Persistence and periodicity of a delayed ratio-dependent predator–prey model with stage structure and prey dispersal

Applied Mathematics and Computation 159 (2004) 823–846 www.elsevier.com/locate/amc

Persistence and periodicity of a delayed ratio-dependent predator–prey model with stage structure and prey dispersal Rui Xu *, M.A.J. Chaplain, F.A. Davidson Department of Mathematics, University of Dundee, 23 Perth Road, Dundee DD1 4HN, UK

Abstract A delayed periodic ratio-dependent predator–prey model with prey dispersal and stage structure for predator is investigated. It is assumed that immature individuals and mature individuals of the predator species are divided by a fixed age, and that immature predators donÕt have the ability to attack prey, and that predator species is confined to one of the patches while the prey species can disperse between two patches. We first discuss the uniform persistence and impermanence of the model. By using Gaines and MawhinÕs continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are derived for the existence of positive periodic solutions of the proposed model. Numerical simulations are presented to illustrate the feasibility of our main results. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Stage structure; Ratio-dependence; Dispersion; Delay; Persistence; Periodic solution

1. Introduction Traditional Lotka–Volterra type predator–prey model with Michaelis– Menten or Holling type-II functional response has received great attention from both theoretical and mathematical biologists, and has been well studied.

* Corresponding author. Address: Department of Mathematics, Institute of Shijiazhuang Mechanical Engineering, Shijiazhuang 050003, PR China. E-mail address: [email protected] (R. Xu).

0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.11.006

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The standard Lotka–Volterra type model is built by assuming that the per capita rate of predation depends on the prey numbers only. Recently, the traditional prey-dependent predator–prey model has been challenged by several biologists (see, for example [4–6,15] and the references cited therein). There is growing explicit biological and physiological evidence [4–6,15] that in some situations, especially when predator have to search for food (and therefore have to share or compete for food), a more suitable general predator–prey model should be based on the ‘‘ratio-dependent’’ theory. This roughly states that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. This is strongly supported by numerous field and laboratory experiments and observations [3,4,6,17]. Based on the Michaelis–Menten or Holling type-II function, Arditi and Ginzburg [3] proposed a ratio-dependent function of the form
 x cðx=yÞ cx P ¼ ¼ y m þ ðx=yÞ my þ x and the following ratio-dependent predator–prey model x_ ¼ xða  bxÞ  cxy=ðmy þ xÞ; y_ ¼ yðd þ fx=ðmy þ xÞÞ:

ð1:1Þ

Here xðtÞ and yðtÞ represent the densities of the prey and the predator at time t, respectively. a=b is the carrying capacity of the prey, d > 0 is the death rate of the predator, and a; c; m and f =c are positive constants that stand for the intrinsic growth rate of the prey, capturing rate, half saturation constant and conversion rate of the predator, respectively. The ratio-dependent predator–prey models with or without time delays have been studied by many researchers recently and very rich dynamics have been observed (see, for example [7,19–21,36–38] and the references cited therein). However, it is assumed in the classical ratio-dependent predator–prey model that each individual predator admits the same ability to attack prey. This assumption is obviously unrealistic for many animals. In the natural world, there are many species whose individuals have a life history that take them through two stages, immature and mature, where immature predators are raised by their parents, and the rate they attack at prey and the reproductive rate can be ignored. Stage structured models have received much attention in recent years (see, for example [1,2,10,13,23,28,29,31,32,34,36]). In [1] a model of single species population growth with stage structure as a reasonable generalization of the classical logistic model was proposed. This model assumes an average age to maturity which appears as a constant time delay reflecting a delayed birth of

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immatures and a reduced survival of immatures to their maturity. The model takes the form x_ i ðtÞ ¼ axm ðtÞ  cxi ðtÞ  aecs xm ðt  sÞ; x_ m ðtÞ ¼ aecs xm ðt  sÞ  bx2m ðtÞ;

t > s;

ð1:2Þ

where xi ðtÞ and xm ðtÞ represents the densities of the immature and mature population, respectively. a > 0 represents the birth rate, c > 0 is the immature death rate, b > 0 is the mature death and overcrowding rate, s is the time to maturity. The term aecs xm ðt  sÞ represents the immature who were born at time t  s and survive at time t (with the immature death rate c), and therefore represents the transformation of immatures to matures. It is shown in [1] that the positive equilibrium of model (1.2) is globally stable, and that the introduction of stage structure does not affect the permanence of the species, but under some hypotheses it may maximize the total carry capacity of the population. Recently, Wang et al. [36] proposed a stage structured ratio-dependent predator–prey model, in which they showed the effect of the duration time of immature predator on the global dynamics of predator–prey system. Sufficient conditions are derived in [36] for the permanence and global stability of a positive equilibrium of the proposed model. Interest has been growing in the study of mathematical models of populations dispersing among patches in a heterogeneous environment ([8,9,12,13,18,24–26,30,33,35,38,39], and references cited therein). Many of the existing models deal with a single population dispersing among patches. Some of them deal with competition and predator–prey interactions in patchy environments. Most of the previous works deal with autonomous population systems. The analysis of these models has been centered around the coexistence of populations and the stability of equilibria. These works indicate that a diffusion process in an ecological system is often considered to have a stabilizing influence on the system [18,33], but is also probably destabilizing the system [25,26]. We note that any biological or environmental parameters are naturally subject to fluctuation in time. As Cushing [11] pointed out that it is necessary and important to consider models with periodic ecological parameters or perturbations which might be quite naturally exposed (for example, those due to seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc.). Thus, the assumption of periodicity of the parameters is a way of incorporating the periodicity of the environment. Motivated by the works of Aiello and Freedman [1] and Wang et al. [36], in this paper, we study the combined effect of stage structure, dispersion and periodicity of the environment on the dynamics of predator–prey systems. To do so, we discuss the following delayed periodic ratio-dependent predator–prey model with prey dispersal and stage structure for predator

