Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism
© Li and Xiong; licensee Springer 2013
Received: 29 November 2012
Accepted: 18 June 2013
Published: 23 July 2013
In this paper, a predator-prey system with sex-structure and sexual favoritism is considered. Firstly, the impact of the sexual favoritism coefficient on the stability of the ordinary differential equation (ODE) model is studied. By choosing sexual favoritism coefficient as a bifurcation parameter, it is shown that a Hopf bifurcation can occur as it passes some critical value, and the stability of the bifurcation is also considered by using an analytical method. Secondly, the impact of the time delay on the stability of the delay differential equation (DDE) model is investigated, where time delay is regarded as a bifurcation parameter. It is found that a Hopf bifurcation can occur as the time delay passes some critical values. Using the normal form theory and center manifold argument, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are derived. Numerical simulations are performed to support theoretical results and some complex dynamic behaviors are observed, including period-halving bifurcations, period-doubling bifurcations, high-order periodic oscillations, chaotic oscillation, fast-slow oscillation, even unbounded oscillation. Finally, a brief conclusion is given.
MSC:34K13, 34K18, 34K60, 37D45, 37N25, 92D25.
1 Introduction and formulation of the model
Sex ratio means the comparison of male and female individual number in populations. Usually, we assume the sex ratio is . However, to some wildlife, the sex ratio of populations will change with the kinds, mate, environment conditions, social behavior, resource, adaptability, heredity, gene structure, etc. (see [1–9]). The animal’s sex ratio will change with different animals in the different life history stage. Along with the growth of the age, the male individuals tend to relatively decrease, but there exists a variety of birds for which, on the contrary, the male individuals relatively increase. In isolated populations, males compete locally for mates and resource, sex ratio will affect the dynamic behavior of the population . Sex ratio is a basic dynamic factor for the analysis of populations, and it has important influence on the dynamic state of populations.
where and are the male, female individuals of the prey population, is the predator population. The parameters a, b, , , , , , , k are positive, , , , are constants of proportionality for male and female prey growth and death (, where ), a is the constants of proportionality for a predator, is the predation coefficient for a predator and () is the rate of conversing prey into a predator, respectively. The authors obtained the conditions for the equilibrium stability of system (1.1).
Moreover, Boukal et al.  considered sex-selective predation using several simple predator-prey models, for example, male-biased predation is frequently related to prey traits shaped by sexual selection, predators and parasitoids are attracted by mating signals of their male prey; female-biased predation is often related to prey traits shaped by fecundity selection since it is easier or more rewarding to detect them than prey. The author found that long-term effects of sex-selective predation depend on the interplay of predation bias and prey mating system, given the conclusion that ‘predation on the “less limiting” prey sex can yield a stable predator-prey equilibrium, while predation on the other sex usually destabilizes the dynamics and promotes population collapses’. For the methods, models, data, results, and more details, see .
where σ is the sexual favoritism coefficient, means that the predator prefers predating male prey to female prey, means that the predator prefers predating female prey to male prey, and means there is no sexual favoritism. The authors obtained the conditions for the equilibrium stability of system (1.2). But, how does the dynamic behavior go when the positive equilibrium loses stability? Does there exist a periodic solution or other rich dynamic behaviors?
To the best of our knowledge, few papers focus on the predator-prey system with sex-structure. Recently, Xiong and Zhang  have studied a predator-prey model with sex-structure. They obtained the sufficient and realistic conditions for the existence of a positive periodic solution by constructing a V functional, and using the result of the existence of positive periodic solutions, the global attractivity of a positive periodic solution was also obtained. Based on system (1.1), Li and Xiong  investigated the discrete periodic sex structure model and obtained sufficient and realistic conditions for the existence and global attractivity of a positive periodic solution for it. The pest management strategy of a prey-predator system model with sexual favoritism was considered by Pei et al. , and the conditions for a global asymptotically stable pest-eradication periodic solution and permanence of the system were established. The local asymptotic stability of system (1.2) has been studied by Liu et al. . However, by choosing σ as a bifurcation parameter, we obtain Hopf bifurcation conditions for system (1.2). In model (1.3), we introduce time delay due to the gestation of the predator. So, we believe that this is the first time that a predator-prey model with sex-structure and time delay has been formulated and analyzed.
This paper is organized as follows. In Section 2, we first focus on the stability of the equilibrium point and the Hopf bifurcation of ODE system (1.2) by choosing σ as a bifurcation parameter. The stability of the bifurcation is also considered by using an analytical method introduced by Kazarinov . In Section 3, we investigate the existence of Hopf bifurcations and the estimation of the length of delay to preserve the stability of DDE system (1.3). By using the normal form theory and center manifold argument introduced by Hassard , we derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions. Finally, in Section 4, numerical simulations are performed to support the theoretical results. Numerical results show that ODE system (1.2) considered has chaotic behavior under some parameter sets of values and the Hopf bifurcation of DDE system (1.3) is subcritical, and the bifurcating periodic solutions are unstable under certain conditions.
