Global attractor and stabilization for a coupled PDE-ODE system

TL;DR

This paper investigates the long-term behavior of a coupled PDE-ODE system modeling forest ecosystems, establishing the existence of a global attractor in the monotone case and proving stabilization to equilibrium in the non-monotone case with small coupling.

Contribution

It introduces new results on the global attractor and stabilization for a coupled PDE-ODE system in biological modeling, addressing both monotone and non-monotone cases.

Findings

Existence of a smooth global attractor with finite Hausdorff and fractal dimension in the monotone case.

Stabilization of solutions to a single equilibrium when the coupling constant is sufficiently small in the non-monotone case.

Complexity of the equilibrium set in the non-monotone scenario, including discontinuous solutions.

Abstract

We study the asymptotic behavior of solutions of one coupled PDE-ODE system arising in mathematical biology as a model for the development of a forest ecosystem. In the case where the ODE-component of the system is monotone, we establish the existence of a smooth global attractor of finite Hausdorff and fractal dimension. The case of the non-monotone ODE-component is much more complicated. In particular, the set of equilibria becomes non-compact here and contains a huge number of essentially discontinuous solutions. Nevertheless, we prove the stabilization of any trajectory to a single equilibrium if the coupling constant is small enough.