above and

Q=d(b,u, Summing

up, we can formulate

- c, Q2 + 4b2cl u, v, .

now the following

result.

4.1. Assume that condition (18) holds. Then the steady-state solution u,, v, of the reaction-diffusion system with abrupt harvesting (8)-(11) is globally asymptotically stable, that means that for any solution of the system

THEOREM

lim u(t, x) = u, ,

t+co on fi.

lim v(t, x) = v,

t4w

215

Impulse parabolic equations

Taking into account that condition (17) means that both roots of the characteristic polynom for the matrix A must be less than -p ln(1 + R), we can also obtain the following sufficient conditions for the asymptotic stability of steady-state solution u, , u, : $ (ln(1 + R))’ - 5 (b, 2.4, + QZI*) ln(1 + R) + (blc2 - &ci)u,

v* > 0 (1%

bi u* + c2v, > pln(1 To conclude, we give some mentioned above the harvesting tion (16) we conclude that R 4 condition (12) implies that the

+ R).

comments concerning conditions (19). Observing that as we rate is rather small due to the nature of the system and condi1, and thus ln(1 + R) = o(R). Further, due to [14, lemma 22.11 upper limit = 4 < o. lim sup [email protected],f+s) s s-co

exists uniformly with respect to t E R. Thus the steady-state solution u, , u* of impulse system (8)-(11) is asymptotically stable if the following conditions hold: (6, c, - bzq)u* u* 2 6 > 0, b,u* + c2u* 2 r/ > 0

(20)

with some 6, q which can be rather small. Evidently the first condition from (20) implies the fulfilment of condition (18), while the second condition is satisfied almost always. It can be said that harvesting at reasonable rate preserves the asymptotic stability of steady-state solution of the competition model without harvesting. Acknowledgements-The first version of this paper was prepared while both authors were visiting the Department of Mathematics of the University of Florence which they thank for its warm hospitality. Research of the second author was supported by a fellowship of the Italian Consiglio Nazionale delle Ricerche. REFERENCES 1. BAINOV

D., KAMONT Z. & MINCHEV E., First order impulsive partial differential inequalities, Appl. Math. 61, 207-230 (1994). BAINOV D. D. & SIMEONOV P. S., Systems with Impulse Effect, Stability Theory and Applications. Ellis Horwood, Chichester (1989). CHAN C. Y., KE L. & VATSALA A. S., Impulsive quenching for reaction-diffusion equtions, Nonlinear Analysis 22, 1323-1328 (1994). ERBE L. H., FREEDMAN H. I. & WU J. H., Comparison principles for impulsive parabolic equations with applications to models of single species growth, J. Austral. math. Sot. Ser. B 32, 382-400 (1991). KAUL S., LAKSHMIKANTHAM V. & LEELA S., Extremal solutions, comparison principle and stability criteria for impulsive differential equations with variable times, Nonlinear Analysis 22, 1263-1270 (1994). LAKSHMIKANTHAM V., BAINOV D. D. & SIMEONOV P. S., Theory of Impulsive Differential Equations. World Scientific, Singapore (1989). LAKSHMIKANTHAM V., LELLA S. & KAUL S., Comparison principle for impulsive differential equations with variable times and stability theory, Nonlinear Analysis 22, 499-503 (1994). ROGOVCHENKO S. P., Periodic solutions of hyperbolic systems with fixed times of impulse action, Preprint of the Institute of Mathematics of the Ukrainian Academy of Sciences No. 88.8 (1988). (In Russian.) ROGOVCHENKO S. P. & ROGOVCHENKO YU. V., Periodic solutions of a weakly nonlinear hyperbolic impulse system, in Asymptotic Integration of Nonlinear Equations (Edited by YU. A. MITROPOLSKII), 97-103. Institute of Mathematics of the Ukrainian Academy of Sciences, Kiev (1992). (In Russian.) Comput.

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M. KIRANE and Y. V. ROGOVCHENKO

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