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J. Math. Anal. Appl. 295 (2004) 15–39 www.elsevier.com/locate/jmaa

Dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response

Meng Fan a,1 and Yang Kuang b,?,2

a School of Mathematics and Statistics and Key Laboratory for Vegetation Ecology of the Education Ministry,

Northeast Normal University, 5268 Renmin Street, Changchun, Jilin 130024, PR China

b Department of Mathematics and Statistics, Arizona state University, Tempe, AZ 85287-1804, USA

Received 18 July 2003 Available online 11 May 2004 Submitted by H.R. Thieme

Abstract In this paper, we systematically study the dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response. The explorations involve the permanence, extinction, global asymptotic stability (general nonautonomous case); the existence, uniqueness and stability of a positive (almost) periodic solution and a boundary (almost) periodic solution for the periodic (almost periodic) case. The paper ends with some interesting numerical simulations that complement our analytical ?ndings. ? 2004 Elsevier Inc. All rights reserved.

Keywords: Beddington–DeAngelis predator–prey system; Permanence; Extinction; Periodic solution; Almost periodic solution

* Corresponding author.

E-mail addresses: mfan@nenu.edu.cn (M. Fan), kuang@asu.edu (Y. Kuang).

1 Supported by the NNSF of PR China (No. 10171010 and 10201005), the Key Project on Science and

Technology of the Education Ministry of PR China (No. Key 01061) and the Science Foundation of Jilin Province of PR China for Distinguished Young Scholars. 2 Work is partially supported by NSF grant DMS-0077790. 0022-247X/$ – see front matter ? 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2004.02.038

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1. Introduction Understanding the dynamical relationship between predator and prey is a central goal in ecology, and one signi?cant component of the predator–prey relationship is the predator’s rate of feeding upon prey, i.e., the so-called predator’s functional response. Functional response is a double rate: it is the average number of prey killed per individual predator per unit of time. In general, the functional response can be classi?ed into two types: preydependent and predator-dependent. Prey dependent means that the functional response is only a function of the prey’s density, while predator-dependent means that the functional response is a function of both the prey’s and the predator’s densities. Functional response equations that are strictly prey-dependent, such as the Holling family, are predominant in the literature. For example, since 1959, Holling’s prey-dependent type II functional response has served as the basis for a very large literature on predator–prey theory [35]. The traditional Kolmogorov type predator–prey model with Holling’s type II functional response x = rx 1 ? x K ? cy x , m+x y = ?dy + fy x , m+x (1.1)

and its various generalized forms have received great attention from both theoretical and mathematical biologists, and have been well studied. However, the prey-dependent functional responses fail to model the interference among predators, and have been facing challenges from the biology and physiology communities (see, e.g., [1,3–6,23]). Some biologists have argued that in many situations, especially when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey–predator model should be predator-dependent. There is much signi?cant evidence to suggest that predator dependence in the functional response occurs quite frequently in laboratory and natural systems [2,5,17,28,30,35]. Given that large numbers of experiments and observations suggest that predators do indeed interfere with one another’s activities so as to result in competition effects and that prey alters its behavior under increased predator-threat, the models with predator-dependent functional response stand as reasonable alternatives to the models with prey-dependent functional response [35]. Starting from this argument and the traditional prey-dependent-only model (1.1), Arditi and Ginzburg [3] ?rst proposed the following ratio-dependent predator–prey model: x = x(a ? bx) ? cxy , my + x y = ?dy + f xy , my + x (1.2)

which incorporates mutual interference by predators. Note that (1.2) is a result of replacing the prey-dependent functional response x/(m + x) in (1.1) by a ratio-dependent one (x/y)/(m + x/y). For detailed justi?cations of (1.2) and its merits versus (1.1), see [3,33]. As for the mathematical aspect of (1.2), since [3], (1.2) has been studied by many authors and seen great progress (e.g., autonomous case [21,24,29,31,32,38]; nonautonomous continuous case [20,37]; nonautonomous discrete time case [19]). Many authors have observed that

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

17

the ratio-dependent models can exhibit much richer, more complicated and more reasonable or acceptable dynamics, but it has somewhat singular behavior at low densities which has been the source of controversy and been criticized on other grounds. Recently, two of the most vocal opponents in this debate collaborated on a very useful summary that clearly delineated the areas of agreement and disagreement [1]. Surprisingly, Abrams and Ginzburg agreed on many more issues than they disagreed on. More importantly, it seems clear that predator density should have a strong effect on predator’s functional response in nature. Skalski and Gilliam [35] p resent statistical evidence from 19 predator–prey systems that three predator-dependent functional responses (Beddington–DeAngelis, Crowley–Martin, and Hassell–Varley) can provide better description of predator feeding over a range of predator–prey abundances. In some cases, the Beddington–DeAngelis type preformed even better. Their most salient ?nding is that predator dependence in the functional response is a nearly ubiquitous property of the published data sets. Although the predator-dependent models that they considered ?t those data reasonably well, no single functional response best describes all of the data sets. Theoretical studies have shown that the dynamics of models with predator-dependent functional responses can differ considerably from those with prey-dependent functional responses. The predator–prey system with the Beddington–DeAngelis functional response x =x r ? x K ? αxy , a + bx + cy y = ?dy + βxy a + bx + cy (1.3)

was originally proposed by Beddington [8] and DeAngelis et al. [16], independently. The Beddington–DeAngelis is similar to the well-known Holling type 2 functional response but has an extra term cy in the denominator modelling mutual interference among predators and has some of the same qualitative features as the ratio-dependent form but avoids some of the singular behaviors of ratio-dependent models at low densities which have been the source of controversy. Mathematically, we may think of both the traditional prey-dependent and ratio-dependent models as limiting cases (c = 0 for the former and a = 0 for the latter) of the general Beddington–DeAngelis type predator–prey system. The Beddington–DeAngelis functional response can be derived mechanistically via considerations of time utilization or spatial limits on predation [8,15,34,36]. System (1.3) and the analogous systems with diffusion in a constant environment have received much attention in the literatures [10–12,15,25,26]. The studies [11,25,26] present a complete classi?cation of the global dynamics of (1.3): if there is no positive steady state, then the boundary steady state (K, 0) is globally attracting; if (1.3) admits a positive steady state but it is unstable, then there is a unique limit cycle; otherwise, the positive steady state is the global attractor. The scenario is similar to the traditional Kolmogorov type predator–prey model with Michaelis–Menten (or Holling type II) functional response. Although much progress has been seen in the study of predator–prey models with the Beddington–DeAngelis functional response, such models are not well studied yet in the sense that all the known results are for models with constant environment. The assumption that the environment is constant is rarely the case in real life. Most natural environments are physically highly variable, and in response, birth rates, death rates, and other vital rates of populations, vary greatly in time. Yet the dominant focus in theoretical models of popula-

