# American Institute of Mathematical Sciences

March  2016, 21(2): 447-470. doi: 10.3934/dcdsb.2016.21.447

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

 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, United States

Received  December 2014 Revised  July 2015 Published  November 2015

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.
Citation: Wen Jin, Horst R. Thieme. An extinction/persistence threshold for sexually reproducing populations: The cone spectral radius. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 447-470. doi: 10.3934/dcdsb.2016.21.447
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