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  8 a12 ðtÞy2 ðtÞ > _ ðtÞ ¼ x ðtÞ r ðtÞ  a ðtÞx ðtÞ  x þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ; > 1 1 1 11 1 mðtÞy2 ðtÞþx1 ðtÞ > > > > x_ ðtÞ ¼ x2 ðtÞðr2 ðtÞ  a22 ðtÞx2 ðtÞÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ; > > < 2 x1 ðtÞy2 ðtÞ y_ 1 ðtÞ ¼ aðtÞ mðtÞy  c1 ðtÞy1 ðtÞ 2 ðtÞþxR 1 ðtÞ t > >  c ðsÞ ds x1 ðtsÞy2 ðtsÞ > ; aðt  sÞe ts 1 > > mðtÞy2 ðtsÞþx1 ðtsÞ > R t > >  c1 ðsÞ ds x1 ðtsÞy2 ðtsÞ :  c2 ðtÞy2 ðtÞ; y_ 2 ðtÞ ¼ aðt  sÞe ts mðtÞy2 ðtsÞþx1 ðtsÞ ð1:3Þ where xi ðtÞ denotes the density of prey species x in patch i at time t, i ¼ 1; 2; y1 ðtÞ and y2 ðtÞ represent the densities of immature and mature individual predators at time t, respectively. Predator species is confined to patch 1 while the prey species can disperse between two patches. ri ðtÞ is the intrinsic growth rate of the prey at patch i, i ¼ 1; 2; aii ðtÞ is the rate of intra-specific competition of the prey in ith patch, i ¼ 1; 2. a12 ðtÞ is the capturing rate of the mature predator, aðtÞ=a12 ðtÞ is the conversion rate of nutrients into the reproduction of the predator; Di ðtÞ is dispersion rate of prey species x; i ¼ 1; 2. mðtÞ is the half saturation rate of the predator. r1 ðtÞ; r2 ðtÞ; a11 ðtÞ; a12 ðtÞ; a22 ðtÞ; aðtÞ; mðtÞ; c1 ðtÞ and c2 ðtÞ are continuously positive periodic functions with period x. The model is derived under the following assumptions. (H1) the death rate of the immature predator is proportional to the existing immature population with a proportionality c1 ðtÞ > 0; c2 ðtÞ is the death rate of mature predators. The term Rt x1 ðt  sÞy2 ðt  sÞ  c ðsÞ ds aðt  sÞe ts 1 mðtÞy2 ðt  sÞ þ x1 ðt  sÞ represents the number of immature predators that were born at time t  s which still survive at time t and are transferred from the immature stage to the mature stage at time t. We refer to the article of Liu et al. [29]. It is assumed in (1.3) that immature individual predators do not feed on prey and do not have the ability to reproduce. The initial conditions for system (1.3) take the form of xi ðhÞ ¼ /i ðhÞ; /i ðhÞ P 0; /i ð0Þ > 0;

yi ðhÞ ¼ wi ðhÞ;

wi ðhÞ > 0; i ¼ 1; 2;

h 2 ½s; 0;

ð1:4Þ

where ð/1 ðhÞ; /2 ðhÞ; w1 ðhÞ; w2 ðhÞÞ 2 Cð½s; 0; R4þ0 Þ, the Banach space of continuous functions mapping the interval ½s; 0 into R4þ0 , where we define R4þ0 ¼ fðx1 ; x2 ; x3 ; x4 Þ : xi P 0; i ¼ 1; 2; 3; 4g and the interior of R4þ0 :

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R4þ ¼ fðx1 ; x2 ; x3 ; x4 Þ : xi > 0; i ¼ 1; 2; 3; 4g: For continuity of initial conditions, we require Z 0 R0 /1 ðsÞw2 ðsÞ  c ðuÞ du ds: y1 ð0Þ ¼ aðsÞe s 1 mðsÞw s 2 ðsÞ þ /1 ðsÞ

ð1:5Þ

We adopt the following notations throughout this paper. Z 1 x  f ¼ f ðtÞ dt; f L ¼ min jf ðtÞj; f M ¼ max jf ðtÞj; t2½0;x ½0;x x 0 where f is a continuous x-periodic function. The organization of this paper is as follows. In the next section, we first prove the positivity of solutions of system (1.3) with initial conditions (1.4) and (1.5). Sufficient conditions are derived for the uniform persistence and impermanence of system (1.3). In Section 3, by using Gaines and MawhinÕs continuation theorem of coincidence degree theory, we show the existence of positive x-periodic solutions of system (1.3) with initial conditions (1.4) and (1.5). Numerical simulations are presented to illustrate the feasibility of our main results. In Section 4, a brief discussion is given to conclude this work.

2. Uniform persistence and impermanence In this section, we will discuss the uniform persistence and impermanence of system (1.3) with initial conditions (1.4) and (1.5). Definition (i) System (1.3) is said to be uniformly persistent if there exists a compact region D Int R4þ such that every solution X ðtÞ of system (1.3) with initial conditions (1.4) and (1.5) eventually enters and remains in the region D. (ii) System (1.3) is said to be impermanent if there is a positive solution ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ of (1.3) satisfying minflim inf x1 ðtÞ; lim inf x2 ðtÞ; lim inf y1 ðtÞ; lim inf y2 ðtÞg ¼ 0: t!þ1

t!þ1

t!þ1

t!þ1

We first in the following prove the positivity of solutions of system (1.3) with initial conditions (1.4) and (1.5). Lemma 2.1. Solutions of system (1.3) with initial conditions (1.4) and (1.5) are positive for all t P 0.

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Proof. Let ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞ be a solution of system (1.3) with initial conditions (1.4) and (1.5). Let us first consider y2 ðtÞ for t 2 ½0; s. Noting that /1 ðhÞ P 0; w2 ðhÞ > 0 for h 2 ½s; 0, it follows from the fourth equation of system (1.3) that Rt /1 ðt  sÞw2 ðt  sÞ  c ðsÞ ds  c2 ðtÞy2 ðtÞ y_ 2 ðtÞ ¼ aðt  sÞe ts 1 mðtÞw2 ðt  sÞ þ /1 ðt  sÞ P  c2 ðtÞy2 ðtÞ:

ð2:1Þ

Therefore, a standard comparison argument shows Rt  c ðsÞ ds y2 ðtÞ P y2 ð0Þe 0 2 >0 for t 2 ½0; s: It follows from the first and second equations of system (1.3) that for t 2 ½0; s, x_ 1 ðtÞjx1 ¼0 ¼ D1 ðtÞx2 > 0

for

x2 > 0;

x_ 2 ðtÞjx2 ¼0 ¼ D2 ðtÞx1 > 0

for

x1 > 0:

Thus, we derive x1 ðtÞ > 0; x2 ðtÞ > 0 for t 2 ½0; s. In a similar way we treat the intervals ½s; 2s; . . . ; ½ns; ðn þ 1Þs; n 2 N . Thus, x1 ðtÞ > 0; x2 ðtÞ > 0 and y2 ðtÞ > 0 for all t P 0. By (1.5) and the third equation of system (1.3), one can derive Z t Rt x1 ðsÞy2 ðsÞ  c ðuÞ du ds: ð2:2Þ y1 ðtÞ ¼ aðsÞe s 1 mðsÞy 2 ðsÞ þ x1 ðsÞ ts Hence the positivity of y1 ðtÞ for t P 0 follows. This completes the proof.

h

Lemma 2.2. Let X ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ be any positive solution of system (1.3) with initial conditions (1.4) and (1.5). Then there exists a T1 > 0 such that xi ðtÞ 6 M;

yi ðtÞ 6 M ði ¼ 1; 2Þ

for

t P T1 ;

where 

M L A þ r1M þ DM 2 þ a =m M >M ¼ 4AaL11

L L A ¼ min c1 ; c2 : 

