Nonlinear Functional Difference Equations with ApplicationsView this Special Issue
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Bifurcation Analysis in a Delayed Diffusive Leslie-Gower Model
We investigate a modified delayed Leslie-Gower model under homogeneous Neumann boundary conditions. We give the stability analysis of the equilibria of the model and show the existence of Hopf bifurcation at the positive equilibrium under some conditions. Furthermore, we investigate the stability and direction of bifurcating periodic orbits by using normal form theorem and the center manifold theorem.
The dynamic relationship between predators and their preys has long been and will continue to be one of dominant themes in both ecology and mathematical ecology due to its universal existence and importance. A major trend in theoretical work on prey-predator dynamics has been to derive more realistic models, trying to keep to maximum the unavoidable increase in complexity of their mathematics . In this optic, recent years, the important Leslie-Gower predator-prey model [2, 3] has been extensively studied in [4–7]. A modified version of Leslie-Gower predator-prey model with Holling-type II functional response takes the form where and represent prey and predator population densities at time , respectively. , and are positive constants. is the growth rate of prey . describes the growth rate of predator . measures the strength of competition among individuals of species . is the maximum value of the per capita reduction of due to , and is the maximum value of the per capita reduction of due to , which is not available in abundance. measures the extent to which environment provides protection to prey . measures the extent to which environment provides protection to the predator .
On the other hand, time delay plays an important role in many biological dynamical systems, being particularly relevant in ecology . For some predator-prey systems, the rate of the prey population depends on the predation of predator in the earlier times [8–14]. The results indicated that delay differential equations exhibit much more complicated dynamics than ordinary differential equations since a time delay could cause a stable equilibrium to become unstable and induce bifurcations.
In this paper, we will focus on the complex dynamics of the delay effect in the extended reaction-diffusion model. The reproduction of the individuals is modeled by diffusion with diffusion coefficients and for the prey and predator, respectively. This basic model is described by a system of two partial differential equations: where , . , is a bounded open domain in with boundary , is the outward unit normal vector on , and homogeneous Neumann boundary conditions reflect the situation where the population cannot move across the boundary of the domain. And we incorporate a single discrete delay in the negative feedback of the predator's density.
The rest of the paper is organized as follows. In Section 2, we give the stability property of the equilibria of model (1). In Section 3, we mainly analyze the distribution of the roots of the characteristic equation and show the occurrence of Hopf bifurcation at the positive equilibrium of model (2) under some conditions. In Section 4, we investigate the stability and direction of bifurcating periodic orbits by using normal form of theorem and the center manifold theorem, corresponding to theorems we also give some numerical simulations.
2. Equilibria Stability
In this section, we consider the existence and stability of the equilibria of model (1).
It is easy to verify that model (1) always has three boundary equilibria:(i) (extinction of prey and predator), which is a nodal source point;(ii) (extinction of the predator), which is a saddle point;(iii) (extinction of the prey), which is a stable node when .
For the positive equilibria, we have which yields For simplicity, we define then (4) can be written as which has two roots given by
Case 1 (, i.e., ). Model (1) has a unique positive equilibrium .
Case 3 (, i.e., ). (i) Suppose that , that is, , then(i1) if , that is, , model (1) has two positive equilibria and ; (i2) if , that is, , (4) has a unique positive root of multiplicity given by , then model (1) has a unique positive equilibrium ; (i3) if , that is, , (4) has no positive root; hence model (1) has no positive equilibrium; (ii) if , that is, , (4) has no positive root; hence model (1) has no positive equilibrium.
We show the bifurcation diagram to display the distribute of the positive roots; in Figure 1, the whole region has been divided into six parts; the number indicates the number of positive equilibria.
Next, we analyze the stability of these positive equilibria. Let be arbitrary positive equilibrium, and the Jacobian matrix for is given by The corresponding characteristic equation is where And the sign of is determined by Thus, Hence, , , , , and . Obviously, the positive equilibrium is a saddle point.
In the following, we study the stability of other positive equilibria. The sign of is determined by Then we can get Hence, if , then , , and are true. Summarizing the above, we can obtain the following theorem.
Theorem 1. For model (1),(i)if holds, the unique positive equilibrium is locally asymptotically stable for ;(ii)if and hold, the unique positive equilibrium is locally asymptotically stable for ;(iii)if and hold, model (1) has two positive equilibria, the positive equilibrium is locally asymptotically stable for , and is a saddle point.
Figure 2 shows the dynamics of model (1). In this case, is a nodal source point; is a saddle point; is a nodal sink point, which is locally asymptotically stable; is locally asymptotically stable; is a saddle point. There exists a separatrix curve determined by the stable manifold of , which divides the behavior of trajectories; that is, the stable manifold of saddle splits the feasible region into two parts such that orbits initiating inside tend to the positive equilibrium , while orbits initiating outside tend to except for the stable manifolds of . This means that, in this situation, the trajectories of the model can have different behavior strongly depending on the initial conditions.
Theorem 2. For model (1), if the unique positive equilibrium exists, and , then it is a cusp of codimension 2.
Proof. The Jacobian matrix at is
we have know that . Moreover, , if and only if
and the associate Jordan matrix is
Hence, following [15, 16], we know that the unique positive equilibrium is a cusp of codimension 2.
