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J Theor Biol. 2019 Dec 24:110142. doi: 10.1016/j.jtbi.2019.110142. [Epub ahead of print]

Density dependence on multiple spatial scales maintains spatial variation in both abundance and traits.

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Department of Theoretical Biology, Bielefeld University, Bielefeld, Germany. Electronic address:
Department of Theoretical Biology, Bielefeld University, Bielefeld, Germany.


Population density affects fitness through various processes, such as mate finding and competition. The fitness of individuals in a population can in turn affect its density, making population density a key quantity linking ecological and evolutionary processes. Density effects are, however, rarely homogeneous. Different life-history processes can be affected by density over different spatial scales. In birds, for example, competition for food may depend on the number of birds nesting in the direct vicinity, while competition for nesting sites may occur over larger areas. Here we investigate how the effects of local density and of density in a nearby patch can jointly affect the emergence of spatial variation in abundance as well as phenotypic diversification. We study a two-patch model that is described by coupled ordinary differential equations. The patches have no intrinsic differences: they both have the same fitness function that describes how an individual's fitness depends on density in its own patch as well as the density in the other patch. We use a phase-space analysis, combined with a mathematical stability analysis to study the long-term behaviour of the system. Our results reveal that the mutual effect that the patches have on each other can lead to the emergence and long-term maintenance of a low and a high density patch. We then add traits and mutations to the model and show that different selection pressures in the high and low density patch can lead to diversification between these patches. Via eco-evolutionary feedbacks, this diversification can in turn lead to changes in the long-term population densities: under some parameter settings, both patches reach the same equilibrium density when mutations are absent, but different equilibrium densities when mutations are allowed. We thus show how, even in the absence of differences between patches, interactions between them can lead to differences in long-term population density, and potentially to trait diversification.


Allee effect; density-dependent selection; diversity; eco-evolutionary dynamics; population density


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