2



A þ r2M þ DM 1 þ 4AaL22

2 ;

ð2:3Þ

Proof. Suppose X ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ is any positive solution of system (1.3) with initial conditions (1.4) and (1.5). Define qðtÞ ¼ x1 ðtÞ þ x2 ðtÞ þ y1 ðtÞ þ y2 ðtÞ:

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Calculating the derivative of qðtÞ along positive solutions of (1.3), we derive
 aM L _ 6  AqðtÞ þ x1 ðtÞ A þ r1M þ DM qðtÞ þ  a x ðtÞ 2 11 1 mL  L þ x2 ðtÞ A þ r2M þ DM 1  a22 x2 ðtÞ   2 M L 2 A þ r1M þ DM A þ r2M þ DM 2 þ a =m 1 þ ; ð2:4Þ 6  AqðtÞ þ 4aL11 4aL22 where A is defined in (2.3). It follows from (2.4) that   2 M L 2 A þ r1M þ DM A þ r2M þ DM 2 þ a =m 1 lim sup qðtÞ 6 þ :¼ M  : 4AaL11 4AaL22 t!þ1

ð2:5Þ

Therefore, there exist a T1 > 0 and an M > M  such that if t P T1 , xi ðtÞ 6 M, yi ðtÞ 6 M, i ¼ 1; 2. This completes the proof. h Theorem 2.1. System (1.3) with initial conditions (1.4) and (1.5) is uniformly persistent provided that L (H2) r1L > aM 12 =m , M (H3) aL ec1 s > cM 2 .

Proof. Suppose X ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ is a solution of system (1.3) with initial conditions (1.4) and (1.5). It follows from the first and second equation of system (1.3) that
 a12 ðtÞ  a11 ðtÞx1 ðtÞ þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ; x_ 1 ðtÞ P x1 ðtÞ r1 ðtÞ  mðtÞ ð2:6Þ x_ 2 ðtÞ ¼ x2 ðtÞðr2 ðtÞ  a22 ðtÞx2 ðtÞÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ: Consider the following auxiliary system  L M u_ 1 ðtÞ ¼ u1 ðtÞ r1L  aM 12 =m  a11 u1 ðtÞ þ D1 ðtÞðu2 ðtÞ  u1 ðtÞÞ;  u_ 2 ðtÞ ¼ u2 ðtÞ r2L  aM 22 u2 ðtÞ þ D2 ðtÞðu1 ðtÞ  u2 ðtÞÞ:

ð2:7Þ

Let V11 ðtÞ ¼ minfu1 ðtÞ; u2 ðtÞg: Using a similar argument in the proof of Theorem 3.2 in [39], along solutions of (2.7) we derive  L  L r  aM r2L 12 =m lim inf V11 ðtÞ P min 1 ; :¼ m1 : M t!þ1 aM a 11 22

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A standard comparison argument shows that lim inf minfx1 ðtÞ; x2 ðtÞg P m1 : t!þ1

Therefore, there exist a T2 P T1 and a positive constant m1 < m1 such that if t P T2 , minfx1 ðtÞ; x2 ðtÞg P m1 :

ð2:8Þ

It follows from (2.8) and the fourth equation of system (1.3) that for t > T2 þ s Rt x1 ðt  sÞy2 ðt  sÞ  c ðsÞ ds y_ 2 ðtÞ ¼ aðt  sÞe ts 1 mðtÞy2 ðt  sÞ þ x1 ðt  sÞ m1 y2 ðt  sÞ M  cM ð2:9Þ  c2 ðtÞy2 ðtÞ P aL ec1 s M 2 y2 ðtÞ: m y2 ðt  sÞ þ m1 Consider the following auxiliary equation duðtÞ m1 uðt  sÞ M ¼ aL ec1 s M  cM 2 uðtÞ: dt m uðt  sÞ þ m1 By Theorem 9.1 in Chapter 4 of [22] we obtain   M m1 aL ec1 s  cM 2 lim uðtÞ ¼ : t!þ1 mM cM 2 By the comparison principle, we derive   M m1 aL ec1 s  cM 2 :¼ m2 : lim inf y2 ðtÞ P t!1 mM cM 2 Hence, there exist a T3 > T2 þ s and a positive constant m2 < m2 such that if t P T3 , y2 ðtÞ > m2 . It follows from (2.2) that for t P T3 þ s Z t Rt x1 ðsÞy2 ðsÞ  c ðuÞ du ds y1 ðtÞ ¼ aðsÞe s 1 mðsÞy2 ðsÞ þ x1 ðsÞ ts Z t m1 m2 M P aL ec1 ðtsÞ M ds m m2 þ m1 ts   aL m1 m2 M 1  ec1 s : ¼ M M c1 ðm m2 þ m1 Þ This completes the proof. L

h

M L M Theorem 2.2. If aM ec1 s < cL2 or aL12 =mM þ DL1 > r1M þ DM 2 þ c2 ; D2 > r2 þ M M D1 þ c2 , then system (1.3) is impermanent.

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831

L

Proof. We first assume aM ec1 s < cL2 . Let ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ be a positive solution of system (1.3) with initial conditions (1.4) and (1.5). Choose a positive L constant p > 1 such that paM ec1 s < cL2 . Thus, if y2 ðt þ hÞ < py2 ðtÞ for s 6 h 6 0, by the fourth equation of system (1.3) we derive 
  x1 ðt  sÞ L L y_ 2 ðtÞ < y2 ðtÞ paM ec1 s L  cL2 < y2 ðtÞ paM ec1 s  cL2 : pm y2 ðtÞ þ x1 ðt  sÞ By Theorem 4.2 in Chapter 5 of [16], we derive limt!1 y2 ðtÞ ¼ 0. In this case, the mature predator species will go to extinction. M L M M M If aL12 =mM þ DL1 > r1M þ DM 2 þ c2 ; D2 > r2 þ D1 þ c2 , then there exists b > 0 such that aL12 L M ¼ r1M þ DM 2  D 1 þ c2 : mM þ b 2 ð0Þ Let d ¼ x1 ð0Þþx < b. We now claim that y2 ð0Þ

x1 ðtÞ þ x2 ðtÞ
for all

t P 0;

and

lim x1 ðtÞ þ x2 ðtÞ ¼ 0:

t!þ1

Otherwise, there is a first time t1 such that x1 ðt1 Þ þ x2 ðt1 Þ ¼ b; y2 ðt1 Þ

x1 ðtÞ þ x2 ðtÞ
for all

t 2 ½0; t1 Þ:

