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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 & 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. doi: 10.1016/j.jde.2004.12.013. Google Scholar [2] M. Boulanouar, The asymptotic behavior of a structured cell population,, J. Evol. Eq., 11 (2011), 531. doi: 10.1007/s00028-011-0100-8. Google Scholar [3] Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups,, North-Holland, (1987). Google Scholar [4] J. M. Cushing, A size-structured model for cannibalism,, Theoret. Population Biol., 42 (1992), 347. doi: 10.1016/0040-5809(92)90020-T. Google Scholar [5] G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, Math. Biosci., 46 (1979), 279. doi: 10.1016/0025-5564(79)90073-7. Google Scholar [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. doi: 10.1137/060659211. Google Scholar [7] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, Fun. Anal. Evol. Eq., (2007), 187. doi: 10.1007/978-3-7643-7794-6_12. Google Scholar [8] K. J. Engel, Operator matrices and systems of evolution equations,, RIMS Kokyuroku, 966 (1996), 61. Google Scholar [9] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). Google Scholar [10] L. R. Fox, Cannibalism in natural populations,, Annu. Rev. Ecol. Syst., 6 (1975), 87. doi: 10.1146/annurev.es.06.110175.000511. Google Scholar [11] J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model,, J. Math. Anal. Appl., 328 (2007), 119. doi: 10.1016/j.jmaa.2006.05.032. Google Scholar [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. doi: 10.1080/00036810701545634. Google Scholar [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. Google Scholar [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. doi: 10.3934/cpaa.2009.8.1825. Google Scholar [15] G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process,, \emph{Discr. Cont. Dyn. Syst. B}, 7 (2007), 735. doi: 10.3934/dcdsb.2007.7.735. Google Scholar [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. doi: 10.3934/dcdsb.2013.18.109. Google Scholar [17] X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model,, Discr. Cont. Dyn. Syst. B, 19 (2014), 391. doi: 10.3934/dcdsb.2014.19.391. Google Scholar [18] M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators,, J. Math. Anal. Appl., 167 (1992), 443. doi: 10.1016/0022-247X(92)90218-3. Google Scholar [19] Ph. Getto, O. Diekmann, and A. M. de Roos, On the (dis)advantages of cannibalism,, J. Math. Biol., 51 (2005), 695. doi: 10.1007/s00285-005-0342-6. Google Scholar [20] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation,, Lect. Notes in Math., 1076 (1984), 86. doi: 10.1007/BFb0072769. Google Scholar [21] G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213. Google Scholar [22] B-Z Guo, W-L Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809. doi: 10.1080/03605308908820630. Google Scholar [23] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393. doi: 10.1006/jmaa.1999.6708. Google Scholar [24] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431. doi: 10.1006/jmaa.2000.7089. Google Scholar [25] A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations,, Springer, (1986). Google Scholar [26] R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, J. Funct. Anal., 89 (1990), 291. doi: 10.1016/0022-1236(90)90096-4. Google Scholar [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [28] S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427. doi: 10.1002/mma.462. Google Scholar [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. doi: 10.1007/s00028-004-0159-6. Google Scholar [30] K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, SIAM J. Appl. Math., 32 (1977), 484. doi: 10.1137/0132040. Google Scholar [31] K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics,, SIAM J. Math. Anal., 11 (1980), 901. doi: 10.1137/0511080. Google Scholar [32] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Marcell Dekker, (1985). Google Scholar

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. doi: 10.1016/j.jde.2004.12.013. Google Scholar [2] M. Boulanouar, The asymptotic behavior of a structured cell population,, J. Evol. Eq., 11 (2011), 531. doi: 10.1007/s00028-011-0100-8. Google Scholar [3] Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups,, North-Holland, (1987). Google Scholar [4] J. M. Cushing, A size-structured model for cannibalism,, Theoret. Population Biol., 42 (1992), 347. doi: 10.1016/0040-5809(92)90020-T. Google Scholar [5] G. Di Blasio, Nonlinear age-dependent population growth with history-dependent birth rate,, Math. Biosci., 46 (1979), 279. doi: 10.1016/0025-5564(79)90073-7. Google Scholar [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. doi: 10.1137/060659211. Google Scholar [7] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, Fun. Anal. Evol. Eq., (2007), 187. doi: 10.1007/978-3-7643-7794-6_12. Google Scholar [8] K. J. Engel, Operator matrices and systems of evolution equations,, RIMS Kokyuroku, 966 (1996), 61. Google Scholar [9] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). Google Scholar [10] L. R. Fox, Cannibalism in natural populations,, Annu. Rev. Ecol. Syst., 6 (1975), 87. doi: 10.1146/annurev.es.06.110175.000511. Google Scholar [11] J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model,, J. Math. Anal. Appl., 328 (2007), 119. doi: 10.1016/j.jmaa.2006.05.032. Google Scholar [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. doi: 10.1080/00036810701545634. Google Scholar [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. Google Scholar [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. doi: 10.3934/cpaa.2009.8.1825. Google Scholar [15] G. Fragnelli, A. Idrissi and L. Maniar, The asymptotic behavior of a population equation with diffusion and delayed birth process,, \emph{Discr. Cont. Dyn. Syst. B}, 7 (2007), 735. doi: 10.3934/dcdsb.2007.7.735. Google Scholar [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. doi: 10.3934/dcdsb.2013.18.109. Google Scholar [17] X. Fu and D. Zhu, Stability analysis for a size-structured juvenile-adult population model,, Discr. Cont. Dyn. Syst. B, 19 (2014), 391. doi: 10.3934/dcdsb.2014.19.391. Google Scholar [18] M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators,, J. Math. Anal. Appl., 167 (1992), 443. doi: 10.1016/0022-247X(92)90218-3. Google Scholar [19] Ph. Getto, O. Diekmann, and A. M. de Roos, On the (dis)advantages of cannibalism,, J. Math. Biol., 51 (2005), 695. doi: 10.1007/s00285-005-0342-6. Google Scholar [20] G. Greiner, A typical Perron-Frobenius theorem with applications to an age-dependent populationequation,, Lect. Notes in Math., 1076 (1984), 86. doi: 10.1007/BFb0072769. Google Scholar [21] G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213. Google Scholar [22] B-Z Guo, W-L Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809. doi: 10.1080/03605308908820630. Google Scholar [23] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393. doi: 10.1006/jmaa.1999.6708. Google Scholar [24] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431. doi: 10.1006/jmaa.2000.7089. Google Scholar [25] A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations,, Springer, (1986). Google Scholar [26] R. Nagel, The spectrum of unbounded operator matrices with non-diagonal domain,, J. Funct. Anal., 89 (1990), 291. doi: 10.1016/0022-1236(90)90096-4. Google Scholar [27] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [28] S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427. doi: 10.1002/mma.462. Google Scholar [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. doi: 10.1007/s00028-004-0159-6. Google Scholar [30] K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, SIAM J. Appl. Math., 32 (1977), 484. doi: 10.1137/0132040. Google Scholar [31] K. E. Swick, Periodic solutions of a nonlinear age-dependent model of single species population dynamics,, SIAM J. Math. Anal., 11 (1980), 901. doi: 10.1137/0511080. Google Scholar [32] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Marcell Dekker, (1985). Google Scholar
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