Abstract
This work deals with the analysis of a delayed diffusive predator-prey system under Neumann boundary conditions. The dynamics are investigated in terms of the stability of the nonnegative equilibria and the existence of Hopf bifurcation by analyzing the characteristic equations. The direction of Hopf bifurcation and the stability of bifurcating periodic solution are also discussed by employing the normal form theory and the center manifold reduction. Furthermore, we prove that the positive equilibrium is asymptotically stable when the delay is less than a certain critical value and unstable when the delay is greater than the critical value.
1. Introduction
The study on the dynamics of predator-prey systems is one of the dominant subjects in ecology and mathematical ecology due to its universal existence and importance [1]. A prototypical predator-prey interaction model is of the following form: where and are the densities of the prey and predator at time , respectively.
Furthermore, the function is growth rate of the prey in the absence of predation, which is given by If , this reduces to the traditional logistic form , see [2] and the references therein. Here, the parameter stands for the specific growth rate of the prey , and for carrying capacity of the prey in the absence of predators.
The product gives the rate at which prey is consumed, and is termed as the functional response [3]. In particular, these functions can be defined by where denotes the capture rate, and the half capturing saturation constant.
The proportionality constant is the rate of prey consumption. And the function is given by where denotes the natural death rate of the predators, and can be used to model predator in traspecific competition that is not direct competition for food, such as some type of territoriality, see [2]. In this paper, we discuss the case of , which is used in a much more traditional case. Based on the above discussions, we can obtain the following model:
Setting , , model (1.5) leads to the following dimensionless equation: where denotes conversion rate.
In recent years, the models involving time delay and spatial diffusion have been extensively studied by many authors and many interesting results have been obtained, including the stability, the existence of Hopf bifurcation, and direction of bifurcating periodic solutions, see [1, 4–18]. In this paper, we mainly focus on the effects of both spatial diffusion and time delay factors on system (1.6) with Neumann boundary conditions as follows: where is a bounded open domain in with a smooth boundary , and denotes the Laplacian operator in . and denote the diffusion coefficients of the prey and predator , respectively. is the outward unit normal vector on . can be regarded as the gestation of the predator. System (1.7) includes not only the dispersal processes, but also some of the past states of the system.
Throughout this paper, we restrict ourselves to the one-dimensional spatial domain for the sake of convenience.
The remaining parts of the paper are structured in the following way. In Section 2, we analyze the distribution of the roots of the characteristic equation and give various conditions on the stability of a unique positive equilibrium and the existence of Hopf bifurcation with time delay. In Section 3, applying the normal form theory [19, 20] and the center manifold reduction of partial functional differential equations [21], we derive the explicit algorithm in order to determine the direction of the Hopf bifurcation, the stability, and other properties on bifurcating periodic solutions. Finally, a brief discussion is given.
2. Stability of Positive Equilibrium and Existence of Hopf Bifurcation
In this section, by analyzing the associated characteristic equation of system (1.7) at the positive equilibrium, we investigate the stability of the positive equilibria of system (1.7).
It is straightforward to see that system (1.7) has the following two boundary equilibria:(i) (total extinct) which is saddle point, hence it is unstable;(ii) (extinct of the predator) which is saddle point if , or stable if .
To find the positive equilibrium, we set which yields
Obviously, system (1.7) has a unique positive equilibrium with , where
Set , and drop the bars for simplicity of notations, then system (1.7) can be transformed into the following equivalent system:
Assume that and is defined by with the inner product .
Denote and . Then system (2.4) can be rewritten as an abstract differential equation in the phase space as follows: where , , , and , are given by respectively, where , .
The linearization of (2.6) is given by and its characteristic equation is where and , .
It is well known that the eigenvalue problem has eigenvalues , with the corresponding eigenfunctions . Substituting into characteristic equation (2.10), we obtain
Hence, we can conclude that the characteristic equation (2.10) is equivalent to the sequence of the following characteristic equations: where The stability of the positive equilibrium can be determined by the distribution of the roots of (2.14) , that is, the equilibrium is locally asymptotically stable if all the roots of (2.14) have negative real parts. Note that is not a root of (2.14) for any . Next, we analyze the behaviour of system (1.7) in two situations: with/without delay effect.
