Using the Lotka-Voletraa equations, researchers present a model for stochastic evolutionary game dynamics that accommodates for changing population size. The presented model is one of the first game theoretic models to adopt a stochastic approach while incorporating change in population size. The key point is that this model has stochastic evolutionary game dynamics and allows for ‘natural’ changes in population size and structure.
Deterministic equations – for a two type species model (x & y)
Stochastic equations – for species x & y
Stochastic equations were implemented using the Gillespie algorithm, and selected game theory games were dominance, coexistence, and coordination.
Comparisons of stochastic and deterministic equations can be viewed in figure 1 (A – C), where stochastic simulations typically fluctuated around their deterministic counterpart. Typically, increasing carrying capacity and population size results in increasing estimated probabilities for population extinction, which can be seen in figure 2.
Stochastic simulation in the coexistence game provided some interesting results as in seen in figure 3. In two scenarios the stochastic extinction of one species led to an increased carrying capacity of the other, which seemed to increase the existence time of the surviving species when compared to true coexisting model.
One last note was that authors indicated that stochastic simulations dramatically reduced the chances of a single dominant individual invading the weaker population.