  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.
