Ratio-dependent predator-prey models have been increasingly favored by fieldecologists where predator-prey interactions have to be taken into account theprocess of predation search. In this paper we study the conditions of theexistence and stability properties of the equilibrium solutions in areaction-diffusion model in which predator mortality is neither a constant noran unbounded function, but it is increasing with the predator abundance. Weshow that analytically at a certain critical value a diffusion driven (Turingtype) instability occurs, i.e. the stationary solution stays stable withrespect to the kinetic system (the system without diffusion). We also show thatthe stationary solution becomes unstable with respect to the system withdiffusion and that Turing bifurcation takes place: a spatially non-homogenous(non-constant) solution (structure or pattern) arises. A numerical scheme thatpreserve the positivity of the numerical solutions and the boundedness of preysolution will be presented. Numerical examples are also included.
展开▼
