The consequences of non-passive advection and directed motion for population dynamics

A. R. Robinson has proposed a method for studying population growth and advection in the sea. The assumption made by Robinson of a non–divergent velocity field for the biological tracer may not be valid for many situations. When the assumption is valid the biological dynamics of an advected patch are no different from those of a stationary patch, unless a greater degree of spatial structure is incorporated than he considers. The generality and applicability of the model are increased if the assumption of non–divergence can be relaxed. If the velocity field associated with a patch is convergent, as can occur for many biological tracers, then new dynamics arise, both in a Lagrangian sense and when working at the whole–patch level. These new dynamics are explored for four fundamental examples, which could provide new paradigms for spatially explicit models in mathematical ecology.