About the minimal time crisis problem in the prey-predator Lotka-Volterra model
Résumé
We consider the classical prey-predator Lotka-Volterra model with a control on the mortality of the predator. We consider that the state of the system is in "crisis" or in danger when the density of the preys is below a given threshold. We first study the viability kernel associated to the constraint to stay above this threshold for any time. We then study the optimal control which consists in minimizing the time spent below the threshold (i.e. minimizing the time crisis) over finite and infinite horizon. We show that the optimal trajectory does not necessarily correspond to the minimal time to reach the viability kernel.