Spatiotemporal pattern formation and selection induced by cross-diffusion in a cancer growth model with Allee effect

In this paper, we study the temporal and spatiotemporal dynamics of a cancer growth model with the Allee effect. For the temporal model, we first prove its well-posedness and analyze its rich local bifurcations, such as the transcritical bifurcation, the saddle-node bifurcation, the Hopf bifurcation, and the Bogdanov-Takens bifurcation. For the diffusion system, we first prove that cross-diffusion is a key mechanism for the formation of spatial patterns. Then, we study the Turing instability and derive the amplitude equation by using the weak nonlinear analysis method. Using the stability of the amplitude equation, we discuss the hexagonal patterns and stripe patterns and its combinations of several kinds of patterns under different parameter conditions. Finally, all possible cases of the system under different parameter sets are investigated, and a large number of numerical simulations are carried out to verify the analytical results. Further, we compare the stochastic model with Gaussian white noise to the deterministic model, and we find that the noise can improve the stability.