Stratified mobility fishery models with harvesting outside of no-take areas

Abstract

Mobility stratification, identifiable from k-means clustering on an appropriate displacement data set, is a common feature of many fish species wherein distinct low-mobility ‘station-keeper’ and high-mobility ‘ranger’ types are recognized. From recapture records of speckled snapper Lutjanus rivulatus, we develop a Gaussian mixture model of the probability density function for random displacements by the two types. This leads to a system of two coupled reaction-diffusion equations. We consider a single no-take area (NTA) in one and two dimensions containing a mobility-structured species. The minimum size of this NTA that leads to species survival is derived and then generalised to a population with n mobility types. Exact non-uniform 1-D steady states are constructed for the full nonlinear mobility-structured model with lethal (zero density boundary condition) harvesting outside of the NTA. This model is then extended to include an array of evenly spaced NTAs with a bounded harvesting rate allowed between them. The minimum size of linear, circular and annular NTAs and the maximum sizes of the surrounding fractionally harvested zones that ensure species survival and connectivity are calculated.