Competition between foraging predators and hiding preys as a nonzero-sum differential game

In this work we investigate a (seasonal) prey-predator model where the system evolves during a season whose length is fixed. Predators have the choice between foraging the food (eating preys) and reproducing (laying eggs at a rate proportional to their energy). Preys can either eat, which would maintain their population in the absence of predators, or hide from the predators but they then suffer a positive mortality rate. In this case the population size can decrease even faster than if they were not hiding and were foraged by the predators. In their own turn they lay eggs at a constant rate whether they are hiding or eating. Following Darwin´s principle that the fittest population will survive we postulate that both populations must maximize the number of their offspring, which yields a nonzero-sum differential game.