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All Journal Jambura Journal of Biomathematics (JJBM)
Mukherjee, Debasis
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Computer Science & IT Decision Sciences, Operations Research & Management Mathematics

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Complex dynamics in a discrete-time model of two competing prey with a shared predator Mukherjee, Debasis
Jambura Journal of Biomathematics (JJBM) Volume 5, Issue 2: December 2024
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.37905/jjbm.v5i2.27453

Abstract

This paper concentrates on the study of a discrete time model of two competing prey with a shared predator. The condition for the existence and local stability of positive fixed point are derived. By using an iteration scheme and the comparison principle of difference equations, it is possible to obtain the sufficient condition for global stability of the positive fixed point. The sufficient criterion for Neimark-Sacker bifurcation and flip bifurcation are established. The system admits chaotic dynamics for a certain choice of the system parameters which is controlled by applying hybrid control method. The intra-specific competition among predators and the intrinsic growth rate of prey species have major impact for different bifurcation. For continuous system, handling time spent for prey population plays an important role for obtaining limit cycle behaviour. The decrease amount of this rate makes the system stable. Global convergence of the solutions to the coexistence equilibrium point is possible for a particular choice of system parameters. The obtained results for discrete system are verified through numerical simulations. Also some diagrams are presented for continuous system.
Fear induced dynamics on Leslie-Gower predator-prey system with Holling-type IV functional response Mukherjee, Debasis
Jambura Journal of Biomathematics (JJBM) Volume 3, Issue 2: December 2022
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v3i2.14348

Abstract

This paper analyzes the effect of fear in a Leslie-Gower predator-prey system with Holling type IV functional response. Firstly, we show positivity and boundedness of the system. Then we discuss the structure of the positive equilibrium point, dynamical behavior of all the steady states and long term survival of all the populations in  the system. It is shown that fear factor has an impact on the prey and predator equilibrium densities. We have shown the occurrence of transcritical bifurcation around the axial steady state. The presence of a Hopf bifurcation near the interior steady state has been developed by choosing the level of fear as a bifurcation parameter. Furthermore, we discuss the character of the limit cycle generated by Hopf bifurcation. A global stability criterion of the positive steady state point is derived. Numerically, we checked our analytical findings.
Impact of predator fear on two competing prey species Mukherjee, Debasis
Jambura Journal of Biomathematics (JJBM) Volume 2, Issue 1: June 2021
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v2i1.9249

Abstract

Predator-prey interaction is a fundamental feature in the ecological system. The majority of studies have addressed how competition and predation affect species coexistence. Recent field studies on vertebrate has shown that fear of predators can influence the behavioural pattern of prey populations and reduce their reproduction. A natural question arises whether species coexistence is still possible or not when predator induce fear on competing species. Based on the above observation, we propose a mathematical model of two competing prey-one predator system with the cost of fear that affect not only the reproduction rate of both the prey population but also the predation rate of predator. To make the model more realistic, we incorporate intraspecific competition within the predator population. Biological justification of the model is shown through positivity and boundedness of solutions. Existence andstability of different boundary equilibria are discussed. Condition for the existence of coexistence equilibrium point is derived from showing uniform persistence. Local as well as a global stability criterion is developed. Bifurcation analysis is performed by choosing the fear effect as the bifurcation parameter of the model system. The nature of the limit cycle emerging through a Hopf bifurcation is indicated. Numerical experiments are carried out to test the theoretical results obtained from this model.
Stability and bifurcation of a two competing prey-one predator system with anti-predator behavior Mukherjee, Debasis
Jambura Journal of Biomathematics (JJBM) Volume 3, Issue 1: June 2022
Publisher : Department of Mathematics, Universitas Negeri Gorontalo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.34312/jjbm.v3i1.13820

Abstract

This article considers the impact of competitive response to interfering time and anti-predator behavior of a three species system in which one predator consumes both the competing prey species. Here one of the competing species shows anti-predator behavior. We have shown that its solutions are non-negative and bounded. Further, we analyze the existence and stability of all the feasible equilibria. Conditions for uniform persistence of the system are derived. Applying Bendixson's criterion for high-dimensional ordinary differential equations, we prove that the coexistence equilibrium point is globally stable under specific conditions. The system admits Hopf bifurcation when anti-predator behavior rate crosses a critical value. Analytical results are verified numerically.
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