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
References:
[1]

M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv:1112.5968 [math.FA], 2012. Google Scholar

[2]

A. C. Ashih and W. G. Wilson, Two-sex population dynamics in space: Effects of gestation time on persistence, Theor. Pop. Biol., 60 (2001), 93-106. Google Scholar

[3]

E. Bohl, Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.  Google Scholar

[4]

E. Bohl, Monotonie: Lösbarkeit Und Numerik Bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974.  Google Scholar

[5]

F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.  Google Scholar

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[7]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[8]

J. M. Cushing, On the relationship between $r$ and $R_0$ and its role in the bifurcation of stable equilibria of Darwinian matrix models, J. Biol. Dyn., 5 (2011), 277-297. doi: 10.1080/17513758.2010.491583.  Google Scholar

[9]

J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333. Google Scholar

[10]

K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[12]

K.-H. Förster and B. Nagy, On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra and its Applications, 120 (1989), 193-205. doi: 10.1016/0024-3795(89)90378-9.  Google Scholar

[13]

G. Gripenberg, On the definition of the cone spectral radius, Proc. Amer. Math. Soc., 143 (2015), 1617-1625. doi: 10.1090/S0002-9939-2014-12375-6.  Google Scholar

[14]

K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649. doi: 10.1007/BF00276145.  Google Scholar

[15]

K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102. doi: 10.1007/BF00046676.  Google Scholar

[16]

K. P. Hadeler, Pair formation models with maturation period, J. Math. Biol., 32 (1993), 1-15. doi: 10.1007/BF00160370.  Google Scholar

[17]

K. P. Hadeler, Homogeneous systems with a quiescent phase, Math. Model. Nat. Phenom., 3 (2008), 115-125. doi: 10.1051/mmnp:2008044.  Google Scholar

[18]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.  Google Scholar

[19]

J. Hofbauer, A unified approach to persistence, Acta Appl. Math., 14 (1989), 11-22. doi: 10.1007/BF00046670.  Google Scholar

[20]

J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.  Google Scholar

[21]

M. Iannelli, M. Martcheva and F. A. Milner, Gender-Structured Population Models: Mathematical Methods, Numerics, and Simulations, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717488.  Google Scholar

[22]

W. Jin, Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces, 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]

W. Jin, H. L. Smith and H. R. Thieme, Persistence and critical domain size for diffusing populations with two sexes and short reproductive season, 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]

W. Jin and H. R. Thieme, Persistence and extinction of diffusing populations with two sexes and short reproductive season, Discrete and Continuous Dynamical Systems - B, 19 (2014), 3209-3218. doi: 10.3934/dcdsb.2014.19.3209.  Google Scholar

[26]

M. A. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.  Google Scholar

[27]

M. A. Krasnosel'skij, J. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989.  Google Scholar

[28]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696.  Google Scholar

[29]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (Russian), Uspehi Mat. Nauk (N.S.), 3 (1948), 3-95, English Translation, AMS Translation, 1950 (1950), 128pp.  Google Scholar

[30]

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]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[32]

B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754. doi: 10.1090/S0002-9939-2013-11520-0.  Google Scholar

[33]

J. Mallet-Paret and R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discr. Cont. Dyn. Sys. (DCDS-A), 8 (2002), 519-562. doi: 10.3934/dcds.2002.8.519.  Google Scholar

[34]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Appl., 7 (2010), 103-143. doi: 10.1007/s11784-010-0010-3.  Google Scholar

[35]

T. E. X. Miller, A. K. Shaw, B. D. Inouye and M. G. Neubert, Sex-biased dispersal and the speed of two-sex invasions, Amer. Nat., 177 (2011), 549-561. Google Scholar

[36]

R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory (E. Fadell and G. Fournier, eds.), Springer, Berlin New York, 886 (1981), 309-330.  Google Scholar

[37]

R. D. Nussbaum, Hilbert's Projective Metric and Iterated Nonlinear Maps, Mem. AMS 75, Number 391, Amer. Math. Soc., Providence, 1988. doi: 10.1090/memo/0391.  Google Scholar

[38]

R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496. doi: 10.1112/S0024610798006425.  Google Scholar

[39]

H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann., 138 (1959), 259-286. doi: 10.1007/BF01342907.  Google Scholar

[40]

H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.  Google Scholar

[41]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc., Providence, 2011.  Google Scholar

[42]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.  Google Scholar

[43]

H. R. Thieme, Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.  Google Scholar

[44]

H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv:1302.3905 [math.FA], 2013. Google Scholar

[45]

H. R. Thieme, Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces, 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]

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

[49]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

show all references

References:
[1]

M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones, arXiv:1112.5968 [math.FA], 2012. Google Scholar

[2]

A. C. Ashih and W. G. Wilson, Two-sex population dynamics in space: Effects of gestation time on persistence, Theor. Pop. Biol., 60 (2001), 93-106. Google Scholar

[3]

E. Bohl, Eigenwertaufgaben bei monotonen Operatoren und Fehlerabschätzungen für Operatorgleichungen, Arch. Rat. Mech. Anal., 22 (1966), 313-332.  Google Scholar

[4]

E. Bohl, Monotonie: Lösbarkeit Und Numerik Bei Operatorgleichungen, Springer, Berlin Heidelberg, 1974.  Google Scholar

[5]

F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.  Google Scholar

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc., 96 (1986), 425-430. doi: 10.1090/S0002-9939-1986-0822433-4.  Google Scholar

[7]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[8]

