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

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December 2014, 19(10): 3209-3218. doi: 10.3934/dcdsb.2014.19.3209

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

Wen Jin ^1, and Horst R. Thieme ^2,

1.	School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, United States
2.	School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287

Received June 2013 Revised October 2013 Published October 2014

Abstract
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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: cone spectral radius, discrete dynamical system, Impulsive reaction diffusion system, ordered Banach space, two-sex population, stability., discrete semiflow, basic reproduction number, eigenvector, homogeneous map, difference equation.

Mathematics Subject Classification: Primary: 39A70, 35B99, 92D25; Secondary: 39A6.

Citation: Wen Jin, Horst R. Thieme. Persistence and extinction of diffusing populations with two sexes and short reproductive season. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3209-3218. doi: 10.3934/dcdsb.2014.19.3209

References:

[1]	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. doi: 10.1006/tpbi.2001.1527.
[2]	F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.
[3]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003. doi: 10.1002/0470871296.
[4]	R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047.
[5]	O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton, 2013.
[6]	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.
[7]	K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102. doi: 10.1007/BF00046676.
[8]	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.
[9]	M. A. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen 1964.
[10]	M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989.
[11]	B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.
[12]	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.
[13]	M. A. Lewis and B. Li, Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models, Bull. Math. Biol., 74 (2012), 2383-2402. doi: 10.1007/s11538-012-9757-6.
[14]	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.
[15]	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.
[16]	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. doi: 10.1086/659628.
[17]	R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory, (eds. E. Fadell and G. Fournier), Springer, Berlin New York, (1981), 309-331.
[18]	R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 2 (1998), 480-496. doi: 10.1112/S0024610798006425.
[19]	H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann., 138 (1959), 259-286. doi: 10.1007/BF01342907.
[20]	H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.
[21]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.
[22]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Providence, 2011
[23]	H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, preprint, arXiv:1302.3905.
[24]	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]	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. doi: 10.1006/tpbi.2001.1527.
[2]	F. F. Bonsall, Linear operators in complete positive cones, Proc. London Math. Soc., 8 (1958), 53-75.
[3]	R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley, Chichester, 2003. doi: 10.1002/0470871296.
[4]	R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497-518. doi: 10.1017/S0308210506000047.
[5]	O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton, 2013.
[6]	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.
[7]	K. P. Hadeler, Pair formation in age-structured populations, Acta Appl. Math., 14 (1989), 91-102. doi: 10.1007/BF00046676.
[8]	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.
[9]	M. A. Krasnosel'skij, Positive Solutions of Operator Equations, Noordhoff, Groningen 1964.
[10]	M. A. Krasnosel'skij, Je. A. Lifshits and A. V. Sobolev, Positive Linear Systems: The Method of Positive Operators, Heldermann Verlag, Berlin, 1989.
[11]	B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory, Cambridge University Press, Cambridge, 2012. doi: 10.1017/CBO9781139026079.
[12]	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.
[13]	M. A. Lewis and B. Li, Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models, Bull. Math. Biol., 74 (2012), 2383-2402. doi: 10.1007/s11538-012-9757-6.
[14]	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.
[15]	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.
[16]	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. doi: 10.1086/659628.
[17]	R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, Fixed Point Theory, (eds. E. Fadell and G. Fournier), Springer, Berlin New York, (1981), 309-331.
[18]	R. D. Nussbaum, Eigenvectors of order-preserving linear operators, J. London Math. Soc., 2 (1998), 480-496. doi: 10.1112/S0024610798006425.
[19]	H. H. Schaefer, Halbgeordnete lokalkonvexe Vektorräume. II, Math. Ann., 138 (1959), 259-286. doi: 10.1007/BF01342907.
[20]	H. H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966.
[21]	H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, 1995.
[22]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Providence, 2011
[23]	H. R. Thieme, Eigenvectors and eigenfunctionals of homogeneous order-preserving maps, preprint, arXiv:1302.3905.
[24]	X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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