Hopf bifurcation of a diffusive Gause-type predator-prey model induced by time fractional-order derivatives

Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional-order derivatives into a diffusive Gause-type predator-prey model, which is time fractional-order reaction-diffusion equations and a generalized form of its corresponding first-derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional-order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause-type predator-prey model in the forms of the time fractional-order ordinary equations and of the time fractional-order reaction-diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional-order derivatives. Some numerical simulations are made to verify our results.