2 ODE model (1.2)
2.1 Stability of equilibrium and the existence of a Hopf bifurcation
By verifying the characteristic roots of Eq. (2.1) at each equilibrium, it is easily seen that the equilibrium is always unstable.
According to the above analysis and the result , we obtain the following theorem.
Theorem 2.1 The positive equilibrium is locally asymptotically stable when and unstable when . There exists a critical value such that a single Hopf bifurcation occurs at for decreasing σ, i.e., there exists a nontrivial orbitally periodic orbit of system (1.2) if .
2.2 Hopf bifurcation analysis
We deal with the Hopf bifurcation of system (1.2) using an analytical method introduced by Kazarinov . We first introduce some definitions. Suppose that is a linear space defined on the complex number field ℂ. For any vectors and , where (), is the inner product of the vectors x and y.
with , .
where is the unit matrix.
, γ, w, are smooth continuations of 0, α and the first Lyapunov coefficient at the Hopf point , respectively. When (), a family of stable (unstable) periodic orbits can be found on this family of center manifolds which shrink to the equilibrium point at the Hopf point.
3 DDE model (1.3)
3.1 Existence of a Hopf bifurcation
where . According to the Hopf bifurcation theorem for functional differential equations , we have the following result.
System (1.3) can undergo a Hopf bifurcation at the positive equilibrium when (), where is defined by (3.7).
Remark 3.1 It must be pointed out that Theorem 3.1 cannot determine the stability and the direction of bifurcating periodic solutions, that is, the periodic solutions may exist either for or for , near . Furthermore, we can investigate the stability of the bifurcating periodic orbits by analyzing higher-order terms according to Hassard et al.  by using normal form theory and center manifold theorem and prove that the Hopf bifurcation is subcritical and bifurcating periodic solutions are unstable.
3.2 Estimation of the length of delay to preserve stability
and , , are the Laplace transforms of , , , respectively.
where , and is the smallest positive root of Eq. (3.10).
Equations (3.11) and (3.12), if satisfied simultaneously, are sufficient conditions to guarantee stability. We shall utilize them to get an estimate on the length of delay. Our aim is to find an upper bound on , independent of τ, and then to estimate τ so that (3.11) holds for all values of η, , and in particular at .
then, clearly, from (3.13) we have .
then stability is preserved for .
3.3 Direction and stability of a Hopf bifurcation
In Section 3.1, we have obtained the conditions under which a family of periodic solutions bifurcates from the positive equilibrium of system (1.3) when the delay crosses through the critical values . In this subsection, we shall study the direction of these Hopf bifurcations and the stability of bifurcated periodic solutions arising through Hopf bifurcations by applying the normal form theory and center manifold theorem introduced by Hassard et al. .
where is a Dirac function, then (3.20) is satisfied.
where , we know that are eigenvalues of . Thus they are also eigenvalues of . To determine the Poincaré normal form of the operator A, we need to calculate the eigenvector q of A belonging to the eigenvalue and the eigenvector of belonging to the eigenvalue .
determines the stability of bifurcating periodic solutions. If (<0), the bifurcating periodic solutions are unstable (stable);
determines the period of bifurcating periodic solutions. If (<0), the period increases (decreases).
4 Numerical simulation
In this paper, we have investigated a predator-prey system with sex-structure and sexual favoritism. Firstly, the impact of the sexual favoritism coefficient σ on the stability of the ODE model is studied. From Theorem 2.1, we know the sexual favoritism coefficient σ would determine the stability of ODE system (1.2). In the ecology, sexual favoritism predation could impact population dynamics differently and affect reduced male and female densities in the prey. The numerical simulations show that ODE system (1.2) has complicated dynamic behaviors when we change the parameter σ, including periodic oscillating, period-doubling bifurcations, period-halving bifurcations and chaos. That is to say, sexual favoritism coefficient σ would be an important factor to affect the dynamic behaviors of the system. Secondly, by analyzing the associated characteristic equation, the impact of the time delay τ on the stability of DDE system (1.3) is obtained and the explicit formulae, which determine the stability, direction, and other properties of bifurcating periodic solutions, are also obtained by the Hassard method.
We have obtained estimated length of gestation delay which does not affect the stable coexistence of both predator and prey species at their equilibrium values. From the numerical simulations, we know that the Hopf bifurcation is subcritical and the bifurcating periodic solutions are unstable. It is clear that the larger values of gestation time delay cause fluctuation in population density, and even a very small time delay would make the system subject to unstable oscillation and extinct. These are harmful delays. How to control the bifurcation arising from the DDE system? How can one do this if the time delays make the system subject to unbounded oscillations? The time-varying control strategies and the impulsive control strategies would be considered , which could both improve the stability of the system and control the amplitude of the bifurcated periodic solution effectively. We will continue to study these problems in the future.
The authors would like to thank the reviewers and the editor for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the Natural Science Foundation of Guizhou Province (No. 2116), Natural Science Foundation of Jiangxi Province (No. 20122BAB201002).
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