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

tion and community dynamics has not been on how populations change in response to the physical environment, but on how populations depend on their own population densities or the population densities of other organisms. Although it has long been recognized that temporal ?uctuations in the physical environment are a major driver of population ?uctuations, there has been scant theoretical attention to predict the characteristic of the resultant population ?uctuations. In some cases, ignoring variation in the physical environmental is seen as the ?rst step, or as adequate for mean tendencies. Many researchers appreciate that it is time to for the next step in which the role of physical environmental variation is a focus in theoretical models. Theoretical evidence to date suggests that many population and community patterns represent intricate interactions between biology and variation in the physical environment (see Chesson [13] and other papers in the same issue). When the environmental ?uctuation is taken into account, a model must be nonautonomous, and hence, of course, more dif?cult to analyze in general. But, in doing so, one can and should also take advantage of the properties of those varying parameters. For example, one may assume the parameters are periodic or almost periodic for seasonal reasons. Theoretical studies have shown that the dynamics of models with predator-dependent functional responses can differ considerably from those with prey-dependent functional responses. Theoretical and statistical studies suggest that the predator-dependent models deserve more attention in the literature than they have received to date [11,35]. In this paper, we shall explore the dynamics of the nonautonomous, spatially homogeneous and continuous time predator–prey system with the Beddington–DeAngelis functional response in a more general form x = x a(t) ? b(t)x ? c(t)xy , α(t) + β(t)x + γ (t)y f (t)xy . y = ?d(t)y + α(t) + β(t)x + γ (t)y

(1.4)

This paper is organized as follows. In Section 2, we examine the dynamics of the general nonautonomous case of (1.4) and establish suf?cient criteria for the boundedness, permanence, predator extinction, and globally asymptotic stability. In Section 3, we will explore the existence, uniqueness, and the global asymptotic stability of positive periodic solutions and boundary periodic solutions of (1.4) when the parameters in (1.4) are periodic. In Section 4, we attack the almost periodic case of (1.3). The paper ends with some interesting numerical simulations that complement our analytical ?ndings.

2. General nonautonomous case In this section, we shall explore the dynamics of the nonautonomous predator–prey system (1.4) and present some results including the positive invariance, ultimate boundedness, permanence, predator extinction and the globally asymptotic stability. In the following discussion, we always assume that a(t), b(t), c(t), d(t), m(t), f (t), β(t) and γ (t) are continuous and bounded above and below by positive constants; α(t) is continuous and nonnegative.

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

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2 := {(x, y) ∈ R 2 | x Let R+ 0, y 0}. For a bounded continuous function g(t) on R , we use the following notations: g u := supt ∈R g(t), g l := inft ∈R g(t).

Lemma 2.1. Both the nonnegative and positive cones of R 2 are positively invariant for (1.4). In the remainder of this paper, for biological reasons, we only consider the solutions (x(t), y(t)) with positive initial values, i.e., x(t0 ) > 0 and y(t0 ) > 0. De?nition 2.1. The solution set of system (1.4) is said to be ultimately bounded if there exist B > 0 such that for every solution (x(t), y(t)) of (1.4), there exists T > 0 such that (x(t), y(t)) B for all t t0 + T , where B is independent of particular solution while T may depend on the solution. De?nition 2.2. System (1.4) is said to be permanent if there exist positive constants δ, ? with 0 < δ < ? such that min lim inf x(t), lim inf y(t)

t →+∞ t →+∞

δ,

max lim sup x(t), lim sup y(t)

t →+∞ t →+∞

?

for all solutions of (1.4) with positive initial values. System (1.4) is said to be nonpersistent if there is a positive solution (x(t), y(t)) of (1.4) satisfying min lim inf x(t), lim inf y(t) = 0.

t →+∞ t →+∞

Similarly to Theorem 2.4 in [20], by a standard comparison argument, one can easily prove that

u u Theorem 2.1. If a l γ l > cu and (f l ? d u β u )mε 1 > d α , then the set Γε , de?ned by

Γε := (x, y) ∈ R 2 | mε 1 au + ε, bl

x

ε M1 , mε 2

y

ε , M2

(2.1)

is positively invariant with respect to system (1.4), where

ε (f u ? d l β l )M1 , dlγ l u u (f l ? d u β u )mε a l γ l ? cu ε 1?d α := ? ε, m := , mε 1 2 buγ l d uγ u and ε 0 is suf?ciently small such that mε 1 > 0. ε M1 := ε M2 :=

(2.2)

Remark 2.1. Theorem 2.1 provides conditions for the so-called practical persistence. This concept is studied systematically in [9,14]. Corollary 2.1. Let (x(t), y(t)) be a solution of (1.4) with x(t0 ) > 0 and y(t0 ) > 0. Then 0 . If a l γ l > c u , then lim inf m0 we have lim supt →+∞ x(t) M1 t →+∞ x(t) 1 ; moreover, if ε l u u u u (f ? d β )m1 > d α , then lim inf y(t)

t →+∞

m0 2,

lim sup y(t)

t →+∞

0 M2 .

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

u u Theorem 2.2. If a l γ l > cu and (f l ? d u β u )mε 1 > d α , then system (1.4) is permanent and the set Γε with ε > 0 de?ned by (2.1) is an ultimately bounded region of system (1.4).

Theorem 2.3. If f u < d l β l , then limt →+∞ y(t) = 0. By the predator equation in (1.4), the conclusion is obvious. Remark 2.2. One can easily see that when α(t) ≡ 0, the above discussions also remain valid. System (1.4) reduces to the nonautonomous ratio-dependent predator–prey system investigated by Fan et al. [20] and Lemmas 2.1–2.4 and Theorems 2.1–2.4 reduce to the corresponding results in [20]. De?nition 2.3. A bounded nonnegative solution (x(t), ? y(t)) ? of (1.4) is said to be globally asymptotically stable (or globally attractive) if any other solution (x(t), y(t))T of (1.4) ? | + |y(t) ? y(t) ? |) = 0 . with positive initial values satis?es limt →+∞ (|x(t) ? x(t) Remark 2.3. One can easily show that if system (1.4) has a bounded positive solution which is globally asymptotically stable, then system (1.4) is globally asymptotically stable, i.e., the above property holds for any two solutions with positive initial values, and vice versa. Lemma 2.2 [7]. Let h be a real number and f be a nonnegative function de?ned on [h, +∞) such that f is integrable on [h, +∞) and is uniformly continuous on [h, +∞), then limt →+∞ f (t) = 0. Theorem 2.4. Let (x ? (t), y ? (t)) be a bounded positive solution of system (1.4). If a l γ l > cu u u and (f l ? d u β u )mε 1 > d α and one of the following conditions holds: ? ε (β(t )c(t )+f (t )γ (t ))M2 α(t )f (t ) ? b(t) ? ?(t,m > 0, ε ,mε ) ? ?(t,mε ,M ε )

1 2 1 2

?

? ? b(t) ? ? ? ? b(t) ? ?

f (t )γ (t )mε 1 ε ?(t,mε 1 ,M2 )

?

α(t )c(t ) ε ?(t,mε 1 ,m2 )

?

ε β(t )c(t )M1 ε ,mε ) ?(t,M1 2

> 0, > 0,

α(t )f (t )+f (t )γ (t )y ?(t ) ε ?(t,mε 1 ,m2 )

?

ε β(t )c(t )M2 ε ε ?(t,m1 ,m2 )

f (t )γ (t )x ?(t ) ε ,M ε ) ?(t,M1 2

?

