Stability and Bifurcation Analysis of Differential Equations and its ApplicationsView this Special Issue
Research Article | Open Access
Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response
A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.
The dynamic relationship between prey and predators has long been and will continue to be one of the dominant subjects in mathematical ecology due to its universal existence and importance [1–12]. In [13, 14], the author proposed the following predator-prey model based on the model in May :where and denote the population of prey and predator, respectively, and and are the intrinsic growth rates of prey and predator, respectively. The parameter represents the carrying capacity of the prey and the ratio represents the carrying capacity of the predator. It has been assumed that both prey and predator populations grow logistically and that the predator consumes the prey according to a functional .
In recent years, models with time delay have been extensively studied by many authors [5, 16–26]. The authors of  discussed model (1) with a discrete delay:and obtained the stability of equilibria, the existence of Hopf bifurcation, and the direction of bifurcating periodic solutions. This paper focuses mainly on the effects of both spatial diffusion and time delay on system (1). It is assumed that the delay affects predation and consumption and that the system has homogeneous Neumann boundary conditions:where and are the diffusion coefficients of prey and predator, respectively, denotes the Laplacian operator, and is the outward unit normal vector on . For convenience, it is assumed that , and that all parameters are positive.
The rest of this paper is structured as follows. In Section 2, the local stability of equilibria is analyzed using the associated characteristic equations, and the occurrence of the Hopf bifurcation with time delays is presented. In Section 3, the global asymptotical stability of the interior equilibrium for any is proved by means of the upper-lower solution method. In Section 4, using normal form theory and the center manifold theorem, the stability and direction of bifurcating periodic orbits are investigated. Finally, numerical simulations and a brief discussion are presented.
2. Local Stability and Hopf Bifurcation Analysis
In this section, the local stability of the equilibria of system (3) is analyzed. Denotefor , ; then is a Hilbert space. In the abstract space , system (3) can be regarded as an abstract functional differential equation.
System (3) has two nonnegative equilibria and , whereand is the positive root of the equationLet ; then , , which guarantee the existence of .
From analysis of the characteristic equation of , it can easily be determined that it always has a saddle point. To analyze the stability of the positive equilibrium , the first step is to linearize system (3) at :where ,and is defined asfor .
The characteristic equation of (7) is
Recall that under the Neumann boundary condition has eigenvalues and , with the corresponding eigenfunctions . Substitutinginto (10), the following expression results:Hence, it can be concluded that the characteristic equation (10) is equivalent to the equationwhere
The stability of the positive equilibrium can be determined by the distribution of the roots of (13). It is locally asymptotically stable if all the roots of (13) have negative real parts for all . Obviously, 0 is not a root of (13) for all . When as well as , (13) can be simplified asIt can be verified thatthat is,which requires thatorTherefore, the following theorem can be stated.
This leads to the following theorem.
Theorem 2. If and hold for , then the interior equilibrium of system (3) is asymptotically stable for all .
Proof. If , then there exists such that, for , (20) has a unique positive real root:From (20), it follows thatThen (13) has a pair of pure imaginary roots whenwhereSubstituting into (13) and taking the derivatives with respect to lead toFor , considering , it follows thatIn other words, .
From the above discussion, the following theorem can be stated.
Theorem 3. If holds, then the following statements are true.(1)If , then the interior equilibrium of system (3) is asymptotically stable.(2)If , then the interior equilibrium of system (3) is unstable.(3)System (3) undergoes a Hopf bifurcation at the interior equilibrium for .
3. Global Stability
Lemma 4 (see ). Assume that is defined byThen .
Theorem 5. If , then for system (3), the positive equilibrium is globally asymptotically stable.
Proof. From the maximum principle of parabolic equations, it is known that for any initial value , the corresponding nonnegative solution is strictly positive for . Because , it is possible to choose satisfyingBecauseaccording to Lemma 4 and the comparison principle of parabolic equations, there exists such that, for any , . This in turn implies thatfor .
Hence there exists such that, for any ,Consequently,for .
Because , it can be easily verified thatHence, there exists such that, for any ,This implies thatfor . Again it can be verified thatand hence there exists such that, for any ,Therefore, for , it is possible to obtainand satisfyThis implies that and are a pair of coupled upper and lower solutions of system (3), as in the definition in , for the reason that (3) is a mixed quasimonotonic system.
It is clear that there exists such that, for any ,Define two iteration sequences and as follows: for ,where , .
Then, for ,and there exist and such that , , , , andBecause is the unique positive constant equilibrium of system (3), it must hold that .
According to the results of [27, 28], the solution of system (3) satisfiesHence, the constant equilibrium is globally asymptotically stable.
4. Direction and Stability of Hopf Bifurcation
Part Two has already shown that system (3) undergoes Hopf bifurcation at the interior equilibrium for . In this section, the direction, stability, and period of the periodic solutions from the steady state will be studied by applying the method introduced by Hassard et al.  and the center manifold theorem due to [31–35].
For convenience, let , , . Then system (3) is equal to a single-delay system:
Let , , . Then is the Hopf bifurcation value of system (48), which can be written as follows:wherewithfor .
From the previous discussion, it is known that is a pair of simple purely imaginary eigenvalues of the linear system (7) and the following linear functional differential equation:By the Riesz representation theorem, there exists a 2 × 2 matrix function , whose elements are bounded variables, such thatIn fact, it is possible to choosewhere is a Dirac delta function satisfyingFor , define asFor , defineThen and are adjoint operators under the bilinear form:Therefore, are eigenvalues of as well as . Next, the eigenvectors of and corresponding to the eigenvalues and can be calculated. LetUnder the condition , that is,it follows thatSimilarly, let , and with , that is,it is possible to obtainAccording to the conditions and ,Therefore,Let , , ; thenTherefore, the center subspace of system (52) is , and the adjoint subspace is . Let , whereUsing the notion from , it is also possible to definefor .
Define for andfor , .
Hence,for . Then the center subspace of linear system (7) is given by , whereand , where is a stable subspace.
According to , it is known that the infinitesimal generator of linear system (7) satisfiesMoreover, if and only ifSetting in (49), the center manifoldcan be obtained in . The flow of system (49) can be written as whereFrom Taylor’s formula,whereLet , whereFrom (77),
In the last formula, , , , and are still unknown. Therefore, it is necessary to compute and , as described below.
From (74), it is possible to obtainIn addition, satisfiesTherefore,For ,and thereforeFrom (84) and (86),Solving for ,Similarly,where and are both two-dimensional vectors and can be determined as follows.
For , according to the definition of and the first two equations of (84),ThenNote thatIt follows that