Changes in habitat of fish populations: An inverse problem

• A collage-based approach is used to solve an inverse problem for a fish population model.
• A non-linear parabolic partial differential equation model is used to model the changes in habitat of fish in a dammed river in Brazil.
• Numerical methods are used to develop and implement several example problems to recover the diffusion coefficient in the model.
• Variations in data collection is simulated and the robustness of the model is shown even with noisy data.

Mathematical modelling applies to a wide variety of application areas, and is an active area of research in many disciplines. It is often the case that accurate depiction of real-world phenomena require increasingly complex models. Unfortunately, this increased complexity in a model causes great difficulty when seeking solutions. What is more, developing a model with known parameters that produces results consistent with observed behaviors may prove to be a difficult or even impossible task. These difficulties have brought about an interest in inverse problems.In this paper we utilize a collage-based approach to solve an inverse problem for a model for the migration of three fish species through floodplain waters. A derivation of the mathematical model is presented and a generalized collage method is discussed and applied to this model to recover diffusion parameters. Theoretical and numerical particulars are discussed and results are presented.