2013, 10(5&6): 1301-1333. doi: 10.3934/mbe.2013.10.1301

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations

1. 

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, Raleigh, NC 27695-8212

2. 

Center for Research in Scientific Computation, Center for Quantitative Sciences in Biomedicine, Raleigh, NC 27695-8212, United States, United States

Received  July 2012 Revised  November 2012 Published  August 2013

In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.
Citation: H.Thomas Banks, Danielle Robbins, Karyn L. Sutton. Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1301-1333. doi: 10.3934/mbe.2013.10.1301
References:
[1]

J. Arino, L. Wang and G. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theo. Bio., 241 (2006), 109-118. doi: 10.1016/j.jtbi.2005.11.007.

[2]

C. Baker and F. Rihan, Sensitivity analysis of parameters in modelling with delay-differential equations, MCCM Tec. Rep., 349 (1999).

[3]

H. T. Banks, Necessary conditions for control problems with variable time lags, SIAM J. Control, 6 (1968), 9-47. doi: 10.1137/0306002.

[4]

H. T. Banks, Representations for solutions of linear functional differential equations, J. Differential Equations, 5 (1969), 399-409. doi: 10.1016/0022-0396(69)90052-7.

[5]

H. T. Banks, Delay systems in biological models: Approximation techniques, in "Nonlinear Systems and Applications" (ed. V. Lakshmikantham), Academic Press, New York, (1977), 21-38.

[6]

H. T. Banks, Approximation of nonlinear functional differential equation control systems, J. Optimiz. Theory Appl., 29 (1979), 383-408. doi: 10.1007/BF00933142.

[7]

H. T. Banks, Identification of nonlinear delay systems using spline methods, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, NY, (1982), 47-55.

[8]

H. T. Banks, "A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering," Chapman and Hall/CRC Press, Boca Raton, FL, 2012. doi: 10.1201/b12209.

[9]

H. T. Banks, J. E. Banks and S. L. Joyner, Estimation in time-delay modeling of insecticide-induced mortality, J. Inverse and Ill-posed Problems, 17 (2009), 101-125. doi: 10.1515/JIIP.2009.012.

[10]

H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625. doi: 10.1007/s00285-004-0299-x.

[11]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics, Math. Biosciences, 183 (2003), 63-91. doi: 10.1016/S0025-5564(02)00218-3.

[12]

H. T. Banks and J. A. Burns, An abstract framework for approximate solutions to optimal control problems governed by hereditary systems, in "International Conference on Differential Equations" (ed. H. Antosiewicz), Academic Press, New York, (1975), 10-25.

[13]

H. T. Banks and J. A. Burns, Hereditary control problems: Numerical methods based on averaging approximations, SIAM J. Control & Opt., 16 (1978), 169-208. doi: 10.1137/0316013.

[14]

H. T. Banks, J. A. Burns and E. M. Cliff, Parameter estimation and identification for systems with delays, SIAM J. Control and Optimization, 19 (1981), 791-828. doi: 10.1137/0319051.

[15]

H. T. Banks, M. Davidian, J. R. Samuels, Jr. and Karyn L. Sutton, An inverse problem statistical methodology summary, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. Gerardo Chowell, et al.), Springer, Netherlands, (2009), 249-302. doi: 10.1007/978-90-481-2313-1_11.

[16]

H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, Journal of Inverse and Ill-Posed Problems, 15 (2007), 683-708. doi: 10.1515/jiip.2007.038.

[17]

H. T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, A new approach to optimal design problems, Journal of Inverse and Ill-Posed Problems, 18 (2010), 25-83. doi: 10.1515/JIIP.2010.002.

[18]

H.T. Banks, S. Dediu and H. K. Nguyen, Time delay systems with distribution dependent dynamics, IFAC Annual Reviews in Control, 31 (2007), 17-26. doi: 10.1016/j.arcontrol.2007.02.002.

[19]

H. T. Banks, S. Dediu and H. K. Nguyen, Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space, Math. Biosci. and Engr., 4 (2007), 403-430. doi: 10.3934/mbe.2007.4.403.

[20]

H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems, Inverse Problems, 27 (2011) 075002, 31 pp. doi: 10.1088/0266-5611/27/7/075002.

