September  2020, 25(9): 3523-3551. doi: 10.3934/dcdsb.2020071

Dynamics of a stage-structured population model with a state-dependent delay

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  August 2019 Revised  November 2019 Published  April 2020

Fund Project: The second author is supported by the NNSF of P.R. China (Grant No. 11671123)

This paper is devoted to the dynamics of a predator-prey model with stage structure for prey and state-dependent maturation delay. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. Secondly, the existence, uniqueness, and local asymptotical stability of (boundary and coexisting) equilibria are investigated by means of degree theory and Routh-Hurwitz criteria. Thirdly, the explicit bounds for the eventual behaviors of the mature population are obtained. Finally, by means of comparison principle and two auxiliary systems, it observed that the local asymptotical stability of either of the positive interior equilibrium and the positive boundary equilibrium implies that it is also globally asymptotical stable if the derivative of the delay function around this equilibrium is small enough.

Citation: Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071
References:
[1]

M. AdimyF. CrausteM. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.  Google Scholar

[2]

W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U.  Google Scholar

[3]

W. G. AielloH. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048.  Google Scholar

[4]

J. F. M. Al-Omari, The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput., 271 (2015), 142-153.  doi: 10.1016/j.amc.2015.08.119.  Google Scholar

[5]

J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Anal. Real World Appl., 6 (2005), 13-33.  doi: 10.1016/j.nonrwa.2004.04.002.  Google Scholar

[6] L. S. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing, 1988.   Google Scholar
[7]

K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426.  doi: 10.1090/S0002-9939-96-03437-5.  Google Scholar

[8]

K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model., 215 (2008), 69-76.  doi: 10.1016/j.ecolmodel.2008.02.019.  Google Scholar

[9]

M. E. Fisher and B. S. Goh, Stability results for delayed-recruitment models in population dynamics, J. Math. Biol., 19 (1984), 147-156.  doi: 10.1007/BF00275937.  Google Scholar

[10]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  doi: 10.1016/S0092-8240(86)90003-0.  Google Scholar

[11]

R. Gambell, Birds and mammals – Antarctic whales, in Antarctica, Pergamon Press, New York, 1985, 223–241. doi: 10.1016/B978-0-08-028881-9.50022-4.  Google Scholar

[12]

S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.  doi: 10.1007/s00285-004-0278-2.  Google Scholar

[13]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.  Google Scholar

[14]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.  Google Scholar

[15]

W. S. C. Gurney and R. M. Nisbet, Fluctuating periodicity, generation separation, and the expression of larval competition, Theoret. Population Biol., 28 (1985), 150-180.  doi: 10.1016/0040-5809(85)90026-7.  Google Scholar

[16]

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4612-9892-2.  Google Scholar

[17]

F. Hartung and J. Turi, On the asymptotic behavior of the solutions of a state-dependent delay equation, Differential Integral Equations, 8 (1995), 1867-1872.   Google Scholar

[18]

A. Hou and S. Guo, Stability and Hopf bifurcation in van der Pol oscillators with state-dependent delayed feedback, Nonlinear Dynam., 79 (2015), 2407-2419.  doi: 10.1007/s11071-014-1821-3.  Google Scholar

[19]

C. Huang, Y. Qiao and L. Huang, et al., Dynamical behaviors of a food-chain model with stage structure and time delays, Adv. Difference Equ., 2018 (2018), 26pp. doi: 10.1186/s13662-018-1589-8.  Google Scholar

[20]

H. F. HuoW. T. Li and R. P. Agarwal, Optimal harvesting and stability for two species stage-structured system with cannibalism, Int. J. App. Math., 6 (2001), 59-79.   Google Scholar

[21]

Y. S. Koselsov, Properties of solutions of a class of equations with delay that describe the dynamics of change of the size of a species with account taken of age structure, Mat. Sb. (N.S.), 117 (1982), 86-94.   Google Scholar

[22]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[23]

H. D. Landahl and B. D. Hansen, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17.  doi: 10.1007/BF02463488.  Google Scholar

[24]

S. Z. Li and S. J. Guo, Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1393-1423.  doi: 10.3934/dcdsb.2017067.  Google Scholar

[25]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type Ⅱ functional response, J. Nonlinear Sci., 28 (2018), 1151-1187.  doi: 10.1007/s00332-018-9444-3.  Google Scholar

[26]

M. Lloyd and H. S. Dybas, The periodical cicada problem, Evolution, 20 (1966), 133-149.  doi: 10.1111/j.1558-5646.1966.tb03350.x.  Google Scholar

[27]

J. D. Murray, Mathematical Biology. I. An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[28]

A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Australian J. Zoology, 2 (1954), 1-8.  doi: 10.1071/ZO9540001.  Google Scholar

[29] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar
[30]

