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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 936952, 14 pages
http://dx.doi.org/10.1155/2013/936952
Research Article

Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

1School of Mathematics and Physics, Jiangsu Teachers University of Technology, Changzhou 213001, China
2Basic Department, Yancheng Institute of Technology, Yangcheng 224003, China
3School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

Received 26 April 2013; Accepted 8 July 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Wenzhen Gan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

1. Introduction

The overall behavior of ecological systems continues to be of great interest to both applied mathematicians and ecologists. Two species predator-prey models have been extensively investigated in the literature. But recently more and more attention has been focused on systems with three or more trophic levels. For example, the predator-prey system for three species with Michaelis-Menten type functional response was studied by many authors [14]. However, the systems in [14] are either with discrete delay or without delay or without diffusion. In view of individuals taking time to move, spatial dispersal was dealt with by introducing diffusion term to corresponding delayed ODE model in previous literatures, namely, adding a Laplacian term to the ODE model. In recent years, it has been recognized that there are modelling difficulties with this approach. The difficulty is that diffusion and time delay are independent of each other, since individuals have not been at the same point at previous times. Britton [5] made a first comprehensive attempt to address this difficulty by introducing a nonlocal delay; that is, the delay term involves a weighted-temporal average over the whole of the infinite domain and the whole of the previous times.

There are many results for reaction-diffusion equations with nonlocal delays [518]. The existence and stability of traveling wave fronts were studied in reaction-diffusion equations with nonlocal delay [59]. The stability of impulsive cellular neural networks with time varying was discussed in [10] by means of new Poincare integral inequality. The asymptotic behavior of solutions of the reaction-diffusion equations with nonlocal delay was investigated in [11, 12] by using an iterative technique and in [1315] by the Lyapunov functional. The stability and Hopf bifurcation were discussed in [16] for a diffusive logistic population model with nonlocal delay effect.

Motivated by the work above, we are concerned with the following food chain model with Michaelis-Menten type functional response: for , with homogeneous Neumann boundary conditions and initial conditions where is bounded, Hölder continuous function and satisfies () on . Here, is a bounded domain in with smooth boundary and is the outward normal derivative on . represents the density of the th species (prey, predator, and top predator resp.) at time and location and thus only nonnegative is of interest. The parameter is the intrinsic growth rate of the prey, and and are the death rates of the predator and top-predator. is the intraspecific competitive rate of the th species. is the maximum predation rate. and are the efficiencies of food utilization of the predator and top predator, respectively. We assume the predator and top predator show the Michaelis-Menten (or Holling type II) functional response with and , respectively, where and are half-saturation constants. For a through biological background of similar models, see [18, 19]. As our most knowledge, the tritrophic food chain model has been found to have many interesting biological properties, such as the coexistence and the Hopf bifurcation. However, the effect of nonlocal time delays on the coexistence has not been reported. Our paper mainly concerns this perspective.

Additionally, () represents the nonlocal delay due to the ingestion of predator; that is, mature adult predator can only contribute to the production of predator biomass. The boundary condition in (2) implies that there is no migration across the boundary of .

The main purpose of this paper is to study the global asymptotic behavior of the solution of system (1)–(3). The preliminary results are presented in Section 2. Section 3 contains sufficient conditions for the global asymptotic behaviors of the equilibria of system (1)–(3) by means of the Lyapunov functional. Numerical simulations are carried out to show the feasibility of the conditions in Theorems 810 in Section 4. Finally, a brief discussion is given to conclude this work.

2. Preliminary Results

In this section, we present several preliminary results that will be employed in the sequel.

Lemma 1 (see [3]). Let and be positive constants. Assume that , and is bounded from below. If and in for some positive constant , then .

The following lemma is the Positivity Lemma in [20].

Lemma 2. Let and satisfy where . If for and for all . Then on . Moreover, if the initial function is nontrivial, then in .

Lemma 3 (see [20]). Let and be a pair of constant vector satisfying and let the reaction functions satisfy local Lipschitz condition with . Then system (1)–(3) admits a unique global solution such that whenever , .

The following result was obtained by the method of upper and lower solutions and the associated iterations in [21].

Lemma 4 (see [21]). Let be the nontrivial positive solution of the system where . The following results hold:(1)if , then uniformly on as ;(2)if , then uniformly on as .

Throughout this paper, we assume that where is a nonnegative function, which is continuous in for each and measurable in for each pair .

Now we prove the following propositions which will be used in the sequel.

Proposition 5. For any nonnegative initial function, the corresponding solution of system (1)–(3) is nonnegative.

Proof. Suppose that is a solution and satisfies with . Choose . Then where It follows from Lemma 2 that . Due to the arbitrariness of , we have for .

Proposition 6. System (1)–(3) with initial functions admits a unique global solution satisfying for , where is defined as

Proof. It follows from standard PDE theory that there exists a such that problem (1)–(3) admits a unique solution in . From Lemma 2 we know that for . Considering the first equation in (1), we have Using the maximum principle gives that . In a similar way, we have for . It is easy to see that and are a pair of coupled upper and lower solutions to system (1)–(3) from the direct computation. In virtue of Lemma 3, system (1)–(3) admits a unique global solution satisfying for . In addition, if , is nontrivial, it follows from the strong maximum principle that for all .

