# American Institute of Mathematical Sciences

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

 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 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.
 [1] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [2] Yacouba Simporé, Oumar Traoré. Null controllability of a nonlinear age, space and two-sex structured population dynamics model. Mathematical Control and Related Fields, 2021  doi: 10.3934/mcrf.2021052 [3] Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031 [4] Agnieszka Ulikowska. An age-structured two-sex model in the space of radon measures: Well posedness. Kinetic and Related Models, 2012, 5 (4) : 873-900. doi: 10.3934/krm.2012.5.873 [5] Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, 2021, 29 (4) : 2599-2618. doi: 10.3934/era.2021003 [6] Howard A. Levine, Yeon-Jung Seo, Marit Nilsen-Hamilton. A discrete dynamical system arising in molecular biology. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2091-2151. doi: 10.3934/dcdsb.2012.17.2091 [7] Shaohong Fang, Jing Huang, Jinying Ma. Stabilization of a discrete-time system via nonlinear impulsive control. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1803-1811. doi: 10.3934/dcdss.2020106 [8] Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170 [9] Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure and Applied Analysis, 2011, 10 (1) : 141-160. doi: 10.3934/cpaa.2011.10.141 [10] Vladimir Müller, Aljoša Peperko. On the Bonsall cone spectral radius and the approximate point spectrum. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5337-5354. doi: 10.3934/dcds.2017232 [11] Mengting Fang, Yuanshi Wang, Mingshu Chen, Donald L. DeAngelis. Asymptotic population abundance of a two-patch system with asymmetric diffusion. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3411-3425. doi: 10.3934/dcds.2020031 [12] Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229 [13] Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367 [14] Zhanyuan Hou. Geometric method for global stability of discrete population models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3305-3334. doi: 10.3934/dcdsb.2020063 [15] Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19 [16] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control and Related Fields, 2021, 11 (4) : 857-883. doi: 10.3934/mcrf.2020049 [17] Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277 [18] Gigi Thomas, Edward M. Lungu. A two-sex model for the influence of heavy alcohol consumption on the spread of HIV/AIDS. Mathematical Biosciences & Engineering, 2010, 7 (4) : 871-904. doi: 10.3934/mbe.2010.7.871 [19] Bedr'Eddine Ainseba, Mostafa Bendahmane, Yuan He. Stability of conductivities in an inverse problem in the reaction-diffusion system in electrocardiology. Networks and Heterogeneous Media, 2015, 10 (2) : 369-385. doi: 10.3934/nhm.2015.10.369 [20] 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

2020 Impact Factor: 1.327