# Well-posedness and qualitative properties of a dynamical model for the ideal free distribution

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## Abstract

Understanding the spatial distribution of populations in heterogeneous environments is an important problem in ecology. In the case of a population of organisms that can sense the quality of their environment and move to increase their fitness, one theoretical description of the expected distribution of the population is the ideal free distribution, where individuals locate themselves to optimize fitness. A model for a dynamical process that allows a population to achieve an ideal free distribution was proposed by the Cosner (Theor Popul Biol 67:101–108, 2005). The model is based on a reaction–diffusion–advection equation with nonlinear diffusion which is similar to a porous medium equation with additional advection and population growth terms. We establish that the model is well-posed, show that solutions stabilize, determine the stationary states, discuss their stability, and describe the biological interpretation of the results.

## Keywords

Porous medium equation Asymptotic behavior Ideal free distribution## Mathematics Subject Classification (2010)

35K55 35K65 92D40## Notes

### Acknowledgments

The research of CC was partially supported by NSF Grant 1118623.

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