Abstract:We will investigate the stability and invariant manifolds of a new discrete host-parasitoid model. It is a generalization of the Beddington-Nicholson-Bailey model. Our study establishes analytically, for the first time, the stability of the coexistence fixed point.
“…This is a generalized Beddington host–parasitoid model to study the interaction of certain plants and herbivores, where x
n and y
n stand for the density of two populations at time n , and r is the growth rate. In [10], Elaydi et al . investigate the stability and invariant manifolds of this model and also the stability of the coexistence fixed point.…”
Section: Modelmentioning
confidence: 99%
“…It is noted that Elaydi et al . [10] obtain the stability region for positive fixed point in parameter space by using a numerical method. Figure 5.…”
In this paper, we study the dynamics behaviour of a stratum of plant–herbivore which is modelled through the following F(x, y)=(f(x, y), g(x, y)) two-dimensional map with four parameters defined by
where x≥0, y≥0, and the real parameters a, b, r, k are all positive. We will focus on the case a≠b. We study the stability of fixed points and do the analysis of the period-doubling and the Neimark–Sacker bifurcations in a standard way.
“…This is a generalized Beddington host–parasitoid model to study the interaction of certain plants and herbivores, where x
n and y
n stand for the density of two populations at time n , and r is the growth rate. In [10], Elaydi et al . investigate the stability and invariant manifolds of this model and also the stability of the coexistence fixed point.…”
Section: Modelmentioning
confidence: 99%
“…It is noted that Elaydi et al . [10] obtain the stability region for positive fixed point in parameter space by using a numerical method. Figure 5.…”
In this paper, we study the dynamics behaviour of a stratum of plant–herbivore which is modelled through the following F(x, y)=(f(x, y), g(x, y)) two-dimensional map with four parameters defined by
where x≥0, y≥0, and the real parameters a, b, r, k are all positive. We will focus on the case a≠b. We study the stability of fixed points and do the analysis of the period-doubling and the Neimark–Sacker bifurcations in a standard way.
“…Then, the positive equilibrium point of the models consisting density-dependent factor instead of a constant reproductive rate of the host can be locally stable [4,9]. Even so, there are many reasons affecting the number of species such as immigration, human activities, and interaction of among species.…”
Section: Introductionmentioning
confidence: 99%
“…The the some forms of density-dependent factor (interspecific interactions) g(H t ) has been studied in [4,6,9,12]. Misra and Mitra (2006) examined dynamics of discrete single-species population for different values of the parameter R and b which depend on Hassel growth function expessed with…”
Abstract:The growth of the populations depending on interspecific interactions may exhibit drastic changes. The stability of populations change not only interspecific interactions but also under external effects such as immigration effect. Thus, population models can show complex dynamics. Immigration effect often simplify the population dynamics; and tends to supress chaotic behavior. This situation which can allow the local stability analysis of population is important for control of a two species interacting system. In this paper, we investigated dynamics of a host-parasite model with the immigration parameter added to the host population under the constant searching efficiency. We conclude that immigration parameter produces certain interesting results.
“…Among various ecological models, the study of host-parasitoid model attracts more and more interest [9,10]. Parasitoids are insect species whose larva develop by feeding on the bodies of other arthropods, usually killing them.…”
In this paper, we study the global dynamics and bifurcations of a two-dimensional discrete time host-parasitoid model with strong Allee effect. The existence of fixed points and their stability are analysed in all allowed parametric region. The bifurcation analysis shows that the model can undergo fold bifurcation and Neimark-Sacker bifurcation. As the parameters vary in a small neighbourhood of the Neimark-Sacker bifurcation condition, the unique positive fixed point changes its stability and an invariant closed circle bifurcates from the positive fixed point. From the viewpoint of biology, the invariant closed curve corresponds to the periodic or quasi-periodic oscillations between host and parasitoid populations. Furthermore, it is proved that all solutions of this model are bounded, and there exist some values of the parameters such that the model has a global attractor. These theoretical results reveal the complex dynamics of the present model.
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