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. doi: 10.1016/j.jtbi.2005.11.007. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

H. T. Banks, Delay systems in biological models: Approximation techniques,, in, (1977), 21. Google Scholar

[6]

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

[7]

H. T. Banks, Identification of nonlinear delay systems using spline methods,, in, (1982), 47. Google Scholar

[8]

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

[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. doi: 10.1515/JIIP.2009.012. Google Scholar

[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. doi: 10.1007/s00285-004-0299-x. Google Scholar

[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. doi: 10.1016/S0025-5564(02)00218-3. Google Scholar

[12]

H. T. Banks and J. A. Burns, An abstract framework for approximate solutions to optimal control problems governed by hereditary systems,, in, (1975), 10. Google Scholar

[13]

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

[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. doi: 10.1137/0319051. Google Scholar

[15]

H. T. Banks, M. Davidian, J. R. Samuels, Jr. and Karyn L. Sutton, An inverse problem statistical methodology summary,, in, (2009), 249. doi: 10.1007/978-90-481-2313-1_11. Google Scholar

[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. doi: 10.1515/jiip.2007.038. Google Scholar

[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. doi: 10.1515/JIIP.2010.002. Google Scholar

[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. doi: 10.1016/j.arcontrol.2007.02.002. Google Scholar

[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. doi: 10.3934/mbe.2007.4.403. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[24]

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

[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, (2011), 11. Google Scholar

[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, (2012), 12. Google Scholar

[27]

R. Bellman and K. L. Cooke, "Differential-Difference Equations,", Mathematics in Science and Engineering, 6 (1963). Google Scholar

[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. doi: 10.1137/0513039. Google Scholar

[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. doi: 10.1515/jiip.2007.013. Google Scholar

[30]

S. N. Busenberg and K. L. Cooke, eds., "Differential Equations and Applications in Ecology, Epidemics, and Population Problems,", Academic Press, (1981). Google Scholar

[31]

V. Capasso, E. Grosso and S. L. Paveri-Fontana, eds., "Mathematics in Biology and Medicine,", Lecture Notes in Biomath., 57 (1985). doi: 10.1007/978-3-642-93287-8. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

K. L. Cooke, Functional differential equations: Some models and perturbation problems,, in, (1967), 167. Google Scholar

[36]

J. M. Cushing, "Integrodifferential Equations and Delay Models in Population Dynamics,", Lec. Notes in Biomath., 20 (1977). Google Scholar

[37]

C. Elton, "Voles, Mice, and Lemmings,", Clarendon Press, (1942). Google Scholar

[38]

P. L. Errington, Predation and vertebrate populations,, Quarterly Review of Biology, 21 (1946), 144. Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,", Mathematics and its Applications, 74 (1992). Google Scholar

[46]

K. P. Hadeler, Delay equations in biology,, in, 730 (1979), 136. Google Scholar

[47]

J. Hale, "Theory of Functional Differential Equations,", Second edition, (1977). Google Scholar

[48]

J. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Applied Math Sciences, 99 (1993). Google Scholar

[49]

F. C. Hoppensteadt, ed., "Mathematical Aspects of Physiology,", Lectures in Applied Math, (1981). Google Scholar

[50]

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

[51]

G. E. Hutchinson, "An Introduction to Population Ecology,", Yale University, (1978). Google Scholar

[52]

F. Kappel, An approximation scheme for delay equations,, in, (1982), 585. Google Scholar

[53]

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

[54]

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

[55]

Y. Kuang, "Delay Differential Equations With Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). Google Scholar

[56]

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

[57]

N. MacDonald, "Time Lags in Biological Models,", Lecture Notes in Biomathematics, 27 (1978). Google Scholar

[58]

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

[59]

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

[60]

N. Minorsky, "Nonlinear Oscillations,", D. Van Nostrand, (1962). Google Scholar

[61]

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

[62]

D. Robbins, "Sensitivity Functions for Delay Differential Equation Models,", Ph. D. Dissertation, (2011). Google Scholar

[63]

R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites Tumor of the mouse,, in, (1995), 335. Google Scholar

[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. doi: 10.3934/mbe.2008.5.681. Google Scholar

[65]

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

[66]

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

[67]

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

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. doi: 10.1016/j.jtbi.2005.11.007. Google Scholar

[2]

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

[3]

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

[4]

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

[5]

H. T. Banks, Delay systems in biological models: Approximation techniques,, in, (1977), 21. Google Scholar

[6]

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

[7]

