Rich Dynamics of Seasonal Carrying Capacity Prey-Predator Models with Crowley–Martin Functional Response

Abstract

In this paper, we present novel seasonal carrying capacity prey–predator models with a general functional response, which is that of Crowley–Martin. Seasonality effects are classified into two categories: sudden and periodic perturbations. Models with sudden perturbations are analytically investigated in terms of good and bad circumstances by addressing the existence, positivity, and boundedness of the solution; obtaining the stability conditions for each equilibrium point and the dynamics involving the existence of a limit cycle; determining the Hopf bifurcation with respect to the carrying capacity; and finding the uniform persistence conditions of the models. Moreover, some numerical simulations are performed to demonstrate and validate our theoretical findings. In contrast, models with periodic perturbations are computationally investigated. In analytical findings, the degree of seasonality and the classification of circumstances play a significant role in the uniqueness of the coexistence equilibrium point, the stability of the system, and the existence of a limit cycle. The model with periodic perturbations shows the presence of different dynamics for prey and predator, such as the doubling of the limit cycle and chaos dynamics, so this influence can have a diverse range of possible solutions, which makes the system more enriched with different dynamics. As a result of these findings, many phenomena and changes can be interpreted in ecosystems from an ecological point of view.

Keywords

• predator–prey model
• Crowley–Martin
• carrying capacity
• seasonality