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Abstract The ideal free distribution in ecology was introduced by Fretwell and Lucas to model the habitat selection of animal populations. In this paper, we revisit the concept via a mean field game system with local coupling, which models a dynamic version of the habitat selection game in ecology. We establish the existence of classical solution of the ergodic mean field game system, including the case of heterogeneous diffusion when the underlying domain is one-dimensional and further show that the population density of agents converges to the ideal free distribution of the underlying habitat selection game, as the cost of control tends to zero. Our analysis provides a derivation of ideal free distribution in a dynamical context. 

Inspired by a recent study associating shifting temperature conditions with changes in the efficiency with which predators convert prey to offspring, we propose a predator prey model of reaction-diffusion type to analyze the consequence of such effects on the population dynamics and spread of {the predator} species. In the model, the predator conversion efficiency is represented by a spatially heterogeneous function depending on the variable $$\xi=x-c_1t$$ for some given $$c_1>0$$. Using the Hamilton-Jacobi approach, we provide explicit formulas for the spreading speed of the predator species. When the conversion function is monotone increasing, the spreading speed is determined in all cases and non-local pulling is possible. When the function is monotone decreasing, we provide formulas for the spreading speed when the rate of shift of the conversion function is sufficiently fast or slow. 

Abstract Habitat loss and fragmentation have independent impacts on biodiversity; thus, field studies are needed to distinguish their impacts. Moreover, species with different locomotion rates respond differently to fragmentation, complicating direct comparisons of the effects of habitat loss and fragmentation across differing taxa and landscapes. To overcome these challenges, we combined mechanistic mathematical modeling and laboratory experiments to compare how species with different locomotion rates were affected by low (∼80% intact) and high (∼30% intact) levels of habitat loss. In our laboratory experiment, we usedCaenorhabditis elegansstrains with different locomotion rates and subjected them to the different levels of habitat loss and fragmentation by placingEscherichia coli(C. elegansfood) over different proportions of the Petri dish. We developed a partial differential equation model that incorporated spatial and biological phenomena to predict the impacts of habitat arrangement on populations. Only species with low rates of locomotion declined significantly in abundance as fragmentation increased in areas with low (p = 0.0270) and high (p = 0.0243) levels of habitat loss. Despite that species with high locomotion rates changed little in abundance regardless of the spatial arrangement of resources, they had the lowest abundance and growth rates in all environments because the negative effect of fragmentation created a mismatch between the population distribution and the resource distribution. Our findings shed new light on incorporating the role of locomotion in determining the effects of habitat fragmentation.