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Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response
Applied Mathematical Modelling 35 (2011) 5564–5578
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Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response q Zijian Liu a,⇑, Shouming Zhong a,b, Chun Yin a, Wufan Chen c a
School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 610054, PR China Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu 610054, PR China c School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 610054, PR China b
a r t i c l e
i n f o
Article history: Received 23 September 2010 Received in revised form 29 April 2011 Accepted 8 May 2011 Available online 17 May 2011 Keywords: Reaction–diffusion Holling type III functional response Predator–prey system Permanence Stability
a b s t r a c t An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Reaction–diffusion equations can be used to model the spatio-temporal distribution and abundance of organisms. A typical form of reaction–diffusion population model is
@u ¼ DDu þ uf ðx; uÞ; @t where u(x, t) is the population density at a space point x and time t; D > 0 is the diffusion constant, Du is the Laplacian of u with respect to the variable x, and f(x, u) is the growth rate per capita, which is affected by the heterogeneous environment. Such an ecological model was first considered by Skellam [1], and similar reaction–diffusion biological models were also studied by Fisher [2] and Kolmogoroff et al. [3] earlier. In the past decade, the reaction–diffusion models, especially in population dynamics, have been studied extensively. For example, Ainseba and Anitßa [4] considered a 2 2 system of semilinear partial differential equations of parabolic-type to describe the interactions between a prey population and a predator population. They obtained some necessary and sufficient conditions for stabilizability. Liu and Huang [5] investigated a diffusive predator–prey model with Holling type III functional response and obtained some sufficient conditions for the ultimate boundedness of solutions and permanence of the system, they also presented the existence of a unique globally stable periodic solution. Recently, Apreutesei and Dimitriu [6] studied a prey–predator model defined by an initial-boundary value q This research was supported by National Basic Research Program of China (2010CB732501), the National Natural Science Foundation of PR China (10961022 and 10901130) and the National Natural Science Foundation of China (Grant No. 30970305). ⇑ Corresponding author. Tel.: +86 15882059791; fax: +86 991 8583556. E-mail addresses: [email protected] (Z. Liu), [email protected] (S. Zhong).
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.019
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problem whose dynamics is described by a Holling type III functional response. They established global existence and uniqueness of the strong solution. Meanwhile, they proved that if the initial data are positive and satisfy a certain regularity condition, the solution of the problem is positive and bounded on the domain Q = (0, T) X and then they deduced the continuous dependence on the initial data. More researches on the reaction–diffusion population dynamics, please see [7–10]. There are many examples of evolutionary systems which at certain instants are subjected to rapid changes. In the simulations of such processes it is frequently convenient and valid to neglect the durations of rapid changes. The perturbations are often treated continuously. In fact, the ecological systems are often affected by environmental changes and other human activities. These perturbations bring sudden changes to the systems, which are referring to impulsive differential equations (see [11,12]). In the past twenty years they have attracted the interest of many researchers, since they provided a natural description of several real processes. Process of this type is often investigated in various fields of science and technology, physics, population dynamics [13,14], epidemics [15], ecology [16], biology, optimal control [17], and so on. Recently, some impulsive reaction–diffusion predator–prey models have been investigated. Especially, Akhmet et al. [18] presented an impulsive ratio-dependent predator–prey system with diffusion; meanwhile, they obtained some conditions for the permanence of the predator–prey system and for the existence of a unique globally stable periodic solution. Wang et al. [19] generalized the above impulsive ratio-dependent system to n + 1 species and got some analogous results. Motivated by the above works, we present and study the following impulsive reaction–diffusion predator–prey system with Holling type III functional response:
@u c1 ðt; xÞu2 v ¼ D1 Du þ u½aðt; xÞ bðt; xÞu 2 ; @t u þ rðt; xÞv 2
ð1:1Þ
@v c2 ðt; xÞu2 v ; ¼ D2 Dv dðt; xÞv þ 2 @t u þ rðt; xÞv 2
ð1:2Þ
uðtþk ; xÞ ¼ uðt k ; xÞfk ðx; uðt k ; xÞ; v ðt k ; xÞÞ;
v ðtþk ; xÞ ¼ v ðtk ; xÞg k ðx; uðtk ; xÞ; v ðtk ; xÞÞ; @u @ v ¼ 0; @n@X
@n @ X
ð1:3Þ k ¼ 1; 2; . . . ;
¼ 0:
ð1:4Þ ð1:5Þ
In this system, it is assumed that the predator and prey species are confined to a fixed bounded space domain X Rn with smooth boundary oX and non uniformly distributed in the domain. Furthermore, they are subjected to short-term external influence at fixed moment of time tk, where {tk}, k = 1, 2, . . . is a sequence of real numbers 0 = t0 < t1 < < tk < with limk?1tk = +1. Denote by @/on the outward derivative, X ¼ X [ @ X, and Du ¼ @ 2 u=@x21 þ þ @ 2 u=@x2n the Laplace operator. In Eqs. (1.1) and (1.2), D1 Du and D2 Dv with positive diffusion coefficients D1 and D2 reflect the non-homogeneous dispersion of populations. The Neumann boundary conditions (1.5) characterize the absence of migration. In the absence of predator, the prey species has a logistic growth rate. We assume that the predator functional response has the form of the Holling type III functional response function c1(t, x)u2/(u2 + r(t, x)v2), which reflects the capture ability of the predator. More discussion on the models with functional responses, please see [20–23]. In this paper, we will investigate the asymptotic behavior of non-negative solutions for impulsive reaction–diffusion system (1.1)–(1.5). Note that according to biological interpretation of the solutions u(t, x) and v(t, x) they must be nonnegative. We will give conditions for the long-term survival of each species in terms of permanence. The permanence of the system indicates that the number of individuals of each species stabilizes on certain boundaries with respect to time. This paper is organized as follows. In Section 2, we will give some basic assumptions and useful auxiliary results. Conditions for the ultimate boundedness of solutions and permanence of the system are obtained in Section 3. In Section 4, we establish conditions for the existence of the unique periodic solution of the system. Examples and numerical simulations are presented in Section 5 to illustrate the feasibility of our results. Finally, we discuss our results obtained and present some interesting problems. 2. Preliminaries Let N and R be the sets of all positive integers and real numbers, respectively, and Rþ ¼ ½0; 1Þ. The following assumptions will be needed throughout the paper. (A1) Functions a(t, x), b(t, x), d(t, x), c1(t, x), c2(t, x) and r(t, x) are bounded positive-valued on R X, continuously differentiable in t and x, periodic in t with a period s > 0; (A2) Functions fk(x, u, v) and gk(x, u, v), k 2 N, are continuously differentiable in all arguments and positive-valued; (A3) There exists a number p 2 N such that tk+p = tk + s for all k P 1; (A4) Sequences fk and gk satisfy the following equalities: fk+p(x, u, v) = fk(x, u, v) and gk+p(x, u, v) = gk(x, u, v) for all k P 1 and x, u, v. Conditions of periodicity are natural because of the seasonal changes and biological rhythms. We introduce the following notations: G ¼ Rþ X; G ¼ Rþ X,
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Rk ¼ fðt; xÞ : t 2 ðt k1 ; t k Þ; x 2 Xg;
k 2 N;
R¼
[
Rk ;
k2N
Rk ¼ fðt; xÞ : t 2 ðt k1 ; t k Þ; x 2 Xg;
k 2 N;
R¼
[
Rk :
k2N
Denote by F a class of functions / : G ! R with the following properties: (B1) /(t, x) is of class C2 in x, x 2 X and of class C1 in ðt; xÞ 2 Rk ; k 2 N; (B2) for all k 2 N; x 2 X, there exist the following limits:
lim /ðs; xÞ ¼ /ðtk ; xÞ;
s!t k
lim /ðs; xÞ ¼ / t þk ; x :
s!tþ k
We shall call a vector-function ðuðt; xÞ; v ðt; xÞÞ 2 F F a solution of Problems (1.1)–(1.5) if it satisfies (1.1), (1.2) on R, (1.5) by x 2 oX, and (1.3), (1.4) for every k 2 N. For a bounded function /(t, x), we denote /L = inf(t,x)/(t, x) and /M = sup(t,x)/(t, x). Consider the following impulsive logistic differential equation: dz dt
¼ azðb zÞ;
t – tk ;
zðt þk Þ ¼ zðt k Þkk ðzðt k ÞÞ;
k 2 N;
ð2:1Þ
where z 2 Rþ ; a and b are positive constants, strictly increasing sequence {tk} satisfies condition (A3), and kk ; k 2 N, are continuous positive-valued functions such that kk+p(z) = kk(z) for all z 2 Rþ ; k 2 N. Condition (A3) implies that tk+1 tk P h = mini=0,1,. . .,p(ti+1 ti) > 0, k P 1. Denote Q 1 ¼ b=ð1 eabh Þ; Q 2 ¼ Q 1 maxk¼1;2;...;p maxz2½0;Q 1 kk ðzÞ; Q 3 ¼ maxðz0 ; Q 1 ; Q 2 Þ, where z0 is given below. Then we have the following useful result. Lemma 2.1. Every solution z(t) = z(t, 0, z0), z0 = z(0) = z(0+) > 0 of system (2.1) satisfies 0 < z(t) 6 Q3 for all t P 0.
Proof. For t 2 [0, t1], we have that
zðtÞ ¼
bz0 eabt Þ
z0 ð1
abt
þ be
ð2:2Þ
:
It is obvious that the solution is positive-valued and no larger than max{z0, b} on the interval. Moreover, if h 6 t 6 t1, then
zðtÞ ¼
bz0 eabt Þ
z0 ð1
abt
6
þ be
b ¼ Q 1: 1 eabh
ð2:3Þ
Particularly, 0 < z(t1) 6 Q1, hence, 0 < z t þ 1 ¼ zðt 1 Þk1 ðzðt 1 ÞÞ 6 Q 2 . It is easy to show that 0 < z(t) 6 max (Q1, Q2) 6 Q3 if t 2 [t1, t2], and, similarly to (2.3) one can verify that 0 < z(t2) 6 Q1. Further, in the same manner we can show that 0 < z(t) 6 Q3 if t 2 (tk, tk+1], k = 2, 3, . . . h Now, let us give another useful lemma. Consider the following vector impulsive differential equation
dw ¼ Aw þ Fðt; wÞ; t – t i ; dt þ w t i ¼ wðti Þ þ Gi ðwðti ÞÞ; i 2 N;
ð2:4Þ ð2:5Þ
Lnþ1 ð p
where w ¼ ðw0 ; w1 ; . . . ; wn Þ 2 XÞ; p > n is a positive integer. The operator A has the domain DðAÞ ¼ n o 2;p @n n : n 2 W ðXÞ; @n j@ X ¼ 0 , where W2,p(X) is the Sobolev space of functions from Lp(X) that have two generalized derivatives. Function F(t, w(t)) satisfies suptkF(t, w(t))k < 1, Gi(w) is q-period in i. For any a > 0, we define the fractional power Aa of the operator A by
Aa ¼
1 CðaÞ
Z
1
esA sa1 ds;
0
where C is the Gamma function. The operators Aa are bounded and bijective. The operator Aa, a > 0, is defined as (Aa)1, and DðAa Þ ¼ RðAa Þ. The operator A0 is the identity operator in X. For 0 6 a 6 1, we introduce the space X a ¼ DðAa Þ with the norm kxka = kAaxk. Here kk is the norm in the space X ¼ Lnþ1 . p We denote by Cm+a(X), where m is a positive integer and 0 < a < 1, the space of m-times continuously differentiable functions f : X ! R, which have m-order derivatives satisfying the Hölder condition with exponent a. By Theorem 9 of paper [18], we have the following lemma immediately.
