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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show all references

References:
[1]

arXiv:1112.5968 [math.FA], 2012. Google Scholar

[2]

Theor. Pop. Biol., 60 (2001), 93-106. Google Scholar

[3]

Arch. Rat. Mech. Anal., 22 (1966), 313-332.  Google Scholar

[4]

Springer, Berlin Heidelberg, 1974.  Google Scholar

[5]

Proc. London Math. Soc., 8 (1958), 53-75.  Google Scholar

[6]

Proc. Amer. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[7]

J. Differential Equations, 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[8]

J. Biol. Dyn., 5 (2011), 277-297. doi: 10.1080/17513758.2010.491583.  Google Scholar

[9]

Nat. Res. Mod., 8 (1994), 297-333. Google Scholar

[10]

Springer, Berlin Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[11]

J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[12]

Linear Algebra and its Applications, 120 (1989), 193-205. doi: 10.1016/0024-3795(89)90378-9.  Google Scholar

[13]

Proc. Amer. Math. Soc., 143 (2015), 1617-1625. doi: 10.1090/S0002-9939-2014-12375-6.  Google Scholar

[14]

J. Math. Biol., 26 (1988), 635-649. doi: 10.1007/BF00276145.  Google Scholar

[15]

Acta Appl. Math., 14 (1989), 91-102. doi: 10.1007/BF00046676.  Google Scholar

[16]

J. Math. Biol., 32 (1993), 1-15. doi: 10.1007/BF00160370.  Google Scholar

[17]

Math. Model. Nat. Phenom., 3 (2008), 115-125. doi: 10.1051/mmnp:2008044.  Google Scholar

[18]

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[19]

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[20]

Proc. Amer. Math. Soc., 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.  Google Scholar

[21]

SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717488.  Google Scholar

[22]

dissertation, Arizona State University, May 2014.  Google Scholar

[23]

W. Jin, H. L. Smith and H. R. Thieme, Persistence versus extinction for a class of discrete-time structured population models,, J. Math. Biology, ().   Google Scholar

[24]

J. Dynamics and Differential Equations, (2015), 17 pages, DOI 10.1007/s10884-015-9434-1. doi: 10.1007/s10884-015-9434-1.  Google Scholar

[25]

Discrete and Continuous Dynamical Systems - B, 19 (2014), 3209-3218. doi: 10.3934/dcdsb.2014.19.3209.  Google Scholar

[26]

Noordhoff, Groningen, 1964.  Google Scholar

[27]

Heldermann Verlag, Berlin, 1989.  Google Scholar

[28]

De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696.  Google Scholar

[29]

Uspehi Mat. Nauk (N.S.), 3 (1948), 3-95, English Translation, AMS Translation, 1950 (1950), 128pp.  Google Scholar

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B. Lemmens, B. Lins, R. D. Nussbaum and M. Wortel, Denjoy-Wolff theorems for Hilbert's and Thompson's metric spaces,, J. Analyse Math., ().   Google Scholar

[31]

Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[32]

Proc. Amer. Math. Soc., 141 (2013), 2741-2754. doi: 10.1090/S0002-9939-2013-11520-0.  Google Scholar

[33]

Discr. Cont. Dyn. Sys. (DCDS-A), 8 (2002), 519-562. doi: 10.3934/dcds.2002.8.519.  Google Scholar

[34]

J. Fixed Point Theory and Appl., 7 (2010), 103-143. doi: 10.1007/s11784-010-0010-3.  Google Scholar

[35]

Amer. Nat., 177 (2011), 549-561. Google Scholar

[36]

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[37]

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J. London Math. Soc., 58 (1998), 480-496. doi: 10.1112/S0024610798006425.  Google Scholar

[39]

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[40]

Macmillan, New York, 1966.  Google Scholar

[41]

Amer. Math. Soc., Providence, 2011.  Google Scholar

[42]

Springer, New York, 1983.  Google Scholar

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arXiv:1302.3905 [math.FA], 2013. Google Scholar

[45]

arXiv:1406.6657 [math.FA], 2014. Google Scholar

[46]

H. R. Thieme, Spectral radii and Collatz-Wielandt numbers for homogeneous order-preserving maps and the monotone companion norm, Ordered Structures and Applications (tentative title),, Positivity VII (Zaanen Centennial Conference) (M. de Jeu, ().   Google Scholar

[47]

H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations,, J. Dynamics and Differential Equations, (): 10884.   Google Scholar

[48]

Second edition. Die Grundlehren der mathematischen Wissenschaften, Band 123 Springer-Verlag New York Inc., New York, 1968.  Google Scholar

[49]

Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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