Lotka-Volterra equations are considered a dynamical game, where
the phenotypes of the predator and of the prey can vary. This differs from the
usual procedure of specifying as a priori laws according to which strategies are
supposed to change. The question at stake is the survival of each of the species,
instead of the maximization of a given pay-off by each player, as it is
commonly discussed in games. The predator needs the prey, while the prey
can survive without the predator.
These obvious and simplistic constraints are enough to shape the regulation
of the system: notably, the largest closed set of initial conditions can be
delineated, from which there exists at least one evolutionary path where the
population can avoid extinction forever. To these so-called viable trajectories,
viable strategies are associated, respectively for the prey or for the predator.
A coexistence set can then be defined.
Within this set and outside the boundary, strategies can vary arbitrarily
within given bounds while remaining viable, whereas on the boundary, only
specific strategies can guarantee the viability of the system. Thus, the largest
set can be determined, outside of which strategies will never be flexible
enough to avoid extinction.