  The spatial arrangement of individuals is thought to overcome the dilemma of
cooperation: When cooperators engage in clusters they might share the benefit
of cooperation while being more protected against non-cooperating individuals,
which benefit from cooperation but save the cost of cooperation. This is
paradigmatically shown by the spatial prisoner's dilemma model. Here, we study
this model in one and two spatial dimensions, but explicitly take into account
that in biological setups fitness collection and selection are separated
processes occurring mostly on vastly different time scales. This separation is
particularly important to understand the impact of mobility on the evolution of
cooperation. We find that even small diffusive mobility strongly restricts
cooperation since it enables non-cooperative individuals to invade cooperative
clusters. Thus, in most biological scenarios, where the mobility of competing
individuals is an irrefutable fact, the spatial prisoner's dilemma alone cannot
explain stable cooperation but additional mechanisms are necessary for spatial
structure to promote the evolution of cooperation. The breakdown of cooperation
is analyzed in detail. We confirm the existence of a phase transition, here
controlled by mobility and costs, which distinguishes between purely
cooperative and non-cooperative absorbing states. While in one dimension the
model is in the class of the Voter Model, it belongs to the Directed
Percolation (DP) universality class in two dimensions.
