An extinction/persistence threshold for sexually reproducing populations:The cone spectral radius

Wen Jin; Horst R. Thieme

doi:10.3934/dcdsb.2016.21.447

Article Contents

2016, Volume 21, Issue 2: 447-470. Doi: 10.3934/dcdsb.2016.21.447

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An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius

Wen Jin^1, and
Horst R. Thieme^2,

1.
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804
2.
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287-1804

Received: November 30, 2014

Revised: June 30, 2015

Published: February 2016

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Abstract

Persistence and local stability of the extinction state are studied for discrete-time population models $x_n = F(x_{n-1})$, $n \in \mathbb{N}$, with a map $F$ on the cone $X_+$ of an ordered normed vector space $X$. Since sexual reproduction is accounted for, the first order approximation of $F$ at 0 is an order-preserving homogeneous map $B$ on $X_+$ that is not additive. The cone spectral radius of $B$ acts as the threshold parameter that separates persistence from local stability of 0, the extinction state. An important ingredient of the persistence theory for the induced semiflow is a homogeneous order-preserving eigenfunctional $\theta:X_+ \to \mathbb{R}_+$ of $B$ that is associated with the cone spectral radius and interacts with an appropriate persistence function $\rho$. Applications are presented for spatially distributed or rank-structured populations that reproduce sexually.

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Mathematics Subject Classification: Primary: 39A70, 92D25; Secondary: 39A60.

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