Stabilization of Age-Structured Competing Populations

Abstract: Age-structured models represent the dynamic behaviors of populations over time and result in integro-partial differential equations (IPDEs). Such processes arise in biotechnology, economics, demography, and other domains. Coupled age-structured IPDE population dynamics with two or more species occur in epidemiology and ecology, but have received little attention thus far. 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. These generalizations open exciting possibilities for future research directions, such as investigating population systems with more than two species.