3. Global Stability

In this section, we study the asymptotic behavior of the equilibrium of system (1)–(3). In the beginning, we show the existence and uniqueness of positive steady state.

Let us consider the following equations: A direct computation shows that the above equations have only one positive solution if and only if is satisfied. Taking into account, we consider the equation , which corresponds to in Figure 1.

936952.fig.001
Figure 1: Graphs of the equations in (12). is the boundary condition and corresponds to , while corresponds to . Intersection point between and gives the positive solution to (12). Parameter values are listed in the example in Section 4.

It is clear that if , the intersection point always lies in . Let Suppose that and hold. We take as a parameter and consider the following system: When the parameter is sufficiently small, the first two equations in (15) can be approximated by (12). Moreover, by continuously increasing the value of goes up, and meanwhile the intersection point between and also goes up. However, goes up faster than , while keeps still. In other words, there exists a critical value such that the intersection point lies in , which implies that there is a unique positive solution to (15), or equivalently (1)–(3).

Lemma 7. Assume that and hold, and then the positive steady state of system (1)–(3) is locally asymptotically stable.

Proof. We get the local stability of by performing a linearization and analyzing the corresponding characteristic equation. Similarly as in [22], let be the eigenvalues of on with the homogeneous Neumann boundary condition. Let be the eigenspace corresponding to in , for . Let be an orthonormal basis of , and . Then Let and . Then the linearization of (1) is , where . The coefficients are defined as follows: Since is invariant under the operator for each , then the operator on is . Let and we can get the characteristic equation , where , , and are defined as follows: It is easy to see that and . It follows from assumption and that . So and for from the direct calculation. According to the Routh-Hurwitz criterion, the three roots of all have negative real parts.
By continuity of the roots with respect to and Routh-Hurwitz criterion, we can conclude that there exists a positive constant such that Consequently, the spectrum of , consisting only of eigenvalues, lies in . It is easy to see that is locally asymptotically stable and follows from Theorem of [23].

Theorem 8. Assume that and hold, and the positive steady state of system (1)–(3) with nontrivial initial function is globally asymptotically stable.

Proof. It is easy to see that the equations in (1) can be rewritten as Define where and are positive constants to be determined. Calculating the derivatives along the positive solution to the system (1)–(3) yields where Applying the inequality , we derive from (25) that According to the property of the Kernel functions , (), we know that Define a new Lyapunov functional as follows: It is derived from (27) and (28) that Since is the unique positive solution of system (1). Using Proposition 5, there exists a constant which does not depend on or such that () for . By Theorem in [24], we have Assume that and denote Then (29) is transformed into If we choose then . Therefore, we have
From Proposition 6 we can see that the solution of system (1) and (3) is bounded, and so are the derivatives of () by the equations in (1). Applying Lemma 1, we obtain that Recomputing gives where /. Using (30) and (1), we obtain that the derivative of is bounded in . From Lemma 1, we conclude that as . Therefore Applying the Poincaré inequality leads to where and is the smallest positive eigenvalue of with the homogeneous Neumann condition. Therefore, So we have as . Similarly, and as . According to (30), there exists a subsequence , and non-negative functions , such that Applying (40) and noting that , we then have , . That is,
Furthermore, the local stability of combining with (43) gives the following global stability.

Theorem 9. Assume that and hold, and then the semi-trivial steady state of system (1)–(3) with non-trivial initial functions is globally asymptotically stable.

Proof. It is obvious that system (1)–(3) always has two non-negative equilibria as follows: and . If and are satisfied, system (1) has the other semitrivial solution denoted by , where
We consider the stability of under condition and . Equation (1) can be rewritten as Similar to the argument of Theorem 8, we have and as uniformly on provided that the following additional condition holds: Next, we consider the asymptotic behavior of . Let Then there exists such that Consider the following two systems: Combining comparison principle with (50), we obtain that By Lemma 4, we obtain which implies that uniformly on .

Theorem 10. Suppose that and hold, and then the semi-trivial steady state of system (1)–(3) with non-trivial initial functions is globally asymptotically stable.

Proof. We study the stability of the semi-trivial solution . Similarly, the equations in (1) can be written as Define Calculating the derivative of along , we get from (54) that Define It is easy to see that Assume that Let Choose Then we get (). Therefore we have uniformly on .

Next, we consider the asymptotic behavior of and . For any , integrating (57) over yields where . It implies that for the constant which is independent of . Now we consider the boundedness of and . From the Green's identity, we obtain Note that Thus, we get . In a similar way, we have . Here and are independent of .

It is easy to see that . These imply that From the Sobolev compact embedding theorem, we know

In the end, we show that the trivial solution is an unstable equilibrium. Similarly to the local stability to , we can get the characteristic equation of as If , then . It is easy to see that this equation admits a positive solution . According to Theorem 5.1 in [23], we have the following result.

Theorem 11. The trivial equilibrium is an unstable equilibrium of system (1)–(3).

4. Numerical Illustrations

In this section, we perform numerical simulations to illustrate the theoretical results given in Section 3.

In the following, we always take , where and However, it is difficult for us to simulate our results directly because of the nonlocal term. Similar to [25], the equations in (1) can be rewritten as follows: where Each component is considered with homogeneous Neumann boundary conditions, and the initial condition of