• Previous Article
    Nonlinear conformation response in the finite channel: Existence of a unique solution for the dynamic PNP model
  • DCDS-B Home
  • This Issue
  • Next Article
    Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition
October  2012, 17(7): 2451-2464. doi: 10.3934/dcdsb.2012.17.2451

Stability conditions for a class of delay differential equations in single species population dynamics

1. 

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

2. 

College of Science and Engineering, Aoyama Gakuin University, Sagamihara, 2525258, Japan

3. 

Department of Systems Engineering, Shizuoka University, Hamamatsu, 4328561, Japan

Received  January 2011 Revised  February 2012 Published  July 2012

We consider a class of nonlinear delay differential equations,which describes single species population growth with stage structure. By constructing appropriate Lyapunov functionals, the global asymptotic stability criteria, which are independent of delay, are established. Much sharper stability conditions than known results are provided. Applications of the results to some population models show the effectiveness of the methods described in the paper.
Citation: Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451
References:
[1]

W. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.

[2]

W. Aiello, H. 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.

[3]

J. Arino, L. Wang and G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theore. Biol., 241 (2006), 109-119.

[4]

J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109-117.

[5]

S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Disc. Cont. Dyn. Syst. Ser. B, 1 (2001), 233-256.

[6]

S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, Instability and complex dynamic behavior in population models with long time delays, Theor. Pop. Biol., 22 (1982), 147-176.

[7]

M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Systems, 15 (2007), 1-19.

[8]

M. Bodnar and U. Foryś, Global stability and Hopf bifurcation for a gerneral class of delay differential equations, Math. Mathods Appl. Sci., 31 (2008), 1197-1207. doi: 10.1002/mma.965.

[9]

F. Brauer and Z. Ma, Stability of stage-structured population models, J. Math. Anal. Appl., 126 (1987), 301-315.

[10]

F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.

[11]

T. A. Burton and G. Makey, Asymptotic stability for functional differential equations, J. Math. Anal. Appl., 126 (1994), 301-315.

[12]

T. A. Burton and G. Makey, Marachkov type stability results for functional-differential equations, E. J. Qualitative Theory of Diff. Equ., 1998, 17 pp.

[13]

K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.

[14]

F. Crauste, Stability and Hopf bifurcation for a first-order delay differential equation with distributed delay, in "Complex Time-Delay Systems," Understanding Complex Systems, Springer, Berlin, (2010), 263-296.

[15]

J. M. Cushing, Time delays in single growth models, J. Math. Biol., 4 (1977), 257-264.

[16]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single-species dynamics, Bull. Math. Biol., 48 (1986), 485-492.

[17]

U. Foryś, Global stability for a class of delay differential equations, Appl. Math. Lett., 17 (2004), 581-584.

[18]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[19]

J. R. Haddock and J. Terjéki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations, J. Diff. Equat., 48 (1983), 95-122.

[20]

J. K. Hale, "Theory of Functional Differential Equations," Second editon, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.

[21]

L. Hatvani, Asymptotic stability conditions for a linear nonautonomous delay differential equation, in "Differential Equations and Applications to Biology and to Industry" (eds. M. Martelli, K. Cooke, E. Cumberbatch, R. Tang and H. Thieme) (Claremont, CA, 1994), World Sci. Publ., River Edge, NJ, (1996), 181-190.

[22]

N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Society, 25 (1950), 226-232.

[23]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global Stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.

[24]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.

[25]

G. Karakostas, Ch. G. Philos and Y. G. Sficas, Stable steady state of some population models, J. Dynam. Differential Equations, 4 (1992), 161-190.

[26]

I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differntial equations of population dynamics, Math. Comput. Model., 35 (2002), 295-301.

[27]

Y. Kuang, Global attractivity in delay defferential equations related to models of physiology and population biology, Japan J. Indust. Appl. Math., 9 (1992), 205-238.

[28]

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

[29]

S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc., 96 (1986), 75-78.

[30]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.

[31]

A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zoo., 2 (1954), 9-65.

[32]

S. Ruan, Delay differential equations in single species dynamics, in "Delay Differential Equations and Applications" (eds. O. Arino, E. Ait Dads and M. Hbid), NATO Sci. Ser. II Math. Phys. Chem., 205, Springer, Dordrecht, (2006), 477-517.

[33]

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.

[34]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011.

[35]

C. E. Taylor and R. R. Sokal, Oscillations in housefly population sizes due to time lags, Ecology, 57 (1976), 1060-1067.

[36]

H.-O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$, Mem. Am. Math. Soc., 113 (1995), vi+76 pp.

show all references

References:
[1]

W. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.

[2]

W. Aiello, H. 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.

[3]

J. Arino, L. Wang and G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theore. Biol., 241 (2006), 109-119.

[4]

J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109-117.

[5]

S. Bernard, J. Bélair and M. C. Mackey, Sufficient conditions for stability of linear differential equations with distributed delay, Disc. Cont. Dyn. Syst. Ser. B, 1 (2001), 233-256.

[6]

S. P. Blythe, R. M. Nisbet and W. S. C. Gurney, Instability and complex dynamic behavior in population models with long time delays, Theor. Pop. Biol., 22 (1982), 147-176.

[7]

M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Systems, 15 (2007), 1-19.

[8]

M. Bodnar and U. Foryś, Global stability and Hopf bifurcation for a gerneral class of delay differential equations, Math. Mathods Appl. Sci., 31 (2008), 1197-1207. doi: 10.1002/mma.965.

