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:
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M. Akian, S. Gaubert and R. D. Nussbaum, A Collatz-Wielandt characterization of the spectral radius of order-preserving homogeneous maps on cones,, , (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. Google Scholar

[3]

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

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E. Bohl, Monotonie: Lösbarkeit Und Numerik Bei Operatorgleichungen,, Springer, (1974). Google Scholar

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

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

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

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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. 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. Google Scholar

[10]

K. D. Deimling, Nonlinear Functional Analysis,, Springer, (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. 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. 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. 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. doi: 10.1007/BF00276145. Google Scholar

[15]

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

[16]

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

[17]

K. P. Hadeler, Homogeneous systems with a quiescent phase,, Math. Model. Nat. Phenom., 3 (2008), 115. 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. doi: 10.1137/0520025. Google Scholar

[19]

J. Hofbauer, A unified approach to persistence,, Acta Appl. Math., 14 (1989), 11. 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. 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, (2005). doi: 10.1137/1.9780898717488. Google Scholar

[22]

W. Jin, Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces,, dissertation, (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), 10884. 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. doi: 10.3934/dcdsb.2014.19.3209. Google Scholar

[26]

M. A. Krasnosel'skij, Positive Solutions of Operator Equations,, Noordhoff, (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, (1989). Google Scholar

[28]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications,, De Gruyter, (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. 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, (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. 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. 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. 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. 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, 886 (1981), 309. Google Scholar

[37]

R. D. Nussbaum, Hilbert's Projective Metric and Iterated Nonlinear Maps,, Mem. AMS 75, 75 (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. doi: 10.1112/S0024610798006425. Google Scholar

[39]

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

[40]

H. H. Schaefer, Topological Vector Spaces,, Macmillan, (1966). Google Scholar

[41]

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

[42]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer, (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. doi: 10.1137/080732870. Google Scholar

[44]

H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps,, , (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,, , (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, (1968). Google Scholar

[49]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (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,, , (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. Google Scholar

[3]

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

[4]

E. Bohl, Monotonie: Lösbarkeit Und Numerik Bei Operatorgleichungen,, Springer, (1974). Google Scholar

[5]

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

[6]

G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems,, Proc. Amer. Math. Soc., 96 (1986), 425. 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. 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. 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. Google Scholar

[10]

K. D. Deimling, Nonlinear Functional Analysis,, Springer, (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. 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. 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. 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. doi: 10.1007/BF00276145. Google Scholar

[15]

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

[16]

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

[17]

K. P. Hadeler, Homogeneous systems with a quiescent phase,, Math. Model. Nat. Phenom., 3 (2008), 115. 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. doi: 10.1137/0520025. Google Scholar

[19]

J. Hofbauer, A unified approach to persistence,, Acta Appl. Math., 14 (1989), 11. 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. 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, (2005). doi: 10.1137/1.9780898717488. Google Scholar

[22]

W. Jin, Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces,, dissertation, (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), 10884. 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. doi: 10.3934/dcdsb.2014.19.3209. Google Scholar

[26]

M. A. Krasnosel'skij, Positive Solutions of Operator Equations,, Noordhoff, (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, (1989). Google Scholar

[28]

U. Krause, Positive Dynamical Systems in Discrete Time. Theory, Models and Applications,, De Gruyter, (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. 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, (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. 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. 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. 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. 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, 886 (1981), 309. Google Scholar

[37]

R. D. Nussbaum, Hilbert's Projective Metric and Iterated Nonlinear Maps,, Mem. AMS 75, 75 (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. doi: 10.1112/S0024610798006425. Google Scholar

[39]

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

[40]

H. H. Schaefer, Topological Vector Spaces,, Macmillan, (1966). Google Scholar

[41]

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

[42]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer, (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. doi: 10.1137/080732870. Google Scholar

[44]

H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps,, , (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,, , (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, (1968). Google Scholar

[49]

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

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