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Lemma 2.2. Assume that the functions Gi are continuously differentiable and there exists a positive-valued function g(M) such that
sup kGk ðwÞka 6 gðMÞ;
k 2 N;
ð2:6Þ
kwka 6M
n for some a 2 ð12 þ 2p ; 1Þ. Let w(t, w0), w0 = (w00, w10, . . . , wn0) 2 Xa, be a bounded solution of Eqs. (2.4) and (2.5), i.e.,
kwðt; w0 ÞkC 6 N;
t > 0:
ð2:7Þ 1þm
nþ1
Then the set {w(t, w0) : t > 0} is relatively compact in C ðX; R Þ for 0 < m < 2a 1 n/p. The following comparison theorems will be needed throughout the paper. Lemma 2.3 (Walter [24]). Suppose that vector-functions v(t, x) = (v1(t, x), . . . , vm(t, x)) and w(t, x) = (w1(t, x), . . . , wm(t, x)), m P 1, satisfy the following conditions: (i) they are of class C2 in x, x 2 X and of class C1 in ðt; xÞ 2 ½a; b X, where X Rn is a bounded domain with smooth boundary; (ii) vt lDv g(t, x, v) 6 wt lDw g(t, x,w), where (t, x) 2 [a, b] X, l = (l1, . . . , lm) > 0 (inequalities between vectors are satisfied coordinate-wise), vector-function g(t, x, u) = (g1(t, x, u), . . . , gm(t, x,u)) is continuously differentiable and quasi-monotonically increasing with respect to u = (u1, . . . , um):
@g i ðt; x; u1 ; . . . ; um Þ P 0; @uj
i; j ¼ 1; . . . ; m;
i – j;
(iii) ov/on = ow/on = 0, (t, x) 2 [a, b] oX. Then v(t, x) 6 w(t, x) for ðt; xÞ 2 ½a; b X. Lemma 2.4 (Smith [25]). Assume that T and d are positive numbers, a function u(t, x) is continuous on ½0; T X, continuously differentiable in x 2 X, with continuous derivatives @ 2u/@xi@xj and @u/@t on (0, T] X, and u(t, x) satisfies the following inequalities:
@u dDu þ cðt; xÞu P 0; ðt; xÞ 2 ð0; T X; @t @u P 0; ðt; xÞ 2 ð0; T @ X; @n uð0; xÞ P 0; x 2 X; where c(t, x) is bounded on (0, T] X. Then u(t, x) P 0 on ð0; T X. Moreover, u(t, x) is strictly positive on ð0; T X if u(0, x) is not identically zero. On the basis of the upper and lower solution method for quasi-monotone systems (see [26]), we can verify that, for continuously differentiable initial functions u0 ðxÞ : X ! Rþ ; v 0 ðxÞ : X ! Rþ ; u0 ðxÞ and v0(x) are not identically zero, there exists a classical solution of system (1.1), (1.2) and (1.5), which can be extended to the semi-axis t > 0. A vector-function (u(t, x), v(t, x)) is the classical solution of system without impulses (1.1), (1.2) and (1.5), if it is of class C2 in x, x 2 X, of class C1 in x; x 2 X, of class C1 in t, t > 0, and satisfies the system. Using the existence of solutions of system (1.1), (1.2) and (1.5), we can verify the existence of solutions for impulsive system (1.1)–(1.5). Indeed, if 0 < t 6 t1, the solutions of the system are well-defined as classical solutions of system without impulses (1.1), (1.2) and (1.5). Impulsive conditions (1.3) and (1.4) imply that the functions ðuðtþ Þ; v ðtþ 1 ÞÞ are continuously þ þ 1 differentiable in x, and satisfy the boundary conditions (1.5). Hence, assuming u t1 ; v t1 as new initial functions we can continue the solution on (t1, t2]. Proceeding in this way, we can construct the solution for all t > 0. According to biological interpretation, we only consider the non-negative solutions of the system. Hence, the following assertion is of major importance. Lemma 2.5. Assume that conditions (A1)–(A4) hold. Then non-negative and positive quadrants of R2 are positively invariant for system (1.1)–(1.5). Proof. Let us consider the first coordinate u(t, x) of a solution since the proof for the second one is very similar. It can be ^ ðt; xÞ and u ðt; xÞ such that simply verified that u
^ @u cM ^ ð0; xÞ ¼ u0 ðxÞ; ^u ^ aL bM u ^ p1ffiffiffiffi ¼ 0; u D1 Du @t 2 rL h i @u u aM bL u ¼ 0; u ð0; xÞ ¼ u0 ðxÞ D1 Du @t are lower and upper solutions of Eq. (1.1). Then, since u0(x) P 0 and u0(x) is not identically zero, by Lemma 2.4, we get ^ ðt; xÞ > 0 and u ðt; xÞ > 0 for t 2 (0, t1]. Since u(t, x) is bounded from below by positive function u ^ ðt; xÞ, we have u(t, x) > 0 u
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for t 2 (0, t1]. Taking into account positiveness of the function f1, we can repeat the same argument to prove the positiveness of u(t, x) for t 2 [t1, t2]. By induction, we have u(t, x) > 0 for t 2 (0, 1). h 3. Permanence In this section, applying the upper and lower solution method and comparison theory of differential equation, we establish some sufficient conditions for the ultimate boundedness and permanence of the system. Before this, two definitions are given firstly. Definition 3.1. Solutions of system (1.1)–(1.5) are said to be ultimately bounded if there exist positive constants N1, N2 such that for every solution (u(t, x, u0, v0), v(t, x, u0, v0)), there exists a moment of time t ¼ tðu0 ; v 0 Þ > 0 such that u(t, x, u0, v0) 6 N1, v(t, x, u0, v0) 6 N2 for all x 2 X, and t P t. Definition 3.2. System (1.1)–(1.5) is called permanent if there exist positive constants m1, m2, N1 and N2 such that for every solution with non-negative initial functions u0(x) and v0(x) that are not identically zero, there exists a moment of time ~t ¼ ~tðu0 ; v 0 Þ such that
m1 6 uðt; x; u0 ; v 0 Þ 6 N1 ;
m2 6 v ðt; x; u0 ; v 0 Þ 6 N2
for all x 2 X, and t P ~t. Theorem 3.1. Assume that conditions (A1)–(A4) hold, and, moreover: (i) there exists a positive-valued function g(M) such that fk(x, u, v) 6 g(M) if k 2 N; u 6 M; (ii) the inequality L
sd þ
p X
v P 0 and x 2 X;
ln g k < 0
ð3:1Þ
k¼1
holds, where gk = sup(x,u,v)gk(x, u, v). Then all solutions of system (1.1)–(1.5) with non-negative initial conditions are ultimately bounded. ðt; x; u0 Þ be a solution of the equation Proof. Let u
@u u ðaM bL u Þ ¼ 0: D1 Du @t
ð3:2Þ
Using inequality
@u c1 ðt; xÞu2 v D1 Du u½aðt; xÞ bðt; xÞu þ 2 @t u þ rðt; xÞv 2 @u L P D1 Du uðaM b uÞ; @t
0¼
we obtain
0¼
@u @u L u ðaM bL u Þ P D1 Du D1 Du uðaM b uÞ: @t @t
ðt; M u Þ, where Mu is such that ku0 ðxÞkC ¼ maxx2X ju0 ðxÞj 6 M u . Note Applying Lemma 2.3, we conclude that uðt; x; u0 ; v 0 Þ 6 u ðt; M u Þ of Eq. (3.2) with initial condition independent of x does that, according to the uniqueness theorem, the solution u ðt; Mu Þ satisfies the ordinary differential equation du =dt ¼ u ðaM bL u Þ. not depend on x for t > 0. Therefore, the function u Hence,
ðt k ; M u Þgðu ðt k ; M u ÞÞ: kuðt þk ; x; u0 ; v 0 ÞkC ¼ kuðtk ; x; u0 ; v 0 Þfk ðx; uðt k ; x; u0 ; v 0 Þ; v ðt k ; x; u0 ; v 0 ÞÞkC 6 u Since all solutions of the impulsive differential equation
du ðaM bL u Þ; ¼u dt
ðtk ÞÞ ðt k Þgðu tþk ¼ u u
are ultimately bounded by Lemma 2.1, we get ultimate boundedness of solutions of Eq. (1.1) with impulses (1.3), i.e., there exists a positive constant N1 such that u(t, x) 6 N1, starting with some moment of time t1 . For the predator population, when t P t1 ,
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@v c2 ðt; xÞu2 v @v cM N 1 L P D2 Dv þ dðt; xÞv 2 D2 Dv þ d v 2pffiffiffiffi ; @t u þ rðt; xÞv 2 @t 2 rL
0¼
pffiffiffiffi L ðt; M v Þ, where v ðt; M v Þ is a solution of the initial value problem dv =dt ¼ dL v þ cM it follows that v ðt; x; u0 ; v 0 Þ 6 v 2 N 1 =2 r ð0; M v Þ ¼ M v . with v Linear periodic impulsive equation
dv cM N 1 L ¼ d v þ 2pffiffiffiffi ; dt 2 rL
v
þ t k ¼ g k v ðt k Þ
ð3:3Þ
ðtÞ ¼ X 0 ðtÞ þ CXðtÞ, where X0(t) is a s-periodic piecewise continuous function, C is a constant and has the general solution v L
XðtÞ ¼ expðd t þ
X
ln g k Þ
0
(see [11]). By (3.1), X(t) ? 0 as t ? 1. All solutions of (3.3) are ultimately bounded, therefore, all solutions of Eqs. (1.2) and (1.4) are ultimately bounded, too. h Theorem 3.2. Assume that conditions (A1)–(A4) hold, and, moreover: (i) Solutions of system 1.1, 1.2, 1.3, 1.4, 1.5 are ultimately bounded, i.e., there exist positive constants N1 and N2 such that for every solution (u(t, x, u0, v0), v(t, x, u0, v0)), there exists t ¼ tðu0 ; v 0 Þ > 0 such that u(t, x, u0, v0) 6 N1 and v(t, x, u0, v0) 6 N2 for all t P t; (ii) the following inequalities p X
ln
k¼1
cM fk ðx; u; v Þ þ s aL p1ffiffiffiffi > 0 x2X;ðu;v Þ2S 2 rL inf
ð3:4Þ
and p X
ln
k¼1
inf
x2X;ðu;v Þ2S
M g k ðx; u; v Þ þ s cL2 d >0
ð3:5Þ
hold, where S = {(u, v):0 < u 6 N1, 0 < v 6 N2}. Then there exist positive constants r1 and r2 such that an arbitrary solution of system (1.1)–(1.5) with non-negative initial conditions not identically equal to zero satisfies
ðuðt; xÞ; v ðt; xÞÞ 2 P ¼ ðu; v Þ : r1 6 uðt; xÞ 6 N1 ;
r2 6 v ðt; xÞ 6 N2 ;
starting with a certain moment of time. Proof. Lemma 2.4 implies that if u0(x) P 0, v0(x) P 0, u0(x) and v0(x) are not identically zero, then u(t, x, u0, v0) > 0, v(t, x, u0, v0) > 0 for all x 2 X and t > 0. Considering the solution on the interval t P e with some small e > 0, we get initial conditions (u(e, x, u0, v0), v(e, x, u0, v0)) separated from zero. Therefore, we can assume, without loss generality, that minx2X u0 ðxÞ ¼ mu > 0 and minx2X v 0 ðxÞ ¼ mv > 0. Using inequality
@u c1 ðt; xÞu2 v D1 Du u½aðt; xÞ bðt; xÞu þ 2 @t u þ rðt; xÞv 2 M @u c M D1 Du u aL b u p1ffiffiffiffi ; 6 @t 2 rL
0¼
we obtain
0¼
^ @u cM @u cM M ^u ^ aL bM u ^ p1ffiffiffiffi 6 D1 Du D1 Du u aL b u p1ffiffiffiffi : @t @t 2 rL 2 rL
^ ðt; mu Þ for t 2 [0, t1]. Applying the last inequality for t = t1, together with Eq. Now, using Lemma 2.3 for m ¼ 1; uðt; x; u0 ; v 0 Þ P u (1.3), we obtain that
^ðt 1 ; mu Þ u tþ1 ; x; u0 ; v 0 P u
inf
x2X;ðu;v Þ2S
f1 ðx; u; v Þ:
Thus, the solution u(t, x, u0, v0) is bounded from below by a solution of periodic logistic equation with impulses
^ du cM ^ aL p1ffiffiffiffi bM u ^ ; ¼u dt 2 rL
^ ðtk Þ ^ ðt þk Þ ¼ u u
inf
x2X;ðu;v Þ2S
fk ðx; u; v Þ:
ð3:6Þ
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By Theorem 2.1 [27] and condition (3.4), Eq. (3.6) has a unique piecewise continuous and strictly positive periodic solution ^ ðtÞ such that every solution u ^ ðt; um Þ of (3.6) with um > 0 has the property u ^ ðt; um Þ ! u ^ ðtÞ as t ? 1. Therefore, there exists a u ^ ðt; um Þðum > 0Þ of Eq. (3.6) we get u ^ ðt; um Þ P r1 , starting with some mopositive constant r1 such that, for every solution u ment of time ^t 1 ¼ ^t 1 ðum Þ > 0. ^ ðt; um Þ of Eq. (3.6), we conclude that Since solution u(t, x, u0, v0) of Eqs. (1.1) and (1.3) is bounded from below by solution u uðt; x; u0 ; v 0 Þ P r1 for t P ^t1 . Now, let us consider the predator population. When t P ^t1 , since uðt; x; u0 ; v 0 Þ P r1 , we have
0¼
@v c2 ðt; xÞu2 v @v cL rM v 3 M 6 : D2 Dv þ dðt; xÞv 2 D2 Dv þ d cL2 v þ 22 2 @t u þ rðt; xÞv @t r þ r M v 2 1
^ ðt; mv Þ, where v ^ ð0; mv Þ ¼ mv is the solution of equation Hence, v ðt; x; u0 ; v 0 Þ P v
dv^ L cL rM v 3 M ; ¼ c2 d v 22 dt r þ rM v 2
v^
þ tk ¼ v^ ðtk Þg^k ;
ð3:7Þ
1
^ ðtÞ 6 r2 for t 2 [0, t1], then where g^k ¼ inf x2X;ðu;v Þ2S g k ðx; u; v Þ. If v
(
M
v^ ðt1 ; mv Þ P mv exp
t1 cL2 d
cL2 r M r22 2 r1 þ rM r22
!)