α(t )c(t ) ε ?(t,mε 1 ,m2 )

?

ε β(t )c(t )M1 ε ,mε ) ?(t,M1 2

> 0, > 0,

(α(t )f (t )+β(t )c(t )y ?(t )) ε ?(t,mε 1 ,m2 )

?

ε f (t )γ (t )M2 ε) ?(t,mε ,M 1 2

? ? b(t) ? ?

f (t )γ (t )mε 1 ε ?(t,mε 1 ,M2 )

?

α(t )f (t )+β(t )c(t )y ?(t )+f (t )γ (t )y ?(t ) ε ?(t,mε 1 ,m2 ) ? f (t )γ (t )x (t ) α(t )c(t )+β(t )c(t )x ?(t ) ε ,M ε ) ε ?(t,M1 ?(t,mε 2 1 ,m2 )

(α(t )c(t )+β(t )c(t )x ?(t )) ε ?(t,mε 1 ,m2 )

> 0, > 0, (2.3)

?

> 0,

ε where mε i , Mi , i = 1, 2, are de?ned in (2.2) and

? t, x(t), y(t) = α(t) + β(t)x ? (t) + γ (t)y ? (t) α(t) + β(t)x(t) + γ (t)y(t) . Then (x ? (t), y ? (t)) is globally asymptotically stable.

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

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Proof. Let (x(t), y(t))T be any solution of (1.4) with positive initial value. Since Γε is an ultimately bounded region of (1.4), there exists T1 > 0 such that (x(t), y(t)) ∈ Γε and (x ? (t), y ? (t)) ∈ Γε for all t t0 + T1 . Consider a Liapunov function de?ned by V (t) = ln x(t) ? ln x ? (t) + ln y(t) ? ln y ? (t) , D + V (t) t t0 . (2.4) A direct calculation of the right derivative produces V (t) ?b(t) x(t) ? x ? (t) + of V (t) along the solutions of (1.4)

α(t)c(t)|y ? (t) ? y(t)| ?(t, x(t), y(t)) ? (t) ? x ? (t)y(t)) β(t)c(t)(x(t)y + sgn x(t) ? x ? (t) ?(t, x(t), y(t)) α(t)f (t)|x(t) ? x ? (t)| + ?(t, x(t), y(t)) f (t)γ (t)(x(t)y ? (t) ? x ? (t)y(t)) . + sgn y(t) ? y ? (t) ?(t, x(t), y(t))

Note that there are two terms containing x(t)y ? (t) ? x ? (t)y(t) in the right-hand side of the above inequality and x(t)y ? (t) ? x ? (t)y(t) = x(t) y ? (t) ? y(t) + y(t) x(t) ? x ? (t) = y ? (t) x(t) ? x ? (t) + x ? (t) y ? (t) ? y(t) . That is to say, each x(t)y ? (t) ? x ? (t)y(t) has two different expressions. In order to determine the sign of V (t), we have four cases to consider. For simplicity, we prefer to carry out detailed discussion for just one of the four cases since the others are similar, V (t) α(t)f (t)|x(t) ? x ? (t)| α(t)c(t)|y(t) ? y ? (t)| + ?(t, x(t), y(t)) ?(t, x(t), y(t)) ? β(t)c(t)x(t)|y(t) ? y (t)| β(t)c(t)y(t)|x(t) ? x ? (t)| + + ?(t, x(t), y(t)) ?(t, x(t), y(t)) f (t)γ (t)x(t)|y(t) ? y ? (t)| f (t)γ (t)y(t)|x(t) ? x ? (t)| + ? ?(t, x(t), y(t)) ?(t, x(t), y(t)) α(t)f (t) | x(t) ? x ? (t)| α(t)c(t)|y(t) ? y ? (t)| + ?b(t) x(t) ? x ? (t) + ε ε ?(t, mε ?(t, mε 1 , m2 ) 1 , m2 ) ε ε β(t)c(t)M1 |y(t) ? y ? (t)| β(t)c(t)M2 |x(t) ? x ? (t)| + + ε , mε ) ε ?(t, M1 ?(t, mε 2 1 , M2 ) ε |x(t) ? x ? (t)| ? f (t)γ (t)M2 f (t)γ (t)mε 1 |y(t) ? y (t)| + ? ε ε ε ?(t, m1 , M2 ) ?(t, mε 1 , M2 ) ε | (β(t)c(t) + f (t)γ (t))M2 α(t)f (t) = ? b(t) ? ? x(t) ? x ? (t) ε) ε, Mε) ?(t, mε , m ?(t, m 1 2 1 2 ε f (t)γ (t)mε β(t)c(t)M1 α(t)c(t) 1 ? ? ? y(t) ? y ? (t) . ε ε ε ε ε ?(t, mε , M ) ?(t, m , m ) ?(t, M , m ) 1 2 1 2 1 2 ?b(t) x(t) ? x ? (t) +

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

From (2.3) it follows that there exists a positive constant ? > 0 such that D + V (t) ?? x(t) ? x ? (t) + y(t) ? y ? (t) , t t0 + T1 . (2.5) Integrating on both sides of (2.5) from t0 + T1 to t produces

t

V (t) + ?

t0 +T1

x(s) ? x ? (s) + y(s) ? y ? (s) ds

V (t0 + T1 ) < +∞,

t Then

t

t0 + T1 .

x(s) ? x ? (s) + y(s) ? y ? (s) ds

t0 +T1

??1 V (t0 + T1 ) < +∞,

t

t0 + T1 ,

and hence, |x(t) ? x ? (t)| + |y(t) ? y ? (t)| ∈ L1 ([t0 + T1 , +∞)). The boundedness of x ? (t) and y ? (t) and the ultimate boundedness of x(t) and y(t) imply that x(t), y(t), x ? (t) and y ? (t) all have bounded derivatives for t t0 + T1 (from the equations satis?ed by them). Then it follows that |x(t) ? x ? (t)|+|y(t) ? y ? (t)| is uniformly continuous on [t0 + T1 , +∞). By Lemma 2.2, we have limt →+∞ (|x(t) ? x ? (t)| + |y(t) ? y ? (t)|) = 0. The proof is completed. 2 Remark 2.4. If α(t) ≡ 0, then Theorem 2.4 reduces to Theorem 2.15 and Corollary 2.16 in [20] for the nonautonomous ratio-dependent predator–prey system. Remark 2.5. The conditions in Theorem 2.4 seem a little bit sophisticated and depend on the positive solution (x ? (t), y ? (t)). But they can be easily satis?ed provided that b(t) and f (t) are appropriately large. In fact, we can replace those conditions by some more easily veri?able but stronger ones, which are independent of (x ? (t), y ? (t)). For example, the conditions in the ?rst group of (2.3) can be replaced by b(t) ? β(t)c(t) 2f (t) ? > 0, α(t) α(t)γ (t) f (t)γ (t)mε 2c(t) 1 ε ε 2 ? α(t) > 0, (α(t) + β(t)M1 + γ (t)M2 )

which are much more easily veri?able but a little bit stronger.