Then for any t 2 ½0; t1 , we derive from (1.3) that 
d a12 ðtÞy2 ðtÞ ðx1 ðtÞ þ x2 ðtÞÞ ¼ x1 ðtÞ r1 ðtÞ  a11 ðtÞx1 ðtÞ  dt mðtÞy2 ðtÞ þ x1 ðtÞ þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ þ x2 ðtÞðr2 ðtÞ  a22 ðtÞx2 ðtÞÞ 
a12 ðtÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ 6 x1 ðtÞ r1 ðtÞ  mðtÞ þ x1 ðtÞ=y2 ðtÞ þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ þ r2 ðtÞx2 ðtÞ þ D2 ðtÞðx1 ðtÞ
 aL þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ  x2 ðtÞÞ 6 x1 ðtÞ r1M  M 12 m þb
M L þ r2 x2 ðtÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ 6 r1M þ DM 2  D1   aL12 L x1 ðtÞ þ r2M þ DM  M 1  D2 x2 ðtÞ m þb 6  cM 2 ðx1 ðtÞ þ x2 ðtÞÞ; which yields M

x1 ðtÞ þ x2 ðtÞ 6 ðx1 ð0Þ þ x2 ð0ÞÞec2 t :

ð2:10Þ

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On the other hand, it follows from the fourth equation of system (1.3) that for all t P 0 y_ 2 ðtÞ P  cM 2 y2 ðtÞ; which implies M

y2 ðtÞ P y2 ð0Þec2 t :

ð2:11Þ

Thus, we derive from (2.10) and (2.11) that x1 ðtÞ þ x2 ðtÞ x1 ð0Þ þ x2 ð0Þ 6 ¼d
for all

t 2 ½0; t1 ;

which leads to a contradiction. Therefore, we obtain M

x1 ðtÞ þ x2 ðtÞ 6 x1 ð0Þ þ x2 ð0Þec2 t

for all

t P 0:

We therefore obtain lim ðx1 ðtÞ þ x2 ðtÞÞ ¼ 0;

t!þ1

which yields lim x1 ðtÞ ¼ lim x2 ðtÞ ¼ 0:

t!þ1

t!þ1

h

The proof is complete.

M L M M M Theorem 2.3. If aL12 =mM þ DL1 > r1M þ DM 2 þ c2 ; D2 > r2 þ D1 þ c2 , then there is a positive solution ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ of system (1.3) satisfying

lim ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ ¼ ð0; 0; 0; 0Þ:

t!þ1

aL

M 12 Proof. Let b ¼ rM þDM D L þcM  m . Assume ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ is a solution 1

2

of system (1.3) with

1

2

x1 ð0Þþx2 ð0Þ y2 ð0Þ

< b. It follows from the proof of Theorem 2.2 that

lim x1 ðtÞ ¼ lim x2 ðtÞ ¼ 0:

t!þ1

t!þ1

Since y2 ðtÞ is positive and bounded, we have 0 6 lim inf y2 ðtÞ 6 lim sup y2 ðtÞ ¼ s < þ1: t!þ1

t!þ1

We now show lim y2 ðtÞ ¼ 0. Otherwise, we have s > 0. Since lim x2 ðtÞ ¼ 0, t!þ1

t!þ1

there is a t2 > s such that x2 ðtÞ <

mL cL2 s L

2aM ec1 s

for all

t P t2 :

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On the other hand, by the definition of s, there is a t3 > t2 þ s such that y2 ðt3 Þ >

s 2

y_ 2 ðt3 Þ > 0:

and

ð2:12Þ

It follows from (2.12) that R t3  c1 ðsÞ ds x1 ðt3  sÞy2 ðt3  sÞ  c2 ðt3 Þy2 ðt3 Þ > 0; aðt3  sÞe t3 s mðt3 Þy2 ðt3  sÞ þ x1 ðt3  sÞ which yields x1 ðt3  sÞ >

mL cL2 aM e

y ðt Þ > cL s 2 3 1

mL cL2 s L

2aM ec1 s

;

which leads to a contradiction. Hence, we must have s ¼ 0, i.e., lim y2 ðtÞ ¼ 0:

t!þ1

cL e

1 Hence, for 8 e > 0, there exists a T  > 0 such that if t > T  , jy2 ðtÞj < . cL s aM ð1e 1 Þ  It follows from (2.2) that for t > T þ s, Z t  Rt   x1 ðsÞy2 ðsÞ  c1 ðuÞ du  s ds jy1 ðtÞj ¼  aðsÞe mðsÞy2 ðsÞ þ x1 ðsÞ ts Z t Z t cL1 e L M cL1 ðtsÞ 6a e jy2 ðsÞj ds < ec1 ðtsÞ ds ¼ e: cL1 s 1e ts ts

We therefore obtain limt!þ1 y1 ðtÞ ¼ 0. The proof is complete.

h

3. Existence of positive periodic solutions In this section, using Gaines and MawhinÕs continuation theorem of coincidence degree theory, we show the existence of positive periodic solutions to system (1.3). For convenience, in the following we shall summarize a few concepts and results from [14] that will be basic for this section. Let X , Y be real Banach spaces, let L: DomL X ! Y be a linear mapping, and N : X ! Y be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dim KerL ¼ codim ImL < þ1 and ImL is closed in Y . If L is a Fredholm mapping of index zero and there exist continuous projectors P : X ! X, and Q:Y !Y such that ImP ¼ KerL, KerQ ¼ ImL ¼ ImðI  QÞ, then the restriction LP of L to DomL\ KerP : ðI  P ÞX ! ImL is invertible. Denote the inverse of LP by KP . If X is an  if open bounded subset of X , the mapping N will be called L-compact on X   QN ðXÞ is bounded and KP ðI  QÞN : X ! X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J : ImQ ! KerL.

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Lemma 3.1. Let X X be an open bounded set. Let L be a Fredholm mapping of  Assume index zero and N be L-compact on X. (a) for each k 2 ð0; 1Þ; x 2 oX \ DomL, Lx 6¼ kNx; (b) for each x 2 oX \ KerL, QNx 6¼ 0; (c) degfJQN ; X \ KerL; 0g 6¼ 0.  \ DomL. Then Lx ¼ Nx has at least one solution in X We are now able to state our result on the existence of positive periodic solutions of system (1.3). Theorem 3.1. Let (H2)–(H3) hold. Then system (1.3) with initial conditions (1.4) and (1.5) has at least one positive x-periodic solution. Proof. We first consider the following subsystem   8 a12 ðtÞy2 ðtÞ > _ x ðtÞ ¼ x ðtÞ r ðtÞ  a ðtÞx ðtÞ  þ D1 ðtÞðx2 ðtÞ  x1 ðtÞÞ; > 1 1 1 11 1 mðtÞy2 ðtÞþx1 ðtÞ > < x_ 2 ðtÞ ¼ x2 ðtÞðr2 ðtÞ  a22 ðtÞx2 ðtÞÞ þ D2 ðtÞðx1 ðtÞ  x2 ðtÞÞ; > Rt > >  c ðsÞ ds x1 ðtsÞy2 ðtsÞ : y_ 2 ðtÞ ¼ aðt  sÞe ts 1  c2 ðtÞy2 ðtÞ mðtÞy2 ðtsÞþx1 ðtsÞ ð3:1Þ with initial conditions x1 ðhÞ ¼ /1 ðhÞ; x2 ðhÞ ¼ /2 ðhÞ; y2 ðhÞ ¼ w2 ðhÞ; /1 ðhÞ P 0; /2 ðhÞ P 0; w2 ðhÞ > 0; h 2 ½s; 0; /1 ð0Þ > 0; /2 ð0Þ > 0:

