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August  2016, 21(6): 1975-1998. doi: 10.3934/dcdsb.2016032

## Asymptotic analysis of a size-structured cannibalism population model with delayed birth process

 1 Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China 2 Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241

Received  November 2014 Revised  February 2016 Published  June 2016

In this paper, we study a size-structured cannibalism model with environment feedback and delayed birth process. Our focus is on the asymptotic behavior of the system, particularly on the effect of cannibalism and time lag on the long-term dynamics. To this end, we formally linearize the system around a steady state and study the linearized system by $C_0$-semigroup framework and spectral analysis methods. These analytical results allow us to achieve linearized stability, instability and asynchronous exponential growth results under some conditions. Finally, some examples are presented and simulated to illustrate the obtained stability conclusions.
Citation: Dongxue Yan, Yu Cao, Xianlong Fu. Asymptotic analysis of a size-structured cannibalism population model with delayed birth process. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1975-1998. doi: 10.3934/dcdsb.2016032
##### References:
 [1] A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Diff. Eq., 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013. [2] M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Eq., 11 (2011), 531-552. doi: 10.1007/s00028-011-0100-8. [3] Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups, North-Holland, Amsterdam, 1987. [4] J. M. Cushing, A size-structured model for cannibalism, Theoret. Population Biol., 42 (1992), 347-361. doi: 10.1016/0040-5809(92)90020-T. [5] G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci.,46 (1979), 279-291. doi: 10.1016/0025-5564(79)90073-7. [6] O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1069. doi: 10.1137/060659211. [7] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., (2007), 187-200. doi: 10.1007/978-3-7643-7794-6_12. [8] K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. [9] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. [10] L. R. Fox, Cannibalism in natural populations, Annu. Rev. Ecol. Syst., 6 (1975), 87-106. doi: 10.1146/annurev.es.06.110175.000511. [11] 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. [12] J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized size-structured Daphnia model with inflow, Appl. Anal., 86 (2007), 1087-1103. doi: 10.1080/00036810701545634. [13] J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249-266. [14] J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Commun. Pure Appl. Anal., 8 (2009), 1825-1839. doi: 10.3934/cpaa.2009.8.1825. [15] G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process, Discr. Cont. Dyn. Syst. B, 7 (2007), 735-754. doi: 10.3934/dcdsb.2007.7.735. [16] X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 18 (2013), 109-131. doi: 10.3934/dcdsb.2013.18.109. [17] X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model, Discr. Cont. Dyn. Syst. B, 19 (2014), 391-417. doi: 10.3934/dcdsb.2014.19.391. [18] M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467. doi: 10.1016/0022-247X(92)90218-3. [19] Ph. Getto, O. Diekmann, and A. M. de Roos, On the (dis)advantages of cannibalism, J. Math. Biol., 51 (2005), 695-712. doi: 10.1007/s00285-005-0342-6. [20] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation, Lect. Notes in Math., 1076 (1984), 86-100. doi: 10.1007/BFb0072769. [21] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [22] B-Z Guo, W-L Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832. doi: 10.1080/03605308908820630. [23] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. doi: 10.1006/jmaa.1999.6708. [24] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443. doi: 10.1006/jmaa.2000.7089. [25] A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. [26] R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302. doi: 10.1016/0022-1236(90)90096-4. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [28] S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462. [29] S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77. doi: 10.1007/s00028-004-0159-6. [30] K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. doi: 10.1137/0132040. [31] K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910. doi: 10.1137/0511080. [32] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.

show all references

##### References:
 [1] A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Diff. Eq., 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013. [2] M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Eq., 11 (2011), 531-552. doi: 10.1007/s00028-011-0100-8. [3] Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups, North-Holland, Amsterdam, 1987. [4] J. M. Cushing, A size-structured model for cannibalism, Theoret. Population Biol., 42 (1992), 347-361. doi: 10.1016/0040-5809(92)90020-T. [5] G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate, Math. Biosci.,46 (1979), 279-291. doi: 10.1016/0025-5564(79)90073-7. [6] O. Diekmann, Ph. Getto and M. Gyllenberg, Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1069. doi: 10.1137/060659211. [7] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, Fun. Anal. Evol. Eq., (2007), 187-200. doi: 10.1007/978-3-7643-7794-6_12. [8] K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80. [9] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. [10] L. R. Fox, Cannibalism in natural populations, Annu. Rev. Ecol. Syst., 6 (1975), 87-106. doi: 10.1146/annurev.es.06.110175.000511. [11] 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. [12] J. Z. Farkas and T. Hagen, Linear stability and positivity results for a generalized size-structured Daphnia model with inflow, Appl. Anal., 86 (2007), 1087-1103. doi: 10.1080/00036810701545634. [13] J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249-266. [14] J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Commun. Pure Appl. Anal., 8 (2009), 1825-1839. doi: 10.3934/cpaa.2009.8.1825. [15] G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process, Discr. Cont. Dyn. Syst. B, 7 (2007), 735-754. doi: 10.3934/dcdsb.2007.7.735. [16] X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discr. Cont. Dyn. Syst. B, 18 (2013), 109-131. doi: 10.3934/dcdsb.2013.18.109. [17] X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model, Discr. Cont. Dyn. Syst. B, 19 (2014), 391-417. doi: 10.3934/dcdsb.2014.19.391. [18] M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467. doi: 10.1016/0022-247X(92)90218-3. [19] Ph. Getto, O. Diekmann, and A. M. de Roos, On the (dis)advantages of cannibalism, J. Math. Biol., 51 (2005), 695-712. doi: 10.1007/s00285-005-0342-6. [20] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation, Lect. Notes in Math., 1076 (1984), 86-100. doi: 10.1007/BFb0072769. [21] G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229. [22] B-Z Guo, W-L Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832. doi: 10.1080/03605308908820630. [23] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. doi: 10.1006/jmaa.1999.6708. [24] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow, J. Math. Anal. Appl., 252 (2000), 431-443. doi: 10.1006/jmaa.2000.7089. [25] A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. [26] R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain, J. Funct. Anal., 89 (1990), 291-302. doi: 10.1016/0022-1236(90)90096-4. [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [28] S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462. [29] S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77. doi: 10.1007/s00028-004-0159-6. [30] K. E. Swick, A nonlinear age-dependent model of single species population dynamics, SIAM J. Appl. Math., 32 (1977), 484-498. doi: 10.1137/0132040. [31] K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics, SIAM J. Math. Anal., 11 (1980), 901-910. doi: 10.1137/0511080. [32] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.
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