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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

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 & 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.  doi: 10.1006/tpbi.2001.1527.  Google Scholar

[2]

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

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (2003).  doi: 10.1002/0470871296.  Google Scholar

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R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497.  doi: 10.1017/S0308210506000047.  Google Scholar

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O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (2013).   Google Scholar

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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

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

[8]

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

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

[10]

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

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

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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

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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.  doi: 10.1007/s11538-012-9757-6.  Google Scholar

[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.  doi: 10.3934/dcds.2002.8.519.  Google Scholar

[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.  doi: 10.1007/s11784-010-0010-3.  Google Scholar

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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.  doi: 10.1086/659628.  Google Scholar

[17]

R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem,, Fixed Point Theory, (1981), 309.   Google Scholar

[18]

R. D. Nussbaum, Eigenvectors of order-preserving linear operators,, J. London Math. Soc., 2 (1998), 480.  doi: 10.1112/S0024610798006425.  Google Scholar

[19]

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

[20]

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

[21]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Amer. Math. Soc., (1995).   Google Scholar

[22]

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

[23]

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

[24]

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]

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.  doi: 10.1006/tpbi.2001.1527.  Google Scholar

[2]

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

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (2003).  doi: 10.1002/0470871296.  Google Scholar

[4]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 497.  doi: 10.1017/S0308210506000047.  Google Scholar

[5]

O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (2013).   Google Scholar

[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.  doi: 10.1007/BF00178324.  Google Scholar

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[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.  doi: 10.1007/s11538-012-9757-6.  Google Scholar

[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.  doi: 10.3934/dcds.2002.8.519.  Google Scholar

[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.  doi: 10.1007/s11784-010-0010-3.  Google Scholar

[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.  doi: 10.1086/659628.  Google Scholar

[17]

R. D. Nussbaum, Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem,, Fixed Point Theory, (1981), 309.   Google Scholar

[18]

R. D. Nussbaum, Eigenvectors of order-preserving linear operators,, J. London Math. Soc., 2 (1998), 480.  doi: 10.1112/S0024610798006425.  Google Scholar

[19]

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

[20]

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

[21]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems,, Amer. Math. Soc., (1995).   Google Scholar

[22]

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

[23]

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

[24]

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

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