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 and 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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[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.

[22]

W. Jin, Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces, dissertation, Arizona State University, May 2014.

[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, (to appear).

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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., (to appear) arXiv:1410.1056v2 [math.DS]

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[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.

[44]

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

[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.

[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, B. de Pagter, O. van Gaans, M. Veraar, eds.), Birkhäuser (to appear).

[47]

H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, (to appear) DOI 10.1007/s10884-015-9463-9.

[48]

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

[49]

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

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.

[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.

[3]

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

[4]

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

[5]

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

[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.

[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.

[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.

[9]

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

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[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.

[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.

[22]

W. Jin, Persistence of Discrete Dynamical Systems in Infinite Dimensional State Spaces, dissertation, Arizona State University, May 2014.

[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, (to appear).

[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.

[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.

[26]

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

[27]

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

[28]

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

[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.

[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., (to appear) arXiv:1410.1056v2 [math.DS]

[31]

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

[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.

[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.

[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.

[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.

[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.

[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.

[38]

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

[39]

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

[40]

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

[41]

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

[42]

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

[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.

[44]

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

[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.

[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, B. de Pagter, O. van Gaans, M. Veraar, eds.), Birkhäuser (to appear).

[47]

H. R. Thieme, Eigenfunctionals of homogeneous order-preserving maps with applications to sexually reproducing populations, J. Dynamics and Differential Equations, (to appear) DOI 10.1007/s10884-015-9463-9.

[48]

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

[49]

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

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