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Open AccessArticle

Persistence for a Two-Stage Reaction-Diffusion System

Department of Mathematics, University of Miami, Coral Gables, FL 33124, USA
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, UMI 2807 CNRS-UChile, Universidad de Chile, 8370456 Santiago, Chile
Author to whom correspondence should be addressed.
Mathematics 2020, 8(3), 396; (registering DOI)
Received: 3 February 2020 / Revised: 2 March 2020 / Accepted: 5 March 2020 / Published: 11 March 2020
(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)
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. View Full-Text
Keywords: reaction-diffusion; spatial ecology; population dynamics; stage structure; dispersal reaction-diffusion; spatial ecology; population dynamics; stage structure; dispersal
MDPI and ACS Style

Cantrell, R.S.; Cosner, C.; Martínez, S. Persistence for a Two-Stage Reaction-Diffusion System. Mathematics 2020, 8, 396.

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