J. M. Cushing, On the relationship between $r$ and $R_0$ and its role in the bifurcation of stable equilibria of Darwinian matrix models, J. Biol. Dyn., 5 (2011), 277-297. doi: 10.1080/17513758.2010.491583.  Google Scholar

[9]

J. M. Cushing and Y. Zhou, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333. Google Scholar

[10]

K. D. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[11]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382. doi: 10.1007/BF00178324.  Google Scholar

[12]

K.-H. Förster and B. Nagy, On the Collatz-Wielandt numbers and the local spectral radius of a nonnegative operator, Linear Algebra and its Applications, 120 (1989), 193-205. doi: 10.1016/0024-3795(89)90378-9.  Google Scholar

[13]

G. Gripenberg, On the definition of the cone spectral radius, Proc. Amer. Math. Soc., 143 (2015), 1617-1625. doi: 10.1090/S0002-9939-2014-12375-6.  Google Scholar

[14]

K. P. Hadeler, R. Waldstätter and A. Wörz-Busekros, Models for pair formation in bisexual populations, J. Math. Biol., 26 (1988), 635-649. doi: 10.1007/BF00276145.  Google Scholar

[15]

K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102. doi: 10.1007/BF00046676.  Google Scholar

[16]

K. P. Hadeler, Pair formation models with maturation period, J. Math. Biol., 32 (1993), 1-15. doi: 10.1007/BF00160370.  Google Scholar

[17]

K. P. Hadeler, Homogeneous systems with a quiescent phase, Math. Model. Nat. Phenom., 3 (2008), 115-125. doi: 10.1051/mmnp:2008044.  Google Scholar

[18]

J. K. Hale and P. Waltman, Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20 (1989), 388-395. doi: 10.1137/0520025.  Google Scholar

[19]

J. Hofbauer, A unified approach to persistence, Acta Appl. Math., 14 (1989), 11-22. doi: 10.1007/BF00046670.  Google Scholar

[20]

J. Hofbauer and J. W.-H. So, Uniform persistence and repellors for maps, Proc. Amer. Math. Soc., 107 (1989), 1137-1142. doi: 10.1090/S0002-9939-1989-0984816-4.  Google Scholar

[21]

M. Iannelli, M. Martcheva and F. A. Milner, Gender-Structured Population Models: Mathematical Methods, Numerics, and Simulations, SIAM, Philadelphia, 2005. doi: 10.1137/1.9780898717488.  Google Scholar

[22]

W. Jin, Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces, 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]

W. Jin, H. L. Smith and H. R. Thieme, Persistence and critical domain size for diffusing populations with two sexes and short reproductive season, 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]

W. Jin and H. R. Thieme, Persistence and extinction of diffusing populations with two sexes and short reproductive season, Discrete and Continuous Dynamical Systems - B, 19 (2014), 3209-3218. doi: 10.3934/dcdsb.2014.19.3209.  Google Scholar

[26]

M. A. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964.  Google Scholar

[27]

M. A. Krasnosel'skij, J. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989.  Google Scholar

[28]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications, De Gruyter, Berlin, 2015. doi: 10.1515/9783110365696.  Google Scholar

[29]

M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space (Russian), Uspehi Mat. Nauk (N.S.), 3 (1948), 3-95, English Translation, AMS Translation, 1950 (1950), 128pp.  Google Scholar

[30]

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]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.  Google Scholar

[32]

B. Lemmens and R. D. Nussbaum, Continuity of the cone spectral radius, Proc. Amer. Math. Soc., 141 (2013), 2741-2754. doi: 10.1090/S0002-9939-2013-11520-0.  Google Scholar

[33]

J. Mallet-Paret and R. D. Nussbaum, Eigenvalues for a class of homogeneous cone maps arising from max-plus operators, Discr. Cont. Dyn. Sys. (DCDS-A), 8 (2002), 519-562. doi: 10.3934/dcds.2002.8.519.  Google Scholar

[34]

J. Mallet-Paret and R. D. Nussbaum, Generalizing the Krein-Rutman theorem, measures of noncompactness and the fixed point index, J. Fixed Point Theory and Appl., 7 (2010), 103-143. doi: 10.1007/s11784-010-0010-3.  Google Scholar

[35]

T. E. X. Miller, A. K. Shaw, B. D. Inouye and M. G. Neubert, Sex-biased dispersal and the speed of two-sex invasions, Amer. Nat., 177 (2011), 549-561. Google Scholar

[36]

R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory (E. Fadell and G. Fournier, eds.), Springer, Berlin New York, 886 (1981), 309-330.  Google Scholar

[37]

R. D. Nussbaum, Hilbert's Projective Metric and Iterated Nonlinear Maps, Mem. AMS 75, Number 391, Amer. Math. Soc., Providence, 1988. doi: 10.1090/memo/0391.  Google Scholar

[38]

R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 58 (1998), 480-496. doi: 10.1112/S0024610798006425.  Google Scholar

[39]

H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann., 138 (1959), 259-286. doi: 10.1007/BF01342907.  Google Scholar

[40]

H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.  Google Scholar

[41]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Amer. Math. Soc., Providence, 2011.  Google Scholar

[42]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1983.  Google Scholar

[43]

H. R. Thieme, Spectral bound and reproduction number for infinite dimensional population structure and time-heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870.  Google Scholar

[44]

H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, arXiv:1302.3905 [math.FA], 2013. Google Scholar

[45]

H. R. Thieme, Comparison of spectral radii and Collatz-Wielandt numbers for homogeneous maps, and other applications of the monotone companion norm on ordered normed vector spaces, 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]

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

[49]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

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