Continuity of the spectral radius, applied to structured semelparous two-sex population models

Abstract

If a structured discrete-time population model $x_n=F(x_{n-1})$, $n\in \mathbb{N}$, takes account of two-sexes, the year-to-year turnover operator F often has a first-order approximation B that is homogeneous and order-preserving rather than linear and positive. Still, one can define the spectral radius $\mathcal{T}=\mathbf{r}(B)$, the basic turnover number, which plays the role of a threshold parameter between extinction and persistence. If $\mathcal{T}<1$, the extinction state is locally stable, and if $\mathcal{T}>1$ the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. This threshold property raises interest in the continuous dependence of the basic turnover number on the parameters in the population model. In this paper, we concentrate on the continuous dependence of $\mathcal{T}=\mathbf{r}(B)$ on mating functions.

Keywords:

• Difference equation
• population projection
• population growth factor
• net reproductive value
• Collatz-Wielandt numbers
• Feller kernel

Mathematics Subject Classifications:

• Primary 92D25
• 47B65
• 47G10
• 47H07
• 47J25
• Secondary 28C15
• 47N60
• 47J10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Acknowledgement

The author thanks two anonymous referees for their constructive comments.