Integrable nonlinear reaction-diffusion population models for fisheries

Sustainable natural resource management strategies are required to maintain a fishery. One common device is to designate a No-Take Area (NTA) or a less restrictive Marine Protected Area (MPA) within a fishery. In this work, exact time-dependent solutions to 2-D nonlinear reaction-diffusion equations of Fisher type are constructed to model the fish population density of a single species within an NTA. The spillover rate from an NTA, important for harvesters, has previously been calculated without allowing for spatial variation within the NTA. Here, that spatial dependence is incorporated in the calculation. Spatially-dependent and density-dependent diffusivity and birth rates are considered for linear, rectangular and circular geometries, the latter being the most efficient with minimal critical domain area. The effect of the spatial dependence of diffusivity and birth rates, common in fish populations where reproduction and protection are favoured towards the centre, on the minimum size of an exclusion zone that ensures species survival, is studied. It is shown that when any one of these factors is taken into account, the size of the critical domain decreases by a small amount.