Analytic left inversion of multivariable Lotka-Volterra models

There is great interest in managing populations of animal species that are vital food sources for humans. A classical population model is the Lotka-Volterra system, which can be viewed as a nonlinear input-output system when time-varying parameters are taken as inputs and the population levels are the outputs. If some of these inputs can be actuated, this sets up an open-loop control problem where a certain population profile as a function of time is desired, and the objective is to determine suitable system inputs to produce this profile. Mathematically, this is a left inversion problem. In this paper, the general left inversion problem is solved for multivariable input-output systems that can be represented in terms of Chen-Fliess series using concepts from combinatorial Hopf algebras. The method is then applied to a three species, two-input, two-output Lotka-Volterra system. The biological goal is to change the population dynamics of the top-level predator species in a food chain in order to prevent extinction.