Population games with instantaneous behavior and the Rosenzweig-MacArthur model

How to determine the spatial distribution and population dynamics of animals are some of the key questions in ecology. These two have been coupled before, but there is no general method for determining spatial distributions based on instantanous behavior coupled with population dynamics. We propose modeling interacting populations with instantaneous habitat choice through mean-field games. By using the framework of variational inequalities, we are able to determine existence and uniqueness for habitat distributions of interacting populations, in both continuous and discrete habitats. With some additional restrictions, we are also able to show existence and uniqueness of fixed-points of the population dynamics along with spatial distributions. We illustrate our theoretical results by studying a Rosenzweig-MacArthur model where predators and consumers inhabit a continuous habitat. This study is conducted both theoretically and numerically. Analyzing the emergent dynamics is possible as viewing the system from the vantage point of variational inequalities allows for applying efficient numerical methods. The generality of our theoretical approach opens up for studying complex ecosystems, e.g. the impact of enrichment on spatial distributions in marine ecosystems.

Automated summary

Determining the spatial distribution and population dynamics of animals is important in ecology. This study proposes a model for determining the existence and uniqueness of habitat distributions and population dynamics of interacting populations, in both continuous and discrete habitats. The theoretical results are illustrated using a Rosenzweig-MacArthur model, and the emergent dynamics are analyzed using efficient numerical methods. The theoretical approach has the potential to study complex ecosystems.