ð3:2Þ

u1 ðtÞ ¼ ln½x1 ðtÞ;

ð3:3Þ

Let u2 ðtÞ ¼ ln½x2 ðtÞ;

u3 ðtÞ ¼ ln½y2 ðtÞ:

On substituting (3.3) into (3.1), we obtain du1 ðtÞ a12 ðtÞeu3 ðtÞ ¼ r1 ðtÞ  a11 ðtÞeu1 ðtÞ  þ D1 ðtÞðeu2 ðtÞu1 ðtÞ  1Þ; dt mðtÞeu3 ðtÞ þ eu1 ðtÞ du2 ðtÞ ¼ r2 ðtÞ  a22 ðtÞeu2 ðtÞ þ D2 ðtÞðeu1 ðtÞu2 ðtÞ  1Þ; dt Rt u1 ðtsÞþu3 ðtsÞu3 ðtÞ du3 ðtÞ  c ðsÞ ds e ¼ aðt  sÞe ts 1  c2 ðtÞ: dt mðtÞeu3 ðtsÞ þ eu1 ðtsÞ ð3:4Þ

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

835

It is easy to see that if system (3.4) has one x-periodic solution T T ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ , then z ðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; y2 ðtÞÞ ¼ ðexp½u1 ðtÞ; exp½u2 ðtÞ; T exp½u3 ðtÞÞ is a positive x-periodic solution of subsystem (3.1). Therefore, in the following it suffices to prove that system (3.4) admits at least one x-periodic solution. To apply Lemma 3.1 to (3.4), we first define X ¼ Y ¼ fðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT 2 CðR; R3 Þ : ui ðt þ xÞ ¼ ui ðtÞ; i ¼ 1; 2; 3g and T

kðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ k ¼

3 X i¼1

max jui ðtÞj;

t2½0;x

here j  j denotes the Euclidean norm. Then it is easy to see that both X and Y are Banach spaces with the norm k  k. Let
T du1 ðtÞ du2 ðtÞ du3 ðtÞ T ; ; L : DomL \ X ! X ; Lðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ ¼ ; dt dt dt T

where DomL ¼ fðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ 2 C 1 ðR; R3 Þg and N : X ! X , 2 3 a12 ðtÞeu3 ðtÞ 2 3 u1 ðtÞ u2 ðtÞu1 ðtÞ ðtÞ  a ðtÞe  þ D ðtÞðe  1Þ r 1 11 1 u ðtÞ u ðtÞ u1 mðtÞe 3 þe 1 6 7 u2 ðtÞ 7: r ðtÞ  a ðtÞe þ D2 ðtÞðeu1 ðtÞu2 ðtÞ  1Þ N 4 u2 5 ¼ 6 2 22 4 5 Rt u ðtsÞþu ðtsÞu ðtÞ  c ðsÞ ds 3 3 u3 e1 aðt  sÞe ts 1  c ðtÞ mðtÞeu3 ðtsÞ þeu1 ðtsÞ

2

Define 3 2 3 2 1 Rx 3 u ðtÞ dt u1 u1 x R0x 1 P 4 u2 5 ¼ Q4 u2 5 ¼ 4 x1 R0 u2 ðtÞ dt 5; x 1 u3 u3 u3 ðtÞ dt x 0 2

2

3 u1 4 u2 5 2 X ¼ Y : u3

It is easy to show that KerL ¼ fxjx 2 X ; x ¼ h; h 2 R3 g; Z x ImL ¼ fyjy 2 Y ; yðtÞ dt ¼ 0g is closed in Y 0

and dim KerL ¼ codim ImL ¼ 3 and P and Q are continuous projectors such that ImP ¼ KerL;

KerQ ¼ ImL ¼ ImðI  QÞ:

It follows that L is a Fredholm mapping of index zero. Furthermore, the inverse KP of LP exists and is given by KP : ImL ! DomL \ KerP ,

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KP ðyÞ ¼

Z

t 0

1 yðsÞ ds  x

Z

Z

x

t

yðsÞ ds dt: 0

0

Then QN : X ! Y and KP ðI  QÞN : X ! X are given by i 3 2 R xh a12 ðtÞeu3 ðtÞ 1 r1 ðtÞ  a11 ðtÞeu1 ðtÞ  mðtÞe þ D1 ðtÞðeu2 ðtÞu1 ðtÞ  1Þ dt u3 ðtÞ x 0 þeu1 ðtÞ 6 7 Rx 1 6 7 ½r ðtÞ  a22 ðtÞeu2 ðtÞ þ D2 ðtÞðeu1 ðtÞu2 ðtÞ  1Þdt QNx ¼ 6 x 0 2 7; Rt 4 5 R u ðtsÞþu ðtsÞu ðtÞ x  c1 ðsÞ ds e 1 3 3 1 aðt  sÞe ts  c2 ðtÞ dt x 0 mðtÞeu3 ðtsÞ þeu1 ðtsÞ Z Z 1 x t KP ðI  QÞNx ¼ NxðsÞ ds  NxðsÞ ds dt x 0 0
Z x 0 t 1  NxðsÞ ds:  x 2 0 Z

t

In order to apply Lemma 3.1, in the following we need to search for an appropriate open, bounded subset X. Corresponding to the operator equation Lx ¼ kNx; k 2 ð0; 1Þ, we have   du1 ðtÞ a12 ðtÞeu3 ðtÞ u1 ðtÞ u2 ðtÞu1 ðtÞ ¼ k r1 ðtÞ  a11 ðtÞe  þ D1 ðtÞðe  1Þ ; dt mðtÞeu3 ðtÞ þ eu1 ðtÞ   du2 ðtÞ ¼ k r2 ðtÞ  a22 ðtÞeu2 ðtÞ þ D2 ðtÞðeu1 ðtÞu2 ðtÞ  1Þ ; dt   Rt u1 ðtsÞþu3 ðtsÞu3 ðtÞ du3 ðtÞ  c ðsÞ ds e ¼ k aðt  sÞe ts 1  c ðtÞ : 2 dt mðtÞeu3 ðtsÞ þ eu1 ðtsÞ ð3:5Þ T

