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Discrete and Continuous Dynamical Systems - Series B (DCDS-B)
 

Persistence and extinction of diffusing populations with two sexes and short reproductive season

Pages: 3209 - 3218, Volume 19, Issue 10, December 2014      doi:10.3934/dcdsb.2014.19.3209

 
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Wen Jin - School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, United States (email)
Horst R. Thieme - School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States (email)

Abstract: A model is considered for a spatially distributed population of male and female individuals that mate and reproduce only once in their life during a very short reproductive season. Between birth and mating, females and males move by diffusion on a bounded domain $\Omega$. Mating and reproduction is described by a (positively) homogeneous function (of degree one). We identify a basic reproduction number $\mathcal{R}_0$ that acts as a threshold between extinction and persistence. If $\mathcal{R}_0 <1$, the population dies out while it persists (uniformly weakly) if $\mathcal{R}_0 > 1$. $\mathcal{R}_0$ is the cone spectral radius of a bounded homogeneous map.

Keywords:  Impulsive reaction diffusion system, two-sex population, discrete semiflow, discrete dynamical system, difference equation, ordered Banach space, homogeneous map, cone spectral radius, basic reproduction number, eigenvector, stability.
Mathematics Subject Classification:  Primary: 39A70, 35B99, 92D25; Secondary: 39A60.

Received: June 2013;      Revised: October 2013;      Available Online: October 2014.

 References