Asymptotic behavior of solutions to some classes of multi-order fractional cooperative systems

TL;DR

This paper investigates the long-term behavior of solutions to multi-order fractional cooperative systems, establishing boundedness, attractivity, and convergence rates, with applications to ecosystem models and numerical validation.

Contribution

It provides new theoretical results on the asymptotic behavior of multi-order fractional systems, including boundedness, attractivity, and convergence rates, which were previously unknown.

Findings

Solutions are bounded under certain conditions.

Solutions converge to equilibrium with sharp rates when derivatives have equal order.

Numerical examples confirm theoretical results.

Abstract

This paper is devoted to the study of the asymptotic behavior of solutions to multi-order fractional cooperative systems. First, we demonstrate the boundedness of solutions to fractional-order systems under certain conditions imposed on the vector field. We then prove the global attractivity and the convergence rate of solutions to such systems (in the case when the orders of fractional derivatives are equal, the convergence rate of solutions is sharp and optimal). To our knowledge, these kinds of results are new contributions to the qualitative theory of multi-order fractional positive systems and they seem to have been unknown before in the literature. As a consequence of this result, we obtain the convergence of solutions toward a non-trivial equilibrium point in an ecosystem model (a particular class of fractional-order Kolmogorov systems). Finally, some numerical examples are also provided to illustrate the obtained theoretical results.