Suppose that ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ 2 X is a solution of (3.5) for a certain T k 2 ð0; 1Þ. Since ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ 2 X , there exist ni ; gi 2 ½0; x such that ui ðni Þ ¼ min ui ðtÞ; t2½0;x

ui ðgi Þ ¼ max ui ðtÞ; i ¼ 1; 2; 3: t2½0;x

ð3:6Þ

In the following we first borrow the technique used by Li and Kuang [27] to estimate the upper and lower bounds of u1 ðtÞ and u2 ðtÞ. Clearly, we have from (3.6) that u0i ðni Þ ¼ 0 and

u0i ðgi Þ ¼ 0;

i ¼ 1; 2:

ð3:7Þ

Corresponding to (3.7) we derive r1 ðn1 Þ  a11 ðn1 Þeu1 ðn1 Þ 

a12 ðn1 Þeu3 ðn1 Þ þ D1 ðn1 Þðeu2 ðn1 Þu1 ðn1 Þ  1Þ ¼ 0; mðn1 Þeu3 ðn1 Þ þ eu1 ðn1 Þ

r2 ðn2 Þ  a22 ðn2 Þeu2 ðn2 Þ þ D2 ðn2 Þðeu1 ðn2 Þu2 ðn2 Þ  1Þ ¼ 0 ð3:8Þ and

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

837

a12 ðg1 Þeu3 ðg1 Þ þ D1 ðg1 Þðeu2 ðg1 Þu1 ðg1 Þ  1Þ ¼ 0; mðg1 Þeu3 ðg1 Þ þ eu1 ðg1 Þ

r1 ðg1 Þ  a11 ðg1 Þeu1 ðg1 Þ 

r2 ðg2 Þ  a22 ðg2 Þeu2 ðg2 Þ þ D2 ðg2 Þðeu1 ðg2 Þu2 ðg2 Þ  1Þ ¼ 0: ð3:9Þ If u1 ðn1 Þ 6 u2 ðn2 Þ, then we have u1 ðn1 Þ 6 u2 ðn1 Þ. We therefore derive from the first equation of (3.8) that a11 ðn1 Þeu1 ðn1 Þ P r1 ðn1 Þ 

a12 ðn1 Þeu3 ðn1 Þ aM L 12 P r  ; 1 mðn1 Þeu3 ðn1 Þ þ eu1 ðn1 Þ mL

which yields u2 ðn2 Þ P u1 ðn1 Þ P ln

L r1L  aM 12 =m : M a11

ð3:10Þ

If u2 ðn2 Þ 6 u1 ðn1 Þ, then we have u2 ðn2 Þ 6 u1 ðn2 Þ. We derive from the second equation of (3.8) that a22 ðn2 Þeu2 ðn2 Þ P r2 ðn2 Þ P r2L ; which leads to u1 ðn1 Þ P u2 ðn2 Þ P ln

r2L : aM 22

ð3:11Þ

If u1 ðg1 Þ 6 u2 ðg2 Þ, we have u1 ðg2 Þ 6 u2 ðg2 Þ. It then follows from the second equation of system (3.9) that a22 ðg2 Þeu2 ðg2 Þ 6 r2 ðg2 Þ 6 r2M ; which yields u1 ðg1 Þ 6 u2 ðg2 Þ 6 ln

r2M : aL22

ð3:12Þ

If u2 ðg2 Þ 6 u1 ðg1 Þ, we have u2 ðg1 Þ 6 u1 ðg1 Þ. It then follows from the first equation of (3.9) that a11 ðg1 Þeu1 ðg1 Þ 6 r1 ðg1 Þ 6 r1M ; which leads to u2 ðg2 Þ 6 u1 ðg1 Þ 6 ln

r1M : aL11

We now derive from (3.10)–(3.13) that

ð3:13Þ

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R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

   L   r1L  aM r1M 12 =m    ; ln max ju1 ðtÞj < max  ln   t2½0;x aM aL 11

11

    r2L ;  ln   aM

22

    r2M ;  ln   aL

22

   :¼ R1  ð3:14Þ

and    L   r L  aM rM 12 =m   ;  ln 1L max ju2 ðtÞj < max  ln 1  M t2½0;x a a 11

11

    r2L ;  ln   aM

22

    r2M ;  ln   aL

22

   :¼ R1 :  ð3:15Þ

Next, we estimate the lower and upper bounds of u3 ðtÞ. Multiplying the third equation of (3.5) by eu3 ðtÞ and integrating over ½0; x gives Z

x

c2 ðtÞeu3 ðtÞ dt ¼

Z

0

x

aðt  sÞe

0 M cL1 s

a e 6 mL

Z



Rt ts

c1 ðsÞ ds

eu1 ðtsÞþu3 ðtsÞ dt mðtÞeu3 ðtsÞ þ eu1 ðtsÞ

x

eu1 ðtsÞ dt;

0

which, together with (3.14), implies L

u3 ðn3 Þ 6 ln

aM eR1 c1 s :¼ ln d1 : mL c2

Integrating (3.5) over the interval ½0; x we obtain Z x Z x Rt u1 ðtsÞþu3 ðtsÞu3 ðtÞ  c1 ðsÞ ds e ts aðt  sÞe dt ¼ c2 ðtÞ dt: mðtÞeu3 ðtsÞ þ eu1 ðtsÞ 0 0

ð3:16Þ

ð3:17Þ

It follows from (3.5) and (3.17) that  Z x Z x Rt u1 ðtsÞþu3 ðtsÞu3 ðtÞ  c1 ðsÞ ds e 0 ts ju3 ðtÞj dt < aðt  sÞe þ c2 ðtÞ dt mðtÞeu3 ðtsÞ þ eu1 ðtsÞ 0 0 ¼ 2c2 x: We therefore derive from (3.16) and (3.18) that Z x u3 ðtÞ 6 u3 ðn3 Þ þ ju03 ðtÞj dt 6 ln d1 þ 2c2 x:

ð3:18Þ

ð3:19Þ

0

Multiplying the third equation of (3.5) by eu3 ðtÞ and integrating over ½0; x again gives

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

Z

x

c2 ðtÞeu3 ðtÞ dt ¼

Z

0

x

aðt  sÞe

0

Z



Rt ts

c1 ðsÞ ds

839

eu1 ðtsÞþu3 ðtsÞ dt mðtÞeu3 ðtsÞ þ eu1 ðtsÞ

x

eu1 ðn1 Þþu3 ðtsÞ dt mM eu3 ðg3 Þ þ eu1 ðn1 Þ 0 Z x M aL ec1 s eu1 ðn1 Þ ¼ M u ðg Þ eu3 ðtsÞ dt m e 3 3 þ eu1 ðn1 Þ 0 Z x M aL ec1 s eu1 ðn1 Þ ¼ M u ðg Þ eu3 ðtÞ dt; m e 3 3 þ eu1 ðn1 Þ 0 M

