A SHORT SURVEY ON DELAY DIFFERENTIAL SYSTEMS WITH PERIODIC COEFFICIENTS

Redouane Qesmi

doi:10.11948/2018.296

2018 Volume 8 Issue 1

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Redouane Qesmi. A SHORT SURVEY ON DELAY DIFFERENTIAL SYSTEMS WITH PERIODIC COEFFICIENTS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 296-330. doi: 10.11948/2018.296

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Redouane Qesmi. A SHORT SURVEY ON DELAY DIFFERENTIAL SYSTEMS WITH PERIODIC COEFFICIENTS[J]. Journal of Applied Analysis & Computation, 2018, 8(1): 296-330. doi: 10.11948/2018.296

A SHORT SURVEY ON DELAY DIFFERENTIAL SYSTEMS WITH PERIODIC COEFFICIENTS

Redouane Qesmi

Ecole Suprieure de Technologie, Sidi Mohamed Ben Abdellah University, Fez 30000, Morocco

Abstract

Abstract

In this survey, we collect a few results scatted in the literature covering advances in systems of periodic delay differential equations including both modeling and theory. We will present the (almost) most oldest and (almost) most recent contributions made for this subject.
- Functional differential equation /
- periodicity /
- delay /
- seasonality /
- dynamic modeling /
- Floquet multiplier /
- oscillation /
- bifurcation
MSC: (39-02)

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References

[1]	R. Agarwal, M. Bohner, A. Domoshnitsky and Y. Goltser, Floquet theory and stability of nonlinear integro-differential equations, Acta. Math. Hungar, 2005, 109(4), 305-330. Google Scholar
[2]	L. K. Andersen and M. C. Mackey, Resonance in periodic chemotherapy:A case study of acute myelogenous leukemia, J. Theor. Biol., 2001, 209, 113-130. Google Scholar
[3]	S. A. Avdonin and O. P. Germanovich, The basis property of a family of Floquet solutions of a linear periodic equation of neutral type in a Hilbert space, Siberian Math. J., 1995, 36, 853-858. Google Scholar
[4]	N. Bacaër, Approximation of the basic reproduction number R₀ for vectorborne diseases with a periodic vector population, Bull. Math. Biol., 2007, 69, 1067-1091. Google Scholar
[5]	N. Bacaër and E. H. Ait Dads, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J. Math. Biol., 2011, 62(5), 741-762. Google Scholar
[6]	N. Bacaër and E. H. Ait Dads, On the biological interpretation of a definition for the parameter R₀ in periodic population models, J. Math. Biol., 2012, 65(4), 601-621. Google Scholar
[7]	N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 2006, 53, 421-436. Google Scholar
[8]	N. Bacaër and R. Ouifki, Growth rate and basic reproduction number for populationmodels with a simple periodic factor, Math. Biosci., 2007, 210, 647-658. Google Scholar
[9]	D. Bachrathy and G. Stepan, Improved Prediction of Stability Lobes with Extended Multi Frequency Solution, CIRP Annals-Manufact. Tech., 2013, 62(1), 411-414. Google Scholar
[10]	M. Bardi, An equation of growth of a single species with realistic dependence on crowding and seasonal factors, J. Math. Biol., 1983, 17, 33-43. Google Scholar
[11]	M. Bardi and A. Schiaffino, Asymptotic behavior of positive solutions of periodic delay logistic equations, J. Math. Biol., 1982, 14, 95-100. Google Scholar
[12]	P. V. Bayly, J. E. Halley, B. P. Mann and M. A. Davies, Stability of interrupted cutting by temporal finite element analysis, Proc. of the ASME Design Eng. Tech. Conf., 2001, 6C, 2361-2370. Google Scholar
