Abstract

Classical Lotka-Volterra model was the basic model formulated to study the prey-predator interactions which was further enhanced by adding ecological concepts like functional response, dispersal and time delays to gain better understanding of the dynamics of population interactions. The migration of populations of prey, the predator or both are mentioned as dispersal in prey-predator models. On the foundation of resource availability and predation risk, prey species will choose to move, while predators be apt to move to an advantageous patch to secure more prey. This paper intends to study the dynamics of prey- predator in a two-patch environment with preys’ dispersal at different migration rates. The discrete fractional order model is obtained from the classical integer model through the process of discretization. The existence, uniqueness and positiveness of the solutions of the model are discussed and the stability is analyzed. Pictorial representations of the dynamics are put forward in terms of time series, bifurcation figures and phase portraits. Numerical examples justify the theoretical findings.

Highlights

  • The essence of mathematical modelling is to develop sets of mathematical expressions that can replicate the core aspects of the real-world phenomena

  • This paper intends to study the dynamics of preypredator in a two-patch environment with preys’ dispersal at different migration rates

  • The discrete fractional order model is obtained from the classical integer model through the process of discretization

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Summary

INTRODUCTION

The essence of mathematical modelling is to develop sets of mathematical expressions that can replicate the core aspects of the real-world phenomena. The Lokta-Volterra model [1] presented by the following differential equations describes dynamics of biological systems between two species namely prey and predator. A model with two populations in two different patches with same intrinsic growth rate r and different carrying capacities K1 and K2 is presented by the following logistic equations [12]. Interesting results and dynamics can be explored when model (3) is further extended for competing species and prey-predator interactions. The model consists of four differential equations with prey dispersal which migrates between the two patches, predators do not migrate and same environment characteristics exist in both the patches. After substituting the fixed point FP in the Variation matrix V , we get a1 Ss1 Ss2 Ss3

Ss8 0 0 1 Ss10
CONCLUSION

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