On inverse problems in predator-prey models

Abstract

In this paper, we consider the inverse problem of determining the coefficients of interaction terms within some Lotka-Volterra models, with support from boundary observation of its non-negative solutions. In the physical background, the solutions to the predator-prey model stand for the population densities for predator and prey and are non-negative, which is a critical challenge in our inverse problem study. We mainly focus on the unique identifiability issue and tackle it with the high-order variation method, a relatively new technique introduced by the second author and his collaborators. This method can ensure the positivity of solutions and has broader applicability in other physical models with non-negativity requirements. Our study improves this method by choosing a more general solution $(u_0,v_0)$ to expand around, achieving recovery for all interaction terms. By this means, we improve on the previous results and apply this to physical models to recover coefficients concerning compression, prey attack, crowding, carrying capacity, and many other interaction factors in the system. Finally, we apply our results to study three specific cases: the hydra-effects model, the Holling-Tanner model and the classic Lotka-Volterra model.