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

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

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.

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. BanksS. 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. BanksF. 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. FengL. 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. BanksS. 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. BanksF. 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. FengL. 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

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