2.1. Case
Equation (2.14) with is equivalent to the following quadratic equation: where , , , and are defined as (2.15).
Let and be two roots of (2.16), then for any , we have Then we can get the following theorem.
Theorem 2.1. If holds, the positive equilibrium of system (1.7) with is asymptotically stable.
In the following, we prove that of system (1.7) is globally stable with .
Theorem 2.2. If holds, the positive equilibrium of system (1.7) with is globally asymptotically stable.
Proof. To prove our statement, we need to construct a Lyapunov function. To this end, we define
where
We claim that is positive definite. In fact, set
we can obtain . And the Hessian Matrix at is given by
Hence is positive definite, which follows that
The time derivative of along the solutions of system (1.7) with , we have
then, we obtain .
It is enough to see that satisfies Lyapunov’s asymptotic stability theorem, hence the positive equilibrium of system (1.7) with is globally asymptotically stable.
2.2. Case
In the following, we prove the stability of the positive equilibrium of system (1.7) and the existence of Hopf bifurcation at the positive equilibrium .
Theorem 2.3. Assume that , and hold, then the positive equilibrium is asymptotically stable for all .
Proof. Let be a root of the characteristic equation (2.14), then we have where , , are functions of . A necessary condition for the stability of is that the characteristic equation has a purely imaginary solution . Let and , then we can reduce (2.24) to which lead to Since and , these imply that (2.26) has no positive roots, that is, all roots of (2.14) have negative real parts.
Theorem 2.4. Assume that , , then there exists a sequence where is defined as (2.32), such that, for system (1.7), the following statements are true. (i)If , then the positive equilibrium is asymptotically stable. (ii)If , then the positive equilibrium is unstable. (iii) are Hopf bifurcation values of system (1.7) and these Hopf bifurcations are all spatially homogeneous.
Proof. Let be a root of the characteristic equation (2.14). By the same way in Theorem 2.3, then satisfies the following equation: Since , (2.28) has a unique positive root satisfying and from (2.25) we obtain then (2.16) has one imaginary root when where satisfies
Let be the root of (2.14) satisfying and when is close to . Now by some simple calculations we obtain since and , and from the expression of in (2.29), we immediately see that Therefore, , that is, . This implies that all the roots that cross the imaginary axis at cross from left to right as increases.
Hence the transversality condition holds and accordingly Hopf bifurcation occurs at , and are Hopf bifurcation values of system (1.7) and these Hopf bifurcations are all spatially homogeneous. This completes the proof.
3. Direction and Stability of Hopf Bifurcation
In the previous section, we have already obtained that system (1.7) undergoes Hopf bifurcation at the positive equilibrium when crosses through the critical value . In this section, we will study the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions by employing the normal form method [19, 20] as well as center manifold theorem [21] for partial differential equations with delay. Then we compute the direction and stability of the Hopf bifurcation when for fixed .
Without loss of generality, we denote the critical value of by and set , then is the Hopf bifurcation value of system (2.6). Rescaling the time by to normalize the delay, system (2.6) can be written in the following form: where for .
From Section 2, we know that are a pair of simple purely imaginary eigenvalues of the liner system and the following liner functional differential equation:
By the Riesz representation theorem, there exists a matrix function of bounded variation for , such that In fact, we can choose where and is the Dirac delta function.
For , we define as For , we define Then and are adjoint operators under the following bilinear form: where .
We note that are the eigenvalues of . Since and are two adjoint operators, are also eigenvalues of . We will first try to obtain eigenvector of and corresponding to the eigenvalue and , respectively.
Let be the eigenvector of corresponding to the eigenvalue . Then we have by the definition of eigenvector. Therefore, from (3.6), (3.10), and the definition of , we can get here,
On the other hand, suppose that is the eigenvector of corresponding to the eigenvalue . By the definition of , we have where and we also assume that . To obtain the value of , from (3.10) we have Thus, we can choose such that and , that is to say that let , , then , where is the unit matrix. Then the center subspace of system (3.4) is , and the adjoint subspace . Let , where by using the notation from [20], we also define for .
