Abstract
Functional differential equations arise in the modeling of hereditary systems such as ecological and biological systems, chemical and mechanical systems and many-many other. The long-term behavior and stability of such systems is an important area for investigation. For example, will a population decline to dangerously low levels? Could a small change in the environmental conditions have drastic consequences on the long-term survival of the population? There is a growing body of works devoted to such type of investigations. Analytical solutions of functional differential equations are generally unavailable and a lot of different numerical methods are adopted for obtaining approximate solutions. A natural question to ask is “do the numerical solutions preserve the stability properties of the exact solution?” Thus, to use numerical investigation of functional differential equations it is very important to know if the considered difference analogue of the original differential equation has the reliability to preserve some general properties of this equation, in particular, property of stability. In this chapter the capability of difference analogues of differential equations to save a property of stability of solutions of considered differential equations is studied. In particular, sufficient conditions for the step of discretization, at which stability of difference analogue solution is saved, are obtained for some known mathematical models, such that as the mathematical model of controlled inverted pendulum, Nicholson blowflies equation, predator-prey model, for difference analogue of an integro-differential equation of convolution type.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Acheson DJ (1993) A pendulum theorem. Proc R Soc Lond Ser A, Math Phys Sci 443(1917):239–245
Acheson DJ, Mullin T (1993) Upside-down pendulums. Nature 366(6452):215–216
Bakke VL, Jackiewicz Z (1986) Boundedness of solutions of difference equations and applications to numerical solutions of Volterra integral equations of the second kind. J Math Anal Appl 115:592–605
Bandyopadhyay M, Chattopadhyay J (2005) Ratio dependent predator-prey model: effect of environmental fluctuation and stability. Nonlinearity 18:913–936
Beretta E, Kuang Y (1998) Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear Anal 32(4):381–408
Beretta E, Takeuchi Y (1994) Qualitative properties of chemostat equations with time delays: boundedness local and global asymptotic stability. Differ Equ Dyn Syst 2(1):19–40
Beretta E, Takeuchi Y (1994) Qualitative properties of chemostat equations with time delays II. Differ Equ Dyn Syst 2(4):263–288
Beretta E, Takeuchi Y (1995) Global stability of an SIR epidemic model with time delays. J Math Biol 33:250–260
Beretta E, Kolmanovskii V, Shaikhet L (1998) Stability of epidemic model with time delays influenced by stochastic perturbations. Math Comput Simul 45(3–4):269–277 (Special Issue “Delay Systems”)
Blackburn JA, Smith HJT, Gronbech-Jensen N (1992) Stability and Hopf bifurcations in an inverted pendulum. Am J Phys 60(10):903–908
Borne P, Kolmanovskii V, Shaikhet L (1999) Steady-state solutions of nonlinear model of inverted pendulum. Theory Stoch Process 5(21)(3–4):203–209. Proceedings of the third Ukrainian–Scandinavian conference in probability theory and mathematical statistics, 8–12 June 1999, Kyiv, Ukraine
Borne P, Kolmanovskii V, Shaikhet L (2000) Stabilization of inverted pendulum by control with delay. Dyn Syst Appl 9(4):501–514
Bradul N, Shaikhet L (2007) Stability of the positive point of equilibrium of Nicholson’s blowflies equation with stochastic perturbations: numerical analysis. Discrete Dyn Nat Soc 2007:92959. 25 pages, doi:10.1155/2007/92959
Braverman E, Kinzebulatov D (2006) Nicholson’s blowflies equation with a distributed delay. Can Appl Math Q 14(2):107–128
Brunner H, Lambert JD (1974) Stability of numerical methods for Volterra integro-differential equations. Computing (Arch Elektron Rechnen) 12:75–89
Brunner H, Van der Houwen PJ (1986) The numerical solution of Volterra equations. CWI monographs, vol 3. North Holland, Amsterdam
Busenberg S, Cooke KL (1980) The effect of integral conditions in certain equations modelling epidemics and population growth. J Math Biol 10(1):1332