[13]	P. V. Bayly, J. E. Halley, B. P. Mann and M. A. Davies, Stability of interrupted cutting by temporal finite element analysis, J. of Manuf. Sci. and Eng., 2003, 125(2), 220-225. Google Scholar
[14]	R. Bellman and K. L. Cooke, Differ. Differ. Eqns., Academic Press, New York, 1963. Google Scholar
[15]	L. Berezansky and L. Idels, Periodic Fox Harvesting Models with Delay, App. Math. Comp., 2008, 195(1), 142-153. Google Scholar
[16]	S. Bernard, B. Cajavec Bernard, F. Levi and H. Herzel, Tumor growth rate determines the timing of optimal chronomodulated treatment schedules, PLoS Comp. Biol., 2010, 6(3):e1000712. Google Scholar
[17]	R. Boucekkinea, F. del Rio and O. Licandroc, Endogenous vs Exogenously Driven Fluctuations in Vintage Capital Models, J. Econ. Theo., 1999, 88(1), 161-187. Google Scholar
[18]	L. Bourouiba, S. Gourley, R. Liu and J. Wu, The interaction of migratory birds and domestic poultry, and its role in sustaining avian influenza, SIAM J. Appl. Math., 2011, 71, 487-516. Google Scholar
[19]	L. Bourouiba, J. Wu, S. H. Newman, J. Y. Takekawa, T. Natdorj, N. Batbayar, C. M. Bishop, L. A. Hawkes, P. J. Butler and M. Wikelski, Spatial dynamics of bar-headed geese migration in the context of H5N1, J. Roy. Soc. Interface, 2010, 7(52), 1627-1639. Google Scholar
[20]	N. F. Britton, Spatial Structures and Periodic Travelling Waves in an IntegroDifferential Reaction-Diffusion Population Model, SIAM J. Appl. Math., 1990, 50(6), 1663-1688. Google Scholar
[21]	E. Bueler, Error Bounds for Approximate Eigenvalues of Periodic-Coefficient Linear Delay Differential Equations, SIAM J. Numer. Anal., 2007, 45(6), 2510-2536. Google Scholar
[22]	S. Busenberg and K. L. Cooke, 2007 Periodic solutions of a periodic nonlinear delay differential equation, SIAM J. Appl. Math., 1978, 35, 704-721. Google Scholar
[23]	E. A. Butcher and O. A. Bobrenkov, On the Chebyshev spectral continuous time approximation for constant and periodic delay differential equations, Commun Nonl. Sci. Numer. Simulat., 2011, 16, 1541-1554. Google Scholar
[24]	E. A. Butcher, H. T. Ma, E. Bueler, V. Averina and Z. Szabo, Stability of linear time periodic delay-differential equations via Chebyshev polynomials, Inter. J. Num.. Engineer, 2004, 59(59), 895-922. Google Scholar
[25]	Y. Chen and J. Wu, Dynamics of scalar linear periodic delay-diffrential equations. In:Innite dimensional dynamical systems, Fields Inst. Commun. 64. Springer, New York, 2013, 269-278. Google Scholar
[26]	D.S. Cohen, S. Rosenblat, A delay logistic equation with variable growth rate, SIAM J. Appl. Math., 1982, 42, 608-624. Google Scholar
[27]	C. E. Elme and E. S. van Vleck, Traveling wave solutions for bistable differential-difference equations with periodic diffusion, SIAM J. Appl. Math., 2001, 61(5), 1648-1679. Google Scholar
[28]	K. L. Cooke, Functional differential systems:Some models and perturbation problems, Inter. Sympos. Diff. Eqns. in Dynamic Systs., Academic Press, New York, 1965, 167-183. Google Scholar
[29]	K. L. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemics and population growth, Math. Biosc., 1976, 31, 87-104. Google Scholar
[30]	J. M. Cushing, Integro-differential equations and delay models in population dynamics, Lect. Notes in Biomath. 20, Springer-Verlag, Heidelberg, 1977. Google Scholar
[31]	F. A. Davidson and S. A. Gourley, The effects of temporal delays in a model for a food-limited diffusing population, J. Math. Anal. Appl., 2001, 261, 633-648. Google Scholar
