In this talk, we apply viability theory to a simple Lotka-Volterra control system, where both prey and predator population seek to remain above a danger threshold. The system is inspired by a paper from Bonneuil and Müllers. We consider two cases :
1) The prey and the predator cooperate to maintain both populations in the safe region. In that case, the study is easy and is merely a computation of a viability kernel. We verify that the intersection of the viability kernels equals the viability kernel of the intersection of the two constraints, a remarkable property since one usually only has one inclusion.
2) The prey assumes the predator is hostile. The controls are not chosen with the same purpose. In that case, one needs to make use of a stronger notion of viability, namely the discriminating kernel. The system therefore becomes a pursuit-evasion game.
In both cases, the kernels are computed by integrating the Hamiltonian system backward from a remarkable point in the boundary of the constraints. We find numerically in both cases that the boundary of the kernels contains a subset of the boundary of the constraints. While a proof is easy enough in case 1), it remains a conjecture in case 2), even though we have strong evidence suggesting it is true. This property suggests that, even when one is close to the danger threshold, it is possible for the system to remain viable.
We end the talk with some opening questions regarding finding a compromise between the two species whenever they both choose to be prudent (i.e. when they assume the worst case where their opponent is hostile).