Pattern formation on coupled map lattices induced by cross-diffusion

This paper investigates a general two-dimensional discrete model with self-diffusion and cross-diffusion characteristics. We construct the model using the method of coupled map lattices. By conducting bifurcation analysis and Turing instability analysis on the model, we reveal the crucial role of cross-diffusion in the formation of Turing patterns. Through this study, we gain a deeper understanding of the importance of cross-diffusion in discrete dynamics and provide new insights and approaches for research in related fields. As an application, we apply the theory to two practical models and get very meaningful conclusions. For predator–prey model, cross diffusion coefficient determines the level of danger and driving force exerted by the predators on the prey. When the predators pose a lower level of danger and exert a weaker driving force on the prey, the prey population can maintain a spatially homogeneous state. However, when the predators pose a higher level of danger and exert a stronger driving force on the prey, the prey population is likely to exhibit a chaotic and disordered state due to continuous disturbances and fleeing. For tree–grass model, the results reveal indicate that the spatial distribution patterns of tree–grass populations are jointly determined by the frequency of fire occurrences and the effects of cross-diffusion. The former influences the direction of evolution low fire frequency leads to forest evolution, while high fire frequency leads to grassland evolution. The latter affects whether the distribution is uniform; weak cross-diffusion effects allow the tree–grass population to maintain a spatially uniform distribution, while strong cross-diffusion effects are likely to lead to a non-uniform and irregular patchy distribution of the tree–grass population.