[32]	V. Deshmukh, H-T. Ma, E. A Butcher and E. Bueler, Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation, Nonl. Dyn., 2008, 52, 137-149. Google Scholar
[33]	V. Deshmukh, H-T. Ma and E. A. Butcher, Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials, Proceedings of the 2003 American Control Conference, 2003, 6, 4-6. Google Scholar
[34]	O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations, J. Math. Biol, 1990, 28, 365-382. Google Scholar
[35]	O. Diekmann, S. A. Van Gils, S. M. Verduyn Lunel and H. O. Walther,Delay Equations. Functional-, Complex-, and Nonlinear Analysis, Appl. Math. Sci. 110 Springer-Verlag, New York, 1995. Google Scholar
[36]	K. Dietz, D. Schenzle, Mathematical models for infectious disease statistics, In a Celebration of Statistics. The ISI Centenary Volume, AC Atkinson, SE Fienberg (eds). Springer Berlin, 1985. Google Scholar
[37]	N. M. Dixit and A. S. Perelson, Complex patterns of viral load decay under antiretroviral therapy:influence of pharmacokineticsand intracellular delay, J. Theo. Biol., 2004, 226(1), 95-109. Google Scholar
[38]	A. Domoshnitsky, On stability of nonautonomous integro-differential equations, Proc. Equadiff (Hasselt, Belgium, 2003), 2003, 1059-1064. Google Scholar
[39]	A. Domoshnitsky and Y. Goltser, Approach to study of bifurcatios and stability of integro-differential equations, Math. Comp. Modell., 2002, 36, 663-678. Google Scholar
[40]	Y. F. Dolgii, Application of self-adjoint boundary value problems to investigation of stability of periodic delay systems, Proc. Steklov Inst. Math. 255 Suppl., 2006, 2, 16-25. Google Scholar
[41]	R. Driver, Existence theory for a delay differential system, Cont. D. Eqns., 1963, 1, 317-336. Google Scholar
[42]	L. H. Erbe, W. Krawcewicz and J. Wu, Leray-Schauder degree for semilinear Fredholm maps and periodic boundary value problems of neutral equations, Nonl. Anal. TMA, 1990, 15, 747-764. Google Scholar
[43]	T. Faria, Normal forms for periodic retarded functional differential equations, Proc. Roy. Soc. Eidinburgh, 1997, 127(A), 21-16. Google Scholar
[44]	T. Faria, Periodic solutions for a non-monotone family of delayed differential equations with applications to Nicholson systems, Journal of Differential Equations, 2017, 263(1), 509-533. Google Scholar
[45]	W. Feng and X. Lu, Asymptotic periodicity in diffusive logistic equations with discrete delays, Nonl. Anal., 1996, 26, 171-178. Google Scholar
[46]	W. Feng, X. Lu, On diffusive population models with toxicants and time delays, J. Math. Anal. Appl., 1999, 233, 373-386. Google Scholar
[47]	N.J. Ford and P.M. Lumb, Theory and numerics for multi-term periodic delay differential equations:small solutions and their detection, Elec. Trans. Num. Anal., 2007, 26, 474-483. Google Scholar
[48]	H. I. Freedman and J. Wu, Periodic solutions of single-species models with periodic delay, SIAM J. Math. Anal., 1992, 23, 689-701. Google Scholar
[49]	S. D. Fretwell, Populations in a seasonal environment, Princeton Univ. Press, Princeton, NJ, 1972. Google Scholar
[50]	F. Gao, S. Lu and M. Yao, Periodic solutions for Linard type equation with time-variable coefficient, Adv. Diff. Eqns., 2015, 125, 1-9. Google Scholar
[51]	K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Environmental periodicity and time delays in a "food-limited" population model, J. Math. Anal. Appl., 1990, 147, 545-555. Google Scholar