P aL ec1 s

which yields  u3 ðg3 Þ P ln

 M u1 ðn1 Þ aL ec1 s  cM 2 e mM cM 2

:

It follows from (3.10), (3.11) and (3.20) that   M L r1L  aM r2L aL ec1 s  cM 12 =m 2 u3 ðg3 Þ P max ln ; ln :¼ d2 : þ ln aM aM mM cM 11 22 2 We derive from (3.18) and (3.21) that Z x u3 ðtÞ P u3 ðg3 Þ  ju03 ðtÞj dt P d2  2c2 x;

ð3:20Þ

ð3:21Þ

ð3:22Þ

0

which, together with (3.19), leads to max ju3 ðtÞj < maxfj ln d1 j þ 2c2 x; jd2 j þ 2c2 xg :¼ R2 :

t2½0;x

ð3:23Þ

Clearly, R1 and R2 in (3.14), (3.15) and (3.23) are independent of k. Denote M ¼ 2R1 þ R2 þ R0 , where R0 is taken sufficiently large such that each solution T ðu ; v ; w Þ of the system of algebraic equations Z ew x a12 ðtÞ u r1  a11 e  dt þ D1 ðevu  1Þ ¼ 0; x 0 mðtÞew þ eu r2  a22 ev þ D2 ðeuv  1Þ ¼ 0; Z R eu x aðt  sÞ  t c1 ðsÞ ds e ts dt  c2 ¼ 0 x 0 mðtÞew þ eu

ð3:24Þ

T

satisfies kðu ; v ; w Þ k ¼ ju j þ jv j þ jw j < M (if it exists) and the following 2R1 þ R2 < M; where

ð3:25Þ

840

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

(

    )  r1   r2  r1  am12 ;  ln  ; ¼ max j ln j;  ln a11 a11   a22    9 8   M < aM eR1 cL1 s   A aL ec1 s  c2 =     ; R2 ¼ max  ln ; ln ; : mM c2 mL c2   

R1

( A ¼ max

ð3:26Þ

)  r1  am12 r2 ; : a11 a22 T

T

We now take X ¼ fðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ 2 X : kðu1 ; u2 ; u3 Þ k < Mg. Thus the condition ðaÞ in Lemma 3.1 holds. When ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞT 2 oX \ KerL ¼ oX \ R3 , ðu1 ; u2 ; u3 ÞT is a constant vector in R3 with ju1 j þ ju2 j þ ju3 j ¼ M. If system (3.24) has at least one solution, then we have 2 3 u Rx a12 ðtÞ 2 3 3 r1  a11 eu1  ex3 0 mðtÞe dt þ D1 ðeu2 u1  1Þ u3 þeu1 0 u1 6 7 u u u 2 þ D ðe 1 2  1Þ 6 7 4 5 4 5 r  a e QN u2 ¼ 4 2 22 5 6¼ 0 : R2 t R u x  c1 ðsÞ ds aðtsÞ e1 0 u3 ts dt  c u u e 2

x

0

mðtÞe 3 þe

2

1

If system (3.24) does not have a solution, then we directly obtain T T QN ðu1 ; u2 ; u3 Þ 6¼ ð0; 0; 0Þ . This proves that condition (b) in Lemma 3.1 is satisfied. In order to prove that condition (c) in Lemma 3.1 holds, we define / : DomX  ½0; 1 ! X by 2 6 /ðu1 ; u2 ; u3 ; lÞ ¼ 4

eu1 x

Rx

2

0

3 r1  a11 eu1 7 r2  a22 eu2 5 Rt  c1 ðsÞ ds aðtsÞ ts e dt  c 2 mðtÞeu3 þeu1

6 D1 ðe 6 þ l6 4

u2 u1

 1Þ

u  ex3 u1 u2

D2 ðe

0

Rx 0

a12 ðtÞ mðtÞeu3 þeu1

 1Þ

3 dt 7 7 7; 5

T

where l 2 ½0; 1 is a parameter. When ðu1 ðtÞ; u2 ðtÞ; u3 ðtÞÞ 2 oX \ KerL ¼ T oX \ R3 , ðu1 ; u2 ; u3 Þ is a constant vector in R3 with ju1 j þ ju2 j þ ju3 j ¼ M. We now claim that when ðu1 ; u2 ; u3 ÞT 2 oX \ KerL, /ðu1 ; u2 ; u3 ; lÞ 6¼ 0. Otherwise, there is a constant vector ðu1 ; u2 ; u3 ÞT with ju1 j þ ju2 j þ ju3 j ¼ M satisfying /ðu1 ; u2 ; u3 ; lÞ ¼ 0, which yields

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

r1  a11 eu1  l

eu3 x

Z

x 0

841

a12 ðtÞ dt þ lD1 ðeu2 u1  1Þ ¼ 0; mðtÞeu3 þ eu1

r2  a22 eu2 þ lD2 ðeu1 u2  1Þ ¼ 0; Z Rt eu1 x aðt  sÞ  c1 ðsÞ ds ts e dt  c2 ¼ 0: x 0 mðtÞeu3 þ eu1 Using similar arguments in (3.14), (3.15) and (3.23), we can show that ju1 j < R1 ;

ju2 j < R1 ;

ju3 j < R2 ;

where R1 ; R2 are defined in (3.26). Clearly, we have ju1 j þ ju2 j þ ju3 j < 2R1 þ R2 < M; which leads to a contradiction. Using the property of topological degree and taking J ¼ I : ImQ ! T T KerL; ðu1 ; u2 ; u3 Þ ! ðu1 ; u2 ; u3 Þ , we have T

T

degðJQN ðu1 ; u2 ; u3 Þ ; X \ KerL; ð0; 0; 0Þ Þ T

¼ degð/ðu1 ; u2 ; u3 ; 1Þ; X \ KerL; ð0; 0; 0Þ Þ ¼ degð/ðu1 ; u2 ; u3 ; 0Þ; X \ KerL; ð0; 0; 0ÞT Þ T

Rt u1 Z x aðt  sÞ  c1 ðsÞ ds u1 u2 e ts r1  a11 e ; r2  a22 e ; e dt  c2 ; ¼ deg x 0 mðtÞeu3 þ eu1  T X \ KerL; ð0; 0; 0Þ :