And the center subspace of linear system (3.4) is given by , where and , where is the stable subspace.
Following Wu [20], we know that the infinitesimal generator of linear system (3.4) satisfies moreover, if and only if
As the formulas to be developed for the bifurcation direction and stability are all relative to only, we set in system (3.4) and can obtain the center manifold with the range in . The flow of system (3.4) in the center manifold can be written as follows: where
We rewrite (3.24) as with
Denote , and let Afterwards, from Taylor formula, we have
From (3.26) and (3.28), we have
Since and for appear in , we need to compute them. It follows from (3.26) that In addition, satisfies where Thus, from (3.24) and (2.18), we can get Note that has only two eigenvalues , therefore, (3.33) has unique solution in given by From (2.18), we know that for Therefore, for From (3.34), (3.36), and the definition of , it follows that Noting that , we have Similarly, from (3.34) and (3.37), we obtain
In what follows, we will seek appropriate and in (3.39) and (3.40). From the definition of and (3.33), we have where , then Substituting (3.43) into (3.41), note that then we deduce Finally, we arrive at where and from the above equation we can find the value of and .
Following the similar steps, we also get then Finally, we arrive at where
In a similar manner, we can calculate and . Then can be expressed. Based on the above analysis, it is enough to see that each is determined by the parameters. Thus, we can compute the following values which determine the direction and stability of the following bifurcating periodic orbits:
Theorem 3.1. For system (1.7),(i) determines the directions of the Hopf bifurcation: if , the direction of the Hopf bifurcation is forward (backward), that is, the bifurcating periodic solutions exist for ; (ii) determines the stability of the bifurcating periodic solutions on the center manifold: if , the bifurcating periodic solutions are orbitally asymptotically stable (unstable); (iii) determines the period of the bifurcating periodic solutions: the period increases (decreases) if .
4. Conclusions and Remarks
In this paper, under homogeneous Neumann boundary conditions, we have analyzed dynamical behaviors of the diffusion predator-prey system (1.7) with and without delay. The value of this study lies in two aspects. First, it presents the stability of positive equilibrium of system (1.7) with and without delay, and the existence of Hopf bifurcation, which indicates that the dynamical behaviors become rich and complex with delay. Second, it shows the analysis of stability of Hopf bifurcation, from which one can find that small sufficiently delays cannot change the stability of the positive equilibrium and large delays cannot only destabilize the positive equilibrium but also induce an oscillation near the positive equilibrium.
Next, numerical simulations are performed to illustrate results with respect to the theoretical facts under the special example. When and are all equal , we demonstrate that the positive equilibrium is globally asymptotically stable (see, Figure 1), which means that if the intraspecific competitions of the prey and the predator dominate the inter-specific interaction between the prey and the predator, then both the prey and the predator populations are permanent [14].
In addition, we consider the dynamics of system (1.7) affecting by both spatial diffusion and time-delay factors with fixed parameters and . In this case, , . If , the positive equilibrium is remain stability (see Figure 2), which indicates that the predators and preys can coexist in stable conditions.
(a)
(b)
While if , the positive equilibrium losses its stability and Hopf bifurcation occurs, which means that a family of stable periodic solutions bifurcate from since , (see Figure 3). The numerical result indicates that the predator coexists with the prey with oscillatory behaviors.
(a)
(b)
In the present paper, we incorporate time delay into biological system due to the gestation of the predator, which causes stable equilibrium to become unstable and causes the populations to oscillate via Hopf bifurcation. That is to say that the effect of delay for the population dynamics is tremendous. From a biological perspective, the time delay of species may be related to the gestation of the predator, mature stage from juvenile to adult, the interaction time between prey and predator and others. In these cases, the methods and results in the present paper may provide a favorable value on controlling ecological population. It would be more accurate to describe the growth rate of population.
Acknowledgments
The authors would like to thank the anonymous referee for very helpful suggestions and comments which led to improvements of our original paper. This paper was supported by the National Basic Research Program of China (2012CB426510) and Natural Science Foundation of Zhejiang province (LY12A010041 & LQ12A01009).