[21]

H. T. Banks and F. Kappel, Spline approximations for functional differential equations, J. Differential Equations, 34 (1979), 496-522. doi: 10.1016/0022-0396(79)90033-0.

[22]

H. T. Banks and J. M. Mahaffy, Global asymptotic stability of certain models for protein synthesis and repression, Quart. Applied Math., 36 (1978), 209-221.

[23]

H. T. Banks and J. M. Mahaffy, Stability of cyclic gene models for systems involving repression, J. Theoretical Biology, 74 (1978), 323-334. doi: 10.1016/0022-5193(78)90079-6.

[24]

H. T. Banks and H. Nguyen, Sensitivity of dynamical systems to Banach space parameters, J. Math. Analysis and Applications, 323 (2006), 146-161. doi: 10.1016/j.jmaa.2005.09.084.

[25]

H. T. Banks, Keri L. Rehm, Karyn L. Sutton, Christine Davis, Lisa Hail, Alexis Kuerbis and Jon Morgenstern, Dynamic modeling of behavior change in problem drinkers, N.C. State University, Center for Research in Scientific Computation technical report CRSC-TR11-08, Raleigh, NC, August, 2011; to appear in Quarterly of Applied Mathematics.

[26]

H. T. Banks, D. Robbins and K. L. Sutton, Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations, N.C. State University, CRSC-TR12-14, Raleigh, NC, July, 2012.

[27]

R. Bellman and K. L. Cooke, "Differential-Difference Equations," Mathematics in Science and Engineering, Vol. 6, Academic Press, New York, NY, 1963.

[28]

D. Brewer, The differentiability with respect to a parameter of the solution of a linear abstract Cauchy problem, SIAM J. Math. Anal., 13 (1982), 607-620. doi: 10.1137/0513039.

[29]

J. A. Burns, E. M. Cliff and S. E. Doughty, Sensitivity analysis and parameter estimation for a model of Chlamydia Trachomatis infection, J. Inverse Ill-Posed Problems, 15 (2007), 19-32. doi: 10.1515/jiip.2007.013.

[30]

S. N. Busenberg and K. L. Cooke, eds., "Differential Equations and Applications in Ecology, Epidemics, and Population Problems," Academic Press, New York, 1981.

[31]

V. Capasso, E. Grosso and S. L. Paveri-Fontana, eds., "Mathematics in Biology and Medicine," Lecture Notes in Biomath., Vol. 57, Springer-Verlag, Berlin, Heidelberg, New York, 1985. doi: 10.1007/978-3-642-93287-8.

[32]

J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.

[33]

A. Casal and A. Somolinos, Forced oscillations for the sunflower equation, entrainment, Nonlinear Analysis: Theory, Methods and Applications, 6 (1982), 397-414. doi: 10.1016/0362-546X(82)90025-6.

[34]

S. Choi and N. Koo, Oscillation theory for delay and neutral differential equations, Trends in Mathematics, 2 (1999), 170-176.

[35]

K. L. Cooke, Functional differential equations: Some models and perturbation problems, in "Differential Equations and Dynamical Systems" (eds. J. K. Hale and J. P. LaSalle), Academic Press, New York, (1967), 167-183.

[36]

J. M. Cushing, "Integrodifferential Equations and Delay Models in Population Dynamics," Lec. Notes in Biomath., Vol. 20, Springer-Verlag, Berlin-New York, 1977.

[37]

C. Elton, "Voles, Mice, and Lemmings," Clarendon Press, London, 1942.

[38]

P. L. Errington, Predation and vertebrate populations, Quarterly Review of Biology, 21 (1946), 144-177, 221-245.

[39]

U. Forys and A. Marciniak-Czochra, Delay logistic equation with diffusion, Proc 8th Nat. Conf. Mathematics Applied to Biology and Medicine, Lajs, Warsaw, Poland, (2002), 37-42.

[40]

U. Forys and A. Marciniak-Czochra, Logistic equations in tumor growth modelling, Int.J. Appl. Math. Comput. Sci., 13 (2003), 317-325.

[41]

J. S. Gibson and L. G. Clark, Sensitivity analysis for a class of evolution equations, J. Mathematical Analysis and Applications, 58 (1977), 22-31. doi: 10.1016/0022-247X(77)90224-4.

