Bazykin’s Predator–Prey Model Includes a Dynamical Analysis of a Caputo Fractional Order Delay Fear and the Effect of the Population-Based Mortality Rate on the Growth of Predators

G. Ranjith Kumar, K. Ramesh*, Aziz Khan, K. Lakshminarayan, Thabet Abdeljawad*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we investigate a system of two differential equations of fractional order for the fear effect in prey-predator interactions, in which the density of predators controls the mortality pace of the prey population. The non-integer order differential equation is interpreted in terms of the Caputo derivative, and the development of the non-integer order scheme is described in terms of the influence of memory on population increase. The primary goal of existing research is to explore how the changing aspects of the current scheme are impacted by various types of parameters, including time delay, fear effect, and fractional order. The solutions’ positivity, existence-uniqueness, and boundedness are established with precise mathematical conclusions. The requirements necessary for the local asymptotic stability of different equilibrium points and the global stability of coexistence equilibrium are established. Hopf bifurcation occurs in the system at various delay times. The model’s fractional-order derivatives enhance the model behaviours and provide stability findings for the solutions. We have observed that fractional order plays an important role in population dynamics. Also, Hopf bifurcation for the proposed system have been observed for certain values of order of derivatives. Thus, the stability conditions of the equilibrium points may be changed by changing the order of the derivatives without changing other parametric values. Finally, a numerical simulation is run to verify our conclusions.

Original languageEnglish
Article number130
JournalQualitative Theory of Dynamical Systems
Volume23
Issue number3
DOIs
Publication statusPublished - Jul 2024
Externally publishedYes

Keywords

  • Bazykin’s
  • Caputo fractional derivative
  • Fear effect
  • Hop bifurcation
  • Predator–prey model
  • Stability
  • Time delay

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