#### Description

Traditional continuous time models in spatial ecology typically describe movement in terms of linear diffusion and advection, which combine with nonlinear population dynamics to produce semi-linear parabolic equations and systems. In environments that are favorable everywhere in the sense that the local population growth rate is always positive, organisms can use linear advection and diffusion to achieve an optimal spatial distribution. (Here optimal means evolutionarily stable.) In regions where there are environmental “sinks” where the local growth rate is negative, it does not seem possible to achieve an optimal distribution via linear dispersal. It is possible for organisms using advection on fitness to achieve a distribution that is expected to be optimal. However, if organisms are assumed to move up gradients of their reproductive fitness, and fitness is density dependent (for example logistic), the resulting models are quasi-linear and may have other novel features. Specifically, they may have degenerate diffusion of the type that occurs in the porous medium equation. That makes it challenging to verify whether the population distributions they produce are actually evolutionarily stable. This talk will describe some models involving fitness dependent dispersal and some results and challenges in the analysis of such models.

#### Date of Event

Tuesday, October 18 from 12:00 PM - 1:00 PM

#### Location

Mailman-Hollywood Auditorium 2nd Floor

Spatial Population Models with Fitness Based Dispersal

Mailman-Hollywood Auditorium 2nd Floor

Traditional continuous time models in spatial ecology typically describe movement in terms of linear diffusion and advection, which combine with nonlinear population dynamics to produce semi-linear parabolic equations and systems. In environments that are favorable everywhere in the sense that the local population growth rate is always positive, organisms can use linear advection and diffusion to achieve an optimal spatial distribution. (Here optimal means evolutionarily stable.) In regions where there are environmental “sinks” where the local growth rate is negative, it does not seem possible to achieve an optimal distribution via linear dispersal. It is possible for organisms using advection on fitness to achieve a distribution that is expected to be optimal. However, if organisms are assumed to move up gradients of their reproductive fitness, and fitness is density dependent (for example logistic), the resulting models are quasi-linear and may have other novel features. Specifically, they may have degenerate diffusion of the type that occurs in the porous medium equation. That makes it challenging to verify whether the population distributions they produce are actually evolutionarily stable. This talk will describe some models involving fitness dependent dispersal and some results and challenges in the analysis of such models.

## Presenter Bio

I was an undergraduate at the University of California San Diego and did my graduate work at the University of California Berkeley. I received my Ph. D. in 1977, specializing in partial differential equations, and was awarded the Bernard Friedman Award for Applied Mathematics for my dissertation. My advisor was Murray Protter. I was on the faculty of Texas A&M University from 1977-1982 and moved to the University of Miami in 1982, where I have been since then. I was promoted to the rank of Professor in 1988 and was awarded a Cooper Fellowship by the College of Arts and Sciences in 2015. During the 1985-1986 academic year I had visiting positions at the Institute for Advanced Study and the University of Tennessee. I have authored over 100 publications on partial differential equations and mathematical biology, and am co-author with Robert Stephen Cantrell of the book Spatial Ecology via Reaction-Diffusion Equations (Wiley 2003). I’ve had six Ph.D. students. I’ve had several NSF grants over the over the years. My current research is still mostly on mathematical models in spatial ecology, partial differential equations, and related topics.