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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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109-117. Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Systems, 15 (2007), 1-19. Google Scholar [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.  Google Scholar [9] F. Brauer and Z. Ma, Stability of stage-structured population models, J. Math. Anal. Appl., 126 (1987), 301-315.  Google Scholar [10] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.  Google Scholar [11] T. A. Burton and G. Makey, Asymptotic stability for functional differential equations, J. Math. Anal. Appl., 126 (1994), 301-315. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. M. Cushing, Time delays in single growth models, J. Math. Biol., 4 (1977), 257-264.  Google Scholar [16] H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single-species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  Google Scholar [17] U. Foryś, Global stability for a class of delay differential equations, Appl. Math. Lett., 17 (2004), 581-584.  Google Scholar [18] K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.  Google Scholar [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.  Google Scholar [20] J. K. Hale, "Theory of Functional Differential Equations," Second editon, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar [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.  Google Scholar [22] N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Society, 25 (1950), 226-232.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar [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.  Google Scholar [30] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. Google Scholar [31] A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zoo., 2 (1954), 9-65. Google Scholar [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.  Google Scholar [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.  Google Scholar [34] H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011.  Google Scholar [35] C. E. Taylor and R. R. Sokal, Oscillations in housefly population sizes due to time lags, Ecology, 57 (1976), 1060-1067. Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109-117. Google Scholar [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.  Google Scholar [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.  Google Scholar [7] M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Systems, 15 (2007), 1-19. Google Scholar [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.  Google Scholar [9] F. Brauer and Z. Ma, Stability of stage-structured population models, J. Math. Anal. Appl., 126 (1987), 301-315.  Google Scholar [10] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001.  Google Scholar [11] T. A. Burton and G. Makey, Asymptotic stability for functional differential equations, J. Math. Anal. Appl., 126 (1994), 301-315. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [15] J. M. Cushing, Time delays in single growth models, J. Math. Biol., 4 (1977), 257-264.  Google Scholar [16] H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single-species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  Google Scholar [17] U. Foryś, Global stability for a class of delay differential equations, Appl. Math. Lett., 17 (2004), 581-584.  Google Scholar [18] K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics," Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992.  Google Scholar [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.  Google Scholar [20] J. K. Hale, "Theory of Functional Differential Equations," Second editon, Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977.  Google Scholar [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.  Google Scholar [22] N. D. Hayes, Roots of the transcendental equation associated with a certain difference-differential equation, J. London Math. Society, 25 (1950), 226-232.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.  Google Scholar [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.  Google Scholar [30] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. Google Scholar [31] A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zoo., 2 (1954), 9-65. Google Scholar [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.  Google Scholar [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.  Google Scholar [34] H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011.  Google Scholar [35] C. E. Taylor and R. R. Sokal, Oscillations in housefly population sizes due to time lags, Ecology, 57 (1976), 1060-1067. Google Scholar [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.  Google Scholar
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