3. Periodic case In this section, we will con?ne ourselves to the case when the parameters in system (1.4) are periodic of some common period. The assumption of periodicity of the parameters is a way of incorporating the periodicity of the environment. The periodic oscillation of the parameters seems reasonable in view of seasonal factors, e.g., mating habits, availability of food, weather conditions, harvesting and hunting, etc. A very basic and important problem in the study of a population growth model with a periodic environment is the global existence and stability of positive periodic solution, which plays a similar role as a globally

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

23

stable equilibrium does in an autonomous model. Thus, it is reasonable to seek conditions under which the resulting periodic nonautonomous system would have a positive periodic solution that is globally asymptotically stable. In the section, we will always assume that the parameters in system (1.4) are ω-periodic in t and will study the existence and stability of a positive periodic solutions of system (1.4).

u u Theorem 3.1. If a l γ l > cu and (f l ? d u β u )mε 1 > d α , then system (1.4) has at least one positive periodic solution of period ω, say (x ? (t), y ? (t)), which lies in Γε .

Proof. De?ne a shift operator, which is also known as a Poincare mapping σ : R 2 → R 2 by σ ((x0 , y0 )) = x t0 + ω, t0 , (x0 , y0 ) , y t0 + ω, t0 , (x0 , y0 ) , (x0 , y0 ) ∈ R 2 ,

where (x(t, t0 , (x0 , y0 )), y(t, t0 , (x0 , y0 ))) denotes the solution of (1.4) through the point (t0 , (x0 , y0 )). Theorem 2.1 tells us that the set Γε de?ned by (2.1) is positive invariant with respect to system (1.4), and hence, the operator σ de?ned above maps Γε into itself, i.e., σ (Γε ) ? Γε . Since the solution of (1.4) is continuous with respect to the initial value, the operator σ is continuous. It is not dif?cult to show that Γε is a bounded, closed, convex set in R 2 . By Brouwer ?xed point theorem, σ has at least one ?xed point in Γε , i.e., there exists (x ? , y ? ) ∈ Γε such that (x ? , y ? ) = (x(ω, t0 , (x ? , y ? )), y(ω, t0 , (x ? , y ? ))). Therefore, there exists at least one positive periodic solution, say (x ? (t), y ? (t)), and the invariance of Γε assures that (x ? (t), y ? (t)) ∈ Γε . The proof is complete. 2 The conditions in Theorem 3.1 are given in terms of supremum and in?mum of the parameters. Next, we will employ an alternative approach, that is, a continuation theorem in coincidence degree theory, to establish some criteria for the same problem but in terms of the averages of the related parameters over an interval of the common period. To this end, we ?rst introduce the continuation theorem in the coincidence degree which will come into play later borrowing notations from [22]. Let X, Z be normed vector spaces, L : Dom L ? X → Z be a linear mapping, N : X → Z be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z . If L is a Fredholm mapping of index zero and there exist continuous projections P : X → X and Q : Z → Z such that Im P = Ker L, Im L = Ker Q = Im(I ? Q), it follows that L| Dom L ∩ Ker P : (I ? P )X → Im L is invertible. We denote the inverse of that map by KP . If ? is an open bounded subset of X , the mapping N will be called L-compact ? if QN(?) ? is bounded and KP (I ? Q)N : ? ? → X is compact. Since Im Q is isoon ? morphic to Ker L, there exists an isomorphism J : Im Q → Ker L. The following lemma is from Gains and Mawhin [22]. Lemma 3.1 (Continuation theorem). Let L be a Fredholm mapping of index zero and N ? . Suppose be L-compact on ? (a) For each λ ∈ (0, 1), every solution x of Lx = λNx is such that x ∈ / ?? ;

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

(b) QNx = 0 for each x ∈ ?? ∩ Ker L and the Brouwer degree deg{J QN, ? ∩ Ker L, 0} = 0. ?. Then the operator equation Lx = Nx has at least one solution lying in Dom L ∩ ? For a continuous and periodic function g(t) with period ω, denote by g ? the average of ω g(t) over an interval of length ω, i.e., g ? := (1/ω) 0 g(t) dt. Theorem 3.2. Assume that a ? > (c/γ ) and ? u) a (f? ? dβ ?? c γ ? u > 0. ? ?1 exp{?aω ? } ? dα b

Then system (1.4) has at least one positive ω periodic solution. Proof. Making the change of variables x(t) = exp{u(t)}, y(t) = exp{v(t)}, then system (1.4) is reformulated as u (t) = a(t) ? b(t) exp u(t) ? c(t) exp{v(t)} , α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)} f (t) exp{u(t)} v (t) = ?d(t) + . α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)}

(3.1)

Let X = Z = (u, v)T ∈ C(R, R 2 ) | u(t + ω) = u(t), v(t + ω) = v(t) , (u, v) = max u(t) + max v(t) ,

t ∈[0,ω] t ∈[0,ω]

(u, v) ∈ X (or Z).

Then X, Z are both Banach spaces when they are endowed with the above norm · . Let a(t) ? b(t) exp{u(t)} ? u N1 (t) N = = v N2 (t) ?d(t) + L Then Ker L = (u, v) ∈ X | (u, v) = (h1 , h2 ) ∈ R 2 ,

ω ω c(t ) exp{v(t )} α(t )+β(t ) exp{u(t )}+γ (t ) exp{v(t )} f (t ) exp{u(t )} α(t )+β(t ) exp{u(t )}+γ (t ) exp{v(t )} 1 ω ω 0 u(t) dt 1 ω ω 0 v(t) dt

,

u u = v v

,

P

u u =Q = v v

,

u ∈ X. v

Im L = (u, v) ∈ Z

0

u(t)dt = 0,

0

v(t) dt = 0 ,

and dim Ker L = 2 = codim Im L. Since Im L is closed in Z , L is a Fredholm mapping of index zero. It is easy to show that P , Q are continuous projections such that Im P = Ker L,

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

25

Im L = Ker Q = Im(I ? Q). Furthermore, the generalized inverse (to L) KP : Im L → Dom L ∩ Ker P exists and is given by KP u = v

t 0 t 0

u(s) ds ? v(s) ds ?

1 ω t ω 0 0 1 ω t ω 0 0

u(s) ds dt v(s) ds dt

.

Obviously, QN and KP (I ? Q)N are continuous. It is trivial to show that N is L-compact ? with any open bounded set ? ? X . on ? Now we reach the position to search for an appropriate open, bounded subset ? for the application of the continuation theorem. Corresponding to the operator equation Lx = λNx , λ ∈ (0, 1), we have c(t) exp{v(t)} , α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)} f (t) exp{u(t)} v (t) = λ ?d(t) + . α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)} u (t) = λ a(t) ? b(t) exp{u(t)} ?

(3.2)

Suppose that (u(t), v(t)) ∈ X is an arbitrary solution of system (3.2) for a certain λ ∈ (0, 1). Integrating on both sides of (3.2) over the interval [0, ω], we obtain

ω

aω ? =

0 ω

b(t) exp{u(t)} +

c(t) exp{v(t)} dt, α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)} (3.3)

? = dω

0

f (t) exp{u(t)} dt. α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)}

It follows from (3.2) and (3.3) that

ω ω ω

u (t) dt

0

λ

0

a(t) dt +

0

b(t) exp u(t) dt

+

c(t) exp{v(t)} α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)}

ω ω

< 2aω, ?

