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Dynamical bifurcation of the two dimensional SwiftHohenberg equation with odd periodic condition
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 
References:
[1] 
W. Aiello and H. I. Freedman, A timedelay model of singlespecies growth with stage structure, Math. Biosci., 101 (1990), 139153. 
[2] 
W. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stagestructured population growth with statedependent time delay, SIAM J. Appl. Math., 52 (1992), 855869. 
[3] 
J. Arino, L. Wang and G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theore. Biol., 241 (2006), 109119. 
[4] 
J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109117. 
[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), 233256. 
[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), 147176. 
[7] 
M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Systems, 15 (2007), 119. 
[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), 11971207. doi: 10.1002/mma.965. 
[9] 
F. Brauer and Z. Ma, Stability of stagestructured population models, J. Math. Anal. Appl., 126 (1987), 301315. 
[10] 
F. Brauer and C. CastilloChávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, SpringerVerlag, New York, 2001. 
[11] 
T. A. Burton and G. Makey, Asymptotic stability for functional differential equations, J. Math. Anal. Appl., 126 (1994), 301315. 
[12] 
T. A. Burton and G. Makey, Marachkov type stability results for functionaldifferential 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), 332352. 
[14] 
F. Crauste, Stability and Hopf bifurcation for a firstorder delay differential equation with distributed delay, in "Complex TimeDelay Systems," Understanding Complex Systems, Springer, Berlin, (2010), 263296. 
[15] 
J. M. Cushing, Time delays in single growth models, J. Math. Biol., 4 (1977), 257264. 
[16] 
H. I. Freedman and K. Gopalsamy, Global stability in timedelayed singlespecies dynamics, Bull. Math. Biol., 48 (1986), 485492. 
[17] 
U. Foryś, Global stability for a class of delay differential equations, Appl. Math. Lett., 17 (2004), 581584. 
[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, LiapunovRazumikhin functions and an invariance principle for functionaldifferential equations, J. Diff. Equat., 48 (1983), 95122. 
[20] 
J. K. Hale, "Theory of Functional Differential Equations," Second editon, Applied Mathematical Sciences, Vol. 3, SpringerVerlag, New YorkHeidelberg, 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), 181190. 
[22] 
N. D. Hayes, Roots of the transcendental equation associated with a certain differencedifferential equation, J. London Math. Society, 25 (1950), 226232. 
[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), 11921207. 
[24] 
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 26932708. 
[25] 
G. Karakostas, Ch. G. Philos and Y. G. Sficas, Stable steady state of some population models, J. Dynam. Differential Equations, 4 (1992), 161190. 
[26] 
I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differntial equations of population dynamics, Math. Comput. Model., 35 (2002), 295301. 
[27] 
Y. Kuang, Global attractivity in delay defferential equations related to models of physiology and population biology, Japan J. Indust. Appl. Math., 9 (1992), 205238. 
[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), 7578. 
[30] 
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287289. 
[31] 
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zoo., 2 (1954), 965. 
[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), 477517. 
[33] 
G. Röst and J. Wu, Domaindecomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 26552669. 
[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), 10601067. 
[36] 
H.O. Walther, The 2dimensional attractor of $x'(t)=\mu x(t)+f(x(t1))$, Mem. Am. Math. Soc., 113 (1995), vi+76 pp. 
show all references
References:
[1] 
W. Aiello and H. I. Freedman, A timedelay model of singlespecies growth with stage structure, Math. Biosci., 101 (1990), 139153. 
[2] 
W. Aiello, H. I. Freedman and J. Wu, Analysis of a model representing stagestructured population growth with statedependent time delay, SIAM J. Appl. Math., 52 (1992), 855869. 
[3] 
J. Arino, L. Wang and G. S. K. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theore. Biol., 241 (2006), 109119. 
[4] 
J. R. Beddington and R. M. May, Time delays are not necessarily destabilizing, Math. Biosci., 27 (1975), 109117. 
[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), 233256. 
[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), 147176. 
[7] 
M. Bodnar and U. Foryś, Three types of simple DDE's describing tumor growth, J. Biol. Systems, 15 (2007), 119. 
[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), 11971207. doi: 10.1002/mma.965. 
[9] 
F. Brauer and Z. Ma, Stability of stagestructured population models, J. Math. Anal. Appl., 126 (1987), 301315. 
[10] 
F. Brauer and C. CastilloChávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, SpringerVerlag, New York, 2001. 
[11] 
T. A. Burton and G. Makey, Asymptotic stability for functional differential equations, J. Math. Anal. Appl., 126 (1994), 301315. 
[12] 
T. A. Burton and G. Makey, Marachkov type stability results for functionaldifferential 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), 332352. 
[14] 
F. Crauste, Stability and Hopf bifurcation for a firstorder delay differential equation with distributed delay, in "Complex TimeDelay Systems," Understanding Complex Systems, Springer, Berlin, (2010), 263296. 
[15] 
J. M. Cushing, Time delays in single growth models, J. Math. Biol., 4 (1977), 257264. 
[16] 
H. I. Freedman and K. Gopalsamy, Global stability in timedelayed singlespecies dynamics, Bull. Math. Biol., 48 (1986), 485492. 
[17] 
U. Foryś, Global stability for a class of delay differential equations, Appl. Math. Lett., 17 (2004), 581584. 
[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, LiapunovRazumikhin functions and an invariance principle for functionaldifferential equations, J. Diff. Equat., 48 (1983), 95122. 
[20] 
J. K. Hale, "Theory of Functional Differential Equations," Second editon, Applied Mathematical Sciences, Vol. 3, SpringerVerlag, New YorkHeidelberg, 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), 181190. 
[22] 
N. D. Hayes, Roots of the transcendental equation associated with a certain differencedifferential equation, J. London Math. Society, 25 (1950), 226232. 
[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), 11921207. 
[24] 
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model for viral infections, SIAM J. Appl. Math., 70 (2010), 26932708. 
[25] 
G. Karakostas, Ch. G. Philos and Y. G. Sficas, Stable steady state of some population models, J. Dynam. Differential Equations, 4 (1992), 161190. 
[26] 
I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differntial equations of population dynamics, Math. Comput. Model., 35 (2002), 295301. 
[27] 
Y. Kuang, Global attractivity in delay defferential equations related to models of physiology and population biology, Japan J. Indust. Appl. Math., 9 (1992), 205238. 
[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), 7578. 
[30] 
M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287289. 
[31] 
A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zoo., 2 (1954), 965. 
[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), 477517. 
[33] 
G. Röst and J. Wu, Domaindecomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 26552669. 
[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), 10601067. 
[36] 
H.O. Walther, The 2dimensional attractor of $x'(t)=\mu x(t)+f(x(t1))$, Mem. Am. Math. Soc., 113 (1995), vi+76 pp. 
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