Abstract
In this paper periodic and quasi-periodic behavior of a food chain model with three trophic levels are studied. Michaelis-Menten type ratio-dependent functional response is considered. There are two equilibrium points of the system. It is found out that at most one of these equilibrium points is stable at a time. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. However for the second equilibrium point the computation is more tedious and bifurcation points can only be found by numerical experiments. It has been found that around these points there are periodic solutions and when this point is unstable, the solution is an enlarging spiral from inside and approaches to a limit cycle.
Highlights
The term “ratio-dependent predation” is introduced in [1] to describe situations in which the feeding rates of predators depend on the ratio of the number of preys to the number of predators rather than on prey density alone, as is the case in most classical models
One advantage of the ratio dependence is that they prevent paradoxes of enrichment and biological control predicted by classical models [2,3]
A ratio-dependent food chain model is analyzed and possible dynamical behavior of this system investigated at equilibrium points
Summary
The term “ratio-dependent predation” is introduced in [1] to describe situations in which the feeding rates of predators depend on the ratio of the number of preys to the number of predators rather than on prey density alone, as is the case in most classical models. The classical food chain models with only two trophic levels are shown to be insufficient to produce realistic dynamics [12,13,14,15,16]. We consider the following three trophic levels food chain model with ratio-dependence which is a simple relation between the populations of the three species: z prey on y and only y, and y prey on x and nutrient recycling is not accounted for. Y, z stand for the non dimensional population density of the prey, predator and top predator respectively.
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