[52]	S. A. Gourley, R. Liu, and J. Wu, Spatiotemporal distributions of migratory birds:Patchy models with delay, SIAM J. Appl. Dyn. Syst., 2010, 9, 589-610. Google Scholar
[53]	W. S. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies (revisited), Nature, 1980, 287, 17-21. Google Scholar
[54]	M. T. Hagan, H. B. Demuth and M. Beale, Neural network design, PWS:Boston, 1996. Google Scholar
[55]	W. Hahn, On difference differential equations with periodic coefficients, J. Math. Anal. Appl., 1961, 3, 70-101. Google Scholar
[56]	A. Halanay, Stability theory of linear periodic systems with delay, Revue Math. Pures Appl., 1961, 6, 633-653(in Russian). Google Scholar
[57]	S. Halanay, Differential equations:stability, Oscillations, Time Lags. Academic, New York, 1966. Google Scholar
[58]	J. K. Hale, Linear functional-differential equations with constant coefficients, Contrib. Diff. Eqns., 1963, 2, 291-317. Google Scholar
[59]	J. K. Hale and S. M. Verduyn Lunel, Introduction to functional differential equations, Springer-Verlag New York, 1993. Google Scholar
[60]	J. K. Hale and J. L. Mawhin, Coincidence degree and periodic solutions of neutral equations, J. Diff. Eqns., 1974, 15, 295-307. Google Scholar
[61]	J. K. Hale and M. Weedermann, On perturbations of delay-differential equations with periodic orbits, J. Diff. Eqns., 2004, 197, 219-246. Google Scholar
[62]	F. Hartung, T. Insperger, G. Stepan and J. Turi, Approximate stability charts for milling processes using semi-discretization, Appl. Math. Comp., 2006, 174, 51-73. Google Scholar
[63]	J. A. P. Heesterbeek, M. G. Roberts, Threshold quantities for helminth infections, J. Math. Biol., 1995, 33(a), 415-434. Google Scholar
[64]	J. A. P. Heesterbeek and M. G. Roberts, Threshold quantities for infectious diseases in periodic environments, J. Biol. Syst., 1995, 3(3), 779-787. Google Scholar
[65]	Q. Huang, Necessary and sufficient conditions for the oscillation of a class of differential difference equations with sign-variable periodic coefficients, Kexue Tongbao, 1988, 33, 156. Google Scholar
[66]	Q. Huang and S. Chen, Oscillations of neutral differential equations with periodic coefficients, Proc. Amer. Math. Soc., 1990, 110(4), 997-1001. Google Scholar
[67]	V. Hutson and K. Schmitt, Permanance and dynamics of biological systems, Math. Biosci., 1992, 111, 1-71. Google Scholar
[68]	H. Inaba, On a new perspective of the basic reproduction number in heterogeneous environments, J. Math. Biol., 2012, 65(2), 309-48. Google Scholar
[69]	H. Inaba, The Malthusian parameter and R₀ for heterogeneous populations in periodic environments, J. Math. Biosc&Engin, 2012, 9(b), 313-346. Google Scholar
[70]	T. Insperger and G. Stepan, Remote control of periodic robot motion, Proc. of the 13th CISM-IFTOMM Sympos. Zakopane, Poland, 2000. Google Scholar
[71]	T. Insperger and G. Stepan, Semi-discretization method for delayed systems, Intern. J. Num. Meth. Engin., 2002, 5, 503-518. Google Scholar
[72]	T. Insperger and G. Stepan, Updated semi-discretization method for periodic delay-differential equations with discrete delay, Intern. J. Num. Meth. Engin., 2004, 61, 117-141. Google Scholar
[73]	T. Insperger, B. P. Mann, G. Stepan and P. V. Bayly, Stability of up-milling and down-milling, Part1:Alternative analytical methods, Int. J. Machine Tools Manufact., 2003, 43(1), 25-34. Google Scholar
