American Institute of Mathematical Sciences

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

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

Xianlong Fu 1, and  Dongmei Zhu 1,

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.
Keywords: Population model, stability., $C_0$-semigroup, delayed boundary condition, size-structured.
Mathematics Subject Classification: 34G20, 92D25, 34D20, 35F3.
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:
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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

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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.  Google Scholar

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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.  Google Scholar

[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), 665-675. doi: 10.1016/j.jmaa.2009.07.005.  Google Scholar

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R. Dilão, T. Domingos and E. M. Shahverdiev, Harvesting in a resource dependent age structured Leslie type population model, Math. Biosci., 189 (2004), 141-151. doi: 10.1016/j.mbs.2004.01.008.  Google Scholar

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[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), 1178-1200. doi: 10.1137/080728512.  Google Scholar

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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), 573-600.  Google Scholar

[16]

A. Ducrot and P. Magal, Traveling wave solution for infection age structured epidemic model with vital dynamics, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[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, J. Mallet-Paret, J. Wu, Y. Yi, and H. Zhu (eds.), Fields Institute Communications, 64 (2013), 353-390.  Google Scholar

[24]

B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832. doi: 10.1080/03605308908820630.  Google Scholar

[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), 735-754. doi: 10.3934/dcdsb.2007.7.735.  Google Scholar

[26]

S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462.  Google Scholar

[27]

S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77. doi: 10.1007/s00028-004-0159-6.  Google Scholar

[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), 109-131. doi: 10.3934/dcdsb.2013.18.109.  Google Scholar

[29]

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.  Google Scholar

[30]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.  Google Scholar

[31]

M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Giardini Editori, Pisa, 1994. Google Scholar

[32]

A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. Google Scholar

[33]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.  Google Scholar

[34]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[36]

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.  Google Scholar

[37]

K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80.  Google Scholar

show all references

References:
[1]

T. Hagen, Eigenvalue asymptotics in isothermal forced elongation, J. Math. Anal. Appl., 224 (2000), 393-407. doi: 10.1006/jmaa.1999.6708.  Google Scholar

[2]

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.  Google Scholar

[3]

T. Hagen and M. Renardy, Studies on the linear equations of melt-spinning of viscous fluids, Diff. Int. Equ., 14 (2001), 19-36.  Google Scholar

[4]

J. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase, Discr. Contin. Dyn. Syst., 33 (2013), 4891-4921. doi: 10.3934/dcds.2013.33.4891.  Google Scholar

[5]

M. Farkas, On the stability of stationary age distributions, Appl. Math. Comp., 131 (2002), 107-123. doi: 10.1016/S0096-3003(01)00131-X.  Google Scholar

[6]

J. Z. Farkas, Stability conditions for a nonlinear size-structured model, Nonl. Anal. (RWA), 6 (2005), 962-969. doi: 10.1016/j.nonrwa.2004.06.002.  Google Scholar

[7]

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

[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), 1087-1103. doi: 10.1080/00036810701545634.  Google Scholar

[9]

J. Z. Farkas and T. Hagen, Asymptotic behavior of size-structured populations via juvenile-adult interaction, Discr. Cont. Dyn. Syst. B, 9 (2008), 249-266.  Google Scholar

[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), 665-675. doi: 10.1016/j.jmaa.2009.07.005.  Google Scholar

[11]

R. Dilão, T. Domingos and E. M. Shahverdiev, Harvesting in a resource dependent age structured Leslie type population model, Math. Biosci., 189 (2004), 141-151. doi: 10.1016/j.mbs.2004.01.008.  Google Scholar

[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), 242-267. doi: 10.1111/j.1939-7445.2011.00090.x.  Google Scholar

[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), 54-70. doi: 10.1016/j.nonrwa.2007.08.014.  Google Scholar

[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), 1178-1200. doi: 10.1137/080728512.  Google Scholar

[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), 573-600.  Google Scholar

[16]

A. Ducrot and P. Magal, Traveling wave solution for infection age structured epidemic model with vital dynamics, Nonlinearity, 24 (2011), 2891-2911. doi: 10.1088/0951-7715/24/10/012.  Google Scholar

[17]

D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populations, J. Franklin Inst., 297 (1974), 345-376. Google Scholar

[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.  Google Scholar

[19]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional analysis and evolution equations, 187-200, Birkhäuser, Basel, 2008. doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[20]

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.  Google Scholar

[21]

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.  Google Scholar

[22]

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.  Google Scholar

[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, J. Mallet-Paret, J. Wu, Y. Yi, and H. Zhu (eds.), Fields Institute Communications, 64 (2013), 353-390.  Google Scholar

[24]

B. Guo and W. Chan, A semigroup approach to age dependent population dynamics with time delay, Comm. PDEs, 14 (1989), 809-832. doi: 10.1080/03605308908820630.  Google Scholar

[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), 735-754. doi: 10.3934/dcdsb.2007.7.735.  Google Scholar

[26]

S. Pizzera, An age dependent population equation with delayed birth press, Math. Meth. Appl. Sci., 27 (2004), 427-439. doi: 10.1002/mma.462.  Google Scholar

[27]

S. Pizzera and L. Tonetto, Asynchronous exponential growth for an age dependent population equation with delayed birth process, J. Evol. Equ., 5 (2005), 61-77. doi: 10.1007/s00028-004-0159-6.  Google Scholar

[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), 109-131. doi: 10.3934/dcdsb.2013.18.109.  Google Scholar

[29]

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.  Google Scholar

[30]

G. Greiner, Perturbing the boundary conditions of a generator, Houston J. Math., 13 (1987), 213-229.  Google Scholar

[31]

M. Iannelli, Mathematical Theory of Age-structured Population Dynamics, Giardini Editori, Pisa, 1994. Google Scholar

[32]

A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. Google Scholar

[33]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.  Google Scholar

[34]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar

[35]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[36]

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.  Google Scholar

[37]

K. J. Engel, Operator matrices and systems of evolution equations, RIMS Kokyuroku, 966 (1996), 61-80.  Google Scholar

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