On the effects of migration and spatial heterogeneity on single and multiple species. (English) Zbl 1097.35079
The paper is organized in two parts. The first one mainly deals with the total size of a population at equilibrium which satisfies a logistic type equation with diffusion in a spatially heterogeneous habitat which results in a nonconstant (and possibly taking negative values somewhere) intrinsic growth rate. There is only one positive equilibrium population distribution, which tends to the positive part of the intrinsic growth rate when the migration rate (the diffusion coefficient) tends to zero and to a homogeneous profile equal to the mean value of the intrinsic growth rate when the migration rate goes to infinity. The author proves that for any value of the migration rate, the total population is larger than the one corresponding to an infinite migration rate and moreover that, if the intrinsic growth rate is nonnegative, then the total population size is maximized for an intermediate value of the migration rate and minimized for values of the migration rate equal to zero and to infinity. These results are used in the second part which is devoted to the study of a Lotka-Volterra system for the competition of two species with diffusion in a heterogeneous habitat and the same intrinsic growth rate.
The main results are related to the possibility of invasion of a species in logistic equilibrium by a very small population of the second species, that is, to the study of the linear stability of the “semi-trivial” equilibrium with zero population of the second species. The author considers the case of intra-specific competition larger than the inter-specific one (when coexistence is assured if mutation rate vanishes, i.e. the semi-trivial equilibrium is unstable and there is an attracting coexistence equilibrium) and proves the very interesting result that when diffusion (migration) is present, the invading species may go extinct for values of its mutation rate larger than the ones of the “resident” species (by proving that the semi-trivial equilibrium is stable). Moreover, if the intrinsic growth rate is nonnegative, the semi-trivial equilibrium may even be globally asymptotically stable proving that in this case there is no a coexistence steady state. The proofs are mainly based in the maximum principle, the characterization of the eigenvalues by the Rayleigh quotient, and regularity and compactness results from the theory of elliptic equations.
The main results are related to the possibility of invasion of a species in logistic equilibrium by a very small population of the second species, that is, to the study of the linear stability of the “semi-trivial” equilibrium with zero population of the second species. The author considers the case of intra-specific competition larger than the inter-specific one (when coexistence is assured if mutation rate vanishes, i.e. the semi-trivial equilibrium is unstable and there is an attracting coexistence equilibrium) and proves the very interesting result that when diffusion (migration) is present, the invading species may go extinct for values of its mutation rate larger than the ones of the “resident” species (by proving that the semi-trivial equilibrium is stable). Moreover, if the intrinsic growth rate is nonnegative, the semi-trivial equilibrium may even be globally asymptotically stable proving that in this case there is no a coexistence steady state. The proofs are mainly based in the maximum principle, the characterization of the eigenvalues by the Rayleigh quotient, and regularity and compactness results from the theory of elliptic equations.
Reviewer: Ángel Calsina (Barcelona)
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B50 | Maximum principles in context of PDEs |
| 92D25 | Population dynamics (general) |
| 92D40 | Ecology |
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