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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:
34F05 Ordinary differential equations and systems with randomness
92D25 Population dynamics (general)
34C25 Periodic solutions to ordinary differential equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
34C23 Bifurcation theory for ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Software:
Matlab
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