Mathematical Study on Immigration and Migration of Prey-Predator Eco System with Unlimited Resources for the Predator

Objectives: To study the stability of two species in a prey-predator eco-system, immigration and migration with unlimited resources for the predator. Method: The system consists of a prey (S1), a predator (S2) that migrates and immigrates to subsist off of S1, and a prey (S1). In this instance, the second species' limitless resources are quantified by the corresponding carrying capacity. A collection of first-order non-linear ordinary differential equations make up the system's model equations. Findings: The stability of each of the three equilibrium points is examined and determined. If all of the characteristic roots are negative (if they are real) and have negative real parts (if they are complex), the system would be stable. To determine the stability criteria, the linearized equations for the perturbations over the equilibrium points are analysed. The global stability of linearized equations can be discussed by constructing a suitable Liaupnov's function. Additionally, the Runge-Kutta fourth order technique is used to compute the numerical solutions for the growth rate equations. Novelty: The connection where one species benefits at the other's expense is called predation. In this mutualistic relationship, one creature takes up the other's biomass. We refer to this as a predator. Because of their tight relationship, one organism in this interaction consumes the other. A tiger and a deer are two examples of predator and prey in the real world. The tiger feeds on other organisms, and the deer are its prey.