Stability and bifurcation for a kind of nonlinear delayed differential equations☆
Introduction
Linear and nonlinear delayed differential equations have been extensively applied in many areas such as biology, network, robot engineering, control of signal and so on. The models in the above areas are almost similar in form which have one or many delays. In this paper, we will study a kind of nonlinear delayed differential equations which can describe a two species predator–prey system with stage structure. The kind of model have been studied in [7], [13], [14], but the stage structure of species has been ignored in many literature. In the natural world, however, there are many species whose individual members have a life history which takes them through two stages: immature and mature in [1], [2], [4], [6], [8], [9], [10], [12]. In particular, we have in mind mammalian populations and some amphibious animals, which exhibit these two stages. In these models, the age to maturity is represented by a time delay, leading to systems of retarded functional differential equations. Cannibalism has been observed in a great variety of species, including a number of fish species. Cannibalism models of various types have also been investigated in [5]. For general models population growth one can see [11].
In this paper, we consider a kind of nonlinear delayed differential equations taken in the following form:where represent respectively the immature and mature prey populations densities; Y(t), represents the density of predator population; , is the transformation coefficient of mature predator population; α represents the immatures who were born at time t − τ and survive at time t (with the immature death rate γ), and τ represents the transformation of immatures to matures; , is the death rate of the immature prey population; , represents the mature death and overcrowding rate. The model is derived under the following assumptions:
- [H1]
The birth rate of the immature prey is proportional to the existing mature population with ; the death rate of the immature prey is proportional to the existing immature population with ; the death rate of the mature is proportional to square of itself with a proportionality constant .
- [H2]
The predator only depend on the prey, that is, the predator will die out with the absence of prey. We also assume that .
Section snippets
Stability and Hopf bifurcation
Note that the quantities and of the system (1.1) are independent of the quantity , so we may only consider the following subsystem to be easy to yield the dynamic behaviors of the system (1.1):If , then the system (2.1) has a unique positive equilibrium withNormalizing the delay τ by the time scaling , then the system (2.1) is
Stability of Hopf bifurcation
From Theorem 2.2, we know that Hopf bifurcation will occurs if , and are satisfied. Next, based on the normal form method and the center manifold theory by Hassard et al. in [3], we study the direction of the bifurcation and stability of the bifurcating periodic solutions.
For convenience, we introduce a new parameter and rewrite (2.2) aswhere and , are given by
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Cited by (2)
Hopf bifurcation analysis and amplitude control of the modified Lorenz system
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Supported by the Natural Science Foundation of China (No. 10671154).