[74]	M. H. Jiang, Y. Shen, and X. X. Liao, Global stability of periodic solution for bidirectional associative memory neural netwroks with varying-time delay, App. Math. Comp., 2006, 182, 509-520. Google Scholar
[75]	Y. Jin and X. Q. Zhao, Spatial dynamics of a nonlocal periodic reactiondiffusion model with stage structure, SIAM J. Math. Anal., 2009, 40(6), 2496-2516. Google Scholar
[76]	W. Just, On the eigenvalue spectrum for time-delayed Floquet problems, Phys. D, 2000, 142, 153-165. Google Scholar
[77]	G. Karakostas, The effect of seasonal variations to the delay population equation, Nonl. Anal. TMA, 1982, 6, 1143-1154. Google Scholar
[78]	F. A. Khasawneh and B. P. Mann, A spectral element approach for the stability analysis of time-periodic delay equations with multiple delays,Commun Nonl. Sci. Numer. Simul., 2013, 18(21), 29-2141. Google Scholar
[79]	I. G. E. Kordonis and Ch. G. Philos, Oscillation and nonoscillation in delay or advanced differential equations and in integro-differential equations, Georgian Math. J., 1999, 6, 263. Google Scholar
[80]	Y. Kuang and B. Tang, Uniform persistence in nonautonomous delay differential Kolomogorov-type population models, Rocky Mountain J. Math., 1993, 24(1), 165-186. Google Scholar
[81]	G. Ladas, Ch. G. Philos and Y. G. Sficas, Oscillations in neutral equations with periodic coefficients, Proc. Amer. Math. Soc., 1991, 113(1), 123-134. Google Scholar
[82]	G. Ladas, Y. G. Sficas and I. P. Stavroulakis, Nonoscillatory functional differential equations, Pacific J. Math., 1984, 115, 391-398. Google Scholar
[83]	D. Lehotzky, T. Insperger and G. Stepan, Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays, Nonl. Sci. Numer. Simul., 2016, 35, 177-189. Google Scholar
[84]	Y. Li and Y. Kuang, Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 2001, 255, 260-280. Google Scholar
[85]	Y. Li and Y. Kuang, Periodic solutions in periodic state-dependent delay equations and population models, Proc. Amer. Math. Soc., 2001, 130(5), 1345-1353. Google Scholar
[86]	Y. Li, Existence and global attractivity of a positive periodic solution of a class of delay differential equation, Science in China Series A:Math., 1998, 3(41), 273-284. Google Scholar
[87]	Y. Li and G. Xu, On a periodic prey-predator system with infinite delays, Appl. Math. J. Chinese Univ. Ser. B, 2000, 15, 267-272. Google Scholar
[88]	X. Liang, Y. Yi, and X. Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Diff. Eqns, 2006, 231(1), 57-77. Google Scholar
[89]	X. Liang and X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Comm. Pure Appl. Math., 2007, 40, 1-40. Google Scholar
[90]	Z. Liu, and L. Liao, Existence and global exponential stability of periodic solution of cellular neural networks with time-varying delays, J. Math. Anal. Appl., 2004, 290, 247-262. Google Scholar
[91]	W. P. London and J. A. Yorke, Recurrent outbreaks of measles, chickenpox and mumps. I. Seasonal variation in contact rates, Amer. J. Epidemiol., 1973, 98(6), 453-468. Google Scholar
[92]	X. H. Long, B. Balachandran and B. P. Mann, Dynamics of milling processes with variable time delays, Nonl. Dyn., 2007, 47, 49-63. Google Scholar
[93]	O. Lopes, Forced oscillations in nonlinear neutral differential equations, SIAM J. Appl. Math., 1975, 29, 196. Google Scholar
[94]	Y. Lou and X. Q. Zhao, Threshold dynamics in a time-delayed periodic SIS epidemic model, Disc. Cont. Dyn. Syst. (Ser. B), 2009, 12, 169-186. Google Scholar
