Prey–Predator Models with Variable Carrying Capacity

Abstract

Prey–predator models with variable carrying capacity are proposed. These models are more realistic in modeling population dynamics in an environment that undergoes changes. In particular, prey–predator models with Holling type I and type II functional responses, incorporating the idea of a variable carrying capacity, are considered. The carrying capacity is modeled by a logistic equation that increases sigmoidally between an initial value κ0>κ1 (a lower bound for the carrying capacity) and a final value κ1+κ2 (an upper bound for the carrying capacity). In order to examine the effect of the variable carrying capacity on the prey–predator dynamics, the two models were analyzed qualitatively using stability analysis and numerical solutions for the prey, and the predator population densities were obtained. Results on global stability and Hopf bifurcation of certain equilibrium points have been also presented. Additionally, the effect of other model parameters on the prey–predator dynamics has been examined. In particular, results on the effect of the handling parameter and the predator’s death rate, which has been taken to be the bifurcation parameter, are presented.