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

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.
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:
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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.

[2]

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

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (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. doi: 10.1017/S0308210506000047.

[5]

O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (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. doi: 10.1007/BF00178324.

[7]

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

[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.

[9]

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

[10]

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

[11]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory,, Cambridge University Press, (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. 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. 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. 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. 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. doi: 10.1086/659628.

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer, (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. doi: 10.1006/tpbi.2001.1527.

[2]

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

[3]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley, (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. doi: 10.1017/S0308210506000047.

[5]

O. Diekmann, J. A. P. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics,, Princeton University Press, (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. doi: 10.1007/BF00178324.

[7]

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

[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.

[9]

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

[10]

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

[11]

B. Lemmens and R. D. Nussbaum, Nonlinear Perron-Frobenius Theory,, Cambridge University Press, (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. 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. 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. 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. 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. doi: 10.1086/659628.

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[24]

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

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