[95]	Y. Lou and X. Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math, 2010, 70, 2023-2044. Google Scholar
[96]	X. Lu and W. Feng, Periodic solution and oscillation in a competition model with diffusion and distributed delay effects, Nonl. Anal., 1996, 27, 699-709. Google Scholar
[97]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 1977, 197, 287-289. Google Scholar
[98]	J. Mallet-Paret and G. R. Sell, Systems of differential delay equations:Floquet multipliers and discrete Lyapunov functions, J. Diff. Eqns., 1996, 125, 385-440. Google Scholar
[99]	B.P. Mann, P. V. Bayly, M. A. Davies and J. E. Halley, Limit cycles, bifurcations, and accuracy of the milling process, J. Sound Vib, 2004, 277, 31-48. Google Scholar
[100]	I. Minis and R. Yanushevsky, A new theoretical approach for the prediction of machine tool chatter in milling, ASME J. of Eng. Ind., 1993, 115, 1-8. Google Scholar
[101]	K. Nah and G. Röst, Stability threshold for scalar linear periodic delay differential equations, Canad. Math. Bull. Vol., 2016, 59(4), 849-857. Google Scholar
[102]	A. J. Nicholson, The self-adjustment of populations to change, Cold Spring Harbor Sympos. on Quant. Biol., 1957, 22, 153-173. Google Scholar
[103]	R. M. Nisbet and W. S. C. Gurney, Population dynamics in a periodically varying environment, J. Theor. Biol., 1976, 56, 459-475. Google Scholar
[104]	R. D. Nussbaum, Periodic solutions of some integral equations from the theory of epidemics. In:V. Lakshmikantham (ed.), Nonl. Syst. Appl. Academic Press, New York, 1977, 235-257. Google Scholar
[105]	R. D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal., 1978, 9, 356-376. Google Scholar
[106]	Y. H. Pang and P. Y. Wang, Time periodic solutions of the diffusive Nicholson blowflies equation with delay, Nonl. Anal.:RWA, 2015, 22, 44-53. Google Scholar
[107]	C. V. Pao, Stability and attractivity of periodic solutions of parabolic systems with time delays, J. Math. Anal. Appl., 2005, 304, 423-450. Google Scholar
[108]	C. Perello, Periodic solutions of differential equations with time lag containing a small parameter, J. Diff. Eqns., 1968, 4, 160-175. Google Scholar
[109]	C. G. Philos, Oscillation for first order linear delay differential equations with variable coefficients, Funkcialaj Ekvacioj, 1992, 35, 307-319. Google Scholar
[110]	R. Qesmi, Oscillations in a class of differential equations with time-dependent delay, Submitted. Google Scholar
[111]	R. Qesmi and M. L. Hbid, Periodic solutions for functional differential equations with periodic delay close to zero, Elec. J. Diff. Eqns, 2006, 141, 1-12. Google Scholar
[112]	R. Qesmi and M. L. Hbid, The effect of oscillations in the dynamics of differential equations with delay, J. Math. Anal. Appl., 2007, 335(1), 543-559. Google Scholar
[113]	G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 1987, 28, 253-256. Google Scholar
[114]	G. Röst, Neimark-Sacker bifurcation for periodic delay differential equations, J. Nonl. Anal., 2005, 60, 1025-1044. Google Scholar
[115]	G. Röst, Bifurcation for periodic delay differential equations at points of 1:4 resonance, Func. Diff. Eqns., 2006, 13, 585-602. Google Scholar
[116]	S. Ruan, Delay differential equations in single species dynamics, In:Arino O, Hbid M.L., and Ait D.E. (eds) Delay differential equations and applications, Springer, Berlin Heidelberg New York, 2006, 477-518. Google Scholar