[42]

L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Ann. N. Y. Acad. Sci., 316 (1979), 214-235. doi: 10.1111/j.1749-6632.1979.tb29471.x.

[43]

B. C. Goodwin, "Temporal Organization in Cells: A Dynamic Theory of Cellular Control Processes," Academic Press, London, 1963.

[44]

B. C. Goodwin, Oscillatory behavior in enzymatic control processes, Adv. Enzyme Reg., 3 (1965), 425-439. doi: 10.1016/0065-2571(65)90067-1.

[45]

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

[46]

K. P. Hadeler, Delay equations in biology, in "Functional Differential Equations and Approximation of Fixed Points" (eds. H. O. Peitgen and H. O. Walther), Lect. Notes Math., 730, Springer-Verlag, Berlin, (1979), 136-156.

[47]

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

[48]

J. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Applied Math Sciences, Vol. 99, Springer-Verlag, New York, 1993.

[49]

F. C. Hoppensteadt, ed., "Mathematical Aspects of Physiology," Lectures in Applied Math, Vol. 19, American Mathematical Society, Providence, 1981.

[50]

G. E. Hutchinson, Circular causal systems in ecology, Annals of the NY Academy of Sciences, 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[51]

G. E. Hutchinson, "An Introduction to Population Ecology," Yale University, New Haven, 1978.

[52]

F. Kappel, An approximation scheme for delay equations, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, NY, (1982), 585-595.

[53]

F. Kappel, Generalized sensitivity analysis in a delay system, Proc. Appl. Math. Mech., 7 (2007), 1061001-1061002. doi: 10.1002/pamm.200700458.

[54]

F. Kappel and W. Schappacher, Autonomous nonlinear functional differential equations and averaging approximations, J. Nonlinear Analysis, 2 (1978), 391-422. doi: 10.1016/0362-546X(78)90048-2.

[55]

Y. Kuang, "Delay Differential Equations With Applications in Population Dynamics," Mathematics in Science and Engineering, Vol. 191, Academic Press, New York, NY, 1993.

[56]

N. MacDonald, Time lag in a model of a biochemical reaction sequence with end-product inhibition, J. Theor. Biol., 67 (1977), 727-734. doi: 10.1016/0022-5193(77)90056-X.

[57]

N. MacDonald, "Time Lags in Biological Models," Lecture Notes in Biomathematics, 27, Springer, Berlin, 1978.

[58]

N. Minorsky, Self-excited oscillations in dynamical systems possessing retarded actions, Journal of Applied Mechanics, 9 (1942), A65-A71.

[59]

N. Minorsky, On non-linear phenomenon of self-rolling, Proceedings of the National Academy of Sciences, 31 (1945), 346-349. doi: 10.1073/pnas.31.11.346.

[60]

N. Minorsky, "Nonlinear Oscillations," D. Van Nostrand, Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962.

[61]

D. M. Pratt, Analysis of population development in Daphnia at different temperatures, Biology Bulletin, 22 (1943), 345-365. doi: 10.2307/1538274.

[62]

D. Robbins, "Sensitivity Functions for Delay Differential Equation Models," Ph. D. Dissertation, North Carolina State University, Raleigh, NC, 2011.

[63]

R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites Tumor of the mouse, in "Mathematical Population Dynamics, Vol. 2" (eds. by O. Arino, D. Axelrod and M. Kimmel), Wuertz, Winnipeg, (1995), 335-348.

[64]

F. R. Sharpe and A. J. Lotka, Contribution to the analysis of malaria epidemiology IV: Incubation lag, supplement to Amer. J. Hygiene, 3 (1923), 96-112. doi: 10.3934/mbe.2008.5.681.

[65]

Alfredo Somolinos, Periodic solutions of the sunflower equation, Quarterly of Applied Mathematics, 35 (1978), 465-478.

[66]

K. Thomaseth and C. Cobelli, Generalized sensitivity functions in physiological system identification, Annals of Biomedical Engineering, 27 (1999), 607-616.

[67]

E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 494 (1955), 66-87.

show all references

References:
[1]

J. Arino, L. Wang and G. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theo. Bio., 241 (2006), 109-118. doi: 10.1016/j.jtbi.2005.11.007.