3. Stability and Hopf Bifurcation Analysis in Delayed Reaction-Diffusion Model (2)
According to the previous section, for model (1), we know that , , and are unstable and is a saddle point, and note that a solution of the model (1) is also a solution of the model (2), so they are also unstable for model (2). In the following, we will focus on the dynamics of the positive equilibria of model (2). As an example, we only give the proof of the unique positive equilibrium of model (2).
Introducing small perturbations , and and dropping the hats for simplicity of notation, then we have
Denote In the abstract space , model (19) can be regarded as the following abstract functional differential equation. where , , , , and , are given by here . Then the linearization of model (19) near is Following , we obtain that the characteristic equation for liner model (23) is It is well known that the eigenvalue problem has eigenvalues , with the corresponding eigenfunctions .
The stability of the positive equilibrium can be determined by the distribution of the roots of (27); that is, the equilibrium is locally asymptotically stable if all the roots of (27) have negative real parts. From the result of , the sum of the multiplicities of the roots of (27) in the open right half plane changes only if a root appears on or crosses the imaginary axis. It can be verified that is not a root of (27) for .
Theorem 3. If holds, then the unique positive equilibrium of model (2) is locally asymptotically stable.
Proof. Let be a pair of roots of (27); substituting into (27), then we have Separating the real part from image part, we have then where Obviously, if , is true. Thus (31) has no positive roots for all . Hence, all the roots of (27) have negative real part. This completes the proof.
Let be the root of (24), where and when is close to . Then we have the following transversality condition.
Lemma 4. For , if(H1) holds, then for .
From this transversality condition, we know that when passes through these critical values , the sum of the multiplicities of the roots of (27) in the open right half plane will increases at least two.
Summarizing the above results, we can obtain the following theorem.
Theorem 5. For if (H1) holds, the following statements are true:(i) if , then the equilibrium point is locally asymptotically stable;(ii) if , then the equilibrium is unstable;(iii) are Hopf bifurcation values of model (2).
4. Direction and Stability of Spatial Hopf Bifurcation
In the previous section, we have obtained the conditions under which model (2) undergoes a Hopf bifurcation at the the equilibrium point when crosses though 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 center manifold theorem and normal form method [17, 19] for partial differential equations with delay. Then we compute the direction and stability of the Hopf bifurcation when for fixed .
Define where is defined in the appendix. Then we can get the following theorem.
Theorem 6. For model (19), if (H1) holds, we have the following:(i) determines the direction of the Hopf bifurcation: if (), then the bifurcating periodic solutions exist for ();(ii) determines the stability of bifurcating periodic solutions: the bifurcating periodic solutions are orbitally asymptotically stable (unstable) if ();(iii) determines the period of the bifurcating periodic solutions: the period increases (decreases) if ().
The proof is deferred to the appendix.
5. Conclusions and Remarks
In this paper, we have considered a modified version of Leslie-Gower model with Holling-type II functional and delayed diffusive predator-prey model under homogeneous Neumann boundary conditions. The value of this study lies in two folds. First, it presents local asymptotic stability of the equilibria of model with and without delay and the existence of Hopf bifurcation, which indicates that the dynamics induced by time delay are rich and complex. Second, it give the analysis of direction and stability of spatial 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 oscillatory behaviors near the positive equilibrium.
In the following, we give some numerical examples to illustrate the dynamical behaviors of model (2). In Figure 3, , the unique positive equilibrium remains the stability; the population of the predator and the prey will tend to a steady state. However, in Figure 4, , the positive equilibrium losses its stability and Hopf bifurcation occurs, which means that a family of stable periodic solutions bifurcate from and the system goes into oscillations; it means that the predator coexists with the prey with oscillatory behaviors.
Our results show that time-delay can make a stable equilibrium to become unstable and induce Hopf bifurcation and the system goes into oscillations; that’s to say, the dynamical behaviors of the delay reaction-diffusion equations are much more complex and rich than reaction-diffusion equations.
A. The Proof of Theorem 6
And the abstract functional differential equation can also be written in the form where for .
From Section 2, we know that are a pair of simple purely imaginary eigenvalues of the liner system where , and is defined by .
By using the Riesz representation theorem , we have a function of bounded variation for such that Because in this paper, we discuss the existence of the Hopf bifurcation when , that is ; here we choose where is the Dirac delta function. For , define as and for , define Then and are adjoint operators under the bilinear form
We know that are eigenvalues of . Since and are two adjoint operators, then are also eigenvalue of ; we shall first try to obtain eigenvector of and corresponding to the eigenvalues and , respectively. Let be the eigenvector of corresponding to the eigenvalue . Then by definition of eigenvector we have . Therefore, from (A.5), (A.6), and definition of we get we choose , and then we get . On the other hand, is the eigenvector of corresponding to the eigenvalue . From the definition of , we have where We also assume that . To obtain the value of , from (A.9) we have then we choose such that and . In other words, let , , then , and is unit matrix.
Then the center subspace of model (A.4) is , and the adjoint subspace is . Let , where Let be defined by for . Hence the center subspace of linear system (A.4) is given by , where and , where is the stable subspace.
First we define the coordinate to describe the center manifold at ; from center manifold we have The flow of model (A.2) in the center manifold can be written as follows: where with
Let From above equations, we have
When , . Therefore, for , and for , , then we obtain