Statement of the Main Results

Here we describe our main results in mathematical details. First of all, we analyze the equilibria of the system, their linear stability, and the presence of a stable or center manifold, in dependence of the parameters. We also characterize the strategies that lead to the victory of the aggressive population. In this analysis, the case of populations exactly with the same fitness to the environment plays a special role, since in this situation the final outcome of the war is determined solely by the initial conditions and the specific strategy followed by the aggressive population does not play a major role. Instead, when the fitness levels of the two populations are different, complex scenarios arise and constant strategies are not sufficient to ensure victory starting from favorable conditions. The set of winning strategies can however be greatly simplified, by reducing it to the case of piecewise constant functions with at most one discontinuity. Finally, among all the possible winning strategies, we aim at detecting the one that minimizes the length of the war. For this, constant strategies are not enough, nor piecewise constant strategies with a jump discontinuity, and the quickest victory could be achieved through a strategy assuming some values along a singular arc.