and
v^
( !) cL r M r22 M : t þ1 ; mv P g^1 mv exp t1 cL2 d 22 r1 þ rM r22
^ ðtÞ 6 r2 for t 2 [0, s], then Therefore, if v
v^ ðs; mv Þ P mv exp
( p X
M
ln g^k þ s cL2 d
k¼1
cL2 rM r22
!)
ðr1 Þ2 þ r M r22
:
Taking into account (3.5), we can take sufficiently small r2 > 0 such that p X k¼1
M
cL2 r M r22 2 r1 þ rM r22
!
ln g^k þ s cL2 d
¼ q > 0:
^ ðk2 s; mv Þ P ek2 q mv P r02 (by the additional condition For r02 2 ð0; r2 Þ, there exists a positive integer k2 such that v v^ ðt; mv Þ < r2 for all t 2 [0, k2s]). ^ ðt; v ^ 0 Þ of (3.7) with v ^ 0 > 0, there exists a moment of time ^t such that v ^ ð^t; v ^ 0 Þ P r02 . Denote by Hence, for every solution v v^ ðt; t0 ; v^ 0 Þ the solution of (3.7) with v^ ðt0 ; t0 ; v^ 0 Þ ¼ v^ 0 and consider a positive number
r2 ¼ inffv^ ðt; t0 ; v^ 0 Þ : t0 2 ½0; s; v^ 0 2 ½r02 ; N2 ; t 2 ½t0 ; 2sg: ^ ðt; t0 ; v ^ 0 Þ P r2 for all t P 2s. Indeed, let us take Then v
rs ¼ inffv^ ðs; t0 ; v^ 0 Þ : t0 2 ½0; s; v^ 0 2 ½r02 ; N2 g P r2 ^ ðs; s; v ^ ðt; s; v ^ 0 Þ with v ^ 0 P rs . If v ^ ðt; s; v ^ 0 Þ 6 r2 for all t 2 [t0, 2s], then v ^ ð2s; s; v ^ 0 Þ P eq v ^ 0 Þ P rs . If and consider a solution v v^ ðt; s; v^ 0 Þ > r2 at some moment of time t 2 [s, 2s], then v^ ð2s; s; v^ 0 Þ P rs by definition of number rs. Therefore, it is enough ^ ðt; 2s; v ^ 0 Þ; t P 2s, with v ^ 0 P rs . By construction, these solutions are bounded from below by positive constant to consider v r2 for t 2 [2s, 3s]. Proceeding in this way we prove the boundedness from below for t P 3s. h Through the above analysis, we got some conditions under which the two species are permanent. Then, we will give some conditions that will lead to extinction of the predator species. Theorem 3.3. Assume that system 1.1, 1.2, 1.3, 1.4, 1.5 satisfies conditions (A1)–(A4), and, further: p X k¼1
L ln sup g k ðx; u; v Þ þ s cM < 0: 2 d ðx;u;v Þ
Then v(t, x) ? 0 as t ? 1. ðt; M v Þ the solution of initial value problem Proof. Fix a positive constant Mv such that Mv P v0(x) and denote by v
dv M L ¼ c2 d v ; dt
v ð0; Mv Þ ¼ Mv :
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From the inequality
0¼
@v c2 ðt; xÞu2 v @v L P D2 Dv þ dðt; xÞv 2 D2 Dv þ d cM 2 v; @t u þ rðt; xÞv 2 @t
ðt; M v Þ for t 6 t1. applying the comparison theorem, we can find that v ðt; x; u0 ; v 0 Þ 6 v Moreover, using impulsive condition (1.4), we obtain that
v
t þ1 ; x; u0 ; v 0 6 v ðt1 ; M v Þ sup g 1 ðx; u; v Þ: ðx;u;v Þ
Proceeding in this fashion, we conclude that every solution of Eqs. (1.2) and (1.4) is bounded from above by the corresponding solution of linear impulsive equation
dv M L ¼ c2 d v ; dt
v
þ t k ¼ v ðtk Þ sup g k ðx; u; v Þ: ðx;u;v Þ
Taking into account (3.8), we see that all solutions of the last equation tend to zero as t ? 1. h 4. Periodic solutions In the following, we study the existence of the periodic solution by constructing an appropriate auxiliary function. We will note that the conditions of the existence of the periodic solution are dependent on the permanence of the system. Theorem 4.1. Assume that conditions (A1)–(A4) and (2.6) hold, and system (1.1)–(1.5) is permanent, i.e., there exist positive constants r and N such that an arbitrary solution of the system with non-negative initial functions not identically equal to zero satisfies the condition
ðuðt; xÞ; v ðt; xÞÞ 2 P ¼ fðu; v Þ : r 6 uðt; xÞ 6 N;
r 6 v ðt; xÞ 6 Ng;
starting with a certain moment of time. Let, additionally, p X
ln K i þ skM < 0;
i¼1
where
( K i ¼ max 2 u;v 2P;x2X
fi2
2 2 ) 2 2 @fi @fi @g i @g i 2 þ N þ N þ gi þ N þ N ; @u @v @u @v
kM is the maximal eigenvalue of the matrix
0
cL r L r4 L 2 aM b r N2 ð1þrM1 ÞðN2 þrL r2 Þ B @ ðcM1 þcM1 rM þ2cM2 rM ÞN4 ð1þrL Þ2 r4
ðcM1 þcM1 rM þ2cM2 rM ÞN4
1
r C
A: cM N4 L 2 2 d þ ðN2 þrL r2 Þ2 ð1þr L Þ2 4
Then system 1.1, 1.2, 1.3, 1.4, 1.5 has a unique globally asymptotically stable strictly positive piecewise continuous s-periodic solution. ðt; xÞ; v ðt; xÞÞ be two solutions of system (1.1)–(1.5) bounded by constants r and N from below Proof. Let (u(t, x), v(t, x)) and ðu and above, respectively. Consider the function
VðtÞ ¼
Z
ðt; xÞÞ2 þ ðv ðt; xÞ v ðt; xÞÞ2 dx: ½ðuðt; xÞ u
X
Its derivative has the form
Z @u @ u @ v @ v Þ dx þ 2 ðv v Þ ðu u dx @t @t @t @t X X Z Z ÞDðu u Þdx þ 2D2 ðv v ÞDðv v Þdx ¼ 2D1 ðu u X X Z 2 v c1 u2 v c u Þ uða buÞ ða bu Þ þ 1 dx þ 2 ðu u u 2 þ r v 2 u2 þ r v 2 u X Z 2 v c2 u2 v c2 u þ 2 ðv v Þ dv þ 2 dx :¼ I1 þ I2 þ I3 þ I4 : þ dv 2 2 2 þ r v u þ rv u X
dVðtÞ ¼2 dt
Z
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It follows from the boundary condition (1.5) that
I1 þ I2 ¼ 2D1
Z
r2 ðu u Þdx 2D2
X
Z
r2 ðv v Þdx 6 2D1
Z
X
Þj2 dx 2D2 jrðu u
X
Z
jrðv v Þj2 dx 6 0: X
For the third and fourth terms I3 and I4, we have
Z
Z
2 ðv v Þ þ c1 r v v ðu2 v u 2 v Þ c1 u2 u dx þ 2 2 2 2 2 ðu þ r v Þðu þ rv Þ X X Z 2 v Þ 2 ðv v Þ þ c2 r v v ðu2 v u c2 u2 u þ 2 ðv v Þ dx 2 2 2 2 ðu þ rv Þðu þ r v Þ X
I3 þ I 4 ¼ 2
Þ2 ½a bðu þ u Þdx 2 ðu u
Þ ðu u
Z
ðv v Þ2 ðdÞdx
X
2 v ¼ ðu u Þðuv þu v Þ uu ðv v Þ, thus u Noting that u2 v
I3 þ I 4 ¼ 2
Þ2 a bðu þ u Þ ðu u
Z
X
Z
Z v Þ ðuu r v v Þ c1 rv v ðuv þ u c2 uu Þ2 d þ dx þ 2 dx ð v v 2 þ rv 2 Þ 2 þ r v 2 Þ ðu2 þ r v 2 Þðu ðu2 þ r v 2 Þðu X
v Þ c 1 uu ðuu r v v Þ c2 rv v ðuv þ u dx 2 þ r v 2 Þ ðu2 þ rv 2 Þðu " # " # Z Z 4 cL1 r L r4 cM L 2 2 N Þ2 aM bL r 2 dx þ 2 dx 6 2 ðu u ð v v Þ d þ N ð1 þ r M ÞðN2 þ rL r2 Þ ðN2 þ r L r2 Þ2 X X Z Z 4 M M M ðcM þ cM 1 r þ 2c 2 r ÞN Þðv v Þj 1 Þ2 þ ðv v Þ2 dx: þ 2 jðu u dx 6 kM ½ðu u 2 ð1 þ r L Þ r4 X X þ2
Þðv v Þ ðu u
X
Hence we have dV(t)/dt 6 kMV(t), therefore, Vðt iþ1 Þ 6 V tþ expðkM ðtiþ1 t i ÞÞ and i
V t þiþ1 ¼
Z
fiþ1 ðu ; v Þ2 dx þ ½ufiþ1 ðu; v Þ u X
Z
X
; v Þ2 dx 6 K iþ1 Vðt iþ1 Þ 6 K iþ1 expðkM ðt iþ1 ti ÞÞV tþi : ½v g iþ1 ðu; v Þ v g iþ1 ðu
Let us estimate the variation of the function over the period. We have
Vðt þ sÞ 6 K VðtÞ ¼
p Y
K i expðkM sÞVðtÞ:
i¼1
According to the conditions of the theorem, we have K⁄ < 1. Therefore, Vðms þ sÞ 6 K m VðsÞ ! 0; m ! 1. We have proved that ðt; xÞk ! 0 and kv ðt; xÞ v ðt; xÞk ! 0 as t ? 1, where kk is the norm of the space L2(X). By Lemma 2.2, solutions kuðt; xÞ u of system (1.1)–(1.5) are bounded in the space C1+m. Therefore
ðt; xÞj ! 0; sup juðt; xÞ u x2X
sup jv ðt; xÞ v ðt; xÞj ! 0;
t ! 1:
x2X
Fig. 1. The permanence of species u.
