Stabilization of Predator–Prey Age-Structured Hyperbolic PDE When Harvesting Both Species is Inevitable

Populations (in ecology, epidemics, biotechnology, economics, social processes) do not only interact over time but also age over time. It is therefore common to model them as “age-structured” partial differential equations (PDEs), where age is the ‘space variable.' Since the models also involve integrals over age, both in the birth process and in the interaction among species, they are in fact integro-partial differential equations (IPDEs) with positive states. To regulate the population densities to desired profiles, harvesting is used as input. But non-discriminating harvesting, where wanting to repress one (overpopulated) species will inevitably repress the other (near-extinct) species as well, the positivity restriction on the input (no insertion of population, only removal), and the multiplicative (nonlinear) nature of harvesting, makes control challenging even for ordinary differential equation (ODE) versions of such dynamics, let alone for their IPDE versions, on an infinite-dimensional nonnegative state space. With this paper, we introduce a design for a benchmark version of such a problem: a two-population predator-prey setup. The model is equivalent to two coupled ODEs, actuated by harvesting which must not drop below zero, and strongly (“exponentially”) disturbed by two autonomous but exponentially stable integral delay equations (IDEs). We develop two control designs. With a modified Volterra-like control Lyapunov function, we design a simple feedback which employs possibly negative harvesting for global stabilization of the ODE model, while guaranteeing regional regulation with positive harvesting. With a more sophisticated, restrained controller we achieve regulation for the ODE model globally, with positive harvesting. For the full IPDE model, with the IDE dynamics acting as large disturbances, for both the simple and saturated feedback laws we provide explicit estimates of the regions of attraction. Simulations illustrate the nonlinear infinite-dimensional solutions under the two feedbacks. The paper charts a new pathway for control designs for infinite-dimensional multi-species dynamics and for nonlinear positive systems with positive controls.