Detection and Continuation of a Border Collision Bifurcation in a Forest Fire Model

The behavior of the simplest forest fire model is studied in this paper through bifurcation analysis. The model is a second-order continuous-time impact model where vegetational growth is described as a continuous and slow dynamic process, while fires are modeled as instantaneous and disruptive events. The transition from Mediterranean forests (characterized by wild chaotic fire regimes) to savannas and boreal forests (where fires are almost periodic) is recognized to be a catastrophic transition known as border collision bifurcation in the context of discrete-tine systems. In the present case such a bifurcation can be easily detected numerically and then continued by solving a standard boundary-value problem. The result of the analysis complements previous simulation studies and are consistent with biological intuition

Dynamic Properties of a Forest Fire Model

The reaction-diffusion equations have been widely used in physics, chemistry, and other areas. Forest fire can also be described by such equations. We here propose a fighting forest fire model. By using the normal form approach theory and center manifold theory, we analyze the stability of the trivial solution and Hopf bifurcation of this model. Finally, we give the numerical simulations to illustrate the effectiveness of our results