[2]

C. Baker and F. Rihan, Sensitivity analysis of parameters in modelling with delay-differential equations, MCCM Tec. Rep., 349 (1999).

[3]

H. T. Banks, Necessary conditions for control problems with variable time lags, SIAM J. Control, 6 (1968), 9-47. doi: 10.1137/0306002.

[4]

H. T. Banks, Representations for solutions of linear functional differential equations, J. Differential Equations, 5 (1969), 399-409. doi: 10.1016/0022-0396(69)90052-7.

[5]

H. T. Banks, Delay systems in biological models: Approximation techniques, in "Nonlinear Systems and Applications" (ed. V. Lakshmikantham), Academic Press, New York, (1977), 21-38.

[6]

H. T. Banks, Approximation of nonlinear functional differential equation control systems, J. Optimiz. Theory Appl., 29 (1979), 383-408. doi: 10.1007/BF00933142.

[7]

H. T. Banks, Identification of nonlinear delay systems using spline methods, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, NY, (1982), 47-55.

[8]

H. T. Banks, "A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering," Chapman and Hall/CRC Press, Boca Raton, FL, 2012. doi: 10.1201/b12209.

[9]

H. T. Banks, J. E. Banks and S. L. Joyner, Estimation in time-delay modeling of insecticide-induced mortality, J. Inverse and Ill-posed Problems, 17 (2009), 101-125. doi: 10.1515/JIIP.2009.012.

[10]

H. T. Banks and D. M. Bortz, A parameter sensitivity methodology in the context of HIV delay equation models, J. Math. Biol., 50 (2005), 607-625. doi: 10.1007/s00285-004-0299-x.

[11]

H. T. Banks, D. M. Bortz and S. E. Holte, Incorporation of variability into the mathematical modeling of viral delays in HIV infection dynamics, Math. Biosciences, 183 (2003), 63-91. doi: 10.1016/S0025-5564(02)00218-3.

[12]

H. T. Banks and J. A. Burns, An abstract framework for approximate solutions to optimal control problems governed by hereditary systems, in "International Conference on Differential Equations" (ed. H. Antosiewicz), Academic Press, New York, (1975), 10-25.

[13]

H. T. Banks and J. A. Burns, Hereditary control problems: Numerical methods based on averaging approximations, SIAM J. Control & Opt., 16 (1978), 169-208. doi: 10.1137/0316013.

[14]

H. T. Banks, J. A. Burns and E. M. Cliff, Parameter estimation and identification for systems with delays, SIAM J. Control and Optimization, 19 (1981), 791-828. doi: 10.1137/0319051.

[15]

H. T. Banks, M. Davidian, J. R. Samuels, Jr. and Karyn L. Sutton, An inverse problem statistical methodology summary, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. Gerardo Chowell, et al.), Springer, Netherlands, (2009), 249-302. doi: 10.1007/978-90-481-2313-1_11.

[16]

H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, Journal of Inverse and Ill-Posed Problems, 15 (2007), 683-708. doi: 10.1515/jiip.2007.038.

[17]

H. T. Banks, S. Dediu, S. L. Ernstberger and F. Kappel, A new approach to optimal design problems, Journal of Inverse and Ill-Posed Problems, 18 (2010), 25-83. doi: 10.1515/JIIP.2010.002.

[18]

H.T. Banks, S. Dediu and H. K. Nguyen, Time delay systems with distribution dependent dynamics, IFAC Annual Reviews in Control, 31 (2007), 17-26. doi: 10.1016/j.arcontrol.2007.02.002.

[19]

H. T. Banks, S. Dediu and H. K. Nguyen, Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space, Math. Biosci. and Engr., 4 (2007), 403-430. doi: 10.3934/mbe.2007.4.403.

[20]

H. T. Banks, K. Holm and F. Kappel, Comparison of optimal design methods in inverse problems, Inverse Problems, 27 (2011) 075002, 31 pp. doi: 10.1088/0266-5611/27/7/075002.

[21]

H. T. Banks and F. Kappel, Spline approximations for functional differential equations, J. Differential Equations, 34 (1979), 496-522. doi: 10.1016/0022-0396(79)90033-0.

