A necessary condition for dispersal driven growth of populations with discrete patch dynamics

•We consider when coupling between discrete sink populations causes overall growth.•A new necessary condition in terms of a common linear Lyapunov function is derived.•The result provides insight into life-histories suitable for dispersal driven growth.•The results apply to classes of both linear and nonlinear coupled models. We revisit the question of when can dispersal-induced coupling between discrete sink populations cause overall population growth? Such a phenomenon is called dispersal driven growth and provides a simple explanation of how dispersal can allow populations to persist across discrete, spatially heterogeneous, environments even when individual patches are adverse or unfavourable. For two classes of mathematical models, one linear and one non-linear, we provide necessary conditions for dispersal driven growth in terms of the non-existence of a common linear Lyapunov function, which we describe. Our approach draws heavily upon the underlying positive dynamical systems structure. Our results apply to both discrete- and continuous-time models. The theory is illustrated with examples and both biological and mathematical conclusions are drawn.