This Demonstration combines the results of  and ;  discusses the impact of nonlinear benefits in public goods games, while  incorporated ecological dynamics into standard evolutionary game dynamics.
Here the nonlinearity is given by the synergy/discounting parameter
, we have linearity. When
, the benefits get added up synergistically, with each additional cooperator contributing more than the previous one. When
, the benefits saturate for a certain number of cooperators. The frequencies of cooperators and defectors in the population are given by
. Since each individual in the population has one of these two strategies, we have
. However, the sum
may be less than the carrying capacity of the environment. Hence if we consider
to be the fractions of the two strategies, then we can have
as the fraction of empty spaces. Tracking the ecological dynamics then amounts to looking at the temporal evolution of the population density
, while the evolutionary dynamics are the temporal evolution of the strategies within the population density. The dynamics of the empty sites and the fraction of cooperators
are given by
is the benefit multiplier;
is the death rate;
is the synergy/discounting factor, and
is the group size.