TY - JOUR

T1 - Persistence for a two-stage reaction-diffusion system

AU - Cantrell, Robert Stephen

AU - Cosner, Chris

AU - Martínez, Salomé

N1 - Funding Information:
Acknowledgments: R.S.C. and C.C. are supported in part by NSF Awards DMS 15-14792 and 18-53478. S.M. is supported by CONICYT + PIA/Concurso de Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal AFB170001.

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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.

AB - 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.

KW - Dispersal

KW - Population dynamics

KW - Reaction-diffusion

KW - Spatial ecology

KW - Stage structure

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U2 - 10.3390/math8030396

DO - 10.3390/math8030396

M3 - Article

AN - SCOPUS:85082420188

VL - 8

JO - Mathematics

JF - Mathematics

SN - 2227-7390

IS - 3

M1 - 396

ER -