Global stability in a modified Leslie-Gower type predation model assuming mutual interference among generalist...

Area of Science:

• Ecology
• Mathematical Biology
• Population Dynamics

Background:

• Predator-prey interactions are fundamental in ecology.
• Mathematical models, including the Leslie-Gower model, are used to study these dynamics.
• Competition among predators (CAP) can significantly influence population stability.

Purpose of the Study:

• To analyze the dynamical properties of a Leslie-Gower type predation model incorporating competition among predators (CAP).
• To compare the ecological outcomes of this model with a standard Leslie-Gower model.
• To investigate the stability of equilibrium points and the existence of periodic orbits.

Main Methods:

• Analysis of a Leslie-Gower model with a specific predator interference function $g(y)=y^{\beta}$, where $0<\beta<1$.
• Application of non-standard methodologies to analyze equilibrium points where the Jacobian matrix is undefined.
• Utilizing Lyapunov functions to prove global asymptotic stability (g.a.s) of positive equilibrium points.
• Employing Dulac functions to demonstrate the absence of periodic orbits.

Main Results:

• The unique positive equilibrium point, when it exists, is globally asymptotically stable (g.a.s) in both the standard and CAP-incorporated models.
• The analysis confirmed the non-existence of periodic orbits in both ecological systems.
• The study highlights the robustness of ecosystem stability under predator competition.

Conclusions:

• The incorporation of competition among predators (CAP) into the Leslie-Gower model does not alter the fundamental stability properties regarding the unique positive equilibrium point.
• The absence of periodic orbits suggests predictable population dynamics in these models.
• The findings contribute to a deeper understanding of ecological stability and predator-prey relationships.