Stability Analysis of Fractional-Order Predator-Prey System with Consuming Food Resource

Abstract

The cardinal element of ecology is the predator-prey relationship. The population of interacting organisms is based on many factors such as food, water, space, and protection. A key component among these factors is food. The presence of food for the organisms shapes the structure of the habitat. The present study considers a predator and two types of prey. It is assumed that one prey species utilizes the same food resource as the predator, whereas the other prey species depends on a different food resource. The existence and uniqueness of the model are studied using the Lipschitz condition. The fixed points for the fractional-order model are sorted out, and the existence of the equilibrium points is discussed. The stability analysis of the model for the biologically important fixed points is provided. These include the coexistence fixed point and the prey-free (using the same food resources as the predator does) fixed point. A fractional-order scheme is implemented to support theoretical results for the stability of equilibrium points. The time series solution of the model is presented in the form of plots. Moreover, the impact of some mathematically and biologically important parameters is presented.

Keywords

• ecology
• fractional-order models
• Lipschitz continuity
• Caputo fractional derivative
• stability