American Institute of Mathematical Sciences

Advanced
May  2017, 22(3): 831-840. doi: 10.3934/dcdsb.2017041

Extinction and uniform strong persistence of a size-structured population model

Keng Deng and  Yixiang Wu ,

Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA

* Corresponding author

Dedicated to Steve Cantrell in honor of his 60th birthday

Received  August 2015 Revised  January 2016 Published  December 2016

Fund Project: Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA.

In this paper, we study the long-time behavior of a size-structured population model. We define a basic reproduction number $\mathcal{R}$ and show that the population dies out in the long run if $\mathcal{R}<1$. If $\mathcal{R}>1$, the model has a unique positive equilibrium, and the total population is uniformly strongly persistent. Most importantly, we show that there exists a subsequence of the total population converging to the positive equilibrium.

Keywords: Size-structured population model, basic reproduction number, extinction, uniform persistence.
Mathematics Subject Classification: 92D25, 35B40, 35L0.
Citation: Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
References:
[1]

A. S. Ackleh and K. Deng, Existence-uniqueness of solutions for a nonlinear nonautonomous size-structured population model: an upper-lower solution approach, Canadian Appl. Math. Quart., 8 (2000), 1-15.  doi: 10.1216/camq/1008957333.  Google Scholar

[2]

H. T. Banks, S. L. Ernstberger and S. Hu, Sensitivity equations for a size-structured population model, Quart. Appl. Math., 67 (2009), 627-660.  doi: 10.1090/S0033-569X-09-01105-1.  Google Scholar

[3]

H. T. Banks and F. Kappel, Transformation semigroups and $L^1$-approximation for size structure population models, Semigroup Forum, 38 (1989), 141-155.  doi: 10.1007/BF02573227.  Google Scholar

[4]

H. T. Banks, F. Kappel and C. Wang, A semigroup formulation of a nonlinear size-structured distributed rate population model, Internat. Ser. Numer. Math., 118 (1994), 1-19.   Google Scholar

[5]

A. Calsina and J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377.  Google Scholar

[6]

A. Calsina and M. Sanchon, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl., 286 (2003), 435-452.  doi: 10.1016/S0022-247X(03)00464-5.  Google Scholar

[7]

K. Deng and Y. Wang, Sensitivity analysis for a nonlinear size-structured population model, Quart. Appl. Math., 73 (2015), 401-417.  doi: 10.1090/qam/1366.  Google Scholar

[8]

J. Z. Farkas, Stability conditions for a non-linear size-structured model, Nonlinear Anal. Real World Appl., 6 (2005), 962-969.  doi: 10.1016/j.nonrwa.2004.06.002.  Google Scholar

[9]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.  doi: 10.1016/j.jmaa.2006.05.032.  Google Scholar

[10]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅰ: Fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.  Google Scholar

[11]

Z. Feng, L. Rong and R. K. Swihart, Dynamics of an age-structured metapopulation model, Natural Resource Modeling, 18 (2005), 415-440.  doi: 10.1111/j.1939-7445.2005.tb00166.x.  Google Scholar

[12]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics Giardini Editori e Stampatori, Pisa, 1995. Google Scholar

[13]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations Lecture Notes in Biomath. , 68 Springer-Verlag, Berlin, 1986. Google Scholar

[14]

H. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Amer. Math. Soc., 127 (1999), 2395-2403.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar

show all references

References:
[1]

A. S. Ackleh and K. Deng, Existence-uniqueness of solutions for a nonlinear nonautonomous size-structured population model: an upper-lower solution approach, Canadian Appl. Math. Quart., 8 (2000), 1-15.  doi: 10.1216/camq/1008957333.  Google Scholar

[2]

H. T. Banks, S. L. Ernstberger and S. Hu, Sensitivity equations for a size-structured population model, Quart. Appl. Math., 67 (2009), 627-660.  doi: 10.1090/S0033-569X-09-01105-1.  Google Scholar

[3]

H. T. Banks and F. Kappel, Transformation semigroups and $L^1$-approximation for size structure population models, Semigroup Forum, 38 (1989), 141-155.  doi: 10.1007/BF02573227.  Google Scholar

[4]

H. T. Banks, F. Kappel and C. Wang, A semigroup formulation of a nonlinear size-structured distributed rate population model, Internat. Ser. Numer. Math., 118 (1994), 1-19.   Google Scholar

[5]

A. Calsina and J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377.  Google Scholar

[6]

A. Calsina and M. Sanchon, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl., 286 (2003), 435-452.  doi: 10.1016/S0022-247X(03)00464-5.  Google Scholar

[7]

K. Deng and Y. Wang, Sensitivity analysis for a nonlinear size-structured population model, Quart. Appl. Math., 73 (2015), 401-417.  doi: 10.1090/qam/1366.  Google Scholar

[8]

J. Z. Farkas, Stability conditions for a non-linear size-structured model, Nonlinear Anal. Real World Appl., 6 (2005), 962-969.  doi: 10.1016/j.nonrwa.2004.06.002.  Google Scholar

[9]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007), 119-136.  doi: 10.1016/j.jmaa.2006.05.032.  Google Scholar

[10]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅰ: Fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.  Google Scholar

[11]

Z. Feng, L. Rong and R. K. Swihart, Dynamics of an age-structured metapopulation model, Natural Resource Modeling, 18 (2005), 415-440.  doi: 10.1111/j.1939-7445.2005.tb00166.x.  Google Scholar

[12]

M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics Giardini Editori e Stampatori, Pisa, 1995. Google Scholar

[13]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations Lecture Notes in Biomath. , 68 Springer-Verlag, Berlin, 1986. Google Scholar

[14]

H. Thieme, Uniform weak implies uniform strong persistence for non-autonomous semiflows, Proc. Amer. Math. Soc., 127 (1999), 2395-2403.  doi: 10.1090/S0002-9939-99-05034-0.  Google Scholar

[1]

Dongxue Yan, Xianlong Fu. Asymptotic behavior of a hierarchical size-structured population model. Evolution Equations & Control Theory, 2018, 7 (2) : 293-316. doi: 10.3934/eect.2018015
[2]

Xianlong Fu, Dongmei Zhu. Stability analysis for a size-structured juvenile-adult population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 391-417. doi: 10.3934/dcdsb.2014.19.391
[3]

Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109
[4]

Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3111-3133. doi: 10.3934/dcdsb.2020053
[5]

Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637
[6]

Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015
[7]

Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032
[8]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233
[9]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[10]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[11]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[12]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2969-3004. doi: 10.3934/dcdsb.2020048
[13]

Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719
[14]

Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37
[15]

József Z. Farkas, Thomas Hagen. Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1825-1839. doi: 10.3934/cpaa.2009.8.1825
[16]

Dan Zhang, Xiaochun Cai, Lin Wang. Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3439-3451. doi: 10.3934/dcdsb.2018327
[17]

Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239
[18]

Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040
[19]

Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu. Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences & Engineering, 2005, 2 (2) : 289-315. doi: 10.3934/mbe.2005.2.289
[20]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (43)
  • HTML views (46)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences