In this paper, we study the asymptotic behavior of the global solution to a degenerate forest kinematic model, under the action of a perturbation modelling the impact of climate change. When the main nonlinearity of the model is assumed to be monotone, we prove that the global solution converges to a stationary solution, by showing that a Lyapunov function deduced from the system satisfies a Lojasiewicz-Simon gradient inequality. Under suitable assumptions on the parameters, we prove the continuity of the flow and of the stationary solutions with respect to the perturbation parameter. Although, due to a lack of compactness, the system does not admit the global attractor, we succeed in proving the robustness of the weak attractors, by establishing the existence of a family of positively invariant regions. We also present numerical simulations of the model and experiment the behavior of the solution under the effect of several types of perturbations. Finally, we show that the forest kinematic model can lead to the emergence of chaotic patterns.