A direct calculation shows T

T

degðJQN ðu1 ; u2 ; u3 Þ ; X \ KerL; ð0; 0; 0Þ Þ 8 9 Rt Z x  c1 ðsÞ ds < 1 = ts aðt  sÞmðtÞe ¼ sgn  a11 a22 e2u1 þu2 þu3 dt ¼ 1: 2 : x ; ðmðtÞeu3 þ eu1 Þ 0  is equicontinFinally, it is easy to show that the set fKP ðI  QÞNxjx 2 Xg uous and uniformly bounded. By using the Arzela-Ascoli Theorem, we see that KP ðI  QÞN : X ! X is compact. Consequently, N is L-compact. By now we have proved that X satisfies all the requirements in Lemma 3.1. Hence, (3.4) has at least one x-periodic solution. Accordingly, system (3.1) has at least one positive x-periodic solution. Let ðx1 ðtÞ; x2 ðtÞ; y2 ðtÞÞT be a positive x-periodic solution of system (3.1). Then it is easy to verify that Z t Rt x1 ðsÞy2 ðsÞ  c ðuÞ du y1 ðtÞ ¼ ds aðsÞe s 1 mðsÞy2 ðsÞ þ x1 ðsÞ ts

842

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846 T

is also x-periodic. Thus, ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ is a positive x-periodic solution of system (1.3) with initial conditions (1.4) and (1.5). This completes the proof. h Finally, we give two examples to illustrate the feasibility of our main results. Example 1. Consider the following predator–prey model with prey dispersal and stage structure for predator 8   2y2 ðtÞ > _ x > ðtÞ ¼ x ðtÞ 5 þ sin t  ð2  sin tÞx ðtÞ  þ 2ðx2 ðtÞ  x1 ðtÞÞ; 1 1 1 > 2y2 ðtÞþx1 ðtÞ > > > > < x_ 2 ðtÞ ¼ x2 ðtÞð3  sin t  ð2 þ cos tÞx2 ðtÞÞ þ x1 ðtÞ  x2 ðtÞ; ðtÞy2 ðtÞ y_ 1 ðtÞ ¼ ð3 þ sin tÞ 2yx21ðtÞþx  0:1y1 ðtÞ 1 ðtÞ > > 0:1s x1 ðtsÞy2 ðtsÞ > ð3 þ sinðt  sÞÞe ; > > 2y2 ðtsÞþx1 ðtsÞ > > : y_ 2 ðtÞ ¼ ð3 þ sinðt  sÞÞe0:1s x1 ðtsÞy2 ðtsÞ  0:5y2 ðtÞ: 2y2 ðtsÞþx1 ðtsÞ

ð3:27Þ It is easy to verify that the coefficients in system (3.27) satisfy all assumptions in Theorems 2.1 and 3.1 if s < 10 ln 4. Thus, if s < 10 ln 4, by Theorem 2.1 we see that system (3.27) is uniformly persistent; by Theorem 3.1 we know that system (3.27) has at least one positive 2p-periodic solution. If s > 10 ln 8, by Theorem 2.2 we see that the mature predator will go to extinction. Numerical integration of system (3.27) can be carried out using standard algorithms. We used the ‘‘dde23’’ package in MATLAB. Numerical simulations also confirm the facts above (see, Figs. 1 and 2). 8 x 1 x 2 y1 y2

7 6

solution

5 4 3 2 1 0

0

10

20

30

40

50

60

70

80

90

100

time t

Fig. 1. The periodic solution found by numerical integration of system (3.27) with s ¼ 0:5.

R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846

843

5 x1 x2

4.5

y1 y2

4

solution

3.5 3 2.5 2 1.5 1 0.5 0 0

20

40

60

80

100

120

140

160

180

200

time t

Fig. 2. The extinction of mature predator found by numerical integration of system (3.27) with s ¼ 21.

Example 2. Consider another predator–prey model with stage structure 8   3y2 ðtÞ > _ > ðtÞ ¼ x ðtÞ 0:5 þ 0:1 sin t  2x ðtÞ  x þ x2 ðtÞ  x1 ðtÞ; 1 1 1 > y2 ðtÞþx1 ðtÞ > > > > < x_ 2 ðtÞ ¼ x2 ðtÞð0:5  0:1 sin t  ð2 þ cos tÞx2 ðtÞÞ þ 2ðx1 ðtÞ  x2 ðtÞÞ; ðtÞy2 ðtÞ y_ 1 ðtÞ ¼ ð4 þ sin tÞ yx21ðtÞþx  0:1y1 ðtÞ 1 ðtÞ > > 0:1s x1 ðtsÞy2 ðtsÞ > ð4 þ sinðt  sÞÞe ; > y2 ðtsÞþx1 ðtsÞ > > > : y_ 2 ðtÞ ¼ ð4 þ sinðt  sÞÞe0:1s x1 ðtsÞy2 ðtsÞ  0:3y2 ðtÞ: y2 ðtsÞþx1 ðtsÞ

ð3:28Þ It is easy to verify that the coefficients of system (3.28) satisfy assumptions of Theorem 2.3. By Theorem 2.3 we see that there is a positive solution of system (3.28) satisfying lim ðx1 ðtÞ; x2 ðtÞ; y1 ðtÞ; y2 ðtÞÞ ¼ ð0; 0; 0; 0Þ:

t!þ1

Numerical simulation also confirms this fact (see Fig. 3).

4. Discussion In this paper, based on the works of Aiello and Freedman [1] and Wang et al. [36], we have incorporated the periodicity of the environment, prey dispersal and stage structure for predator into a ratio-dependent predator–prey model. By some comparison arguments, a set of easily verifiable sufficient

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R. Xu et al. / Appl. Math. Comput. 159 (2004) 823–846 1.2 x1 x 2 y1 y2

1

solution

0.8

0.6

0.4

0.2

0

-0.2 0

5

10

15

20

25

30

35

40

45

50

time t

Fig. 3. The impermanence found by numerical integration of system (3.28) with s ¼ 0:5.

conditions are established for the uniform persistence and impermanence of system (1.3). By using Gaines and MawhinÕs continuation theorem of coincidence degree theory, we have shown the existence of positive periodic solutions to system (1.3). By Theorem 2.1, we see that system (1.3) is uniformly persistent if the intrinsic growth rate of prey, the half saturation rate of predator and the conversion rate of prey biomass into juvenile predators are high, and the capturing rate of the adult predator, the death rates of both immature and mature predator are low and the time delay due to maturity for juvenile predator is small. By Theorem 2.2, we see that the mature predator will go to extinction if the conversion rate of prey biomass into juvenile predators is low, and the death rates of both immature and mature predator are high and the time delay due to maturity is large. By Theorem 3.1 we see that if system (1.3) is uniformly persistent, then system (1.3) will have at least one positive periodic solution. We note that the dispersal rates are harmless for the uniform persistence and the existence of positive periodic solutions of system (1.3). We would like to mention here that an interesting but challenging problem associated with the study of system (1.3) should be the uniqueness and global stability of positive periodic solutions. We leave this for future work. References [1] W.G. Aiello, H.I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci. 101 (1990) 139–153.

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