Bush AW, Cook AE (1976) The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater. J Theor Biol 63:385–395
Cai L, Li X, Song X, Yu J (2007) Permanence and stability of an age-structured predator–prey system with delays. Discrete Dyn Nat Soc 2007:54861. 15 pages
Carletti M (2002) On the stability properties of a stochastic model for phage–bacteria interaction in open marine environment. Math Biosci 175:117–131
Chen F (2005) Periodicity in a ratio-dependent predator–prey system with stage structure for predator. J Appl Math 2:153–169
Colliugs JB (1997) The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model. J Math Biol 36:149–168
Ding X, Jiang J (2008) Positive periodic solutions in delayed Gause-type predator–prey systems. J Math Anal Appl 339(2):1220–1230. doi:10.1016/j.jmaa.2007.07.079
Ding X, Li W (2006) Stability and bifurcation of numerical discretization Nicholson blow-flies equation with delay. Discrete Dyn Nat Soc 2006:1–12. Article ID 19413
Edwards JT, Ford NJ, Roberts JA, Shaikhet LE (2000) Stability of a discrete nonlinear integro-differen-tial equation of convolution type. Stab Control: Theory Appl 3(1):24–37
Edwards JT, Ford NJ, Roberts JA (2002) The numerical simulation of the qualitative behavior of Volterra integro-differential equations. In: Levesley J, Anderson IJ, Mason JC (eds) Proceedings of algorithms for approximation IV. University of Huddersfield, pp 86–93
El-Owaidy HM, Ammar A (1988) Stable oscillations in a predator–prey model with time lag. J Math Anal Appl 130:191–199
Elaydi SN (1993) Stability of Volterra difference equations of convolution type. In: Shan-Tao L (ed) Proceedings of the special program at Nankai Institute of Mathematics. World Scientific, Singapore, pp 66–73
Elaydi SN (1994) Periodicity and stability of linear Volterra difference systems. J Differ Equ Appl 181:483–492
Elaydi SN (1995) Global stability of nonlinear Volterra difference systems. Differ Equ Dyn Syst 2:237–345
Elaydi SN, Kocic V (1994) Global stability of a nonlinear Volterra difference equations. Differ Equ Dyn Syst 2:337–345
Elaydi SN, Murakami S (1996) Asymptotic stability versus exponential stability in linear Volterra difference equations of convolution type. J Differ Equ Appl 2:401–410
Fan M, Wang Q, Zhou X (2003) Dynamics of a nonautonomous ratio-dependent predator–prey system. Proc R Soc Edinb A 133:97–118
Fan YH, Li WT, Wang LL (2004) Periodic solutions of delayed ratio-dependent predator–prey model with monotonic and no-monotonic functional response. Nonlinear Anal RWA 5(2):247–263
Farkas M (1984) Stable oscillations in a predator–prey model with time lag. J Math Anal Appl 102:175–188
Feng QX, Yan JR (2002) Global attractivity and oscillation in a kind of Nicholson’s blowflies. J Biomath 17(1):21–26
Ford NJ, Baker CTH (1996) Qualitative behavior and stability of solutions of discretised nonlinear Volterra integral equations of convolution type. J Comput Appl Math 66:213–225
Ford NJ, Baker CTH, Roberts JA (1997) Preserving qualitative behaviour and transience in numerical solutions of Volterra integro-differential equations of convolution type: Lyapunov functional approaches. In: Proceeding of 15th World congress on scientific computation, modelling and applied mathematics (IMACS97), Berlin, August 1997. Numerical mathematics, vol 2, pp 445–450
Ford NJ, Edwards JT, Roberts JA, Shaikhet LE (1997) Stability of a difference analogue for a nonlinear integro differential equation of convolution type. Numerical Analysis Report, University of Manchester 312
Ford NJ, Baker CTH, Roberts JA (1998) Nonlinear Volterra integro-differential equations—stability and numerical stability of θ-methods, MCCM numerical analysis report. J Integral Equ Appl 10:397–416
Garvie M (2007) Finite-difference schemes for reaction–diffusion equations modelling predator–prey interactions in MATLAB. Bull Math Biol 69(3):931–956
Ge Z, He Y (2008) Diffusion effect and stability analysis of a predator–prey system described by a delayed reaction–diffusion equations. J Math Anal Appl 339(2):1432–1450. doi:10.1016/j.jmaa.2007.07.060
Golec J, Sathananthan S (1999) Sample path approximation for stochastic integro-differential equations. Stoch Anal Appl 17(4):579–588
Golec J, Sathananthan S (2001) Strong approximations of stochastic integro-differential equations. Dyn Contin Discrete Impuls Syst 8(1):139–151
