Pattern Dynamical Behaviors of One Type of Tree-Grass Model with Cross-Diffusion

Abstract

One type of tree-grass competition model with cross-diffusion is analyzed in this paper. Firstly, according to the distribution theory of the roots of the characteristic equation, the existence and stability conditions of the positive equilibrium point are obtained, and the global stability of the system is proved by using the Bendixson–Dulac theorem. Secondly, we select the fire factor as the control parameter, and obtain the critical value of Turing bifurcation and the conditions of Turing pattern. Then, we derive the amplitude equation near the Turing bifurcation point by using the standard multiscale analysis method. Finally, we do a sensitivity analysis on the tree-grass system with the fire factor, and then a series of numerical simulations based on these analysis results are carried out to verify our theoretical analysis and Turing pattern structure, which is consistent with the biological significance of the tree-grass model.

The Diffusion-Driven Instability for a General Time-Space Discrete Host-Parasitoid Model

“…Te reaction-difusion models can be further subdivided into continuous reaction-difusion models [12][13][14][15] and discrete reaction-difusion models [16,17]. In biomathematical models, continuous models can be used to describe organisms that are based on a large amount of data, but for organisms in which many movements and changes occur in discrete forms and the data collected and recorded are also in discrete forms; constructing discrete models to describe them will be more in line with objective reality such as host-parasitoid models.…”