Asymmetry Demystified: Strict CLFs and Feedbacks for Predator-Prey Interconnections

TL;DR

This paper develops new strict control Lyapunov functions for predator-prey population models, enabling global stabilization with positive states and inputs, using generalized Volterra functions and tailored feedback design methods.

Contribution

It introduces generalized Volterra-style Lyapunov functions and feedback design techniques for strict stabilization of predator-prey systems with positivity constraints.

Findings

New class of strict CLFs for predator-prey dynamics

Generalization of classical Volterra Lyapunov functions

Examples of feedback and CLF design using forwarding and backstepping

Abstract

The difficulty with control of population dynamics, besides the states being positive and the control having to also be positive, is the extreme difference in the dynamics near extinction and at overpopulated states. As hard as global stabilization is, even harder is finding CLFs that are strict, don't require LaSalle arguments, and permit quantification of convergence. Among the three canonical types of two-population dynamics (mutualism, which borders on trivial, predator-prey, and competition, which makes global stabilization with positive harvesting impossible), predator-prey is the ``sweet spot'' for the study of stabilization. Even when the predator-prey interaction is neutrally stable, global asymptotic stabilization with strict CLFs has proven very difficult, except by conservative, hard-to-gain-insight-from Matrosov-like techniques. In this little note we show directions for the design of clean, elegant, insight-bearing, majorization-free strict CLFs. They generalize the classical Volterra-style Lyapunov functions for population dynamics to non-separable Volterra-style constructions. As a bonus to strictification as an analysis activity, we provide examples of concurrent designs of feedback and CLFs, using customized versions of forwarding and backstepping (note that, in suitable coordinates, predator-prey is both strict-feedforward and strict-feedback), where the striking deviations from these methods' conventional forms is necessitated by the predator-prey's states and inputs needing to be kept positive.