ð4:1Þ
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Z. Liu et al. / Applied Mathematical Modelling 35 (2011) 5564–5578
Now let us consider the sequence ðuðks; x; u0 ; v 0 Þ; v ðks; x; u0 ; v 0 ÞÞ ¼ wðks; w0 Þ; k 2 N. By Lemma 2.2, it is compact in the be a limit point of this sequence, w ¼ limn!1 wðkn s; w0 Þ. Then wðs; wÞ ¼ w. Indeed, since space CðXÞ CðXÞ. Let w w(s, w(kns, w0)) = w(kns, w(s, w0)) and w(kns, w(s, w0)) w(kns, w0) ? 0 as kn ? 1, we get
wk C 6 kwðs; wÞ wðs; wðkn s; w0 ÞÞkC þ kwðs; wðkn s; w0 ÞÞ wðkn s; w0 ÞkC þ kwðkn s; w0 Þ wk C ! 0; kwðs; wÞ
n ! 1:
The sequence wðks; w0 Þ; k 2 N, has a unique limit point. On the contrary, let the sequence has two limit points ¼ limn!1 wðkn s; w0 Þ and w ~ ¼ limn!1 wðkn s; w0 Þ. Then, taking into account (4.1) and w ~ ¼ wðkn s; wÞ, ~ we have w
wk ~ C 6 kw wðkn s; w0 ÞkC þ kwðkn s; w0 Þ wk ~ C ! 0; kw
n ! 1;
¼ w. ~ The solution ðuðt; x; u ; v Þ; v ðt; x; u ; v ÞÞ is the unique periodic solution of system (1.1)–(1.5). By (4.1), it is asymphence w totically stable. h
Fig. 2. The permanence of species v.
5.5 5 4.5 4 3.5
u
3 2.5 2 1.5 1 0.5 0 0
5
10
15
20 t
25
Fig. 3. The section of Fig. 1 with x = 0.
30
35
40
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Z. Liu et al. / Applied Mathematical Modelling 35 (2011) 5564–5578
5. Numerical illustrations In order to verify the validity of our results, we consider system (1.1)–(1.5) with fk(x, u, v) 0.8 and gk(x, u, v) 0.7 for all k = 1, 2, . . . Choose D1 ¼ D2 ¼ 1, a(t, x) = sin(t) + 2 cos(x) + 8, b(t, x) = cos(t) + 2 cos(x) + 4, c1(t, x) = 1.9 sin(t) + sin(x) + 3, c2(t, x) = 0.4 sin(t) + 0.1 sin(x) + 3, d(t, x) = 0.45 cos(t) cos(x) + 0.5, and r(t, x) 1. Obviously, all the parameters have a common period s = 2p in t. Choosing an appropriate per length in the matlab procedure, we can calculate p = 26. Then, we have that the parameters satisfy all the conditions of Theorem 3.1, further
cM fk ðx; u; v Þ þ s aL p1ffiffiffiffi ¼ 7:0788 > 0 x2X;ðu;v Þ2S 2 rL inf
5.5 5 4.5 4 3.5 3 v
k¼1
ln
2.5 2 1.5 1 0.5 0 0
5
10
15
20 t
25
30
35
40
30
35
40
Fig. 4. The section of Fig. 2 with x = 0.
7 6 5 4 u
p X
3 2 1 0 −1
0
5
10
15
20 t
25
Fig. 5. The projection of species u in plane t u.
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Z. Liu et al. / Applied Mathematical Modelling 35 (2011) 5564–5578
and p X k¼1
ln
inf
x2X;ðu;v Þ2S
M g k ðx; u; v Þ þ s cL2 d ¼ 0:4654 > 0;
which satisfy the conditions (3.4) and (3.5) of Theorem 3.2. Hence, all the conditions of Theorem 3.2 are satisfied, then, system (1.1)–(1.5) is permanent. See Figs. 1–7. Here, we choose x 2 X = [2, 2] and the initial conditions u(0, x) = 4, v(0, x) = 3 for all x 2 X. However, if we choose c1(t, x) = 2.9 sin(t) + sin(x) + 4, c2(t, x) = 0.2 sin(t) + 0.1 cos(x) + 3, d(t, x) = 0.1 cos(t) cos(x) + 2.6 and other parameters are unchanged, then the parameters satisfy all the conditions of Theorem 3.1. Furthermore,
7 6 5
v
4 3 2 1 0 −1
0
5
10
15
20 t
Fig. 6. The projection of species
25
30
35
40
v in plane t v.
4.5
4
v
3.5
3
2.5
2
1
1.5
2
2.5 u
3
Fig. 7. The phase of species u and v.
3.5
4
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Z. Liu et al. / Applied Mathematical Modelling 35 (2011) 5564–5578
3 2.5
v
2
1.5 1
0.5
0
1
1.5
2
2.5
3 u
3.5
4
4.5
5
Fig. 8. The phase of species u and v.
Fig. 9. The extinction of species v.
p X k¼1
L ln sup g k ðx; u; v Þ þ s cM ¼ 4:2470 < 0; 2 d ðx;u;v Þ
which satisfies the condition (3.8) of Theorem 3.3. Hence, all the conditions of Theorem 3.3 are satisfied, so that species v goes to extinction. See Figs. 8–10. From Fig. 8, we know that species u is also permanent when species v approaches to zero. Here, we also choose x 2 X = [2, 2] and the initial conditions u(0, x) = 4, v(0, x) = 3 for all x 2 X. 6. Conclusion and discussion If prey and predator populations are in a remote patchy forest or an isolated island or a lake ecosystem, an appropriate reaction–diffusion model is described by (1.1), (1.2) and (1.5), which is the model proposed by Liu and Huang in paper [5]. However, the ecological systems are often affected by environmental changes, these perturbations bring sudden changes to the system. Systems with such sudden perturbations referring to impulsive differential equations. Hence, in this paper, we investigated an impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response. We
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3
v
2 1 0 −1
0
5
10 t
15
20
5
10 t
15
20
3
v
2 1 0 −1
0
Fig. 10. The section and projection of species v.
gave some sufficient conditions for the permanence and the existence of a unique globally stable positive periodic solution of the system. If there are not any impulses in Problems (1.1)–(1.5) (fk and gk are identically equal to zero, this model is investigated in paper [5]), then conditions (3.4), (3.5) take a form of the following inequalities:
cL aL p1ffiffiffiffi > 0; 2 rL
M
cL2 d > 0;
which are sufficient to have the permanence phenomenon for the system. However, in paper [5], the non-impulses system is M permanent only need the condition cL2 d > 0. An interesting problem is to look for conditions under which the impulsive M system is permanent, but the permanent condition is cL2 d > 0 if there are not any impulses. If the system is permanent, by Theorem 4.1, we conclude that it has a unique globally asymptotically stable strictly poP sitive piecewise continuous s-periodic solution if pi¼1 ln K i þ skM < 0. However, there is an interesting problem. If this system has a unique globally asymptotically stable strictly positive s-periodic solution, it should be permanent. It seems that P Theorem 4.1 may imply Theorem 3.2. But unfortunately, we cannot claim it is true. Since the condition pi¼1 ln K i þ skM < 0 in Theorem 4.1 is dependent on r and N, which are decided by the bound of solutions when the system is permanent. Whether is there a unique globally asymptotically stable s-periodic solution only if conditions (3.4) and (3.5) hold (the system is permanent) or other conditions independent of r and N? We will continue to study these problems in the future. In this paper, we only studied the system with Holling type III functional response, whether other type functional responses such as Holling type II and Beddington–DeAngelis functional response can be discussed with the same methods or not, still remain open problems. References [1] J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951) 196–218. [2] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 353–369. [3] A. Kolmogoroff, I. Petrovsky, N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Moscow Univ. Bull. Math. 1 (1937) 1–25. [4] B. Ainseba, S. Anitßa, Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system, Nonlinear Anal. 61 (2005) 491–501. [5] X. Liu, L. Huang, Permanence and periodic solutions for a diffusive ratio-dependent predator–prey system, Appl. Math. Model. 33 (2009) 683–691. [6] N. Apreutesei, G. Dimitriu, On a prey–predator reaction–diffusion system with Holling type III functional response, J. Comput. Appl. Math. 235 (2010) 366–379. [7] J. Shi, R. Shivaji, Persistence in reaction–diffusion models with weak allee effect, J. Math. Biol. 52 (2006) 807–829. [8] C. Duque, K. Kiss, M. Lizana, On the dynamics of an n-dimensional ratio-dependent predator–prey system with diffusion, Appl. Math. Comput. 208 (2009) 98–105. [9] R. Xu, A reaction–diffusion predator–prey model with stage structure and nonlocal delay, Appl. Math. Comput. 175 (2006) 984–1006. [10] Z. Ge, Y. He, Diffusion effect and stability analysis of a predator–prey system described by a delayed reaction–diffusion equations, J. Math. Anal. Appl. 339 (2008) 1432–1450. [11] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [12] J.J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. RWA 10 (2009) 680–690. [13] S. Ahmad, I.M. Stamova, Asymptotic stability of competitive systems with delays and impulsive perturbations, J. Math. Anal. Appl. 334 (2007) 686–700.
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[14] B. Liu, Z. Teng, W. Liu, Dynamic behaviors of the periodic Lotka–Volterra competing system with impulsive perturbations, Chaos Soliton. Fract. 31 (2007) 356–370. [15] H. Zhang, L. Chen, J. Nieto, A delayed epidemic model with stage-structure and pulses for management strategy, Nonlinear Anal. RWA 9 (2008) 1714– 1726. [16] L. He, A. Liu, Existence and uniqueness of solutions for nonlinear impulsive partial differential equations with delay, Nonlinear Anal. RWA 11 (2010) 952–958. [17] V. Lakshmikantham, D. Bainov, P. Simenov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [18] M.U. Akhmet, M. Beklioglu, T. Ergenc, V.I. Tkachenko, An impulsive ratio-dependent predator–prey system with diffusion, Nonlinear Anal. RWA 7 (2006) 1255–1267. [19] Q. Wang, Z. Wang, Y. Wang, H. Zhang, M. Ding, An impulsive ratio-dependent n + 1-species predator–prey model with diffusion, Nonlinear Anal. RWA 11 (2010) 2164–2174. [20] Y. Huang, F. Chen, L. Zhong, Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge, Appl. Math. Comput. 182 (2006) 672–683. [21] T.K. Kar, H. Matsuda, Global dynamics and controllability of a harvested prey–predator system with Holling type III functional response, Nonlinear Anal.: Hybrid Syst. 1 (2007) 59–67. [22] L. Wang, W. Li, Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator–prey model with Holling type functional response, J. Comput. Appl. Math. 162 (2004) 341–357. [23] H. Huo, W. Li, J.J. Nieto, Periodic solutions of delayed predator–prey model with the Beddington–DeAngelis functional response, Chaos Soliton. Fract. 33 (2007) 505–512. [24] W. Walter, Differential inequalities and maximum principles; theory, new methods and applications, Nonlinear Anal. Appl. 30 (1997) 4695–4711. [25] L.H. Smith, Dynamics of Competition, Lecture Notes in Mathematics, vol. 1714, Springer, Berlin, 1999, pp. 192–240. [26] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum, 1992. [27] X. Liu, L. Chen, Global dynamics of the periodic logistic system with periodic impulsive perturbations, J. Math. Anal. Appl. 289 (2004) 279–291.
