Analysis and simulation of an integro-differential Lotka–Volterra model with variable reproduction rates and optimal control

In this work, we present an integro-differential system that generalizes the classical Lotka–Volterra model of competition. The model considers population heterogeneity with respect to reproduction rates, local diffusion in the aspect space and control interventions. We perform a rigorous mathematical analysis proving results on existence and uniqueness of solutions and on existence of optimal controls. We also perform numerical simulations illustrating different behaviors and the role of model parameters on survival and extinction of each population.