Hopf bifurcation in a fractional diffusion food-limited models with feedback control

Abstract

In this paper, we consider the direction and stability of Hopf bifurcation induced by time delay in a food-limited models with feedback control and fractional diffusion. By means of analyzing eigenvalue spectrum, we show that the positive equilibrium is locally asymptotically stable in the absence of time delay, but loses its stability via the Hopf bifurcation when the time delay increases beyond a threshold. Using the norm form and the center manifold theory, we investigate the stability and direction of the Hopf bifurcation. The stability of the Hopf bifurcation leads to the emergence of spatial spiral patterns. Numerical calculations are performed to illustrate our theoretical results.