Contents lists available at ScienceDirect
Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm
Dynamics of impulsive reaction–diffusion predator–prey system with Holling type III functional response q Zijian Liu a,⇑, Shouming Zhong a,b, Chun Yin a, Wufan Chen c a
School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 610054, PR China Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu 610054, PR China c School of Automation Engineering, University of Electronic Science and Technology of China, Chengdu 610054, PR China b
a r t i c l e
i n f o
Article history: Received 23 September 2010 Received in revised form 29 April 2011 Accepted 8 May 2011 Available online 17 May 2011 Keywords: Reaction–diffusion Holling type III functional response Predator–prey system Permanence Stability
a b s t r a c t An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Reaction–diffusion equations can be used to model the spatio-temporal distribution and abundance of organisms. A typical form of reaction–diffusion population model is
@u ¼ DDu þ uf ðx; uÞ; @t where u(x, t) is the population density at a space point x and time t; D > 0 is the diffusion constant, Du is the Laplacian of u with respect to the variable x, and f(x, u) is the growth rate per capita, which is affected by the heterogeneous environment. Such an ecological model was first considered by Skellam [1], and similar reaction–diffusion biological models were also studied by Fisher [2] and Kolmogoroff et al. [3] earlier. In the past decade, the reaction–diffusion models, especially in population dynamics, have been studied extensively. For example, Ainseba and Anitßa [4] considered a 2 2 system of semilinear partial differential equations of parabolic-type to describe the interactions between a prey population and a predator population. They obtained some necessary and sufficient conditions for stabilizability. Liu and Huang [5] investigated a diffusive predator–prey model with Holling type III functional response and obtained some sufficient conditions for the ultimate boundedness of solutions and permanence of the system, they also presented the existence of a unique globally stable periodic solution. Recently, Apreutesei and Dimitriu [6] studied a prey–predator model defined by an initial-boundary value q This research was supported by National Basic Research Program of China (2010CB732501), the National Natural Science Foundation of PR China (10961022 and 10901130) and the National Natural Science Foundation of China (Grant No. 30970305). ⇑ Corresponding author. Tel.: +86 15882059791; fax: +86 991 8583556. E-mail addresses: [email protected] (Z. Liu), [email protected] (S. Zhong).
0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.05.019
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problem whose dynamics is described by a Holling type III functional response. They established global existence and uniqueness of the strong solution. Meanwhile, they proved that if the initial data are positive and satisfy a certain regularity condition, the solution of the problem is positive and bounded on the domain Q = (0, T) X and then they deduced the continuous dependence on the initial data. More researches on the reaction–diffusion population dynamics, please see [7–10]. There are many examples of evolutionary systems which at certain instants are subjected to rapid changes. In the simulations of such processes it is frequently convenient and valid to neglect the durations of rapid changes. The perturbations are often treated continuously. In fact, the ecological systems are often affected by environmental changes and other human activities. These perturbations bring sudden changes to the systems, which are referring to impulsive differential equations (see [11,12]). In the past twenty years they have attracted the interest of many researchers, since they provided a natural description of several real processes. Process of this type is often investigated in various fields of science and technology, physics, population dynamics [13,14], epidemics [15], ecology [16], biology, optimal control [17], and so on. Recently, some impulsive reaction–diffusion predator–prey models have been investigated. Especially, Akhmet et al. [18] presented an impulsive ratio-dependent predator–prey system with diffusion; meanwhile, they obtained some conditions for the permanence of the predator–prey system and for the existence of a unique globally stable periodic solution. Wang et al. [19] generalized the above impulsive ratio-dependent system to n + 1 species and got some analogous results. Motivated by the above works, we present and study the following impulsive reaction–diffusion predator–prey system with Holling type III functional response:
@u c1 ðt; xÞu2 v ¼ D1 Du þ u½aðt; xÞ bðt; xÞu 2 ; @t u þ rðt; xÞv 2
ð1:1Þ
@v c2 ðt; xÞu2 v ; ¼ D2 Dv dðt; xÞv þ 2 @t u þ rðt; xÞv 2
ð1:2Þ
uðtþk ; xÞ ¼ uðt k ; xÞfk ðx; uðt k ; xÞ; v ðt k ; xÞÞ;
v ðtþk ; xÞ ¼ v ðtk ; xÞg k ðx; uðtk ; xÞ; v ðtk ; xÞÞ; @u @ v ¼ 0; @n@X
@n @ X
ð1:3Þ k ¼ 1; 2; . . . ;
¼ 0:
ð1:4Þ ð1:5Þ
In this system, it is assumed that the predator and prey species are confined to a fixed bounded space domain X Rn with smooth boundary oX and non uniformly distributed in the domain. Furthermore, they are subjected to short-term external influence at fixed moment of time tk, where {tk}, k = 1, 2, . . . is a sequence of real numbers 0 = t0 < t1 < < tk < with limk?1tk = +1. Denote by @/on the outward derivative, X ¼ X [ @ X, and Du ¼ @ 2 u=@x21 þ þ @ 2 u=@x2n the Laplace operator. In Eqs. (1.1) and (1.2), D1 Du and D2 Dv with positive diffusion coefficients D1 and D2 reflect the non-homogeneous dispersion of populations. The Neumann boundary conditions (1.5) characterize the absence of migration. In the absence of predator, the prey species has a logistic growth rate. We assume that the predator functional response has the form of the Holling type III functional response function c1(t, x)u2/(u2 + r(t, x)v2), which reflects the capture ability of the predator. More discussion on the models with functional responses, please see [20–23]. In this paper, we will investigate the asymptotic behavior of non-negative solutions for impulsive reaction–diffusion system (1.1)–(1.5). Note that according to biological interpretation of the solutions u(t, x) and v(t, x) they must be nonnegative. We will give conditions for the long-term survival of each species in terms of permanence. The permanence of the system indicates that the number of individuals of each species stabilizes on certain boundaries with respect to time. This paper is organized as follows. In Section 2, we will give some basic assumptions and useful auxiliary results. Conditions for the ultimate boundedness of solutions and permanence of the system are obtained in Section 3. In Section 4, we establish conditions for the existence of the unique periodic solution of the system. Examples and numerical simulations are presented in Section 5 to illustrate the feasibility of our results. Finally, we discuss our results obtained and present some interesting problems. 2. Preliminaries Let N and R be the sets of all positive integers and real numbers, respectively, and Rþ ¼ ½0; 1Þ. The following assumptions will be needed throughout the paper. (A1) Functions a(t, x), b(t, x), d(t, x), c1(t, x), c2(t, x) and r(t, x) are bounded positive-valued on R X, continuously differentiable in t and x, periodic in t with a period s > 0; (A2) Functions fk(x, u, v) and gk(x, u, v), k 2 N, are continuously differentiable in all arguments and positive-valued; (A3) There exists a number p 2 N such that tk+p = tk + s for all k P 1; (A4) Sequences fk and gk satisfy the following equalities: fk+p(x, u, v) = fk(x, u, v) and gk+p(x, u, v) = gk(x, u, v) for all k P 1 and x, u, v. Conditions of periodicity are natural because of the seasonal changes and biological rhythms. We introduce the following notations: G ¼ Rþ X; G ¼ Rþ X,
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Rk ¼ fðt; xÞ : t 2 ðt k1 ; t k Þ; x 2 Xg;
k 2 N;
R¼
[
Rk ;
k2N
Rk ¼ fðt; xÞ : t 2 ðt k1 ; t k Þ; x 2 Xg;
k 2 N;
R¼
[
Rk :
k2N
Denote by F a class of functions / : G ! R with the following properties: (B1) /(t, x) is of class C2 in x, x 2 X and of class C1 in ðt; xÞ 2 Rk ; k 2 N; (B2) for all k 2 N; x 2 X, there exist the following limits:
lim /ðs; xÞ ¼ /ðtk ; xÞ;
s!t k
lim /ðs; xÞ ¼ / t þk ; x :
s!tþ k
We shall call a vector-function ðuðt; xÞ; v ðt; xÞÞ 2 F F a solution of Problems (1.1)–(1.5) if it satisfies (1.1), (1.2) on R, (1.5) by x 2 oX, and (1.3), (1.4) for every k 2 N. For a bounded function /(t, x), we denote /L = inf(t,x)/(t, x) and /M = sup(t,x)/(t, x). Consider the following impulsive logistic differential equation: dz dt
¼ azðb zÞ;
t – tk ;
zðt þk Þ ¼ zðt k Þkk ðzðt k ÞÞ;
k 2 N;
ð2:1Þ
where z 2 Rþ ; a and b are positive constants, strictly increasing sequence {tk} satisfies condition (A3), and kk ; k 2 N, are continuous positive-valued functions such that kk+p(z) = kk(z) for all z 2 Rþ ; k 2 N. Condition (A3) implies that tk+1 tk P h = mini=0,1,. . .,p(ti+1 ti) > 0, k P 1. Denote Q 1 ¼ b=ð1 eabh Þ; Q 2 ¼ Q 1 maxk¼1;2;...;p maxz2½0;Q 1 kk ðzÞ; Q 3 ¼ maxðz0 ; Q 1 ; Q 2 Þ, where z0 is given below. Then we have the following useful result. Lemma 2.1. Every solution z(t) = z(t, 0, z0), z0 = z(0) = z(0+) > 0 of system (2.1) satisfies 0 < z(t) 6 Q3 for all t P 0.
Proof. For t 2 [0, t1], we have that
zðtÞ ¼
bz0 eabt Þ
z0 ð1
abt
þ be
ð2:2Þ
:
It is obvious that the solution is positive-valued and no larger than max{z0, b} on the interval. Moreover, if h 6 t 6 t1, then
zðtÞ ¼
bz0 eabt Þ
z0 ð1
abt
6
þ be
b ¼ Q 1: 1 eabh
ð2:3Þ
Particularly, 0 < z(t1) 6 Q1, hence, 0 < z t þ 1 ¼ zðt 1 Þk1 ðzðt 1 ÞÞ 6 Q 2 . It is easy to show that 0 < z(t) 6 max (Q1, Q2) 6 Q3 if t 2 [t1, t2], and, similarly to (2.3) one can verify that 0 < z(t2) 6 Q1. Further, in the same manner we can show that 0 < z(t) 6 Q3 if t 2 (tk, tk+1], k = 2, 3, . . . h Now, let us give another useful lemma. Consider the following vector impulsive differential equation
dw ¼ Aw þ Fðt; wÞ; t – t i ; dt þ w t i ¼ wðti Þ þ Gi ðwðti ÞÞ; i 2 N;
ð2:4Þ ð2:5Þ
Lnþ1 ð p
where w ¼ ðw0 ; w1 ; . . . ; wn Þ 2 XÞ; p > n is a positive integer. The operator A has the domain DðAÞ ¼ n o 2;p @n n : n 2 W ðXÞ; @n j@ X ¼ 0 , where W2,p(X) is the Sobolev space of functions from Lp(X) that have two generalized derivatives. Function F(t, w(t)) satisfies suptkF(t, w(t))k < 1, Gi(w) is q-period in i. For any a > 0, we define the fractional power Aa of the operator A by
Aa ¼
1 CðaÞ
Z
1
esA sa1 ds;
0
where C is the Gamma function. The operators Aa are bounded and bijective. The operator Aa, a > 0, is defined as (Aa)1, and DðAa Þ ¼ RðAa Þ. The operator A0 is the identity operator in X. For 0 6 a 6 1, we introduce the space X a ¼ DðAa Þ with the norm kxka = kAaxk. Here kk is the norm in the space X ¼ Lnþ1 . p We denote by Cm+a(X), where m is a positive integer and 0 < a < 1, the space of m-times continuously differentiable functions f : X ! R, which have m-order derivatives satisfying the Hölder condition with exponent a. By Theorem 9 of paper [18], we have the following lemma immediately.