Gopalsamy K (1992) Stability and oscillations in delay differential equations of population dynamics. Mathematics and its applications, vol 74. Kluwer Academic, Dordrecht
Gourley SA, Kuang Y (2004) A stage structured predator–prey model and its dependence on maturation delay and death rate. J Math Biol 4:188–200
Gurney WSC, Blythe SP, Nisbet RM (1980) Nicholson’s blowflies revisited. Nature 287:17–21
Gyori I, Ladas G (1991) Oscillation theory of delay differential equations with applications. Oxford mathematical monographs. Oxford University Press, New York
Gyori I, Trofimchuck SI (2002) On the existence of rapidly oscillatory solutions in the Nicholson’s blowflies equation. Nonlinear Anal 48:1033–1042
Hastings A (1983) Age dependent predation is not a simple process. I. Continuous time models. Theor Popul Biol 23:347–362
Hastings A (1984) Delays in recruitment at different trophic levels effects on stability. J Math Biol 21:35–44
Higham DJ, Mao XR, Stuart AM (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. SIAM J Numer Anal 40(3):1041–1063
Higham DJ, Mao XR, Stuart AM (2003) Exponential mean-square stability of numerical solutions to stochastic differential equations. LMS J Comput Math 6:297–313
Hsu SB, Huang TW (1995) Global stability for a class of predator–prey systems. SIAM J Appl Math 55(3):763–783
Huo HF, Li WT (2004) Periodic solution of a delayed predator–prey system with Michaelis–Menten type functional response. J Comput Appl Math 166:453–463
Imkeller P, Lederer Ch (2001) Some formulas for Lyapunov exponents and rotation numbers in two dimensions and the stability of the harmonic oscillator and the inverted pendulum. Dyn Syst 16:29–61
Kapitza PL (1965) Dynamical stability of a pendulum when its point of suspension vibrates, and pendulum with a vibrating suspension. In: ter Haar D (ed) Collected papers of P.L. Kapitza, vol 2. Pergamon Press, London, pp 714–737
Kloeden PE, Platen E (2000) Numerical solution of stochastic differential equations, vol 23. Springer, Berlin
Kocic VL, Ladas G (1990) Oscillation and global attractivity in discrete model of Nicholson’s blowflies. Appl Anal 38:21–31
Kolmanovskii VB, Shaikhet LE (1993) Method for constructing Lyapunov functionals for stochastic systems with aftereffect. Differ Uravn (Minsk) 29(11):1909–1920, (in Russian). Translated in Differential Equations 29(11):1657–1666 (1993)
Kolmanovskii VB, Shaikhet LE (1994) New results in stability theory for stochastic functional differential equations (SFDEs) and their applications. In: Proceedings of dynamic systems and applications, Atlanta, GA, 1993, vol 1, 167–171. Dynamic, Atlanta
Kolmanovskii VB, Shaikhet LE (1995) Method for constructing Lyapunov functionals for stochastic differential equations of neutral type. Differ Uravn (Minsk) 31(11):1851–1857 (in Russian). Translated in Differential Equations 31(11):1819–1825 (1995)
Kolmanovskii VB, Shaikhet LE (1997) Matrix Riccati equations and stability of stochastic linear systems with nonincreasing delays. Funct Differ Equ 4(3–4):279–293
Kolmanovskii VB, Shaikhet LE (1998) Riccati equations and stability of stochastic linear systems with distributed delay. In: Bajic V (ed) Advances in systems, signals, control and computers. IAAMSAD and SA branch of the Academy of Nonlinear Sciences, Durban, pp 97–100. ISBN:0-620-23136-X
Kolmanovskii VB, Shaikhet LE (2002) Construction of Lyapunov functionals for stochastic hereditary systems: a survey of some recent results. Math Comput Model 36(6):691–716
Kolmanovskii VB, Kosareva NP, Shaikhet LE (1999) A method for constructing Lyapunov functionals. Differ Uravn (Minsk) 35(11):1553–1565. Translated in Differential Equations 35(11):1573–1586 (1999) (in Russian)
Kuang Y (1993) Delay differential equations with application in population dynamics. Academic Press, New York
Kulenovic MRS, Ladas G (1987) Linearized oscillations in population dynamics. Bull Math Biol 49(5):615–627
Kulenovic MRS, Nurkanovic M (2006) Asymptotic behavior of a competitive system of linear fractional difference equations. Adv Differ Equ 2006(5):19756. 13 pages, doi:10.1155/ADE/2006/19756
Kulenovic MRS, Ladas G, Sficas YG (1989) Global attractivity in population dynamics. Comput Math Appl 18(10–11):925–928
Kulenovic MRS, Ladas G, Sficas YG (1992) Global attractivity in Nicholson’s blowflies. Appl Anal 43:109–124