[22]

H. T. Banks and J. M. Mahaffy, Global asymptotic stability of certain models for protein synthesis and repression, Quart. Applied Math., 36 (1978), 209-221.

[23]

H. T. Banks and J. M. Mahaffy, Stability of cyclic gene models for systems involving repression, J. Theoretical Biology, 74 (1978), 323-334. doi: 10.1016/0022-5193(78)90079-6.

[24]

H. T. Banks and H. Nguyen, Sensitivity of dynamical systems to Banach space parameters, J. Math. Analysis and Applications, 323 (2006), 146-161. doi: 10.1016/j.jmaa.2005.09.084.

[25]

H. T. Banks, Keri L. Rehm, Karyn L. Sutton, Christine Davis, Lisa Hail, Alexis Kuerbis and Jon Morgenstern, Dynamic modeling of behavior change in problem drinkers, N.C. State University, Center for Research in Scientific Computation technical report CRSC-TR11-08, Raleigh, NC, August, 2011; to appear in Quarterly of Applied Mathematics.

[26]

H. T. Banks, D. Robbins and K. L. Sutton, Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations, N.C. State University, CRSC-TR12-14, Raleigh, NC, July, 2012.

[27]

R. Bellman and K. L. Cooke, "Differential-Difference Equations," Mathematics in Science and Engineering, Vol. 6, Academic Press, New York, NY, 1963.

[28]

D. Brewer, The differentiability with respect to a parameter of the solution of a linear abstract Cauchy problem, SIAM J. Math. Anal., 13 (1982), 607-620. doi: 10.1137/0513039.

[29]

J. A. Burns, E. M. Cliff and S. E. Doughty, Sensitivity analysis and parameter estimation for a model of Chlamydia Trachomatis infection, J. Inverse Ill-Posed Problems, 15 (2007), 19-32. doi: 10.1515/jiip.2007.013.

[30]

S. N. Busenberg and K. L. Cooke, eds., "Differential Equations and Applications in Ecology, Epidemics, and Population Problems," Academic Press, New York, 1981.

[31]

V. Capasso, E. Grosso and S. L. Paveri-Fontana, eds., "Mathematics in Biology and Medicine," Lecture Notes in Biomath., Vol. 57, Springer-Verlag, Berlin, Heidelberg, New York, 1985. doi: 10.1007/978-3-642-93287-8.

[32]

J. Caperon, Time lag in population growth response of Isochrysis Galbana to a variable nitrate environment, Ecology, 50 (1969), 188-192.

[33]

A. Casal and A. Somolinos, Forced oscillations for the sunflower equation, entrainment, Nonlinear Analysis: Theory, Methods and Applications, 6 (1982), 397-414. doi: 10.1016/0362-546X(82)90025-6.

[34]

S. Choi and N. Koo, Oscillation theory for delay and neutral differential equations, Trends in Mathematics, 2 (1999), 170-176.

[35]

K. L. Cooke, Functional differential equations: Some models and perturbation problems, in "Differential Equations and Dynamical Systems" (eds. J. K. Hale and J. P. LaSalle), Academic Press, New York, (1967), 167-183.

[36]

J. M. Cushing, "Integrodifferential Equations and Delay Models in Population Dynamics," Lec. Notes in Biomath., Vol. 20, Springer-Verlag, Berlin-New York, 1977.

[37]

C. Elton, "Voles, Mice, and Lemmings," Clarendon Press, London, 1942.

[38]

P. L. Errington, Predation and vertebrate populations, Quarterly Review of Biology, 21 (1946), 144-177, 221-245.

[39]

U. Forys and A. Marciniak-Czochra, Delay logistic equation with diffusion, Proc 8th Nat. Conf. Mathematics Applied to Biology and Medicine, Lajs, Warsaw, Poland, (2002), 37-42.

[40]

U. Forys and A. Marciniak-Czochra, Logistic equations in tumor growth modelling, Int.J. Appl. Math. Comput. Sci., 13 (2003), 317-325.

[41]

J. S. Gibson and L. G. Clark, Sensitivity analysis for a class of evolution equations, J. Mathematical Analysis and Applications, 58 (1977), 22-31. doi: 10.1016/0022-247X(77)90224-4.