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Lemma 2.2. Assume that the functions Gi are continuously differentiable and there exists a positive-valued function g(M) such that
sup kGk ðwÞka 6 gðMÞ;
k 2 N;
ð2:6Þ
kwka 6M
n for some a 2 ð12 þ 2p ; 1Þ. Let w(t, w0), w0 = (w00, w10, . . . , wn0) 2 Xa, be a bounded solution of Eqs. (2.4) and (2.5), i.e.,
kwðt; w0 ÞkC 6 N;
t > 0:
ð2:7Þ 1þm
nþ1
Then the set {w(t, w0) : t > 0} is relatively compact in C ðX; R Þ for 0 < m < 2a 1 n/p. The following comparison theorems will be needed throughout the paper. Lemma 2.3 (Walter [24]). Suppose that vector-functions v(t, x) = (v1(t, x), . . . , vm(t, x)) and w(t, x) = (w1(t, x), . . . , wm(t, x)), m P 1, satisfy the following conditions: (i) they are of class C2 in x, x 2 X and of class C1 in ðt; xÞ 2 ½a; b X, where X Rn is a bounded domain with smooth boundary; (ii) vt lDv g(t, x, v) 6 wt lDw g(t, x,w), where (t, x) 2 [a, b] X, l = (l1, . . . , lm) > 0 (inequalities between vectors are satisfied coordinate-wise), vector-function g(t, x, u) = (g1(t, x, u), . . . , gm(t, x,u)) is continuously differentiable and quasi-monotonically increasing with respect to u = (u1, . . . , um):
@g i ðt; x; u1 ; . . . ; um Þ P 0; @uj
i; j ¼ 1; . . . ; m;
i – j;
(iii) ov/on = ow/on = 0, (t, x) 2 [a, b] oX. Then v(t, x) 6 w(t, x) for ðt; xÞ 2 ½a; b X. Lemma 2.4 (Smith [25]). Assume that T and d are positive numbers, a function u(t, x) is continuous on ½0; T X, continuously differentiable in x 2 X, with continuous derivatives @ 2u/@xi@xj and @u/@t on (0, T] X, and u(t, x) satisfies the following inequalities:
@u dDu þ cðt; xÞu P 0; ðt; xÞ 2 ð0; T X; @t @u P 0; ðt; xÞ 2 ð0; T @ X; @n uð0; xÞ P 0; x 2 X; where c(t, x) is bounded on (0, T] X. Then u(t, x) P 0 on ð0; T X. Moreover, u(t, x) is strictly positive on ð0; T X if u(0, x) is not identically zero. On the basis of the upper and lower solution method for quasi-monotone systems (see [26]), we can verify that, for continuously differentiable initial functions u0 ðxÞ : X ! Rþ ; v 0 ðxÞ : X ! Rþ ; u0 ðxÞ and v0(x) are not identically zero, there exists a classical solution of system (1.1), (1.2) and (1.5), which can be extended to the semi-axis t > 0. A vector-function (u(t, x), v(t, x)) is the classical solution of system without impulses (1.1), (1.2) and (1.5), if it is of class C2 in x, x 2 X, of class C1 in x; x 2 X, of class C1 in t, t > 0, and satisfies the system. Using the existence of solutions of system (1.1), (1.2) and (1.5), we can verify the existence of solutions for impulsive system (1.1)–(1.5). Indeed, if 0 < t 6 t1, the solutions of the system are well-defined as classical solutions of system without impulses (1.1), (1.2) and (1.5). Impulsive conditions (1.3) and (1.4) imply that the functions ðuðtþ Þ; v ðtþ 1 ÞÞ are continuously þ þ 1 differentiable in x, and satisfy the boundary conditions (1.5). Hence, assuming u t1 ; v t1 as new initial functions we can continue the solution on (t1, t2]. Proceeding in this way, we can construct the solution for all t > 0. According to biological interpretation, we only consider the non-negative solutions of the system. Hence, the following assertion is of major importance. Lemma 2.5. Assume that conditions (A1)–(A4) hold. Then non-negative and positive quadrants of R2 are positively invariant for system (1.1)–(1.5). Proof. Let us consider the first coordinate u(t, x) of a solution since the proof for the second one is very similar. It can be ^ ðt; xÞ and u ðt; xÞ such that simply verified that u
^ @u cM ^ ð0; xÞ ¼ u0 ðxÞ; ^u ^ aL bM u ^ p1ffiffiffiffi ¼ 0; u D1 Du @t 2 rL h i @u u aM bL u ¼ 0; u ð0; xÞ ¼ u0 ðxÞ D1 Du @t are lower and upper solutions of Eq. (1.1). Then, since u0(x) P 0 and u0(x) is not identically zero, by Lemma 2.4, we get ^ ðt; xÞ > 0 and u ðt; xÞ > 0 for t 2 (0, t1]. Since u(t, x) is bounded from below by positive function u ^ ðt; xÞ, we have u(t, x) > 0 u
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for t 2 (0, t1]. Taking into account positiveness of the function f1, we can repeat the same argument to prove the positiveness of u(t, x) for t 2 [t1, t2]. By induction, we have u(t, x) > 0 for t 2 (0, 1). h 3. Permanence In this section, applying the upper and lower solution method and comparison theory of differential equation, we establish some sufficient conditions for the ultimate boundedness and permanence of the system. Before this, two definitions are given firstly. Definition 3.1. Solutions of system (1.1)–(1.5) are said to be ultimately bounded if there exist positive constants N1, N2 such that for every solution (u(t, x, u0, v0), v(t, x, u0, v0)), there exists a moment of time t ¼ tðu0 ; v 0 Þ > 0 such that u(t, x, u0, v0) 6 N1, v(t, x, u0, v0) 6 N2 for all x 2 X, and t P t. Definition 3.2. System (1.1)–(1.5) is called permanent if there exist positive constants m1, m2, N1 and N2 such that for every solution with non-negative initial functions u0(x) and v0(x) that are not identically zero, there exists a moment of time ~t ¼ ~tðu0 ; v 0 Þ such that
m1 6 uðt; x; u0 ; v 0 Þ 6 N1 ;
m2 6 v ðt; x; u0 ; v 0 Þ 6 N2
for all x 2 X, and t P ~t. Theorem 3.1. Assume that conditions (A1)–(A4) hold, and, moreover: (i) there exists a positive-valued function g(M) such that fk(x, u, v) 6 g(M) if k 2 N; u 6 M; (ii) the inequality L
sd þ
p X
v P 0 and x 2 X;
ln g k < 0
ð3:1Þ
k¼1
holds, where gk = sup(x,u,v)gk(x, u, v). Then all solutions of system (1.1)–(1.5) with non-negative initial conditions are ultimately bounded. ðt; x; u0 Þ be a solution of the equation Proof. Let u
@u u ðaM bL u Þ ¼ 0: D1 Du @t
ð3:2Þ
Using inequality
@u c1 ðt; xÞu2 v D1 Du u½aðt; xÞ bðt; xÞu þ 2 @t u þ rðt; xÞv 2 @u L P D1 Du uðaM b uÞ; @t
0¼
we obtain
0¼
@u @u L u ðaM bL u Þ P D1 Du D1 Du uðaM b uÞ: @t @t
ðt; M u Þ, where Mu is such that ku0 ðxÞkC ¼ maxx2X ju0 ðxÞj 6 M u . Note Applying Lemma 2.3, we conclude that uðt; x; u0 ; v 0 Þ 6 u ðt; M u Þ of Eq. (3.2) with initial condition independent of x does that, according to the uniqueness theorem, the solution u ðt; Mu Þ satisfies the ordinary differential equation du =dt ¼ u ðaM bL u Þ. not depend on x for t > 0. Therefore, the function u Hence,
ðt k ; M u Þgðu ðt k ; M u ÞÞ: kuðt þk ; x; u0 ; v 0 ÞkC ¼ kuðtk ; x; u0 ; v 0 Þfk ðx; uðt k ; x; u0 ; v 0 Þ; v ðt k ; x; u0 ; v 0 ÞÞkC 6 u Since all solutions of the impulsive differential equation
du ðaM bL u Þ; ¼u dt
ðtk ÞÞ ðt k Þgðu tþk ¼ u u
are ultimately bounded by Lemma 2.1, we get ultimate boundedness of solutions of Eq. (1.1) with impulses (1.3), i.e., there exists a positive constant N1 such that u(t, x) 6 N1, starting with some moment of time t1 . For the predator population, when t P t1 ,
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@v c2 ðt; xÞu2 v @v cM N 1 L P D2 Dv þ dðt; xÞv 2 D2 Dv þ d v 2pffiffiffiffi ; @t u þ rðt; xÞv 2 @t 2 rL
0¼
pffiffiffiffi L ðt; M v Þ, where v ðt; M v Þ is a solution of the initial value problem dv =dt ¼ dL v þ cM it follows that v ðt; x; u0 ; v 0 Þ 6 v 2 N 1 =2 r ð0; M v Þ ¼ M v . with v Linear periodic impulsive equation
dv cM N 1 L ¼ d v þ 2pffiffiffiffi ; dt 2 rL
v
þ t k ¼ g k v ðt k Þ
ð3:3Þ
ðtÞ ¼ X 0 ðtÞ þ CXðtÞ, where X0(t) is a s-periodic piecewise continuous function, C is a constant and has the general solution v L
XðtÞ ¼ expðd t þ
X
ln g k Þ
0
(see [11]). By (3.1), X(t) ? 0 as t ? 1. All solutions of (3.3) are ultimately bounded, therefore, all solutions of Eqs. (1.2) and (1.4) are ultimately bounded, too. h Theorem 3.2. Assume that conditions (A1)–(A4) hold, and, moreover: (i) Solutions of system 1.1, 1.2, 1.3, 1.4, 1.5 are ultimately bounded, i.e., there exist positive constants N1 and N2 such that for every solution (u(t, x, u0, v0), v(t, x, u0, v0)), there exists t ¼ tðu0 ; v 0 Þ > 0 such that u(t, x, u0, v0) 6 N1 and v(t, x, u0, v0) 6 N2 for all t P t; (ii) the following inequalities p X
ln
k¼1
cM fk ðx; u; v Þ þ s aL p1ffiffiffiffi > 0 x2X;ðu;v Þ2S 2 rL inf
ð3:4Þ
and p X
ln
k¼1
inf
x2X;ðu;v Þ2S
M g k ðx; u; v Þ þ s cL2 d >0
ð3:5Þ
hold, where S = {(u, v):0 < u 6 N1, 0 < v 6 N2}. Then there exist positive constants r1 and r2 such that an arbitrary solution of system (1.1)–(1.5) with non-negative initial conditions not identically equal to zero satisfies
ðuðt; xÞ; v ðt; xÞÞ 2 P ¼ ðu; v Þ : r1 6 uðt; xÞ 6 N1 ;
r2 6 v ðt; xÞ 6 N2 ;