Lakshmikantham V, Trigiante D (1988) Theory of difference equations: numerical methods and applications. Academic Press, New York
Levi M (1988) Stability of the inverted pendulum—a topological explanation. SIAM Rev 30(4):639–644
Levi M, Weckesser W (1995) Stabilization of the inverted linearized pendulum by high frequency vibrations. SIAM Rev 37(2):219–223
Levin JJ, Nohel JA (1963) Note on a nonlinear Volterra equation. Proc Am Math Soc 14(6):924–929
Levin JJ, Nohel JA (1965) Perturbations of a non-linear Volterra equation. Mich Math J 12:431–444
Li J (1996) Global attractivity in a discrete model of Nicholson’s blowflies. Ann Differ Equ 12(2):173–182
Li J (1996) Global attractivity in Nicholson’s blowflies. Appl Math, Ser 11(4):425–434
Li X (2005) Global behavior for a fourth-order rational difference equation. J Math Anal Appl 312(2):555–563
Li YK, Kuang Y (2001) Periodic solutions of periodic delay Lotka–Volterra equations and systems. J Math Anal Appl 255:260–280
Li M, Yan J (2000) Oscillation and global attractivity of generalized Nicholson’s blowfly model. In: Differential equations and computational simulations, Chengdu, 1999. World Scientific, Singapore, pp 196–201
Liao X, Zhou Sh, Ouyang Z (2007) On a stoichiometric two predators on one prey discrete model. Appl Math Lett 20:272–278
Liz E (2007) A sharp global stability result for a discrete population model. J Math Anal Appl 330:740–743
Lubich Ch (1983) On the stability of linear multistep methods for Volterra convolution equations. IMA J Numer Anal 3:439–465
Luo J, Shaikhet L (2007) Stability in probability of nonlinear stochastic Volterra difference equations with continuous variable. Stoch Anal Appl 25(6):1151–1165. doi:10.1080/07362990701567256
Mao X, Shaikhet L (2000) Delay-dependent stability criteria for stochastic differential delay equations with Markovian switching. Stab Control: Theory Appl 3(2):88–102
Marotto F (1982) The dynamics of a discrete population model with threshold. Math Biosci 58:123–128
Maruyama G (1955) Continuous Markov processes and stochastic equations. Rend Circ Mat Palermo 4:48–90
Mata GJ, Pestana E (2004) Effective Hamiltonian and dynamic stability of the inverted pendulum. Eur J Phys 25:717–721
Milstein GN (1988) The numerical integration of stochastic differential equations. Urals University Press, Sverdlovsk (in Russian)
Mitchell R (1972) Stability of the inverted pendulum subjected to almost periodic and stochastic base motion—an application of the method of averaging. Int J Non-Linear Mech 7:101–123
Muroya Y (2007) Persistence global stability in discrete models of Lotka–Volterra type. J Math Anal Appl 330:24–33
Nicholson AJ (1954) An outline of the dynamics of animal populations. Aust J Zool 2:9–65
Ovseyevich AI (2006) The stability of an inverted pendulum when there are rapid random oscillations of the suspension point. Int J Appl Math Mech 70:762–768
Paternoster B, Shaikhet L (1999) Stability in probability of nonlinear stochastic difference equations. Stab Control: Theory Appl 2(1–2):25–39
Paternoster B, Shaikhet L (2000) About stability of nonlinear stochastic difference equations. Appl Math Lett 13(5):27–32
Paternoster B, Shaikhet L (2008) Stability of equilibrium points of fractional difference equations with stochastic perturbations. Adv Differ Equ 2008:718408. 21 pages, doi:10.1155/2008/718408
Peschel M, Mende W (1986) The predator–prey model: do we live in a Volterra world? Akademie Verlag, Berlin
Resnick SI (1992) Adventures in stochastic processes. Birkhauser, Boston
Rodkina A, Schurz H (2005) Almost sure asymptotic stability of drift-implicit theta-methods for bilinear ordinary stochastic differential equations in R 1. J Comput Appl Math 180:13–31
Ruan S, Xiao D (2001) Global analysis in a predator–prey system with nonmonotonic functional response. SIAM J Appl Math 61:1445–1472
Saito Y, Mitsui T (1996) Stability analysis of numerical schemes for stochastic differential equations. SIAM J Numer Anal 33(6):2254–2267
Sanz-Serna JM (2008) Stabilizing with a hammer. Stoch Dyn 8:47–57
Schurz H (1996) Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise. Stoch Anal Appl 14:313–354
Schurz H (1997) Stability, stationarity, and boundedness of some implicit numerical methods for stochastic differential equations and applications. Logos, Berlin
