Asymptotic Behavior of a Degenerate Forest Kinematic Model With a Perturbation

Abstract

In this paper, we study the asymptotic behavior of the global solutions to a degenerate forest kinematic model, under the action of a perturbation modeling the impact of climate change. In the case where the main nonlinear term of the model is monotone, we prove that the global solutions converge to a stationary solution, by showing that the Lyapunov function derived from the system satisfies a Łojasiewicz–Simon gradient inequality. We also present an original algorithm, based on the Statistical Model Checking framework, to estimate the probability of convergence toward nonconstant equilibria. Furthermore, under suitable assumptions on the parameters, we prove the continuity of the flow and of the stationary solutions with respect to the perturbation parameter. Then, we succeed in proving the robustness of the weak attractors, by considering a weak topology phase space and establishing the existence of a family of positively invariant regions. At last, we present numerical simulations of the model and explore the behavior of the solutions under the effect of several types of perturbations. We also show that the forest kinematic model can lead to the emergence of chaotic patterns.