Abstract
In this paper, we prove the existence of a limit cycle for a given system of differential equations corresponding to an asymmetrical intraguild food web model with functional responses Holling type II for the middle and top predators and logistic grow for the (common) prey. The existence of such limit cycle is guaranteed, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, under certain conditions for the parameters involved in the system.
Highlights
We prove the existence of a limit cycle for a given system of differential equations corresponding to an asymmetrical intraguild food web model with functional responses Holling type II for the middle and top predators and logistic grow for the prey
The existence of such limit cycle is guaranteed, via the first Lyapunov coefficient and the Andronov-Hopf bifurcation theorem, under certain conditions for the parameters involved in the system
It is well known that interaction between three species, in which predation and competition occurs, is called intraguild predation
Summary
It is well known that interaction between three species, in which predation and competition occurs, is called intraguild predation (see [1]). There are some recent papers where food chain models between three species have been studied in which the authors have obtained results about the coexistence of the species by looking at the existence of limit cycles for the corresponding model systems, for instance tritrophic models with linear growth prey (see [4] [5] [6]) and logistic growth prey (see [7]) These models do not consider predation of the top predator to the resource (the prey). Hopf in 1942 (see [9] and for a proof in the bidimensional case see [10] and the general case see [[11], Section 5], and [[8], Section 5.4]) This theorem guarantees the existence of a Hopf’s bifurcation at an equilibrium point of a system of ordinary differential equations x = F ( x, μ ). For ξ0 > 0, the equilibrium point p0 is locally unstable point for μ > μ0 and locally stable point for μ < μ0
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