A predator - Prey model with the nonlinear self interaction coupling xky

Abstract

A class of Predator – Prey Models suggested by the continuous form of the two dimensional map of the form (Formula presented.). After passing to the continuous time form of this map that generalizes the classical Lotka Volterra model by a quadratic self interaction term; an additional coupling of the form xky in the prey equation is added. The motivation for this is the fact that there is a simple relation between quadratic and cubic self interactions. If one lets x = u2 and y = v2, we get the interaction coupling uv2 in the prey and u2v in predator equation. It would therefore be of interest to study the simplest nontrivial generalization before this, namely couplings of the form xyk in the prey equation. The predator equation is that of the classical Lotka Volterra. The prototype form of the model involving this generalization, after making the variable changes x ? ax and y ? ay is: (Formula presented.). The model is shown to have resonant normal forms corresponding to the added coupling terms for integer values of its parameter a. For a given value of the number k we define the following family of systems of differential equations. Stability and bifurcation properties of these models are examined. It is also shown that they have limit cycles irrespective of the etailed form of the coupling. Time series derived from these models are examined for invariant parameters such as Lyapunov exponents, fractal dimension as a function of its parameters. The techniques used for this analysis include time series analysis, rescaled range analysis and detrended fluctuation analysis. In order to examine the robustness and the stability of our results; the effects of noise on the analyzed systems are evaluated. © 2020 Elsevier B.V., All rights reserved.