Research Article | Open Access
Bogdanov-Takens and Triple Zero Bifurcations of a Delayed Modified Leslie-Gower Predator Prey System
A delayed modified Leslie-Gower predator prey system with nonlinear harvesting is considered. The existence conditions that an equilibrium is Bogdanov-Takens (BT) or triple zero singularity of the system are given. By using the center manifold reduction, the normal form theory, and the formulae developed by Xu and Huang, 2008 and Qiao et al., 2010, the normal forms and the versal unfoldings for this singularity are presented. The Hopf bifurcation of the system at another interior equilibrium is analyzed by taking delay (small or large) as bifurcation parameter.
For a more detailed study on the properties of the predator prey systems, the multiple bifurcations for some systems (ODE) with more interior equilibria are investigated by many authors, see [1–4] for example. To deal with this type of systems, the more difficult problem is how to obtain the normal form of the system at its degenerate equilibrium, that is, BT bifurcation.
When introducing the time delay into this type of systems, using the methods developed by , the authors in [6, 7] have researched the BT bifurcation of some predator prey systems (DDE). Their results show that time delay may have an effect or not on the BT bifurcation.
Recently, papers [8–10] have considered the triple zero bifurcation of some delay differential equations, depending on the parameters in the original system; some interesting bifurcation results are obtained. But we find that there are few results about the triple zero bifurcation for predator prey systems.
Summarizing the above references, we will consider the following predator prey system with Michaelis-Menten type (nonlinear) prey harvesting: where and denote the prey and predator populations, respectively. represents the negative feedback of the predator’s density. For complete reason, we give the biological meaning of the parameters, one can be seeing them in . and are intrinsic growth rate and environmental carrying capacity for the prey, respectively. is the maximum value of the per capita reduction rate of prey, measures the extent to which the environment provides protection to prey and predator respectively, measures the growth rate of the predator species, and is the maximum value of the per capita reduction rate of predator. represents Michaelis-Menten type harvesting, is the catchability coefficient, is the effort applied to harvest the prey species, and and are suitable constants.
The authors in [12, 13] have studied system (1) without prey harvesting , respectively, in , the global stability and persistence of the system are investigated. In , by using the Hopf bifurcation theorem and taking the delay as a parameter of bifurcation for small and large cases, the existence of the bifurcated limit cycle around a boundary equilibrium or an interior equilibrium is mainly considered.
For computation simplicity, we first rescale system (1).
Let , , , and ; then dropping the bars we obtain where , , , , , and .
System (2) with initial conditions is where and are all continuous bounded functions in the interval .
In this paper, for system (2), we will mainly consider the BT and triple zero bifurcations at and the Hopf bifurcation at . It is easy to see that this system is with six parameters which will let our work become more challenging. When dealing with the BT and triple bifurcations of the delay systems, the core problem is to change delay systems as ordinary differential systems (ODEs).
The concrete organization of the paper is as follows: in Section 2, we will give the conditions under which the equilibrium is a BT singularity, and a universal unfolding will be exhibited; in Section 3, when is a triple zero, the universal unfolding will be presented, and in Section 4, some Hopf bifurcation results at will be obtained.
2. Bogdanov-Takens Bifurcation
System (2) also can be written as Linearizing system (5) at yields the following linear system the corresponding characteristic equation is Evidently, is a double zero eigenvalue if is a triple zero eigenvalue if It is easy to prove that in the above two cases the rest eigenvalues all have negative real parts.
In the following, we first give the normal form of the system (10) at the singularity . Reference , we first rewrite system (10) as , here , , and . By the normal form theory developed by Faria and Magalhaes , one can obtain the center manifold of this system at the origin which is two-dimensional and system can be reduced to an ODE in the plane.
Define to be the infinitesimal generator of system. Consider and let denote the invariant space of associated with the eigenvalue , using the formal adjoint theory in , the phase space can be decomposed by as . Let and be the bases for and , respectively, and be let them normalized such that , ,and , where and are matrices, where .
Next, we will find the and based on the techniques developed by .
Lemma 1 (see Xu and Huang ). The bases of and their dual space have the following representations: where , , and , , , which satisfy(1), (2), (3), (4), (5), (6).
By system (6), we know that
By Lemma 1, we have where , , therefore,
Next, we are interested in giving a versal unfolding for system (5) at BT singularity. Choosing and as bifurcation parameters and incorporating and to system (5), where and vary in a small neighborhood of , we obtain
Then, the normal form of system (21) at is ; that is, where
Then, system (5) exists in the following bifurcation curves in a small neighborhood of the origin in the plane.
3. Triple-Zero Bifurcation
From Section 2, we know that under the conditions (4) and (9) the equilibrium of system (5) is a triple zero singularity. In the following reference, from the work of [8, 9] we will give the triple zero bifurcation at .
Let and ; then, system (5) becomes
To determine a versal unfolding for the original system (28) at , we choose , , and as bifurcation parameters, and let them become , and , respectively, where , and vary in a small neighborhood of ; then (28) can be written as
Next, we need to find the expressions of and based on the techniques developed by .
Lemma 4 (see Qiao et al. ). The bases of and their dual space have the following representations: where , , , , and , , , , which satisfy (1), (2), (3), (4), (5), (6), (7), (8), (9) + + .
Following the formula of Theorem 3.1 in , the normal form with versal unfolding of system (29) on the center manifold takes the following form: where Referring , we know that if that is, for the unfolding normal form (36) then there exist the following results.
Theorem 5. Let (4), (9), and (39) hold. For the parameters , and are sufficiently small,(i)system (36) undergoes a transcritical bifurcation at the origin on the curve (ii)system (36) undergoes a Hopf bifurcation at the origin on the curve (iii)system (36) undergoes a Hopf-bifurcation at the nontrivial equilibrium point on the curve (iv)system (36) undergoes a Bogdanov-Takens bifurcation at the origin on the curve (v)system (36) undergoes a zero-Hopf bifurcation at the origin on the curve
4. Hopf Bifurcation
The Jacobian matrix of system (2) at takes the following form: then, the characteristic equation is When , by Routh-Hurwitz criterion, all roots of (46) have negative real part if Using the formula in Theorem 2.4 of  or Lemma 2.2 of , we have the following theorem.
In the following, using the Hopf bifurcation theorem for a retarded differential system introduced by , the Hopf bifurcations at for small delay and large delay are presented.
Theorem 7. Let , , and . Then, there exists such that for each , system (2) near has a family or periodic solutions with period for such that , , and .
For large delay, by  we have the following results.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper is supported by NSFC (11226142), the Foundation of Henan Educational Committee (2012A110012), the Foundation of Henan Normal University (2011QK04, 2012PL03), and the Scientific Research Foundation for Ph.D. of Henan Normal University (no. 1001).
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