Abstract
In this work, an attempt is made to understand the dynamics of a modified Leslie–Gower model with nonlinear harvesting and Holling type-IV functional response. We study the model system using qualitative analysis, bifurcation theory and singular optimal control. We show that the interior equilibrium point is locally asymptotically stable and the system under goes a Hopf bifurcation with respect to the ratio of intrinsic growth of the predator and prey population as bifurcation parameter. The existence of bionomic equilibria is analyzed and the singular optimal control strategy is characterized using Pontryagin’s maximum principle. The existence of limit cycles appearing through local Hopf bifurcation and its stability is also examined and validated numerically by computing the first Lyapunov number. Optimal singular equilibrium points are obtained numerically for various discount rates.
Highlights
The Leslie–Gower model [1, 2] (LG model) shows how asymptotic solutions converge to a stable equilibrium state
Korobeinikov [4] established the global stability of a positive equilibrium point and showed that the limit cycle could be admitted by the model system
A mathematical model to study the dynamical behavior of a modified Leslie–Gower model with Holling type-IV functional response and nonlinear prey harvesting has been proposed and analyzed
Summary
The Leslie–Gower model [1, 2] (LG model) shows how asymptotic solutions converge to a stable equilibrium (independent of the initial conditions) state. Korobeinikov [4] established the global stability of a positive equilibrium point and showed that the limit cycle could be admitted by the model system This limit cycle exists if we take the Holling type-II or -III functional response. Xu et al [12] studied the stability and Hopf bifurcation of a population model with Holling type-IV functional response and time delay. Gupta et al [24] studied the bifurcation analysis and control of the LG model with Michaelis–Menten type prey harvesting and observed that for a wide range of initial values the system goes to extinction. The parameter i is a measure of the predator’s immunity from or tolerance of the prey We have analyzed this model for its rich dynamics and studied the bio-economic equilibria using singular optimal control strategies. Intrinsic growth rate of the prey Intrinsic growth rates of the predator The environmental carrying capacity for prey The maximum per capita predation rate A direct measure of the predator’s immunity from or tolerance of the prey The half saturation constant in the absence of any inhibitory effect Number of prey required to support one predator at equilibrium
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