ω

v (t) dt

0

λ

0

d(t) dt +

0

f (t) exp{u(t)} dt α(t) + β(t) exp{u(t)} + γ (t) exp{v(t)} (3.4)

? < 2dω. Since (u(t), v(t)) ∈ X , there exist ξi , ηi ∈ [0, ω], i = 1, 2, such that u(ξ1 ) = min u(t),

t ∈[0,ω]

u(η1 ) = max u(t),

t ∈[0,ω]

v(ξ2 ) = min v(t),

t ∈[0,ω]

v(η2 ) = max v(t).

t ∈[0,ω]

(3.5)

26

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

From the ?rst equation of (3.3) and (3.5), we obtain

ω

aω ?

0

? exp u(ξ1 ) , b(t) exp u(ξ1 ) dt = bω ? } := l1 , and hence ln{a/ ? b u (t) dt

0 ω

which reduces to u(ξ1 )

ω

u(t)

u(ξ1 ) +

ln

a ? + 2aω ? := H1 . ? b

(3.6)

On the other hand, from the ?rst equation of (3.3) and (3.5), we also have aω ?

0

b(t) exp u(η1 ) dt +

c(t) dt = γ (t)

c ? exp u(η1 ) dt. ω + bω γ

Then

ω

u(t)

u(η1 ) ?

0

u (t) dt

ln

a ? ? (c/γ ) ? 2aω ? := H2 , ? b max{|H1 |, |H2 |} := B1 . From the ? exp{2aω f (t)(a/ ? b) ? } ? exp{2aω β l (a/ ? b) ? } + γ l exp{v(ξ2 )}

which, together with (3.6), leads to maxt ∈[0,ω] |u(t)| second equation of (3.3) and (3.5), we obtain

ω

? dω

0

f (t) exp{u(t)} dt l β exp{u(t)} + γ l exp{v(t)}

ω

ω

dt.

0

Then v(t) v(ξ2 ) +

0

v (t) dt

ln

? l )a (f? ? dβ ? exp{2aω ? } ? := H3 . + 2dω l ? dγ ? b

(3.7)

The second equation of (3.3) also produces

ω

? dω

0 ω

f (t) exp{u(t)} dt α u + β u exp{u(t)} + γ u exp{v(η2 )}}

? ?(c/γ ) f (t) a exp{?2aω ? } ? b

0

αu

? ?(c/γ ) + βu a ? b ω

dt.

exp{?2aω ? } + γ u exp{v(η2 )}

It follows that v(t) v(η2 ) ?

0

v (t) dt

ln

? ?(c/γ ) ? u) a ? u (f? ? dβ exp{?2aω ? } ? dα ? b

? u dγ

? := H4 , ? 2dω

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

27

which, together with (3.7), leads to maxt ∈[0,ω] |v(t)| max{|H3 |, |H4 |} := B2 . Clearly, B1 and B2 are independent of λ. Take B = B1 + B2 + B3 where B3 > 0 is taken suf?ciently large such that B3 > |l1 | + |L1 | + |l2 | + |L2 |. Now let us consider the algebraic equations ? exp{u} ? 1 a ? ?b ω ?+ ?d 1 ω

ω ω

0

?c(t) exp{v } dt = 0, α(t) + β(t) exp{u} + γ (t) exp{v } (3.8)

0

f (t) exp{u} dt = 0 α(t) + β(t) exp{u} + γ (t) exp{v }

for (u, v) ∈ R 2 , where ? ∈ [0, 1] is a parameter. By carrying out similar arguments as above, one can easily show that any solution (u? , v ? ) of (3.8) with ? ∈ [0, 1] satis?es l1 u? L1 , l2 v? L2 . (3.9)

Now we de?ne ? = {(u, v)T ∈ X | (u, v) < B }, then it is clear that ? veri?es the requirement (a) of Lemma 3.1. When (u, v) ∈ ?? ∩ Ker L = ?? ∩ R 2 , (u, v) is a constant vector in R 2 with (u, v) = |u| + |v | = B . Then from (3.9) and the de?nition of B , one has c(t ) exp{v } ? exp{u} ? 1 ω a ? ?b ω 0 α(t )+β(t ) exp{u}+γ (t ) exp{v } dt u 0 QN = = , f (t ) exp{u} 1 ω v 0 ? ?d + ω 0 α(t )+β(t ) exp{u}+γ (t ) exp{v } dt that is, the ?rst part of (b) of Lemma 3.1 is valid. In order to compute the Brouwer degree, let us consider the homotopy H? (u, v)T = ?QN (u, v)T + (1 ? ?)G (u, v)T , where G (u, v)T = ? exp{u} a ? ?b ?? d

f (t ) exp{u} 1 ω ω 0 α(t )+β(t ) exp{u}+γ (t ) exp{v }

? ∈ [0 , 1 ],

dt

.

From (3.9), it follows that 0 ∈ / H? (?? ∩ Ker L) for ? ∈ [0, 1]. In addition, one can easily show that the algebraic equation G((u, v)T ) = 0 has a unique solution in R 2 . Note that J = I since Im Q = Ker L, by the invariance property of homotopy, direct calculation produces deg(J QN, ? ∩ Ker L, 0) = deg(QN, ? ∩ Ker L, 0) = deg(G, ? ∩ Ker L, 0) = 0, where deg(· , · , ·) is the Brouwer degree. By now we have proved that ? veri?es all require? , i.e., (3.1) has ments of Lemma 3.1, then Lx = Nx has at least one solution in Dom L ∩ ? ? , say (u? (t), v ? (t))T . Set x ? (t) = exp{u? (t)}, at least one ω periodic solution in Dom L ∩ ? y ? (t) = exp{v ? (t)}, then (x ? (t), y ? (t))T is an ω periodic solution of system (1.4) with strictly positive components. The proof is complete. 2

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

Remark 3.1. It is not dif?cult to show that a l γ l > cu implies a ? > (c/γ ) and (f l ? ε u u u u u ? 1 ? ? ? ? u > 0 if ε is chosen ? ? (c/γ ) )b exp{?aω ? } ? dα d β )m1 > d α implies (f ? dβ )(a appropriately, i.e., a l γ l ? cu a l γ l ? cu 1 ? exp{?aω ? } <ε< . u l b γ buγ l In this sense, Theorem 3.2 is better than Theorem 3.1. By Remark 3.1 and Theorem 2.4, one can easily reach the following claim. Corollary 3.1. Let all the assumptions in Theorem 2.4 hold. Then system (1.4) has a unique positive ω periodic solution, which is globally asymptotically stable. Remark 3.2. If in (1.4) α(t) ≡ 0, then (1.4) reduces to the nonautonomous ratio dependent predator–prey system and Theorems 3.1, 3.2 and Corollary 3.1 are the corresponding theorems in [20]. If all the parameters in system (1.4) are positive constants, then (1.4) is the system considered in [11,25], and the assumptions in Theorems 3.1 and 3.2 both reduce to aγ > c, (f ? dβ)(a ? c/γ )b?1 exp{?aω} ? dα > 0, which ensure the autonomous version of (1.4) has a unique positive equilibrium E ? = (x ? , y ? ), where x ? and y ? are positive and satisfy a ? bx ? ? cy ? = 0, α + βx ? + γ y ? ?d + f x? = 0. α + βx ? + γ y ?

