Persistence for a Two-Stage Reaction-Diffusion System

Robert Stephen Cantrell, Chris Cosner and Salomé Martínez
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Robert Stephen Cantrell: Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA
Chris Cosner: Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA
Salomé Martínez: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Universidad de Chile, 8370456 Santiago, Chile

Mathematics, 2020, vol. 8, issue 3, 1-16

Abstract: In this article, we study how the rates of diffusion in a reaction-diffusion model for a stage structured population in a heterogeneous environment affect the model’s predictions of persistence or extinction for the population. In the case of a population without stage structure, faster diffusion is typically detrimental. In contrast to that, we find that, in a stage structured population, it can be either detrimental or helpful. If the regions where adults can reproduce are the same as those where juveniles can mature, typically slower diffusion will be favored, but if those regions are separated, then faster diffusion may be favored. Our analysis consists primarily of estimates of principal eigenvalues of the linearized system around ( 0 , 0 ) and results on their asymptotic behavior for large or small diffusion rates. The model we study is not in general a cooperative system, but if adults only compete with other adults and juveniles with other juveniles, then it is. In that case, the general theory of cooperative systems implies that, when the model predicts persistence, it has a unique positive equilibrium. We derive some results on the asymptotic behavior of the positive equilibrium for small diffusion and for large adult reproductive rates in that case.

Keywords: reaction-diffusion; spatial ecology; population dynamics; stage structure; dispersal (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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