To connect or not to connect isolated patches

Empirical evidence suggests that dispersal can have different effects on the time evolution of a spatially structured population. In this study, we explored the impact of the migration rate in a coupled map lattice system. To capture this impact, we assumed symmetric dispersal and simple dynamics in the local populations. However, we allowed heterogeneity between the patches, including both source-source and source-sink systems. Our results show that this simple theoretical setting has the potential to unify the diversity of behaviours of the total population size observed in previous studies. Indeed, we found that the response of the total population size to migration was non-monotone in source-source and many source-sink situations, thereby suggesting that an increase in the dispersal rate could be related to either an increase or a decrease in the total population size. As we will illustrate, this response provides a possible theoretical explanation of some benefits of control strategies involving spatial considerations as no-take zones. Our study also analyses the impact of the migration rate on persistence, spatial coherence, and initial transients. This was motivated by previous theoretical observations in coupled systems that the rate of migration affects these three aspects. Related to persistence, we rigorously extended a previous result from the linear to the non-linear case. This result essentially states that persistence depends on the stability of the origin. On the other hand, we stress that negative effects due to an increase of the spatial coherence could be neutralised by the unimodal response of the total population size. Finally, the study of the initial transients, which is relevant for interpreting experimental results, highlights that the relationship between the rate of migration and the total population size remains the same even in the transient phase.