Dynamic analysis of a modified Leslie-Gower model with Michaelis-Menten prey harvesting and additive allee effect on predator population Md Golam Mortuja, Mithilesh Kumar Chaube, Santosh Kumar Physica Scripta, 2024 In the current study, the dynamics of a Leslie-Gower model have been studied considering the Allee effect in the growth of the predator and the Michaelis-Menten type harvesting effect on the prey from biological and theoretical perspectives. Key findings highlight how these effects influence the existence of positive equilibria, with local stability properties and potential bifurcation behaviors examined using the center manifold theorem and linearization. Criteria for saddle-node bifurcation are established through Sotomayor’s theorem, while the direction and stability of Hopf bifurcation are explored via the first Lyapunov exponent. An optimal control model is developed to promote sustainable ecosystem development, utilizing the harvesting rate as a control parameter. Further, a few numerical simulations validate our significant findings using phase portrait diagrams.
Dynamic analysis of a modified Leslie-Gower model with nonlinear prey harvesting and prey herd behavior Md Golam Mortuja, Mithilesh Kumar Chaube, Santosh Kumar Physica Scripta, 2023 In this study, a modified Leslie-Gower model with square root functional response has been used to describe prey group defense mechanism and nonlinear predator harvesting. Two equilibrium points are always present and feasible, whereas the predator-free equilibrium point and the interior equilibrium point are only present and feasible under a parametric condition. The equilibria’s local stability has been investigated. The saddle-node bifurcation at the axial equilibrium point is investigated using the harvesting coefficient as the bifurcation parameter. The maximum sustainable yield has been established discovering that if the value of harvesting rate is lower than the maximum sustainable yield, both populations will cohabit and the ecological balance will be maintained. By establishing harvesting rate control parameters with the goal of achieving sustainable development of people and ecosystems as the starting point, an optimal control model of harvesting rate mechanisms. Fisheries management will be aware of the rate at which little fish species (preys) must be taken in order to maintain ecological balance based on the findings of this study. Additional numerical simulations are run to validate the findings.
Bifurcation analysis of a discrete type prey-predator model with Michaelis-Menten harvesting in predator Md Golam Mortuja, Mithilesh Kumar Chaube, Santosh Kumar Zeitschrift Fur Naturforschung Section A Journal of Physical Sciences, 2023 A discrete predator–prey model with square root functional response describing prey herd behavior and nonlinear predator harvesting has been considered in the present work. Three equilibria of the system have been found and observed that two equilibrium points always exist and are feasible, but the interior equilibrium point is feasible under a parametric condition. The local stability of the three equilibria has been analyzed. The interior equilibrium point is locally asymptotically stable under a parametric condition. It is examined that a flip and Neimark–Sacker bifurcations have occurred in the system at the axial equilibrium point. The flip and Neimark–Sacker bifurcations have been analyzed by the center manifold theorem and bifurcation theory, considering the harvesting coefficient as the bifurcation parameter. The proposed discrete model with a nonlinear Michaelis–Menten type harvesting effect on the predator population exhibits rich dynamics; for instance, bifurcations, chaos, and more complex dynamical behaviors. The discrete-time model also produced few numerical simulation results that are more accurate than the continuous model. The proposed discrete model will be performed better than the continuous model in populations with non-overlapping generations or smaller densities. The harvesting coefficient’s optimal value has finally been identified, and an optimal harvesting policy has been introduced. To verify the results, further numerical simulations have been performed.