starting with a certain moment of time. Proof. Lemma 2.4 implies that if u0(x) P 0, v0(x) P 0, u0(x) and v0(x) are not identically zero, then u(t, x, u0, v0) > 0, v(t, x, u0, v0) > 0 for all x 2 X and t > 0. Considering the solution on the interval t P e with some small e > 0, we get initial conditions (u(e, x, u0, v0), v(e, x, u0, v0)) separated from zero. Therefore, we can assume, without loss generality, that minx2X u0 ðxÞ ¼ mu > 0 and minx2X v 0 ðxÞ ¼ mv > 0. Using inequality
@u c1 ðt; xÞu2 v D1 Du u½aðt; xÞ bðt; xÞu þ 2 @t u þ rðt; xÞv 2 M @u c M D1 Du u aL b u p1ffiffiffiffi ; 6 @t 2 rL
0¼
we obtain
0¼
^ @u cM @u cM M ^u ^ aL bM u ^ p1ffiffiffiffi 6 D1 Du D1 Du u aL b u p1ffiffiffiffi : @t @t 2 rL 2 rL
^ ðt; mu Þ for t 2 [0, t1]. Applying the last inequality for t = t1, together with Eq. Now, using Lemma 2.3 for m ¼ 1; uðt; x; u0 ; v 0 Þ P u (1.3), we obtain that
^ðt 1 ; mu Þ u tþ1 ; x; u0 ; v 0 P u
inf
x2X;ðu;v Þ2S
f1 ðx; u; v Þ:
Thus, the solution u(t, x, u0, v0) is bounded from below by a solution of periodic logistic equation with impulses
^ du cM ^ aL p1ffiffiffiffi bM u ^ ; ¼u dt 2 rL
^ ðtk Þ ^ ðt þk Þ ¼ u u
inf
x2X;ðu;v Þ2S
fk ðx; u; v Þ:
ð3:6Þ
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By Theorem 2.1 [27] and condition (3.4), Eq. (3.6) has a unique piecewise continuous and strictly positive periodic solution ^ ðtÞ such that every solution u ^ ðt; um Þ of (3.6) with um > 0 has the property u ^ ðt; um Þ ! u ^ ðtÞ as t ? 1. Therefore, there exists a u ^ ðt; um Þðum > 0Þ of Eq. (3.6) we get u ^ ðt; um Þ P r1 , starting with some mopositive constant r1 such that, for every solution u ment of time ^t 1 ¼ ^t 1 ðum Þ > 0. ^ ðt; um Þ of Eq. (3.6), we conclude that Since solution u(t, x, u0, v0) of Eqs. (1.1) and (1.3) is bounded from below by solution u uðt; x; u0 ; v 0 Þ P r1 for t P ^t1 . Now, let us consider the predator population. When t P ^t1 , since uðt; x; u0 ; v 0 Þ P r1 , we have
0¼
@v c2 ðt; xÞu2 v @v cL rM v 3 M 6 : D2 Dv þ dðt; xÞv 2 D2 Dv þ d cL2 v þ 22 2 @t u þ rðt; xÞv @t r þ r M v 2 1
^ ðt; mv Þ, where v ^ ð0; mv Þ ¼ mv is the solution of equation Hence, v ðt; x; u0 ; v 0 Þ P v
dv^ L cL rM v 3 M ; ¼ c2 d v 22 dt r þ rM v 2
v^
þ tk ¼ v^ ðtk Þg^k ;
ð3:7Þ
1
^ ðtÞ 6 r2 for t 2 [0, t1], then where g^k ¼ inf x2X;ðu;v Þ2S g k ðx; u; v Þ. If v
(
M
v^ ðt1 ; mv Þ P mv exp
t1 cL2 d
cL2 r M r22 2 r1 þ rM r22
!)
and
v^
( !) cL r M r22 M : t þ1 ; mv P g^1 mv exp t1 cL2 d 22 r1 þ rM r22
^ ðtÞ 6 r2 for t 2 [0, s], then Therefore, if v
v^ ðs; mv Þ P mv exp
( p X
M
ln g^k þ s cL2 d
k¼1
cL2 rM r22
!)
ðr1 Þ2 þ r M r22
:
Taking into account (3.5), we can take sufficiently small r2 > 0 such that p X k¼1
M
cL2 r M r22 2 r1 þ rM r22
!
ln g^k þ s cL2 d
¼ q > 0:
^ ðk2 s; mv Þ P ek2 q mv P r02 (by the additional condition For r02 2 ð0; r2 Þ, there exists a positive integer k2 such that v v^ ðt; mv Þ < r2 for all t 2 [0, k2s]). ^ ðt; v ^ 0 Þ of (3.7) with v ^ 0 > 0, there exists a moment of time ^t such that v ^ ð^t; v ^ 0 Þ P r02 . Denote by Hence, for every solution v v^ ðt; t0 ; v^ 0 Þ the solution of (3.7) with v^ ðt0 ; t0 ; v^ 0 Þ ¼ v^ 0 and consider a positive number
r2 ¼ inffv^ ðt; t0 ; v^ 0 Þ : t0 2 ½0; s; v^ 0 2 ½r02 ; N2 ; t 2 ½t0 ; 2sg: ^ ðt; t0 ; v ^ 0 Þ P r2 for all t P 2s. Indeed, let us take Then v
rs ¼ inffv^ ðs; t0 ; v^ 0 Þ : t0 2 ½0; s; v^ 0 2 ½r02 ; N2 g P r2 ^ ðs; s; v ^ ðt; s; v ^ 0 Þ with v ^ 0 P rs . If v ^ ðt; s; v ^ 0 Þ 6 r2 for all t 2 [t0, 2s], then v ^ ð2s; s; v ^ 0 Þ P eq v ^ 0 Þ P rs . If and consider a solution v v^ ðt; s; v^ 0 Þ > r2 at some moment of time t 2 [s, 2s], then v^ ð2s; s; v^ 0 Þ P rs by definition of number rs. Therefore, it is enough ^ ðt; 2s; v ^ 0 Þ; t P 2s, with v ^ 0 P rs . By construction, these solutions are bounded from below by positive constant to consider v r2 for t 2 [2s, 3s]. Proceeding in this way we prove the boundedness from below for t P 3s. h Through the above analysis, we got some conditions under which the two species are permanent. Then, we will give some conditions that will lead to extinction of the predator species. Theorem 3.3. Assume that system 1.1, 1.2, 1.3, 1.4, 1.5 satisfies conditions (A1)–(A4), and, further: p X k¼1
L ln sup g k ðx; u; v Þ þ s cM < 0: 2 d ðx;u;v Þ
Then v(t, x) ? 0 as t ? 1. ðt; M v Þ the solution of initial value problem Proof. Fix a positive constant Mv such that Mv P v0(x) and denote by v
dv M L ¼ c2 d v ; dt
v ð0; Mv Þ ¼ Mv :
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From the inequality
0¼
@v c2 ðt; xÞu2 v @v L P D2 Dv þ dðt; xÞv 2 D2 Dv þ d cM 2 v; @t u þ rðt; xÞv 2 @t
ðt; M v Þ for t 6 t1. applying the comparison theorem, we can find that v ðt; x; u0 ; v 0 Þ 6 v Moreover, using impulsive condition (1.4), we obtain that
v
t þ1 ; x; u0 ; v 0 6 v ðt1 ; M v Þ sup g 1 ðx; u; v Þ: ðx;u;v Þ
Proceeding in this fashion, we conclude that every solution of Eqs. (1.2) and (1.4) is bounded from above by the corresponding solution of linear impulsive equation
dv M L ¼ c2 d v ; dt
v
þ t k ¼ v ðtk Þ sup g k ðx; u; v Þ: ðx;u;v Þ
Taking into account (3.8), we see that all solutions of the last equation tend to zero as t ? 1. h 4. Periodic solutions In the following, we study the existence of the periodic solution by constructing an appropriate auxiliary function. We will note that the conditions of the existence of the periodic solution are dependent on the permanence of the system. Theorem 4.1. Assume that conditions (A1)–(A4) and (2.6) hold, and system (1.1)–(1.5) is permanent, i.e., there exist positive constants r and N such that an arbitrary solution of the system with non-negative initial functions not identically equal to zero satisfies the condition
ðuðt; xÞ; v ðt; xÞÞ 2 P ¼ fðu; v Þ : r 6 uðt; xÞ 6 N;
r 6 v ðt; xÞ 6 Ng;
starting with a certain moment of time. Let, additionally, p X
ln K i þ skM < 0;
i¼1
where
( K i ¼ max 2 u;v 2P;x2X
fi2
2 2 ) 2 2 @fi @fi @g i @g i 2 þ N þ N þ gi þ N þ N ; @u @v @u @v
kM is the maximal eigenvalue of the matrix
0
cL r L r4 L 2 aM b r N2 ð1þrM1 ÞðN2 þrL r2 Þ B @ ðcM1 þcM1 rM þ2cM2 rM ÞN4 ð1þrL Þ2 r4
ðcM1 þcM1 rM þ2cM2 rM ÞN4
1
r C
A: cM N4 L 2 2 d þ ðN2 þrL r2 Þ2 ð1þr L Þ2 4
Then system 1.1, 1.2, 1.3, 1.4, 1.5 has a unique globally asymptotically stable strictly positive piecewise continuous s-periodic solution. ðt; xÞ; v ðt; xÞÞ be two solutions of system (1.1)–(1.5) bounded by constants r and N from below Proof. Let (u(t, x), v(t, x)) and ðu and above, respectively. Consider the function
VðtÞ ¼
Z
ðt; xÞÞ2 þ ðv ðt; xÞ v ðt; xÞÞ2 dx: ½ðuðt; xÞ u
X
Its derivative has the form
Z @u @ u @ v @ v Þ dx þ 2 ðv v Þ ðu u dx @t @t @t @t X X Z Z ÞDðu u Þdx þ 2D2 ðv v ÞDðv v Þdx ¼ 2D1 ðu u X X Z 2 v c1 u2 v c u Þ uða buÞ ða bu Þ þ 1 dx þ 2 ðu u u 2 þ r v 2 u2 þ r v 2 u X Z 2 v c2 u2 v c2 u þ 2 ðv v Þ dv þ 2 dx :¼ I1 þ I2 þ I3 þ I4 : þ dv 2 2 2 þ r v u þ rv u X
dVðtÞ ¼2 dt
Z
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It follows from the boundary condition (1.5) that
I1 þ I2 ¼ 2D1
Z
r2 ðu u Þdx 2D2
X
Z
r2 ðv v Þdx 6 2D1
Z
X
Þj2 dx 2D2 jrðu u
X
Z
jrðv v Þj2 dx 6 0: X
For the third and fourth terms I3 and I4, we have
Z
Z
2 ðv v Þ þ c1 r v v ðu2 v u 2 v Þ c1 u2 u dx þ 2 2 2 2 2 ðu þ r v Þðu þ rv Þ X X Z 2 v Þ 2 ðv v Þ þ c2 r v v ðu2 v u c2 u2 u þ 2 ðv v Þ dx 2 2 2 2 ðu þ rv Þðu þ r v Þ X
I3 þ I 4 ¼ 2
Þ2 ½a bðu þ u Þdx 2 ðu u
Þ ðu u
Z
ðv v Þ2 ðdÞdx
X
2 v ¼ ðu u Þðuv þu v Þ uu ðv v Þ, thus u Noting that u2 v
I3 þ I 4 ¼ 2
Þ2 a bðu þ u Þ ðu u
Z
X
Z
Z v Þ ðuu r v v Þ c1 rv v ðuv þ u c2 uu Þ2 d þ dx þ 2 dx ð v v 2 þ rv 2 Þ 2 þ r v 2 Þ ðu2 þ r v 2 Þðu ðu2 þ r v 2 Þðu X
v Þ c 1 uu ðuu r v v Þ c2 rv v ðuv þ u dx 2 þ r v 2 Þ ðu2 þ rv 2 Þðu " # " # Z Z 4 cL1 r L r4 cM L 2 2 N Þ2 aM bL r 2 dx þ 2 dx 6 2 ðu u ð v v Þ d þ N ð1 þ r M ÞðN2 þ rL r2 Þ ðN2 þ r L r2 Þ2 X X Z Z 4 M M M ðcM þ cM 1 r þ 2c 2 r ÞN Þðv v Þj 1 Þ2 þ ðv v Þ2 dx: þ 2 jðu u dx 6 kM ½ðu u 2 ð1 þ r L Þ r4 X X þ2
Þðv v Þ ðu u
X
Hence we have dV(t)/dt 6 kMV(t), therefore, Vðt iþ1 Þ 6 V tþ expðkM ðtiþ1 t i ÞÞ and i
V t þiþ1 ¼
Z
fiþ1 ðu ; v Þ2 dx þ ½ufiþ1 ðu; v Þ u X
Z
X
; v Þ2 dx 6 K iþ1 Vðt iþ1 Þ 6 K iþ1 expðkM ðt iþ1 ti ÞÞV tþi : ½v g iþ1 ðu; v Þ v g iþ1 ðu
Let us estimate the variation of the function over the period. We have
Vðt þ sÞ 6 K VðtÞ ¼
p Y
K i expðkM sÞVðtÞ:
i¼1
According to the conditions of the theorem, we have K⁄ < 1. Therefore, Vðms þ sÞ 6 K m VðsÞ ! 0; m ! 1. We have proved that ðt; xÞk ! 0 and kv ðt; xÞ v ðt; xÞk ! 0 as t ? 1, where kk is the norm of the space L2(X). By Lemma 2.2, solutions kuðt; xÞ u of system (1.1)–(1.5) are bounded in the space C1+m. Therefore
ðt; xÞj ! 0; sup juðt; xÞ u x2X
sup jv ðt; xÞ v ðt; xÞj ! 0;
t ! 1:
x2X
Fig. 1. The permanence of species u.
