“This paper concerns a mathematical model for population dynamics.”

This paper concerns a mathematical model for population dynamics. Since the seminal work, independently, of Volterra and Lotka in the 1920s, there has been much interest in population dynamics models both from the mathematical and the application point of view. It is striking how these simple mathematical models manage to capture some of the salient features of ecosystems, such as the existence of equilibria and their sensitivity to even minute perturbations, oscillatory behavior, etc. In the past few decades, the original work of Volterra and Lotka has gone through many refinements, incorporating more and more complexities. This work fits into this scheme.

This paper incorporates two recent advances in population dynamics models. First, it takes into account searching processes involved in predation (leading to the so-called ratio-dependent theory). Second, it takes into account natural movements of both predator and prey populations (resulting in diffusion). Our paper analyzes equilibria (in particular so-called "stationary patterns") arising in such models. The main mathematical tool is applying a well-known topological degree argument to a nonlinear elliptic partial differential system.

This paper analyzes how various equilibria arise for different predator-prey pairs and for different external circumstances, as captured by various parameters in the model. By incorporating more realistic features, the model gives useful information from the application point of view.

I have been interested in geometric and topological methods for nonlinear parabolic and elliptic partial differential systems. This particular work is built upon previous works of many authors on population dynamics models, but is directly inspired by the work of my co-author Mingxin Wang on the Sel'kov model of a biochemical process known as glycolysis.