A mathematical analysis on public goods games in the continuous space

We consider the population dynamics of two competing species sharing the same resource, which is modeled by the carrying capacity term of logistic equation. One species (farmer) increases the carrying capacity in exchange for a decreased survival rate, while the other species (exploiter) does not. As the carrying capacity is shared by both species, farmer is altruistic. The effect of continuous spatial structure on the performance of such strategies is studied using the reaction diffusion equations. Mathematical analysis on the traveling wave solution of the system revealed; (1) Farmers can never expel exploiters in any traveling wave solution. (2) The expanding velocity of the exploiter population invading the farmer population can be analytically determined and it depends only on a cost of altruism and the diffusion coefficients while it is independent of the benefit of altruism. (3) When the effect of altruism is small, the dynamics of the invasion of exploiters obeys the Fisher-KPP equation. Numerical calculations confirm these results.