The assumption (2.7) in Theorem 2.4 guarantees that E ? is globally asymptotically stable. That is to say, when the parameters in (1.4) reduce to positive constants, the positive periodic solution claimed above, degenerates to a trivial positive periodic solution, i.e., the positive equilibrium E ? = (x ? , y ? ). Now we go ahead with exploring the boundary dynamics of (1.4), that is to establish suf?cient criteria for the existence and global stability of the boundary ω-periodic solution. Theorem 3.3. System (1.4) always has a boundary ω-periodic solution (x? (t), 0), where

ω t +ω t ?1

x? (t) = exp

0

a(s) ds ? 1

t

b(s) exp ?

s

a(τ ) dτ ds

.

(3.10)

Moreover, (i) if d(t) ? c(t) f (t) ? > 0 for t ∈ [0, ω], α(t) β(t)

then (x? (t), 0) is globally asymptotically stable, i.e., (x? (t), 0) attracts all the solutions of (1.4) with positive initial values;

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

29

(ii) if d(t) > f (t) β(t) for t ∈ [0, ω], x? (t), y(t) 0.

then (x? (t), 0) attracts all solution (x(t), y(t)) of (1.4) with x(t)

Proof. One can easily show that (x? (t), 0) is a solution of (1.4) and x? (t + ω) = x? (t), i.e., (x? (t), 0) is a periodic solution of (1.4). Let (x(t), y(t)) be any solution of (1.4) with x(t0 ) > 0 and y(t0 ) > 0. In order to show that (x? (t), 0) is globally asymptotically stable, consider the Liapunov function de?ned by V (t) = ln x(t) ? ln x? (t) + y(t), t t0 . Calculating the upper right derivative of V (t) along the solution of (1.4) produces D + V (t) ?b(t) x(t) ? x ? (t) + c(t)y(t) α(t) + β(t)x(t) + γ (t)y(t) f (t)x(t)y(t) ? d(t)y(t) + α(t) + β(t)x(t) + γ (t)y(t) c(t) f (t) ? ?b(t) x(t) ? x? (t) ? d(t) ? y(t) α(t) β(t) ??1 x(t) ? x ? (t) + y(t) , t t0 , c(t) f (t) ? α(t) β(t)

where ?1 = min b l , min If x(t) x? (t), then c(t)y(t) α(t) + β(t)x(t) + γ (t)y(t) f (t)x(t)y(t) ? d(t)y(t) + α(t) + β(t)x(t) + γ (t)y(t) f (t) ?b(t) x(t) ? x? (t) ? d(t) ? y(t) β(t) ??2 x(t) ? x? (t) + y(t) , t t0 , f (t) β(t)

t ∈[0,ω]

d(t) ?

> 0.

D + V (t) = ?b(t) x(t) ? x? (t) ?

where ?2 = min b l , min

t ∈[0,ω]

d(t) ?

> 0.

The rest of the proof is exactly the same as those carried out in Theorem 2.4, the details are omitted here. 2 Remark 3.3. If the parameters in (1.4) are positive constants, then the boundary periodic solution (x? (t), 0) reduces to the boundary equilibrium (a/b, 0) of the autonomous version of (1.4).

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

Remark 3.4. In fact, the ?rst component x? (t) of the boundary periodic solution of (1.4) is the ω-periodic solution of the logistic equation x(t) ˙ = x(t)(a(t) ? b(t)x(t)). As such, it is globally asymptotically stable, which is proved by Fan and Wang [18] under weaker assumptions on a(t) and b(t). Next, we present a necessary condition for the existence of positive ω-periodic solution. Theorem 3.4. If system (1.4) admits a positive ω-periodic solution (x ? (t), y ? (t)), then ? < (f?/β). d The conclusion directly follows from the predator equation in (1.2) and the periodicity of y ? (t).

4. Almost periodic case As we all know, the predator–prey interactions in the real world are affected by many factors and undergo all kinds of perturbation, among which some are periodic. When the periods of the periodic perturbations are rationally dependent, the system sustains periodic perturbations while if the periods are rationally independent, the effect on the predator– prey system caused by the periodic perturbations are not periodic but quasi periodic or more generally almost periodic. In this sense, it is more appropriate to assume that the parameters in the model system are almost periodic in the time t . In this section, we devote ourselves to the existence, uniqueness and stability of positive almost periodic solutions of (1.4) under the assumption that a(t), b(t), c(t), d(t), f (t), α(t), β(t), γ (t) are almost periodic functions in t . In addition to the assumptions in Section 2, it is clear that Theorems 2.1–2.4 remain valid for system (1.4). Let x(t) = exp{x(t) ? }, y(t) = exp{y(t) ? }. Then (1.4) becomes x ? (t) = a(t) ? b(t) exp x(t) ? ? c(t) exp{y(t) ? } , α(t) + β(t) exp{x(t) ? } + γ (t) exp{y(t) ? } f (t) exp{x(t) ? } . y ? (t) = ?d(t) + α(t) + β(t) exp{x(t) ? } + γ (t) exp{y(t) ? }

(4.1)

By Theorems 2.2–2.4, it is not dif?cult to show that

u u Theorem 4.1. If a l γ l > cu and (f l ? d u β u )mε 1 > d α , then the set

Γε? := (x, y)T ∈ R 2 | ln mε 1

x

ε ln M1 , ln mε 2

y

ε ln M2

ε is the positively invariant and ultimately bounded region of system (4.1), where mε i , Mi , i = 1, 2, are de?ned in (2.2).

Following the clues and discussion in the proof of Theorem 4.3 in [20], one can easily reach the following theorem. For simplicity, the details are omitted here.

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

31

u u Theorem 4.2. Assume that a l γ l > cu , (f l ? d u β u )mε 1 > d α , and one of the conditions ? ε (β(t )c(t )+f (t )γ (t ))M2 α(t )f (t ) ? b(t) ? ?(t,m > 0, ε ,mε ) ? ?(t,mε ,M ε )

1 2 1 2

?

? ? b(t) ? ? ? ? b(t) ? ?

f (t )γ (t )mε 1 ε ?(t,mε 1 ,M2 )

?

α(t )c(t ) ε ?(t,mε 1 ,m2 )

?

ε β(t )c(t )M1 ε ,mε ) ?(t,M1 2

> 0, > 0,

ε α(t )f (t )+f (t )γ (t )M2 ε) ?(t,mε ,m 1 2

?

ε β(t )c(t )M2 ε) ?(t,mε ,m 1 2

f (t )γ (t )mε 1 ε ,M ε ) ?(t,M1 2

?

α(t )c(t ) ε ?(t,mε 1 ,m2 )

?

ε β(t )c(t )M1 ε ,mε ) ?(t,M1 2

> 0, > 0,

ε) (α(t )f (t )+β(t )c(t )M2 ε ?(t,mε 1 ,m2 )

?

