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Extinction and uniform strong persistence of a size-structured population model
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, 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.
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
M. Iannelli,
Mathematical Theory of Age-Structured Population Dynamics Giardini Editori e Stampatori, Pisa, 1995. |
[13] |
J. A. J. Metz and O. Diekmann,
The Dynamics of Physiologically Structured Populations
Lecture Notes in Biomath. , 68 Springer-Verlag, Berlin, 1986. |
[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. |
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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[12] |
M. Iannelli,
Mathematical Theory of Age-Structured Population Dynamics Giardini Editori e Stampatori, Pisa, 1995. |
[13] |
J. A. J. Metz and O. Diekmann,
The Dynamics of Physiologically Structured Populations
Lecture Notes in Biomath. , 68 Springer-Verlag, Berlin, 1986. |
[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. |
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