Stabilization of Age-Structured Competing Populations

TL;DR

This paper develops a control strategy for stabilizing unstable, age-structured competing predator populations modeled by nonlinear IPDEs, transforming the problem into coupled ODEs and IDEs, and proving stability via backstepping.

Contribution

It introduces a novel backstepping control design for stabilizing unstable age-structured competition models with only partial actuation, extending stability analysis to IPDE systems.

Findings

Global stabilization of the model achieved with backstepping.

Region of attraction for the stabilized equilibrium characterized.

Full IPDE system shown to be locally exponentially stable.

Abstract

Age-structured models capture the dynamic behavior of populations over time and result in nonlinear integro-partial differential equations (IPDEs). These processes arise in various fields such as biotechnology, economics, or demography. While coupled age-structured IPDEs modeling two or more interacting species occur naturally in epidemiology and ecology, they remain relatively underexplored. Prior work has primarily addressed stable and marginally stable dynamics. In constrast, this work considers an exponentially unstable model of two competing predator populations, formally referred to in the literature as ``competition'' dynamics. If one were to apply an input that simultaneously harvests both predator species, one would have control over only the product of the densities of the species, not over their ratio. Therefore, it is necessary to design a control input that directly harvests only one of the two predator species, while indirectly influencing the other via a backstepping approach. The model is transformed into a system of two coupled ordinary differential equations (ODEs), of which only one is actuated, and two autonomous, exponentially stable integral delay equations (IDEs) which enter the ODEs as nonlinear disturbances. The ODEs are globally stabilized with backstepping and an estimate of the region of attraction of the asymptotically stabilized equilibrium of the full IPDE system is provided, under a positivity restriction on control. Additionally, the full IPDE system is also shown to be local exponential stable. Such generalizations of competition dynamics open exciting possibilities for future research directions for systems with more than two species.