[9]

F. Brauer and Z. Ma, Stability of stage-structured population models, J. Math. Anal. Appl., 126 (1987), 301-315.

[10]

F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.

[11]

T. A. Burton and G. Makey, Asymptotic stability for functional differential equations, J. Math. Anal. Appl., 126 (1994), 301-315.

[12]

T. A. Burton and G. Makey, Marachkov type stability results for functional-differential equations, E. J. Qualitative Theory of Diff. Equ., 1998, 17 pp.

[13]

K. Cooke, P. van den Driessche and X. Zou, Interaction of maturation delay and nonlinear birth in population and epidemic models, J. Math. Biol., 39 (1999), 332-352.

[14]

F. Crauste, Stability and Hopf bifurcation for a first-order delay differential equation with distributed delay, in "Complex Time-Delay Systems," Understanding Complex Systems, Springer, Berlin, (2010), 263-296.

[15]

J. M. Cushing, Time delays in single growth models, J. Math. Biol., 4 (1977), 257-264.

[16]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single-species dynamics, Bull. Math. Biol., 48 (1986), 485-492.

[17]

U. Foryś, Global stability for a class of delay differential equations, Appl. Math. Lett., 17 (2004), 581-584.

[18]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.

[19]

J. R. Haddock and J. Terjéki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations, J. Diff. Equat., 48 (1983), 95-122.

[20]

J. K. Hale, "Theory of Functional Differential Equations," Second editon, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.

[21]

L. Hatvani, Asymptotic stability conditions for a linear nonautonomous delay differential equation, in "Differential Equations and Applications to Biology and to Industry" (eds. M. Martelli, K. Cooke, E. Cumberbatch, R. Tang and H. Thieme) (Claremont, CA, 1994), World Sci. Publ., River Edge, NJ, (1996), 181-190.

[22]

N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Society, 25 (1950), 226-232.

[23]

G. Huang, Y. Takeuchi, W. Ma and D. Wei, Global Stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.

[24]

G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 2693-2708.

[25]

G. Karakostas, Ch. G. Philos and Y. G. Sficas, Stable steady state of some population models, J. Dynam. Differential Equations, 4 (1992), 161-190.

[26]

I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differntial equations of population dynamics, Math. Comput. Model., 35 (2002), 295-301.

[27]

Y. Kuang, Global attractivity in delay defferential equations related to models of physiology and population biology, Japan J. Indust. Appl. Math., 9 (1992), 205-238.

[28]

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

[29]

S. M. Lenhart and C. C. Travis, Global stability of a biological model with time delay, Proc. Amer. Math. Soc., 96 (1986), 75-78.

[30]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289.

[31]

A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zoo., 2 (1954), 9-65.

[32]

S. Ruan, Delay differential equations in single species dynamics, in "Delay Differential Equations and Applications" (eds. O. Arino, E. Ait Dads and M. Hbid), NATO Sci. Ser. II Math. Phys. Chem., 205, Springer, Dordrecht, (2006), 477-517.

[33]

G. Röst and J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.

[34]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011.

[35]

C. E. Taylor and R. R. Sokal, Oscillations in housefly population sizes due to time lags, Ecology, 57 (1976), 1060-1067.

[36]

H.-O. Walther, The 2-dimensional attractor of $x'(t)=-\mu x(t)+f(x(t-1))$, Mem. Am. Math. Soc., 113 (1995), vi+76 pp.

[1]

Elena Trofimchuk, Sergei Trofimchuk. Admissible wavefront speeds for a single species reaction-diffusion equation with delay. Discrete and Continuous Dynamical Systems, 2008, 20 (2) : 407-423. doi: 10.3934/dcds.2008.20.407

[2]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

[3]

Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for multistrain models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 507-536. doi: 10.3934/dcdsb.2017025

[4]

Eduardo Liz. Local stability implies global stability in some one-dimensional discrete single-species models. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 191-199. doi: 10.3934/dcdsb.2007.7.191

[5]

Yoji Otani, Tsuyoshi Kajiwara, Toru Sasaki. Lyapunov functionals for virus-immune models with infinite delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 3093-3114. doi: 10.3934/dcdsb.2015.20.3093

[6]

Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048

[7]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[8]

Peter Giesl. Construction of a global Lyapunov function using radial basis functions with a single operator. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 101-124. doi: 10.3934/dcdsb.2007.7.101

[9]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

[10]

Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6669-6688. doi: 10.3934/dcds.2016090

[11]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations and Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493

[12]

Majid Bani-Yaghoub, Chunhua Ou, Guangming Yao. Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2509-2535. doi: 10.3934/dcdss.2020195

[13]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092

[14]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139

[15]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 245-263. doi: 10.3934/dcdss.2020468

[16]

Guillaume Bal, Olivier Pinaud, Lenya Ryzhik. On the stability of some imaging functionals. Inverse Problems and Imaging, 2016, 10 (3) : 585-616. doi: 10.3934/ipi.2016013

[17]

Chuangxia Huang, Lihong Huang, Jianhong Wu. Global population dynamics of a single species structured with distinctive time-varying maturation and self-limitation delays. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2427-2440. doi: 10.3934/dcdsb.2021138

[18]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361

[19]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215

[20]

Deqiong Ding, Wendi Qin, Xiaohua Ding. Lyapunov functions and global stability for a discretized multigroup SIR epidemic model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1971-1981. doi: 10.3934/dcdsb.2015.20.1971

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (168)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]