Abstract
The existence of an Andronov–Hopf and Bautin bifurcation of a given system of differential equations is shown. The system corresponds to a tritrophic food chain model with Holling functional responses type IV and II for the predator and superpredator, respectively. The linear and logistic growth is considered for the prey. In the linear case, the existence of an equilibrium point in the positive octant is shown and this equilibrium exhibits a limit cycle. For the logistic case, the existence of three equilibrium points in the positive octant is proved and two of them exhibit a simultaneous Hopf bifurcation. Moreover the Bautin bifurcation on these points are shown.
Highlights
In the task of understanding the complexity presented by the interactions among the different populations living in a habitat, the mathematical modeling has been a very important rule in ecology in the last decades
In Ref. [1], the authors analyzed the case when h(x) is a linear map, and the functional responses f and g are Holling type III and Holling type II, respectively. They proved that there is a domain in the parameter space where the system (1.1) has a stable periodic orbit which results from an Andronov–Hopf bifurcation
In lemma we show the existence of an equilibrium point in the positive octant Ω under certain conditions on the parameters involved in the system of differential equations
Summary
In the task of understanding the complexity presented by the interactions among the different populations living in a habitat, the mathematical modeling has been a very important rule in ecology in the last decades. [2], when h(x) is a linear map, and the functional responses f and g are Holling type III They proved the existence of two equilibrium points which exhibit simultaneously a zero-Hopf bifurcation in Ω. [1], the authors analyzed the case when h(x) is a linear map, and the functional responses f and g are Holling type III and Holling type II, respectively. They proved that there is a domain in the parameter space where the system (1.1) has a stable periodic orbit which results from an Andronov–Hopf bifurcation.
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