# American Institute of Mathematical Sciences

March  2014, 19(2): 391-417. doi: 10.3934/dcdsb.2014.19.391

## Stability analysis for a size-structured juvenile-adult population model

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

Received  April 2013 Revised  October 2013 Published  February 2014

In this paper, we discuss the asymptotic behavior of a size-structured juvenile-adult population equation with resource-dependent and delayed birth process. The linearization about stationary solutions is analyzed by using semigroup and spectral methods. The juvenile-adult interaction, resource-dependent and delayed boundary condition are considered deliberately for the system to investigate their influences on the asymptotic behavior of solutions. We obtain the stability and instability of the stationary solutions by given some biologically meaningful conditions in two important cases. Finally, two examples are presented and simulated to illustrate the obtained results.
Citation: 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
##### References:
 [1] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393. doi: 10.1006/jmaa.1999.6708. [2] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431. doi: 10.1006/jmaa.2000.7089. [3] T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids,, Diff. Int. Equ., 14 (2001), 19. [4] J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase,, Discr. Contin. Dyn. Syst., 33 (2013), 4891. doi: 10.3934/dcds.2013.33.4891. [5] M. Farkas, On the stability of stationary age distributions,, Appl. Math. Comp., 131 (2002), 107. doi: 10.1016/S0096-3003(01)00131-X. [6] J. Z. Farkas, Stability conditions for a nonlinear size-structured model,, Nonl. Anal. (RWA), 6 (2005), 962. doi: 10.1016/j.nonrwa.2004.06.002. [7] 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. [8] 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. [9] J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction,, Discr. Cont. Dyn. Syst. B, 9 (2008), 249. [10] Y. Liu and Z. He, Stability results for a size-structured population model with resources-dependence and inflow,, J. Math. Anal. Appl., 360 (2009), 665. doi: 10.1016/j.jmaa.2009.07.005. [11] R. Dilão, T. Domingos and E. M. Shahverdiev, Harvesting in a resource dependent age structured Leslie type population model,, Math. Biosci., 189 (2004), 141. doi: 10.1016/j.mbs.2004.01.008. [12] J. B. Shukla, K. Lata and A. K. Misra, Modeling the depletion of a renewable resource by population and industrialization: effect of technology on its conservation,, Nat. Resour. Model., 24 (2011), 242. doi: 10.1111/j.1939-7445.2011.00090.x. [13] J. B. Shukla, S. Sharma, B. Dubey and P. Sinha, Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors,, Nonl. Anal. (RWA), 10 (2009), 54. doi: 10.1016/j.nonrwa.2007.08.014. [14] J. Xia, Z. Liu, R. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with holling type II functional response,, SIAM J. Appl. Math., 70 (2009), 1178. doi: 10.1137/080728512. [15] E. M. C. D'Agata, P. Magal, S. Ruan and G. webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance,, Diff. Int. Equ., 19 (2006), 573. [16] A. Ducrot and P. Magal, Traveling wave solution for infection age structured epidemic model with vital dynamics,, Nonlinearity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. [17] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations,, J. Franklin Inst., 297 (1974), 345. [18] 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 (): 1023. doi: 10.1137/060659211. [19] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, in Functional analysis and evolution equations, (2008), 187. doi: 10.1007/978-3-7643-7794-6_12. [20] K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, SIAM J. Appl. Math., 32 (1977), 484. doi: 10.1137/0132040. [21] 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. [22] 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. [23] A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay,, in Infinite Dimensional Dynamical Systems, 64 (2013), 353. [24] B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809. doi: 10.1080/03605308908820630. [25] 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. doi: 10.3934/dcdsb.2007.7.735. [26] S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427. doi: 10.1002/mma.462. [27] 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. [28] 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. [29] 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. [30] G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213. [31] M. Iannelli, Mathematical Theory of Age-structured Population Dynamics,, Giardini Editori, (1994). [32] A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations,, Springer, (1986). [33] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Marcell Dekker, (1985). [34] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. [36] 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. [37] K. J. Engel, Operator matrices and systems of evolution equations,, RIMS Kokyuroku, 966 (1996), 61.