[117]	S. H. Saker and S. Agarwal, Oscillation and global attractivity in a nonlinear delay periodic model of respiratory dynamics, Computers Math. Appl., 2002, 44, 623-632. Google Scholar
[118]	S. Saker and S. Agarwal, Oscillation and global attractivity in a periodic Nicholson's blowflies model, Math. Comput. Modell., 2002, 35, 719-731. Google Scholar
[119]	D. Schely and S. A. Gourley, Linear stability criteria for population models with periodic perturbed delays, J. Math. Biol., 2000, 40, 500-524. Google Scholar
[120]	S. N. Shimanov, On the theory of linear differential equations with periodic coefficients and time lag, Prikl. Mat. 1Meh., 1963, 27, 450-458[English transl.:J. Appl. Math. Meek., 1963, 27, 674-687]. Google Scholar
[121]	E. Schmitt, Uber eine Klasse linearer funktionaler differential-gleichungen, Math. Ann., 1911, 70, 499-524. Google Scholar
[122]	J. Sieber and R. Szalai, Characteristic matrices for linear periodic delay differential equations, SIAM J. App. Dyn. Sys., 2011, 10(1), 129-147. Google Scholar
[123]	H. C. Simpson, Stability of periodic solutions of nonlinear integro-differential systems, SIAM J. Appl. Math., 1980, 38, 341-363. Google Scholar
[124]	A. L. Skubachevskii and H. O. Walther, On Floquet multipliers for slowly oscillating periodic solutions of nonlinear functional differential equations (Russian), Tr. Mosk. Mat. Obs., 2003, 64, 3-53. Google Scholar
[125]	A. L. Skubachevskii and H. O. Walther, On the Floquet multipliers of periodic solutions to non-linear functional differential equations, J. Dyn. Diff. Eqns., 2006, 18(2), 257-355. Google Scholar
[126]	H. L. Smith, On periodic solutions of a delay integral equation modelling epidemics, J. Math. Biol., 1977, 4, 69-80. Google Scholar
[127]	H. L. Smith, Monotone semiflows generated by functional differential equations, J. Diff. Eqns., 1987, 66, 420-442. Google Scholar
[128]	A. Stokes, A Floquet theory for functional differential equations, Proc. Nad. Acad. Sci. U. S., 1962, 48, 1330-1334. Google Scholar
[129]	R. Szalai, G. Stepan, Period doubling bifurcation and center manifold reduction in a time-periodic and time-delayed model of machining, J. Vibr. Cont., 2010, 16, 1169-1187. Google Scholar
[130]	R. Szalai, G. Stepan and S. J. Hogan, Continuation of bifurcations in periodic delay differential equations using characteristic matrices, SIAM J. Sci. Comp., 2006, 28(4), 1301-1317. Google Scholar
[131]	S. Tanaka, Oscillation of solutions of first-order neutral differential equations, Hiroshima Math. J., 2002, 32, 79-85. Google Scholar
[132]	X. Tang, D. Cao and X. Zou, Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay, J. Diff. Eqns., 2006, 228, 580-610. Google Scholar
[133]	B. Tang and Y. Kuang, Existence, Uniqueness and Asymptotic Stability of Periodic Solutions of Periodic Functional Differential Systems, Tohoku Math. J., 1997, 49, 217-239. Google Scholar
[134]	H. R. Thieme, Persistence under relaxed point-dissipativity:with application to an endemic model, SIAM J. Math Anal., 1993, 24, 407-435. Google Scholar
[135]	H. R. Thieme, Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosc., 2000, 166(2), 173-201. Google Scholar
[136]	H. R. Thieme, Spectral bound and reproduction number for infinitedimensional population structure and time heterogeneity, SIAM. J. Appl. Math., 2009, 70(1), 188-211. Google Scholar
[137]	S. T. Trickey, L. N. Virgin and E. H. Dowell, The stability of limit-cycle oscillations in a nonlinear aeroelastic system, Proc. Roy. Soc. A, 2002, 458(2025), 2203-2226. Google Scholar
