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Ratio-dependent predator-prey model: effect of environmental fluctuation and stability. (English) Zbl 1078.34035
A well known predator-prey model is investigated. A few different settings are under consideration. Firstly, the classical prey-predator model is analyzed with a ratio-dependent functional response. The dynamical behavior depending on the parametric restrictions is discussed. It is shown that under some conditions, the system exhibits Hopf-bifurcation and there exists a small amplitude periodic solution near a nonzero equilibrium point. A numerical example is presented. A sufficient condition providing global stability is derived. The last part of the paper is concerned with the effect of environmental fluctuation on the model system and its stochastic stability. In doing so, the authors introduce stochastic perturbation terms into the growth equations of both prey and predator populations. The equations are proposed to be ItĂ´ stochastic differential equations. Mean square stability is analyzed by means of a Lyapunov function. Necessary and sufficient conditions for the stability of an interior equilibrium point for the model system are obtained. Using a stochastic numerical scheme and MATLAB software, a numerical simulation is performed.
MSC:
34F05ODE with randomness
92D25Population dynamics (general)
34C25Periodic solutions of ODE
60H35Computational methods for stochastic equations
34C23Bifurcation (ODE)
34D23Global stability of ODE
60H10Stochastic ordinary differential equations
Software:
Matlab