show all references

##### References:
 [1] T. Hagen, Eigenvalue asymptotics in isothermal forced elongation,, J. Math. Anal. Appl., 224 (2000), 393. doi: 10.1006/jmaa.1999.6708. [2] T. Hagen and M. Renardy, Eigenvalue asymptotics in nonisothermal elongational flow,, J. Math. Anal. Appl., 252 (2000), 431. doi: 10.1006/jmaa.2000.7089. [3] T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids,, Diff. Int. Equ., 14 (2001), 19. [4] J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase,, Discr. Contin. Dyn. Syst., 33 (2013), 4891. doi: 10.3934/dcds.2013.33.4891. [5] M. Farkas, On the stability of stationary age distributions,, Appl. Math. Comp., 131 (2002), 107. doi: 10.1016/S0096-3003(01)00131-X. [6] J. Z. Farkas, Stability conditions for a nonlinear size-structured model,, Nonl. Anal. (RWA), 6 (2005), 962. doi: 10.1016/j.nonrwa.2004.06.002. [7] 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. [8] 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. [9] J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction,, Discr. Cont. Dyn. Syst. B, 9 (2008), 249. [10] Y. Liu and Z. He, Stability results for a size-structured population model with resources-dependence and inflow,, J. Math. Anal. Appl., 360 (2009), 665. doi: 10.1016/j.jmaa.2009.07.005. [11] R. Dilão, T. Domingos and E. M. Shahverdiev, Harvesting in a resource dependent age structured Leslie type population model,, Math. Biosci., 189 (2004), 141. doi: 10.1016/j.mbs.2004.01.008. [12] J. B. Shukla, K. Lata and A. K. Misra, Modeling the depletion of a renewable resource by population and industrialization: effect of technology on its conservation,, Nat. Resour. Model., 24 (2011), 242. doi: 10.1111/j.1939-7445.2011.00090.x. [13] J. B. Shukla, S. Sharma, B. Dubey and P. Sinha, Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors,, Nonl. Anal. (RWA), 10 (2009), 54. doi: 10.1016/j.nonrwa.2007.08.014. [14] J. Xia, Z. Liu, R. Yuan and S. Ruan, The effects of harvesting and time delay on predator-prey systems with holling type II functional response,, SIAM J. Appl. Math., 70 (2009), 1178. doi: 10.1137/080728512. [15] E. M. C. D'Agata, P. Magal, S. Ruan and G. webb, Asymptotic behavior in nosocomial epidemic models with antibiotic resistance,, Diff. Int. Equ., 19 (2006), 573. [16] A. Ducrot and P. Magal, Traveling wave solution for infection age structured epidemic model with vital dynamics,, Nonlinearity, 24 (2011), 2891. doi: 10.1088/0951-7715/24/10/012. [17] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations,, J. Franklin Inst., 297 (1974), 345. [18] 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 (): 1023. doi: 10.1137/060659211. [19] O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics,, in Functional analysis and evolution equations, (2008), 187. doi: 10.1007/978-3-7643-7794-6_12. [20] K. E. Swick, A nonlinear age-dependent model of single species population dynamics,, SIAM J. Appl. Math., 32 (1977), 484. doi: 10.1137/0132040. [21] 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. [22] 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. [23] A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay,, in Infinite Dimensional Dynamical Systems, 64 (2013), 353. [24] B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay,, Comm. PDEs, 14 (1989), 809. doi: 10.1080/03605308908820630. [25] 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. doi: 10.3934/dcdsb.2007.7.735. [26] S. Pizzera, An age dependent population equation with delayed birth press,, Math. Meth. Appl. Sci., 27 (2004), 427. doi: 10.1002/mma.462. [27] 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. [28] 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. [29] 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. [30] G. Greiner, Perturbing the boundary conditions of a generator,, Houston J. Math., 13 (1987), 213. [31] M. Iannelli, Mathematical Theory of Age-structured Population Dynamics,, Giardini Editori, (1994). [32] A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations,, Springer, (1986). [33] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Marcell Dekker, (1985). [34] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations,, Springer, (2000). [35] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer, (1983). doi: 10.1007/978-1-4612-5561-1. [36] 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. [37] K. J. Engel, Operator matrices and systems of evolution equations,, RIMS Kokyuroku, 966 (1996), 61.
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