Y. Qu and J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynam., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x.  Google Scholar

[31]

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49 (1987), 253-255.  doi: 10.1016/S0092-8240(87)80045-9.  Google Scholar

[32]

Y. Satio and Y. Takeuchi, A time-delay model for prey-predator growth with stage structure, Can. Appl. Math. Q., 11 (2003), 293-302.   Google Scholar

[33]

X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 423-430.  doi: 10.1007/s102550200042.  Google Scholar

[34]

X. Song and L. Chen, A predator-prey system with stage structure and harvesting for predator, Ann. Differential Equations, 18 (2002), 264-277.   Google Scholar

[35]

K. Tognetti, The two stage stochastic model, Math. Biosci., 25 (1975), 195-204.  doi: 10.1016/0025-5564(75)90002-4.  Google Scholar

[36]

P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.  doi: 10.2307/1932137.  Google Scholar

[37]

P. J. Wangersky and W. J. Cunningham, On time lags in equations of growth, Proc. Natl. Acad. Sci. U.S.A., 42 (1956), 699-702.  doi: 10.1073/pnas.42.9.699.  Google Scholar

[38]

S. N. WoodS. P. BlytheW. S. C. Gurney and R. M. Nisbet, Instability in mortality estimation schemes related to stage-structure population models, IMA J. Math. Appl. Med. Biol., 6 (1989), 47-68.  doi: 10.1093/imammb/6.1.47.  Google Scholar

[39]

S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059.  Google Scholar

[40]

Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay, Appl. Math. Comput., 214 (2009), 228-235.  doi: 10.1016/j.amc.2009.03.078.  Google Scholar

[41] Q. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.   Google Scholar
[42]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for a nonlinear second order differential equation with state-dependent delay, Proc. Amer. Math. Soc., 145 (2017), 4893-4903.  doi: 10.1090/proc/13714.  Google Scholar

[43]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state-dependent delay, Math. Methods Appl. Sci., 40 (2017), 5307-5331.  doi: 10.1002/mma.4388.  Google Scholar

show all references

References:
[1]

M. AdimyF. CrausteM. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.  Google Scholar

[2]

W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U.  Google Scholar

[3]

W. G. AielloH. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048.  Google Scholar

[4]

J. F. M. Al-Omari, The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput., 271 (2015), 142-153.  doi: 10.1016/j.amc.2015.08.119.  Google Scholar

[5]

J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Anal. Real World Appl., 6 (2005), 13-33.  doi: 10.1016/j.nonrwa.2004.04.002.  Google Scholar

[6] L. S. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing, 1988.   Google Scholar
[7]

K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426.  doi: 10.1090/S0002-9939-96-03437-5.  Google Scholar

[8]

K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model., 215 (2008), 69-76.  doi: 10.1016/j.ecolmodel.2008.02.019.  Google Scholar

[9]

M. E. Fisher and B. S. Goh, Stability results for delayed-recruitment models in population dynamics, J. Math. Biol., 19 (1984), 147-156.  doi: 10.1007/BF00275937.  Google Scholar

[10]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  doi: 10.1016/S0092-8240(86)90003-0.  Google Scholar

[11]

R. Gambell, Birds and mammals – Antarctic whales, in Antarctica, Pergamon Press, New York, 1985, 223–241. doi: 10.1016/B978-0-08-028881-9.50022-4.  Google Scholar

[12]

S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.  doi: 10.1007/s00285-004-0278-2.  Google Scholar

[13]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.  Google Scholar

[14]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.  Google Scholar

[15]

W. S. C. Gurney and R. M. Nisbet, Fluctuating periodicity, generation separation, and the expression of larval competition, Theoret. Population Biol., 28 (1985), 150-180.  doi: 10.1016/0040-5809(85)90026-7.  Google Scholar

[16]

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4612-9892-2.  Google Scholar

[17]

F. Hartung and J. Turi, On the asymptotic behavior of the solutions of a state-dependent delay equation, Differential Integral Equations, 8 (1995), 1867-1872.   Google Scholar

[18]

A. Hou and S. Guo, Stability and Hopf bifurcation in van der Pol oscillators with state-dependent delayed feedback, Nonlinear Dynam., 79 (2015), 2407-2419.  doi: 10.1007/s11071-014-1821-3.  Google Scholar

[19]

C. Huang, Y. Qiao and L. Huang, et al., Dynamical behaviors of a food-chain model with stage structure and time delays, Adv. Difference Equ., 2018 (2018), 26pp. doi: 10.1186/s13662-018-1589-8.  Google Scholar

[20]

H. F. HuoW. T. Li and R. P. Agarwal, Optimal harvesting and stability for two species stage-structured system with cannibalism, Int. J. App. Math., 6 (2001), 59-79.   Google Scholar

[21]

