Normal forms of double Hopf bifurcation for a reaction-diffusion system with delay and nonlocal spatial average and applications.

In this paper, we are concerned with a reaction-diffusion model incorporating delay and nonlocal effects. The normal form of double Hopf bifurcation is derived. The diffusive model of pollen tube tip growth is discussed and numerical simulations show that spatially homogeneous and inhomogeneous periodic solutions can be both stable or connected by a heteroclinic orbit under certain conditions. In addition, the diffusive Lotka-Volterra model with delay and nonlocality is considered and spatially inhomogeneous quasi-periodic solution is obtained. • Derive the algorithm of normal form of double Hopf bifurcation for reaction-diffusion system with delay and spatial average. • Investigate the dynamics near the double Hopf bifurcation point for two biological models. • Find bistability of homogeneous and inhomogeneous periodic solutions and pattern transitions for pollen tube tip growth model. • Find stable inhomogeneous quasi-periodic solutions and pattern transitions of periodic and quasi-periodic solutions for Lotka-Volterra model. [ABSTRACT FROM AUTHOR]