ð4:1Þ
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Now let us consider the sequence ðuðks; x; u0 ; v 0 Þ; v ðks; x; u0 ; v 0 ÞÞ ¼ wðks; w0 Þ; k 2 N. By Lemma 2.2, it is compact in the be a limit point of this sequence, w ¼ limn!1 wðkn s; w0 Þ. Then wðs; wÞ ¼ w. Indeed, since space CðXÞ CðXÞ. Let w w(s, w(kns, w0)) = w(kns, w(s, w0)) and w(kns, w(s, w0)) w(kns, w0) ? 0 as kn ? 1, we get
wk C 6 kwðs; wÞ wðs; wðkn s; w0 ÞÞkC þ kwðs; wðkn s; w0 ÞÞ wðkn s; w0 ÞkC þ kwðkn s; w0 Þ wk C ! 0; kwðs; wÞ
n ! 1:
The sequence wðks; w0 Þ; k 2 N, has a unique limit point. On the contrary, let the sequence has two limit points ¼ limn!1 wðkn s; w0 Þ and w ~ ¼ limn!1 wðkn s; w0 Þ. Then, taking into account (4.1) and w ~ ¼ wðkn s; wÞ, ~ we have w
wk ~ C 6 kw wðkn s; w0 ÞkC þ kwðkn s; w0 Þ wk ~ C ! 0; kw
n ! 1;
¼ w. ~ The solution ðuðt; x; u ; v Þ; v ðt; x; u ; v ÞÞ is the unique periodic solution of system (1.1)–(1.5). By (4.1), it is asymphence w totically stable. h
Fig. 2. The permanence of species v.
5.5 5 4.5 4 3.5
u
3 2.5 2 1.5 1 0.5 0 0
5
10
15
20 t
25
Fig. 3. The section of Fig. 1 with x = 0.
30
35
40
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5. Numerical illustrations In order to verify the validity of our results, we consider system (1.1)–(1.5) with fk(x, u, v) 0.8 and gk(x, u, v) 0.7 for all k = 1, 2, . . . Choose D1 ¼ D2 ¼ 1, a(t, x) = sin(t) + 2 cos(x) + 8, b(t, x) = cos(t) + 2 cos(x) + 4, c1(t, x) = 1.9 sin(t) + sin(x) + 3, c2(t, x) = 0.4 sin(t) + 0.1 sin(x) + 3, d(t, x) = 0.45 cos(t) cos(x) + 0.5, and r(t, x) 1. Obviously, all the parameters have a common period s = 2p in t. Choosing an appropriate per length in the matlab procedure, we can calculate p = 26. Then, we have that the parameters satisfy all the conditions of Theorem 3.1, further
cM fk ðx; u; v Þ þ s aL p1ffiffiffiffi ¼ 7:0788 > 0 x2X;ðu;v Þ2S 2 rL inf
5.5 5 4.5 4 3.5 3 v
k¼1
ln
2.5 2 1.5 1 0.5 0 0
5
10
15
20 t
25
30
35
40
30
35
40
Fig. 4. The section of Fig. 2 with x = 0.
7 6 5 4 u
p X
3 2 1 0 −1
0
5
10
15
20 t
25
Fig. 5. The projection of species u in plane t u.
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and p X k¼1
ln
inf
x2X;ðu;v Þ2S
M g k ðx; u; v Þ þ s cL2 d ¼ 0:4654 > 0;
which satisfy the conditions (3.4) and (3.5) of Theorem 3.2. Hence, all the conditions of Theorem 3.2 are satisfied, then, system (1.1)–(1.5) is permanent. See Figs. 1–7. Here, we choose x 2 X = [2, 2] and the initial conditions u(0, x) = 4, v(0, x) = 3 for all x 2 X. However, if we choose c1(t, x) = 2.9 sin(t) + sin(x) + 4, c2(t, x) = 0.2 sin(t) + 0.1 cos(x) + 3, d(t, x) = 0.1 cos(t) cos(x) + 2.6 and other parameters are unchanged, then the parameters satisfy all the conditions of Theorem 3.1. Furthermore,
7 6 5
v
4 3 2 1 0 −1
0
5
10
15
20 t
Fig. 6. The projection of species
25
30
35
40
v in plane t v.
4.5
4
v
3.5
3
2.5
2
1
1.5
2
2.5 u
3
Fig. 7. The phase of species u and v.
3.5
4
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3 2.5
v
2
1.5 1
0.5
0
1
1.5
2
2.5
3 u
3.5
4
4.5
5
Fig. 8. The phase of species u and v.
Fig. 9. The extinction of species v.
p X k¼1
L ln sup g k ðx; u; v Þ þ s cM ¼ 4:2470 < 0; 2 d ðx;u;v Þ
which satisfies the condition (3.8) of Theorem 3.3. Hence, all the conditions of Theorem 3.3 are satisfied, so that species v goes to extinction. See Figs. 8–10. From Fig. 8, we know that species u is also permanent when species v approaches to zero. Here, we also choose x 2 X = [2, 2] and the initial conditions u(0, x) = 4, v(0, x) = 3 for all x 2 X. 6. Conclusion and discussion If prey and predator populations are in a remote patchy forest or an isolated island or a lake ecosystem, an appropriate reaction–diffusion model is described by (1.1), (1.2) and (1.5), which is the model proposed by Liu and Huang in paper [5]. However, the ecological systems are often affected by environmental changes, these perturbations bring sudden changes to the system. Systems with such sudden perturbations referring to impulsive differential equations. Hence, in this paper, we investigated an impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response. We
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3
v
2 1 0 −1
0
5
10 t
15
20
5
10 t
15
20
3
v
2 1 0 −1
0
Fig. 10. The section and projection of species v.
gave some sufficient conditions for the permanence and the existence of a unique globally stable positive periodic solution of the system. If there are not any impulses in Problems (1.1)–(1.5) (fk and gk are identically equal to zero, this model is investigated in paper [5]), then conditions (3.4), (3.5) take a form of the following inequalities:
cL aL p1ffiffiffiffi > 0; 2 rL
M
cL2 d > 0;
which are sufficient to have the permanence phenomenon for the system. However, in paper [5], the non-impulses system is M permanent only need the condition cL2 d > 0. An interesting problem is to look for conditions under which the impulsive M system is permanent, but the permanent condition is cL2 d > 0 if there are not any impulses. If the system is permanent, by Theorem 4.1, we conclude that it has a unique globally asymptotically stable strictly poP sitive piecewise continuous s-periodic solution if pi¼1 ln K i þ skM < 0. However, there is an interesting problem. If this system has a unique globally asymptotically stable strictly positive s-periodic solution, it should be permanent. It seems that P Theorem 4.1 may imply Theorem 3.2. But unfortunately, we cannot claim it is true. Since the condition pi¼1 ln K i þ skM < 0 in Theorem 4.1 is dependent on r and N, which are decided by the bound of solutions when the system is permanent. Whether is there a unique globally asymptotically stable s-periodic solution only if conditions (3.4) and (3.5) hold (the system is permanent) or other conditions independent of r and N? We will continue to study these problems in the future. In this paper, we only studied the system with Holling type III functional response, whether other type functional responses such as Holling type II and Beddington–DeAngelis functional response can be discussed with the same methods or not, still remain open problems. References [1] J.G. Skellam, Random dispersal in theoretical populations, Biometrika 38 (1951) 196–218. [2] R.A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics 7 (1937) 353–369. [3] A. Kolmogoroff, I. Petrovsky, N. Piscounoff, Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem, Moscow Univ. Bull. Math. 1 (1937) 1–25. [4] B. Ainseba, S. Anitßa, Internal stabilizability for a reaction–diffusion problem modeling a predator–prey system, Nonlinear Anal. 61 (2005) 491–501. [5] X. Liu, L. Huang, Permanence and periodic solutions for a diffusive ratio-dependent predator–prey system, Appl. Math. Model. 33 (2009) 683–691. [6] N. Apreutesei, G. Dimitriu, On a prey–predator reaction–diffusion system with Holling type III functional response, J. Comput. Appl. Math. 235 (2010) 366–379. [7] J. Shi, R. Shivaji, Persistence in reaction–diffusion models with weak allee effect, J. Math. Biol. 52 (2006) 807–829. [8] C. Duque, K. Kiss, M. Lizana, On the dynamics of an n-dimensional ratio-dependent predator–prey system with diffusion, Appl. Math. Comput. 208 (2009) 98–105. [9] R. Xu, A reaction–diffusion predator–prey model with stage structure and nonlocal delay, Appl. Math. Comput. 175 (2006) 984–1006. [10] Z. Ge, Y. He, Diffusion effect and stability analysis of a predator–prey system described by a delayed reaction–diffusion equations, J. Math. Anal. Appl. 339 (2008) 1432–1450. [11] A.M. Samoilenko, N.A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995. [12] J.J. Nieto, D. O’Regan, Variational approach to impulsive differential equations, Nonlinear Anal. RWA 10 (2009) 680–690. [13] S. Ahmad, I.M. Stamova, Asymptotic stability of competitive systems with delays and impulsive perturbations, J. Math. Anal. Appl. 334 (2007) 686–700.
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