American Institute of Mathematical Sciences

Advanced
June  2018, 7(2): 293-316. doi: 10.3934/eect.2018015

Asymptotic behavior of a hierarchical size-structured population model

Dongxue Yan and  Xianlong Fu ,

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

* Corresponding author: Xianlong Fu

Received  October 2017 Revised  December 2017 Published  May 2018

Fund Project: This work is supported by NSF of China (Nos. 11671142 and 11771075), STCSM (No. 18dz2271000) and Shanghai Leading Academic Discipline Project (No. B407).

We study in this paper a hierarchical size-structured population dynamics model with environment feedback and delayed birth process. We are concerned with the asymptotic behavior, particularly on the effects of hierarchical structure and time lag on the long-time dynamics of the considered system. We formally linearize the system around a steady state and study the linearized system by $C_0-{\rm{semigroup}}$ framework and spectral analysis methods. Then we use the analytical results to establish the linearized stability, instability and asynchronous exponential growth conclusions under some conditions. Finally, some examples are presented and simulated to illustrate the obtained results.

Keywords: Hierarchical size-structured population, delayed boundary condition, C0-semigroup, linearized stability, asynchronous exponential growth.
Mathematics Subject Classification: Primary: 35F30, 92D25; Secondary: 34D20.
Citation: Dongxue Yan, Xianlong Fu. Asymptotic behavior of a hierarchical size-structured population model. Evolution Equations & Control Theory, 2018, 7 (2) : 293-316. doi: 10.3934/eect.2018015
References:
[1]

A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical size-structured model: Well-posedness and approximation, Appl. Math. Optim., 51 (2005), 35-59.  doi: 10.1007/s00245-004-0806-2.  Google Scholar

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Diff. Equ., 217 (2005), 431-455.  doi: 10.1016/j.jde.2004.12.013.  Google Scholar

[3]

A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20.  doi: 10.1006/jmaa.2001.6705.  Google Scholar

[4]

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

[5]

M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Equ., 11 (2011), 531-552.  doi: 10.1007/s00028-011-0100-8.  Google Scholar

[6]

À. Calsina and J. Saldaña, Asymptotic behavior of a model of hierarchically structured population dynamics, J. Math. Biol., 35 (1997), 967-987.  doi: 10.1007/s002850050085.  Google Scholar

[7]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729.  doi: 10.1007/BF00163023.  Google Scholar

[8]

J. M. Cushing and S. R.-J. Jang, Dynamics of hierarchical models in discrete time, J. Diff. Equ. Appl., 11 (2005), 95-115.  doi: 10.1080/10236190512331328343.  Google Scholar

[9]

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.  Google Scholar

[10]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations (Eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser, 2008, 187-200. doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[11]

O. Diekmann, M. Gyllenberg and J. A. J. Metz, Steady state analysis of structured population models, Theoret. Population Biol., 63 (2003), 309-338.  doi: 10.1016/S0040-5809(02)00058-8.  Google Scholar

[12]

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

[13]

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

[14]

J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discr. Impuls. Syst. Ser. A, 17 (2010), 639-657.   Google Scholar

[15]

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

[16]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Comm. Pure Appl. Anal., 8 (2009), 1825-1839.  doi: 10.3934/cpaa.2009.8.1825.  Google Scholar

[17]

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

[18]

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

[19]

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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Mathematics Applied to Science(New Orleans, La., 1986), Academic Press, 1988, 79-105.  Google Scholar

[23]

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.  Google Scholar

[24]

E. A. Kraev, Existence and uniqueness results for height structured hierarchical population models, Natur. Resource Modeling, 14 (2001), 45-70.  doi: 10.1111/j.1939-7445.2001.tb00050.x.  Google Scholar

[25]

A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar

[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.  Google Scholar

[27]

R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math., 19 (1996), 83-100.  doi: 10.1080/16073606.1996.9631827.  Google Scholar

[28]

R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. 1184, Springer-Verlag, 1986. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in L1, Discr. Cont. Dyn. Syst., 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663.  Google Scholar

[33]

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

[34]

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

[35]

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90 (1980), 153-161.  doi: 10.1007/BF01303264.  Google Scholar

[36]

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

[37]

