Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation

Abstract

The standard model for the dynamics of a fragmented density-dependent population is built from several local logistic models coupled by migrations. First introduced in the 1970s and used in innumerable articles, this standard model applied to a two-patch situation has never been completely analysed. Here, we complete this analysis and we delineate the conditions under which fragmentation associated to dispersal is either beneficial or detrimental to total population abundance. Therefore, this is a contribution to the SLOSS question. Importantly, we also show that, depending on the underlying mechanism, there is no unique way to generalize the logistic model to a patchy situation. In many cases, the standard model is not the correct generalization. We analyse several alternative models and compare their predictions. Finally, we emphasize the shortcomings of the logistic model when written in the r-K parameterization and we explain why Verhulst's original polynomial expression is to be preferred.

Carrying capacity in a heterogeneous environment with habitat connectivity

“…Importantly, though, Arditi et al . () and DeAngelis et al . () proved using the Pearl–Verhulst model that diffusion increases TRAPA only if there exists a positive relationship between the phenomenological logistic parameters r and K (see Mallet () for review of evidence for a positive r‐K relationship).…”

“…These surprising results were called ‘somewhat paradoxical’ by Arditi et al . (). Lou () extended these results to a continuous spatial setting by using a reaction–diffusion model, and again arrived at the same paradoxical conclusion (see also DeAngelis et al .…”

“…With a negative r‐K relationship, diffusion is mathematically predicted to decrease the TRAPA (Arditi et al . ). This requirement can be understood as follows (see also Fig.…”

“…For spatially distributed species, the determinants of total realised asymptotic population abundance (Arditi et al . ) (abbreviated TRAPA here for convenience) are, surprisingly, more complex than simply the sum of the local carrying capacities over every point in space.…”

“…, Supporting Information and Arditi et al . ; DeAngelis et al . ): with a positive r‐K correlation, highly productive patches (high K ) are able to compensate for much of the emigration (high r ).…”