[138]	D. J. Tweten, G. M. Lipp, F. A. Khasawneh and B. P. Mann, On the comparison of semi-analytical methods for the stability analysis of delay differential equations, J. Sound Vib., 2012, 331(17), 4057-71. Google Scholar
[139]	C. Wang, Existence and stability of periodic solutions for parabolic systems with time delays, J. Math. Anal. Appl., 2008, 339, 1354-1361. Google Scholar
[140]	J. Wang, L. Zhou and Y. Tang, Asymptotic periodicity of a food-limited diffusive population model with time-delay, J. Math. Anal. Appl., 2006, 313, 381-399. Google Scholar
[141]	X. S. Wang and J. Wu, Seasonal migration dynamics:periodicity, transition delay, and finite dimensional reduction, Proc. Roy. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2012, 468(2139), 634-650. Google Scholar
[142]	X. S. Wang and J. Wu, periodic systems of delay differential equations and avian influenza dynamics, J. Math. Sci., 2014, 201(5), 693-704. Google Scholar
[143]	Y. Wang and J. Yin, Periodic solutions of a class of degenerate parabolic system with delays, J. Math. Anal. Appl., 2011, 380, 57-68. Google Scholar
[144]	W. Wang and X. Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Differ. Eqns., 2008, 20, 699-717. Google Scholar
[145]	G. F. Webb, Theory of nonlinear age-dependent population dynamics, Marcel Dekker:New York, 1985. Google Scholar
[146]	X. Wu, F. M. G. Magpantay, J. Wu and X. Zou, Stage-structured population systems with temporally periodic delay, Math. Methods Appl. Sci., 2015, 38(16), 3464-3481. Google Scholar
[147]	D. Xu and X. Q. Zhao, Dynamics in a periodic competitive model with stage structure, J. Math. Anal. Appl, 2005, 311, 417-438. Google Scholar
[148]	Y. Yang, R. Zhang, J. Yin and C. Jin, Existence of time periodic solutions for the Nicholson's blowflies model with Newtonian diffusion, Math. Methods Appl. Sci., 2010, 33, 922-934. Google Scholar
[149]	A. Yilmaz, E. AL-Regib and J. Ni, Machine tool chatter suppression by multilevel random spindle speed variation, ASME J. of Manuf. Sci. Engin., 2002, 124, 208-216. Google Scholar
[150]	T. Zhang, J. Liu and Z. Teng, A non-autonomous epidemic model with time delay and vaccination, Math. Methods Appl. Sci., 2010, 33, 1-11. Google Scholar
[151]	M. X. Zhao and D. Balachandran, Dynamics and stability of milling process, Inter. J. Solids Structs., 2001, 38, 2233-2248. Google Scholar
[152]	X. Q. Zhao, Uniform persistence and periodic coexistence states in infinitedimensional periodic semiflows with applications, Canad. Appl. Math. Quart, 1995, 3, 473-495. Google Scholar
[153]	X. Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. Google Scholar
[154]	X. Q. Zhao, Permanence implies the existence of interior periodic solutions for FDEs, Qual. Theo. Differ. Eqns. Appl, 2008, 2, 125-137. Google Scholar
[155]	X. Q. Zhao, Basic reproduction ratios for periodic compartmental models with time delay, J. Dyn. Diff. Eqns., 2015, 1-16. Google Scholar
[156]	Q. Zhou, J. Sun and G. Chen, Global exponential stability and periodic oscillations of reaction-diffusion BAM neural networks with periodic coefficients and general delays, Inter. J. Bif. Chaos, 2007, 17, 129-142. Google Scholar
[157]	A. M. Zver'kin, Series expansion of solutions of linear differential-difference equations, Proc. of Workshop on the Theory of Differential Equations with Deviated Argument in Russian, 1967, 4, 3-50. Google Scholar