Chaos in a Hybrid Food Chain Model

Abstract

In this paper, chaotic and periodic dynamics in a hybrid food chain system with Holling type IV and Lotka-Volterra responses are discussed. The system is observed to be dissipative. The global stability of the equilibrium points is analyzed using Routh-Hurwitz criterion and Lyapunov direct method. Chaos phenomena is characterized by attractors and bifurcation diagram. The effect of the controlling parameter of the model is investigated theoretically and numerically.

On the Stability of Four Dimensional Lotka-Volterra Prey-Predator System

“…More details on the modified Lotka-Volterra models and prey-predator models can be found in [10][11][12][13][14]. In [15], a modified for Lotka-Volterra model was proposed and studied, and it represents a food chain consisting of three species.…”

“…In the third-place are hyenas; they live on less powerful animals such as deer or zebra. In our work, a modification of the last mathematical model was introduced such that another predator was added to the model that studied in [14]. The present model has four firstorder nonlinear differential equations describing the dynamic behavior for four species, in which all the species grow logistically.…”

“…The present model has four firstorder nonlinear differential equations describing the dynamic behavior for four species, in which all the species grow logistically. The current model is considered more comprehensive than the model studied in [14]. Because of the increase in the number of parameters as well as the number of differential equations, and because of that this model contains sixteen possible equilibrium points, while the number of the possible equilibrium points in the previous model is eight.…”