Modeling fish population movements: from an individual-based representation to an advection-diffusion equation.

In this paper, we address the problem of modeling fish population movements. We first consider a description of movements at the level of individuals. An individual-based model is formulated as a biased random walk model in which the velocity of each fish has both a deterministic and a stochastic component. These components are function of a habitat suitability index, h, and its spatial gradient nablah. We derive an advection-diffusion partial differential equation (PDE) which approximates this individual-based model (IBM). The approximation process enables us to obtain a mechanistic representation of the advection and diffusion coefficients which improves the heuristic approaches of former studies. Advection and diffusion are linked and exhibit antagonistic behaviors: strong advection goes with weak diffusion leading to a directed movement of fish. On the contrary weak advection goes with strong diffusion corresponding to a searching behavior. Simulations are conducted for both models which are compared by computing spatial statistics. It is shown that the PDE model is a good approximation to the IBM.