D. Yan and X. Fu, The asymptotic behavior of an age-cycle structured cell model with delay, J. Dyn. Cont. Syst., 22 (2016), 441-458.  doi: 10.1007/s10883-015-9285-4.  Google Scholar

[38]

D. Yan and X. Fu, Asymptotic analysis of a spatially and size-structured population model with delyaed birth process, Comm. Pure Appl. Anal., 15 (2016), 637-655.  doi: 10.3934/cpaa.2016.15.637.  Google Scholar

show all references

References:
[1]

A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical size-structured model: Well-posedness and approximation, Appl. Math. Optim., 51 (2005), 35-59.  doi: 10.1007/s00245-004-0806-2.  Google Scholar

[2]

A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Diff. Equ., 217 (2005), 431-455.  doi: 10.1016/j.jde.2004.12.013.  Google Scholar

[3]

A. Bátkai and S. Piazzera, Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl., 264 (2001), 1-20.  doi: 10.1006/jmaa.2001.6705.  Google Scholar

[4]

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

[5]

M. Boulanouar, The asymptotic behavior of a structured cell population, J. Evol. Equ., 11 (2011), 531-552.  doi: 10.1007/s00028-011-0100-8.  Google Scholar

[6]

À. Calsina and J. Saldaña, Asymptotic behavior of a model of hierarchically structured population dynamics, J. Math. Biol., 35 (1997), 967-987.  doi: 10.1007/s002850050085.  Google Scholar

[7]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729.  doi: 10.1007/BF00163023.  Google Scholar

[8]

J. M. Cushing and S. R.-J. Jang, Dynamics of hierarchical models in discrete time, J. Diff. Equ. Appl., 11 (2005), 95-115.  doi: 10.1080/10236190512331328343.  Google Scholar

[9]

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.  Google Scholar

[10]

O. Diekmann and M. Gyllenberg, Abstract delay equations inspired by population dynamics, in Functional Analysis and Evolution Equations (Eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise and J. von Below), Birkhäuser, 2008, 187-200. doi: 10.1007/978-3-7643-7794-6_12.  Google Scholar

[11]

O. Diekmann, M. Gyllenberg and J. A. J. Metz, Steady state analysis of structured population models, Theoret. Population Biol., 63 (2003), 309-338.  doi: 10.1016/S0040-5809(02)00058-8.  Google Scholar

[12]

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

[13]

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

[14]

J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discr. Impuls. Syst. Ser. A, 17 (2010), 639-657.   Google Scholar

[15]

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

[16]

J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Comm. Pure Appl. Anal., 8 (2009), 1825-1839.  doi: 10.3934/cpaa.2009.8.1825.  Google Scholar

[17]

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

[18]

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

[19]

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.  Google Scholar

[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.  Google Scholar

[21]

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

[22]

G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive operators, Mathematics Applied to Science(New Orleans, La., 1986), Academic Press, 1988, 79-105.  Google Scholar

[23]

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.  Google Scholar

[24]

E. A. Kraev, Existence and uniqueness results for height structured hierarchical population models, Natur. Resource Modeling, 14 (2001), 45-70.  doi: 10.1111/j.1939-7445.2001.tb00050.x.  Google Scholar

[25]

A. J. Metz and O. Diekmann, The Dynamics of Psyiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar

[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.  Google Scholar

[27]

R. Nagel, G. Nickel and S. Romanelli, Identification of extrapolation spaces for unbounded operators, Quaest. Math., 19 (1996), 83-100.  doi: 10.1080/16073606.1996.9631827.  Google Scholar

[28]

R. Nagel, (ed.), One-Parameter Semigroups of Positive Operators, Lect. Notes in Math. 1184, Springer-Verlag, 1986. Google Scholar

[29]

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

[30]

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

[31]

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

[32]

A. Rhandi and R. Schnaubelt, Asymptotic behavior of a non-autonomous population equation with diffusion in L1, Discr. Cont. Dyn. Syst., 5 (1999), 663-683.  doi: 10.3934/dcds.1999.5.663.  Google Scholar

[33]

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

[34]

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

[35]

J. Voigt, A perturbation theorem for the essential spectral radius of strongly continuous semigroups, Monatsh. Math., 90 (1980), 153-161.  doi: 10.1007/BF01303264.  Google Scholar

