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.
Funding Information:
This research received external funding as noted in the subsequent 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 Cient?ficos y Tecnol?gicos de Excelencia con Financiamiento Basal AFB170001.
Publisher Copyright:
© 2020 by the authors.
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 -