ε f (t )γ (t )M2 ε ?(t,mε 1 ,M2 )

? ? b(t) ? ?

ε) (α(t )c(t )+β(t )c(t )M1 ε) ?(t,mε ,m 1 2 ε+f (t )γ (t )M ε α(t )f (t )+β(t )c(t )M2 2 ε ?(t,m1 ,mε 2) ε f (t )γ (t )mε α(t )c(t )+β(t )c(t )M1 1 ε ,M ε ) ε ,mε ) ?(t,M1 ?(t,m 1 2 2

f (t )γ (t )mε 1 ε ?(t,mε 1 ,M2 )

?

> 0, > 0, (4.2)

?

> 0,

ε holds, where mε i , Mi , i = 1, 2, are de?ned in (2.2) and

? t, x(t), y(t) = α(t) + β(t)x ? (t) + γ (t)y ? (t) α(t) + β(t)x(t) + γ (t)y(t) . Then system (1.4) has a unique positive almost periodic solution which is uniformly asymptotically stable in Γ0 and is globally asymptotically stable. Now we turn to attack the “boundary dynamics" of (1.4) in the almost periodic case, i.e., the existence and stability of the boundary almost periodic solution (x ? (t), 0). First, we introduce a result on the existence of almost periodic solution to the logistic equation due to Jiang [27]. Lemma 4.1. Assume that a(t) and b(t) are almost periodic in t and that inf b(t) > 0, 1 lim t →+∞ t

t

t ∈R +

a(τ ) dτ > 0.

0

Then the logistic equation x(t) ˙ = x(t)(a(t) ? b(t)x(t)) admits a unique almost periodic uniformly asymptotic stable solution x (t), where

t t ?1

x (t) =

?∞

exp ?

s

a(τ ) dτ b(s) ds

.

(4.3)

Theorem 4.3. System (1.4) always has a boundary almost periodic solution (x (t), 0), where x (t) is de?ned by (4.3). Moreover, (i) if

t ∈R +

inf d(t) ?

c(t) f (t) ? > 0, α(t) β(t)

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

then (x (t), 0) is globally asymptotically stable, i.e., (x (t), 0) attracts all the solutions of (1.4) with positive initial values; (ii) if

t ∈R +

inf d(t) ?

f (t) , β(t) x (t), y(t) 0.

then (x (t), 0) attracts all solution (x(t), y(t)) of (1.4) with x(t)

5. Discussion In this paper, we made a systematic effort toward analyzing the global dynamics of a nonautonomous predator–prey system with the Beddington–DeAngelis functional response. The novel aspect of the system is the incorporation of environmental ?uctuations due to seasonal periodic and almost periodic changes. In this section, we will further discuss the dynamics of (1.4) based on our extensive numerical simulations. A natural question is: what are the necessary and suf?cient conditions for the existence and global stability of the positive (almost) periodic solution? In this paper, we have established some suf?cient criteria (Theorems 3.1 and 3.2) for the existence of positive periodic solutions, but we would like to point out that the conditions there are just suf?cient ones. See, for example, Fig. 1(a). One can easily show that the conditions in Theorems 3.1 and 3.2 failed. However, our numerical simulation (Fig. 1(a)) shows that (1.4) admits a positive 1-periodic solution, which is globally asymptotically stable. That is to say, in Theorems 3.1 and 3.2, there is room for improvement. ? < (f/β) is necessary for the existence of a positive ω-periodic Theorem 3.4 states that d solution. The numerical simulations strongly support this. For example, let the parameters ? (f/β). Our numerical simulations show that all the solutions of (1.4) in (1.4) vary and d tend to a globally asymptotically stable boundary periodic solution. However, the numer? < (f/β) is not suf?cient for the existence of positive ical simulations also indicate that d ? < (f/β) and a ω-periodic solution. See, for example, Fig. 1(b), where d ? > (c/β), and a solution of (1.4) tends to a globally asymptotically stable boundary 1-periodic solution (x? (t), 0). Theorem 3.3 provides suf?cient criteria for the existence of a globally asymptotically stable boundary ω-periodic solution. However, the criteria also have room for further improvement since, for d(t) f (t)/β(t), the numerical simulations indicate that system (1.4) may admit a globally asymptotically stable boundary periodic solution (x? (t), 0), see, for example, Fig. 1(b) (in which d(t) < f (t)/β(t)) and Fig. 1(c) (in which d(t) = f (t)/β(t)). In addition, we would like to present a numerical simulation to show an interesting phenomena: the global attractor of the nonautonomous predator–prey system (1.4) can be a boundary equilibrium, see Fig. 1(d). In fact, under appropriate assumptions, system (1.4) can also have an positive steady state as its global attractor. Now we turn to another more interesting question: how does the dynamics of the autonomous version of (1.4) evolve under the periodic or almost periodic perturbation when system (1.4) admits a limit cycle? Usually, the periodic (almost periodic) predator–prey system (1.4) can be viewed as a periodic (almost periodic) perturbation of its autonomous

M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

33

Fig. 1. (a) A solution of (1.4) with a(t) = 3 + 2 sin(2π t), b(t) = 0.2(2 + sin(2π t)), c(t) = 5 + cos(2π t), d(t) = 2 + sin(2π t), f (t) = 0.8(5 + cos(2π t)), α(t) = 5 + sin(2π t), β(t) = 1 + 0.5 sin(2π t), γ (t) = 1 and initial conditions x(0) = 2, y(0) = 4. The solution tends to a globally asymptotically stable positive 1-periodic solution (x ? (t), y ? (t)). (b) A solution of (1.4) with a(t) = 3 + sin(2π t), b(t) = 7 + 0.5 sin(2π t), c(t) = 10 + cos(2π t), d(t) = 1.2 + cos(2π t), f (t) = 10 + cos(2π t), α(t) = 10 + 0.1 sin(2π t), β(t) = 1, γ (t) = 5 and initial conditions x(0) = 4, y(0) = 2. The solution tends to a globally asymptotically stable boundary 1-periodic solution (x? (t), 0). (c) A solution of (1.4) with a(t) = 3 + cos(2π t), b(t) = 0.2(2 + 0.5 cos (2π t)), c(t) = 2 + sin(2π t), d(t) = f (t) = 0.8(2 + sin(2π t)), α(t) = 1 + sin(2π t), β(t) = γ (t) = 1 and initial conditions x(0) = 4, y(0) = 2. The solution tends to a globally asymptotically stable boundary 1-periodic solution (x? (t), 0). (d) A solution of (1.4) with a(t) = 2(3 + 2 sin(2π t)), b(t) = 0.5(3 + 2 sin(2π t)), c(t) = 1 + 0.2 cos(2π t), d(t) = 2.5 + sin(2π t), f (t) = 1 + 0.2 cos (2π t), α(t) = 5 + sin(2π t), β(t) = 1 + 0.2 cos (2π t), γ (t) = 1 and initial conditions x(0) = 1, y(0) = 5. The solution tends to a globally asymptotically stable trivial boundary periodic solution (x? (t), 0), i.e., the boundary equilibrium (4, 0).

