“When the sun comes up, you better be running.”
The Fable of the Lion and the Gazelle,
popular quotation by Undetermined Author,
Abstract
We study the viability of a nonlocal dispersal strategy in a reaction-diffusion system with a fractional Laplacian operator. We show that there are circumstances—namely, a precise condition on the distribution of the resource—under which the introduction of a new nonlocal dispersal behavior is favored with respect to the local dispersal behavior of the resident population. In particular, we consider the linearization of a biological system that models the interaction of two biological species, one with local and one with nonlocal dispersal, that are competing for the same resource. We give a simple, concrete example of resources for which the equilibrium with only the local population becomes linearly unstable. In a sense, this example shows that nonlocal strategies can invade an environment in which purely local strategies are dominant at the beginning, provided that the resource is sufficiently sparse. Indeed, the example considered presents a high variance of the distribution of the dispersal, thus suggesting that the shortage of resources and their unbalanced supply may be some of the basic environmental factors that favor nonlocal strategies.
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Notes
By “overall principles” we mean the availability of a general method, depending on the measurement of some parameters in the environment, which allows a population to choose an optimal strategy. We are referring to the impossibility of having a satisfactory and complete model for population dynamics, due to the complexity of the biological world.
Up to Sect. 2.3 we investigate the opposite situation, too, that is, when the resident population has a nonlocal dispersal strategy and the mutant population has a local one.
In our model, we do not even take into account different dispersal rates \(\mu \) and \(\nu \).
The fact that the Dirichlet boundary condition is not homogeneous reflects mathematically the practical condition of performing an effective distribution plan for the population outside the strategic region.
For the sake of simplicity, we omit the multiplicative normalization constants.
The case \(n\geqslant 3\) is simpler because the Sobolev conjugated exponent \(2^*=2n/(n-2)\) is not critical. Indeed, in this case the parameter \(s'\) does not play much role.
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This work has been supported by ERC Grant 277749 “EPSILON Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities”.
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Massaccesi, A., Valdinoci, E. Is a nonlocal diffusion strategy convenient for biological populations in competition?. J. Math. Biol. 74, 113–147 (2017). https://doi.org/10.1007/s00285-016-1019-z
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DOI: https://doi.org/10.1007/s00285-016-1019-z