[36]

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

[37]

D. Yan and X. Fu, The asymptotic behavior of an age-cycle structured cell model with delay, J. Dyn. Cont. Syst., 22 (2016), 441-458.  doi: 10.1007/s10883-015-9285-4.  Google Scholar

[38]

D. Yan and X. Fu, Asymptotic analysis of a spatially and size-structured population model with delyaed birth process, Comm. Pure Appl. Anal., 15 (2016), 637-655.  doi: 10.3934/cpaa.2016.15.637.  Google Scholar

Figure 1.  Choosing the parameters with $m = 10, ~~T = 120$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.18}{1+s^3}$; (c) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Choosing the parameters with $m = 10, ~~T = 160$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.18}{1+s^3}$; (c) $u_0(s) = \frac{0.09}{1+s^3}+e^{-3s}$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Choosing the parameter $m = 100, ~~T = 1050$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.01}{0.01+s^2}+0.0215$; (c) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Choosing the parameter $m = 80, ~~T = 1200$, "a" represents $u_\ast$ and $U_\ast$, the initial conditions corresponding to curves $b$ and $c$ are: (b) $u_0(s) = \frac{0.01}{0.01+s^2}+0.0215$; (c) $u_0(s) = \frac{0.05}{2+s^3}+0.0365$
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Choosing the parameters with $m = 40$, $T = 100$. The initial condition corresponding to the curve is $u_0(s) = s(16.5-s), ~0\leq s \leq 16.5$, $-1\leq t \leq100$; $u_0(s) = (s-16.5)(39-s), ~16.5\leq s \leq 39$, $-1\leq t \leq100$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Xianlong Fu, Dongmei Zhu. Stability results for a size-structured population model with delayed birth process. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 109-131. doi: 10.3934/dcdsb.2013.18.109
[2]

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
[3]

Dongxue Yan, Xianlong Fu. Asymptotic analysis of a spatially and size-structured population model with delayed birth process. Communications on Pure & Applied Analysis, 2016, 15 (2) : 637-655. doi: 10.3934/cpaa.2016.15.637
[4]

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
[5]

L. M. Abia, O. Angulo, J.C. López-Marcos. Size-structured population dynamics models and their numerical solutions. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1203-1222. doi: 10.3934/dcdsb.2004.4.1203
[6]

Keng Deng, Yixiang Wu. Extinction and uniform strong persistence of a size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 831-840. doi: 10.3934/dcdsb.2017041
[7]

Yunfei Lv, Yongzhen Pei, Rong Yuan. On a non-linear size-structured population model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020053
[8]

Jacek Banasiak, Wilson Lamb. Coagulation, fragmentation and growth processes in a size structured population. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 563-585. doi: 10.3934/dcdsb.2009.11.563
[9]

Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015
[10]

Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233
[11]

Mustapha Mokhtar-Kharroubi, Quentin Richard. Spectral theory and time asymptotics of size-structured two-phase population models. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020048
[12]

Yu-Xia Liang, Ze-Hua Zhou. Supercyclic translation $C_0$-semigroup on complex sectors. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 361-370. doi: 10.3934/dcds.2016.36.361
[13]

H. L. Smith, X. Q. Zhao. Competitive exclusion in a discrete-time, size-structured chemostat model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 183-191. doi: 10.3934/dcdsb.2001.1.183
[14]

Azmy S. Ackleh, H.T. Banks, Keng Deng, Shuhua Hu. Parameter Estimation in a Coupled System of Nonlinear Size-Structured Populations. Mathematical Biosciences & Engineering, 2005, 2 (2) : 289-315. doi: 10.3934/mbe.2005.2.289
[15]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[16]

Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery. Mathematical Biosciences & Engineering, 2005, 2 (4) : 719-741. doi: 10.3934/mbe.2005.2.719
[17]

Jacek Banasiak, Wilson Lamb. The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth. Kinetic & Related Models, 2012, 5 (2) : 223-236. doi: 10.3934/krm.2012.5.223
[18]

Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481
[19]

Horst R. Thieme. Positive perturbation of operator semigroups: growth bounds, essential compactness and asynchronous exponential growth. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 735-764. doi: 10.3934/dcds.1998.4.735
[20]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (62)
  • HTML views (237)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences