Bifurcation analysis of a diffusive radio-dependent model with spatial memory

The cognitive abilities of animals, such as memory, have a significant impact on their movement in space. In this paper, we consider a radio-dependent model with memory-based diffusion under the conditions of Neumann boundary. The stability of a positive equilibrium and the existence of the Turing–Hopf bifurcation induced by memory diffusion and memory delay are carried out in details. Notably, our findings indicate that with a relatively short average memory period, the large memory diffusion can stabilize an otherwise unstable equilibrium. In addition, the third-order truncated normal form for the Turing–Hopf bifurcation restricted to the central manifold is derived, which can reveal the generation of some steady-state and time-periodic solutions with spatial heterogeneity. The coefficients within the normal form are systematically determined through matrix operations, and these results can also be applied to other models with memory diffusion. Ultimately, leveraging the theoretical findings, we elucidated the intricate spatial-temporal dynamics and their associated parameter scopes caused by Turing–Hopf bifurcation through numerical simulations.