H. T. Banks, Identification of nonlinear delay systems using spline methods,, in, (1982), 47. Google Scholar

[8]

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

[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. doi: 10.1515/JIIP.2009.012. Google Scholar

[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. doi: 10.1007/s00285-004-0299-x. Google Scholar

[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. doi: 10.1016/S0025-5564(02)00218-3. Google Scholar

[12]

H. T. Banks and J. A. Burns, An abstract framework for approximate solutions to optimal control problems governed by hereditary systems,, in, (1975), 10. Google Scholar

[13]

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

[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. doi: 10.1137/0319051. Google Scholar

[15]

H. T. Banks, M. Davidian, J. R. Samuels, Jr. and Karyn L. Sutton, An inverse problem statistical methodology summary,, in, (2009), 249. doi: 10.1007/978-90-481-2313-1_11. Google Scholar

[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. doi: 10.1515/jiip.2007.038. Google Scholar

[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. doi: 10.1515/JIIP.2010.002. Google Scholar

[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. doi: 10.1016/j.arcontrol.2007.02.002. Google Scholar

[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. doi: 10.3934/mbe.2007.4.403. Google Scholar

[20]

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

[21]

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

[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. Google Scholar

[23]

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

[24]

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

[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, (2011), 11. Google Scholar

[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, (2012), 12. Google Scholar

[27]

R. Bellman and K. L. Cooke, "Differential-Difference Equations,", Mathematics in Science and Engineering, 6 (1963). Google Scholar

[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. doi: 10.1137/0513039. Google Scholar

[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. doi: 10.1515/jiip.2007.013. Google Scholar

[30]

S. N. Busenberg and K. L. Cooke, eds., "Differential Equations and Applications in Ecology, Epidemics, and Population Problems,", Academic Press, (1981). Google Scholar

[31]

V. Capasso, E. Grosso and S. L. Paveri-Fontana, eds., "Mathematics in Biology and Medicine,", Lecture Notes in Biomath., 57 (1985). doi: 10.1007/978-3-642-93287-8. Google Scholar

[32]

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

[33]

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

[34]

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

[35]

K. L. Cooke, Functional differential equations: Some models and perturbation problems,, in, (1967), 167. Google Scholar

[36]

J. M. Cushing, "Integrodifferential Equations and Delay Models in Population Dynamics,", Lec. Notes in Biomath., 20 (1977). Google Scholar

[37]

C. Elton, "Voles, Mice, and Lemmings,", Clarendon Press, (1942). Google Scholar

[38]

P. L. Errington, Predation and vertebrate populations,, Quarterly Review of Biology, 21 (1946), 144. Google Scholar

[39]

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

[40]

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

[41]

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

[42]

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

[43]

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

[44]

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

[45]

K. Gopalsamy, "Stability and Oscillations in Delay Differential Equations of Population Dynamics,", Mathematics and its Applications, 74 (1992). Google Scholar

[46]

K. P. Hadeler, Delay equations in biology,, in, 730 (1979), 136. Google Scholar

[47]

J. Hale, "Theory of Functional Differential Equations,", Second edition, (1977). Google Scholar

[48]

J. Hale and S. M. V. Lunel, "Introduction to Functional Differential Equations,", Applied Math Sciences, 99 (1993). Google Scholar

[49]

F. C. Hoppensteadt, ed., "Mathematical Aspects of Physiology,", Lectures in Applied Math, (1981). Google Scholar

[50]

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

[51]

G. E. Hutchinson, "An Introduction to Population Ecology,", Yale University, (1978). Google Scholar

[52]

F. Kappel, An approximation scheme for delay equations,, in, (1982), 585. Google Scholar

[53]

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

[54]

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

[55]

Y. Kuang, "Delay Differential Equations With Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). Google Scholar

[56]

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

[57]

N. MacDonald, "Time Lags in Biological Models,", Lecture Notes in Biomathematics, 27 (1978). Google Scholar

[58]

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

[59]

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

[60]

N. Minorsky, "Nonlinear Oscillations,", D. Van Nostrand, (1962). Google Scholar

[61]

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

[62]

D. Robbins, "Sensitivity Functions for Delay Differential Equation Models,", Ph. D. Dissertation, (2011). Google Scholar

[63]

R. Schuster and H. Schuster, Reconstruction models for the Ehrlich Ascites Tumor of the mouse,, in, (1995), 335. Google Scholar

[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. doi: 10.3934/mbe.2008.5.681. Google Scholar

[65]

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

[66]

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

[67]

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

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