version, so it is very interesting to know how the dynamics of its autonomous version evolve under periodic or almost periodic perturbation. Cantrell and Cosner [11] has proved that the autonomous version of (1.4) posses a limit cycle if the positive equilibrium is unstable while there is no limit cycle when the positive equilibrium is locally asymptotically stable since the locally asymptotic stability is equivalent to the globally asymptotic stability as shown by Hwang [25]. There are three natural scenarios. Let T be the period of the limit cycle and ω be the period of the periodic external perturbations. If T = ω, then limit

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M. Fan, Y. Kuang / J. Math. Anal. Appl. 295 (2004) 15–39

Fig. 2. The dynamics of the limit cycle evolves under various periodic perturbations. (a) the limit cycle; (b) subharmonic periodic solution with period close to that of the limit cycle; (c) subharmonic periodic solution with double period of the limit cycle; (d) quasi (almost) periodic solution.

cycle may evolve into a positive harmonic periodic solution of period ω; if T and ω are not equal but rationally dependent, the limit cycle maybe evolve into a positive subharmonic or harmonic periodic solution with the least common multiple of T and ω as its period; if T and ω are rationally independent, the limit cycle maybe evolve to a quasi periodic solution or an almost periodic solution. Our numerical simulations strongly support these claims (Figs. 2–4). As an example, let us consider the following perturbed predator–prey system with the Beddington–DeAngelis functional response x = x(t) 1 ? x(t) ? c(t, ?)x(t)y(t) , 1 + β(t, ?)x(t) + 0.0001y(t) f (t, ?)x(t)y(t) y = ?1.5y(t) + , 1 + β(t, ?)x(t) + 0.0001y(t)

(5.1)

where ? is a parameter. By Lemma 3.2 in [11], one can easily show that system (5.1) with c(t, ?) = 20, f (t, ?) = 18, β(t, ?) = 2, i.e., the autonomous version of (5.1), has a limit . cycle. The numerical simulations show that the period of the limit cycle is T = 11.

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√ Fig. 3. Under periodic perturbation with period (2/ 3), the limit cycle undergoes harmonic periodic solution to quasi periodic or almost periodic solution.

Now we assume that (5.1) subjects to periodic perturbation. First, let c(t, ?) = 20 1 + ? sin(2πt) , β(t, ?) = 2 + ? sin(2πt), that is to say, the period of the periodic perturbation is 1. By intuition, system (1.4) should admit a globally asymptotically stable subharmonic periodic solution, however, the numerical simulations do not completely support the intuition. For ? = 0.3, system (5.1) admits . a globally asymptotically stable subharmonic periodic solution with T = 11, whose period is different from that of the external periodic perturbation but the same as that of the limit cycle; when ? = 0.6 system (5.1) admits a globally asymptotically stable subharmonic periodic solution with double period of the limit cycle; when ? = 0.9, (5.1) has a globally asymptotically stable almost periodic solution (see Fig. 2). √ √ Let√ c(t, ?) = 20(1 + ? sin( 3πt)), f (t, ?) = 18(1 + ? sin( 3πt)), β(t, ?) = 2 + ? sin( 3πt), the period ω of the periodic perturbation is different from that of the limit cycle and they are rationally independent. Numerical simulations show that, for small ?, for example, ? = 0.2, system (5.1) admits a globally asymptotically stable harmonic posif (t, ?) = 18 1 + ? sin(2πt) ,

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Fig. 4. The limit cycle of (5.1) evolves under almost periodic perturbation.

tive periodic solution, but when ? creases, (5.1) has a globally asymptotically stable quasi periodic or almost periodic solution (Fig. 3). Now we are at the position to talk about the evolution of the limit cycle of (1.3) under almost periodic perturbation. In system (5.1), if √ √ c(t, ?) = 20 1 + ? sin( 13πt) , f (t, ?) = 18 1 + ? sin( 13πt) , √ β(t, ?) = 2 + ? sin( 17πt), then when ? = 0 (5.1) admits a limit cycle. When ? increases, the limit cycle evolves into an almost periodic solution (Fig. 4). In general, the omega limit set of a positive trajectory of system (1.4) in the phase plane occupies an annular region (Fig. 5). We fail to perform the numerical simulation showing that (1.4) can admit a harmonic periodic solution when the limit cycle undergoes periodic perturbation with period ω = T since it is almost impossible to determine the accurate period T of the limit cycle. In this paper, we have analytically shown that, under periodic and almost periodic perturbation the boundary equilibrium (a/b, 0) evolves into the boundary ω-periodic solution

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Fig. 5. The phase plane diagram of a chaotic shaped trajectory of Eq. (1.4) with periodic ((a) the parameters are same as those in Fig. 4(d)) and almost periodic coef?cients ((b) the parameters are same as those in Fig. 2(d)).

and the boundary almost periodic solution (x? (t), 0), respectively, while the interior positive equilibrium (x ? , y ? ) evolve into the positive ω-periodic solution and the positive almost periodic solution (x ? (t), y ? (t)), respectively. It is obvious that the trivial equilibrium (0, 0) (unstable) remains itself. Our numerical simulation studies strongly con?rm the analytical results thus derived. For simplicity, we omit the details of such numerical simulations. Combining the analytical analysis and the numerical simulations, we can conclude that the global attractor of the nonautonomous predator–prey system with the Beddington– DeAngelis functional response, in the periodic case, can be positive periodic solution, boundary periodic solution, harmonic periodic solution, subharmonic periodic solution and quasi or almost periodic solution while, in the almost periodic case, can be positive almost periodic solution or boundary almost periodic solution (see Table 1 for details). At the end of this paper, we would like to point out that it is more interesting but more challenging to investigate whether the Beddington–DeAngelis predator–prey system can admit chaotic behavior under periodic or almost periodic perturbations, especially by theoretical analyses. Although we fail to catch such chaotic behavior, we do believe that the answer is af?rmative.

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Table 1 The dynamics of the autonomous predator–prey system with the Beddington–DeAngelis functional response evolve under the periodic and almost periodic perturbation Autonomous case (x? , 0) (x ? , y ? ) limit cycle Periodic case √ √ ps(+, 0) ( , ) √ √ ps(+, +) ( , ) ps(+, +) (?, ?) √ sps(+, +) (?, ) √ ap(+, +) (?, ) Almost periodic case √ √ ap(+, 0) ( , ) √ √ ap(+, +) ( , ) √ ap(+, +) (?, )

ps(+, 0): boundary harmonic periodic solution; ap(+, 0): boundary almost periodic solution; ps(+, +): positive harmonic periodic solution; ap(+, +): positive almost periodic solution; sps(+, +): positive subharmonic √ √ √ periodic solution; ( , ): con?rmed both theoretically and numerically; (?, ): con?rmed numerically but not theoretically; (?, ?): con?rmed neither theoretically nor numerically.

This study shows that the environmental variation has a signi?cant effect on the global dynamics of populations. Therefore, it is very important for ecological models to incorporate both the nonlinear feedback of population interactions and the environmental ?uctuations.

Acknowledgment

We thank the referees for their many thoughtful suggestions that lead to an improved exposition of our manuscript.

References

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