[42]

L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Ann. N. Y. Acad. Sci., 316 (1979), 214-235. doi: 10.1111/j.1749-6632.1979.tb29471.x.

[43]

B. C. Goodwin, "Temporal Organization in Cells: A Dynamic Theory of Cellular Control Processes," Academic Press, London, 1963.

[44]

B. C. Goodwin, Oscillatory behavior in enzymatic control processes, Adv. Enzyme Reg., 3 (1965), 425-439. doi: 10.1016/0065-2571(65)90067-1.

[45]

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

[46]

K. P. Hadeler, Delay equations in biology, in "Functional Differential Equations and Approximation of Fixed Points" (eds. H. O. Peitgen and H. O. Walther), Lect. Notes Math., 730, Springer-Verlag, Berlin, (1979), 136-156.

[47]

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

[48]

J. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations," Applied Math Sciences, Vol. 99, Springer-Verlag, New York, 1993.

[49]

F. C. Hoppensteadt, ed., "Mathematical Aspects of Physiology," Lectures in Applied Math, Vol. 19, American Mathematical Society, Providence, 1981.

[50]

G. E. Hutchinson, Circular causal systems in ecology, Annals of the NY Academy of Sciences, 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[51]

G. E. Hutchinson, "An Introduction to Population Ecology," Yale University, New Haven, 1978.

[52]

F. Kappel, An approximation scheme for delay equations, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, NY, (1982), 585-595.

[53]

F. Kappel, Generalized sensitivity analysis in a delay system, Proc. Appl. Math. Mech., 7 (2007), 1061001-1061002. doi: 10.1002/pamm.200700458.

[54]

F. Kappel and W. Schappacher, Autonomous nonlinear functional differential equations and averaging approximations, J. Nonlinear Analysis, 2 (1978), 391-422. doi: 10.1016/0362-546X(78)90048-2.

[55]

Y. Kuang, "Delay Differential Equations With Applications in Population Dynamics," Mathematics in Science and Engineering, Vol. 191, Academic Press, New York, NY, 1993.

[56]

N. MacDonald, Time lag in a model of a biochemical reaction sequence with end-product inhibition, J. Theor. Biol., 67 (1977), 727-734. doi: 10.1016/0022-5193(77)90056-X.

[57]

N. MacDonald, "Time Lags in Biological Models," Lecture Notes in Biomathematics, 27, Springer, Berlin, 1978.

[58]

N. Minorsky, Self-excited oscillations in dynamical systems possessing retarded actions, Journal of Applied Mechanics, 9 (1942), A65-A71.

[59]

N. Minorsky, On non-linear phenomenon of self-rolling, Proceedings of the National Academy of Sciences, 31 (1945), 346-349. doi: 10.1073/pnas.31.11.346.

[60]

N. Minorsky, "Nonlinear Oscillations," D. Van Nostrand, Co., Inc., Princeton, N.J.-Toronto-London-New York, 1962.

[61]

D. M. Pratt, Analysis of population development in Daphnia at different temperatures, Biology Bulletin, 22 (1943), 345-365. doi: 10.2307/1538274.

[62]

D. Robbins, "Sensitivity Functions for Delay Differential Equation Models," Ph. D. Dissertation, North Carolina State University, Raleigh, NC, 2011.

[63]

R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites Tumor of the mouse, in "Mathematical Population Dynamics, Vol. 2" (eds. by O. Arino, D. Axelrod and M. Kimmel), Wuertz, Winnipeg, (1995), 335-348.

[64]

F. R. Sharpe and A. J. Lotka, Contribution to the analysis of malaria epidemiology IV: Incubation lag, supplement to Amer. J. Hygiene, 3 (1923), 96-112. doi: 10.3934/mbe.2008.5.681.

[65]

Alfredo Somolinos, Periodic solutions of the sunflower equation, Quarterly of Applied Mathematics, 35 (1978), 465-478.

[66]

K. Thomaseth and C. Cobelli, Generalized sensitivity functions in physiological system identification, Annals of Biomedical Engineering, 27 (1999), 607-616.

[67]

E. M. Wright, A non-linear difference-differential equation, J. Reine Angew. Math., 494 (1955), 66-87.

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