Adapting the steepness parameter from stock–recruit curves for use in spatially explicit models

To inform the design of MPA networks and ensure that they will meet stated goals, spatially explicit metapopulation models are often used to simulate the response of fished species to MPA implementation. Typically, such models are simply spatial extensions of traditional, nonspatial population models used in fisheries management. A common assumption used in making this transition is that R′(0), the slope at the origin of the nonspatial, population-wide egg–recruit relationship (often termed the steepness or compensation ratio), can be used to estimate α, the slope at the origin of the small-scale settler–recruit relationship used in spatially explicit models. This assumption is not always correct. In particular, the value of R′(0) often implicitly accounts for a variety of processes spanning the egg–recruit transition, including larval mortality and advection away from suitable habitat. If a spatial model accounts for some of those loss processes explicitly, such as by using an oceanographically realistic dispersal matrix, it becomes necessary to adjust the estimate of α upwards to avoid double-counting those losses. Here I present a simple correction involving the dominant eigenvalue of the dispersal matrix that adjusts the value of R′(0) to avoid this error. Applying this correction factor ensures that a spatially explicit model will predict population collapse at the same level of fishing implied by a large-scale estimate of R′(0).