Y. S. Koselsov, Properties of solutions of a class of equations with delay that describe the dynamics of change of the size of a species with account taken of age structure, Mat. Sb. (N.S.), 117 (1982), 86-94.   Google Scholar

[22]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar

[23]

H. D. Landahl and B. D. Hansen, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17.  doi: 10.1007/BF02463488.  Google Scholar

[24]

S. Z. Li and S. J. Guo, Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1393-1423.  doi: 10.3934/dcdsb.2017067.  Google Scholar

[25]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type Ⅱ functional response, J. Nonlinear Sci., 28 (2018), 1151-1187.  doi: 10.1007/s00332-018-9444-3.  Google Scholar

[26]

M. Lloyd and H. S. Dybas, The periodical cicada problem, Evolution, 20 (1966), 133-149.  doi: 10.1111/j.1558-5646.1966.tb03350.x.  Google Scholar

[27]

J. D. Murray, Mathematical Biology. I. An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.  Google Scholar

[28]

A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Australian J. Zoology, 2 (1954), 1-8.  doi: 10.1071/ZO9540001.  Google Scholar

[29] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar
[30]

Y. Qu and J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynam., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x.  Google Scholar

[31]

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49 (1987), 253-255.  doi: 10.1016/S0092-8240(87)80045-9.  Google Scholar

[32]

Y. Satio and Y. Takeuchi, A time-delay model for prey-predator growth with stage structure, Can. Appl. Math. Q., 11 (2003), 293-302.   Google Scholar

[33]

X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 423-430.  doi: 10.1007/s102550200042.  Google Scholar

[34]

X. Song and L. Chen, A predator-prey system with stage structure and harvesting for predator, Ann. Differential Equations, 18 (2002), 264-277.   Google Scholar

[35]

K. Tognetti, The two stage stochastic model, Math. Biosci., 25 (1975), 195-204.  doi: 10.1016/0025-5564(75)90002-4.  Google Scholar

[36]

P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.  doi: 10.2307/1932137.  Google Scholar

[37]

P. J. Wangersky and W. J. Cunningham, On time lags in equations of growth, Proc. Natl. Acad. Sci. U.S.A., 42 (1956), 699-702.  doi: 10.1073/pnas.42.9.699.  Google Scholar

[38]

S. N. WoodS. P. BlytheW. S. C. Gurney and R. M. Nisbet, Instability in mortality estimation schemes related to stage-structure population models, IMA J. Math. Appl. Med. Biol., 6 (1989), 47-68.  doi: 10.1093/imammb/6.1.47.  Google Scholar

[39]

S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059.  Google Scholar

[40]

Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay, Appl. Math. Comput., 214 (2009), 228-235.  doi: 10.1016/j.amc.2009.03.078.  Google Scholar

[41] Q. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990.   Google Scholar
[42]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for a nonlinear second order differential equation with state-dependent delay, Proc. Amer. Math. Soc., 145 (2017), 4893-4903.  doi: 10.1090/proc/13714.  Google Scholar

[43]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state-dependent delay, Math. Methods Appl. Sci., 40 (2017), 5307-5331.  doi: 10.1002/mma.4388.  Google Scholar

Figure 1.  Simulations of system (41) with $ \tau(x) = 4-2e^{-0.1x} $ and $ (r, c) = (0.1, 1) $ illustrate that the positive equilibrium is globally asymptotically stable
Figure 2.  Simulations of system (41) with $ \tau(x) = 4-2e^{-0.1x} $ and $ (r, c) = (0.6, 0.1) $ illustrate that the positive equilibrium is globally asymptotically stable
Figure 3.  Simulations of system (41) with $ \tau(x)\equiv 4 $ and $ (r, c) = (0.1, 1) $ illustrate that the positive equilibrium is globally asymptotically stable
[1]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

[2]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445

[3]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[4]

Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365

[5]

Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074

[6]

Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143

[7]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039

[8]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[9]

Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801

[10]

Odo Diekmann, Karolína Korvasová. Linearization of solution operators for state-dependent delay equations: A simple example. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 137-149. doi: 10.3934/dcds.2016.36.137

[11]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993

[12]

Qingwen Hu, Bernhard Lani-Wayda, Eugen Stumpf. Preface: Delay differential equations with state-dependent delays and their applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (1) : ⅰ-ⅰ. doi: 10.3934/dcdss.20201i

[13]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167

[14]

A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701

[15]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

[16]

Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56

[17]

Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167

[18]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931

[19]

Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913

[20]

Ferenc Hartung, Janos Turi. Linearized stability in functional differential equations with state-dependent delays. Conference Publications, 2001, 2001 (Special) : 416-425. doi: 10.3934/proc.2001.2001.416

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (86)
  • HTML views (148)
  • Cited by (0)

Other articles
by authors

[Back to Top]