A Mathematical Modeling Approach to Analyse the Effect of Additional Food in a Predator-Prey Interactions with a White Gaussian Noise in Prey’s Growth Rate
Tóm tắt
In this paper a prey-predator mathematical model including the additional food for predator and refuge for prey is considered with a Holling functional response of kind II. The mathematical analyses for positivity, existence and stability of the system around the equilibria are presented. In the first part, it is shown that an increase in the amount of additional food for predator causes a substantial change in the dynamics of both species, where Hopf bifurcation takes place around positive coexisting state with a stable limit cycle and periodic oscillations are initiated at a critical point. In the second part of the paper, a more realistic case is considered and the model is analysed with noise term incorporated in the linear growth rate of prey population. The presence of noise term turns the given prey-predator model into a stochastic differential equation, and non-periodic noise related oscillations, leading to chaotic dynamics, can be observed in both prey and predator densities. It is presented that the criteria for the stability of deterministic model do not ensure the stability of stochastic model. Besides, the conditions for extinction of the species with noise is also numerically and mathematically investigated. The numerical and analytical results of this paper may give useful insights into the persistence and extinction of species for real world applications in ecology.
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