Monday, July 1st, 2024 — 3:45 pm — Aula M1.6, edificio Matematica
On the dynamics of two competing populations through a mixed hyperbolic–parabolic system
Abstract: We focus on the coupling of a hyperbolic balance law and a parabolic equation to model the behaviour of two competing populations. The mixed system is derived adding space distribution to Lotka–Volterra system of ODEs. The main idea is that one population, the prey, diffuses according to a parabolic equation, while the movement of the other population, the predators, is directed towards the regions where the concentration of prey is greater. This is modelled by a non local, and possibly non linear, function on the prey’s density in the flux of the balance law. We allow for more general interaction terms in the sources and also include control functions. Aim of the study of this class of mixed hyperbolic–parabolic system is to provide a usable structure for the search of an optimal control strategy in biological pest control problems.
We present analytical results regarding the well posedness of this class of systems both on the whole space $\mathbb{R}^n$ and on bounded domains, under suitable boundary conditions. In particular, existence and uniqueness are obtained by an application of Banach Fixed Point Theorem. Moreover, we obtain a full set of a priori and stability estimates in view of the interest about control problems based on these equations. A key feature is the fact that the parabolic equation is settled in $\mathbf{L}^1$ , the choice being motivated by the clear physical meaning of total population attached to this norm, whenever solutions are positive.
We conclude the presentation with some numerical integrations, to give an insight into the qualitative properties of the solutions.
Joint works with Rinaldo M. Colombo and Abraham Sylla (University of Brescia).