Schurz H (1998) Partial and linear-implicit numerical methods for nonlinear SDEs. Unpublished Manuscript, Universidad de Los Andes, Bogota
Schurz H (2002) Numerical analysis of SDE without tears. In: Kannan D, Lakshmikantham V (eds) Handbook of stochastic analysis and applications, pp 237–359. Marcel Dekker, Basel
Schurz H (2005) Stability of numerical methods for ordinary SDEs along Lyapunov-type and other functions with variable step sizes. Electron Trans Numer Anal 20:27–49
Schurz H (2006) An axiomatic approach to numerical approximations of stochastic processes. Int J Numer Anal Model 3:459–480
Schurz H (2007) Applications of numerical methods and its analysis for systems of stochastic differential equations. Bull Kerala Math Assoc 4:1–85
Shaikhet L (1995) Stability in probability of nonlinear stochastic hereditary systems. Dyn Syst Appl 4(2):199–204
Shaikhet L (1995) Stability in probability of nonlinear stochastic systems with delay. Mat Zametki 57(1):142–146 (in Russian). Translated in Mathematical Notes, 57(1):103–106 (1995)
Shaikhet L (1996) Stability of stochastic hereditary systems with Markov switching. Theory Stoch Process 2(18)(3–4):180–185
Shaikhet L (1996) Modern state and development perspectives of Lyapunov functionals method in the stability theory of stochastic hereditary systems. Theory Stoch Process 2(18)(1–2):248–259
Shaikhet L (1998) Stability of predator–prey model with aftereffect by stochastic perturbations. Stab Control: Theory Appl 1(1):3–13
Shaikhet L (2005) Stability of difference analogue of linear mathematical inverted pendulum. Discrete Dyn Nat Soc 2005(3):215–226
Shaikhet L (2008) Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations. Dyn Syst Appl 17:235–253
Shaikhet L (2009) Improved condition for stabilization of controlled inverted pendulum under stochastic perturbations. Discrete Contin Dyn Syst 24(4):1335–1343. doi:10.3934/dcds.2009.24.
Shaikhet L, Roberts J (2004) Stochastic Volterra differential integral equation: stability and numerical analysis. University of Manchester. MCCM. Numerical Analysis Report 450, 38p
Shaikhet L, Roberts J (2006) Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations. Adv Differ Equ 2006:73897. 22 pages
Sharp R, Tsai Y-H, Engquist B (2005) Multiple time scale numerical methods for the inverted pendulum problem. In: Multiscale methods in science and engineering. Lecture notes computing science and engineering, vol 44. Springer, Berlin, pp 241–261
So JW-H, Yu JS (1994) Global attractivity and uniformly persistence in Nicholson’s blowflies. Differ Equ Dyn Syst 2:11–18
So JW-H, Yu JS (1995) On the stability and uniform persistence of a discrete model of Nicholson’s blowflies. J Math Anal Appl 193(1):233–244
Stuart S, Humphries AR (1998) Dynamical systems and numerical analysis. Cambridge monographs on applied and computational mathematics, vol 2. Cambridge University Press, Cambridge
Volterra V (1931) Lesons sur la theorie mathematique de la lutte pour la vie. Gauthier-Villars, Paris
Volz R (1982) Global asymptotic stability of a periodical solution to an epidemic model. J Math Biol 15:319–338
Wang LL, Li WT (2003) Existence and global stability of positive periodic solutions of a predator–prey system with delays. Appl Math Comput 146(1):167–185
Wang LL, Li WT (2004) Periodic solutions and stability for a delayed discrete ratio-dependent predator–prey system with Holling-type functional response. Discrete Dyn Nat Soc 2004(2):325–343
Wang Q, Fan M, Wang K (2003) Dynamics of a class of nonautonomous semi-ratio-dependent predator–prey system with functional responses. J Math Anal Appl 278:443–471
Wangersky PJ, Cunningham WJ (1957) Time lag in predator–prey population models. Ecology 38(1):136–139
Wei J, Li MY (2005) Hopf bifurcation analysis in a delayed Nicholson blowflies equation. Nonlinear Anal 60(7):1351–1367
Xiao D, Ruan S (2001) Multiple bifurcations in a delayed predator–prey system with nonmonotonic functional response. J Differ Equ 176:494–510
Zeng X (2007) Non-constant positive steady states of a predator–prey system with cross-diffusions. J Math Anal Appl 332(2):989–1009
Zhang X, Chen L, Neumann UA (2000) The stage-structured predator–prey model and optimal harvesting policy. Math Biosci 168:201–210
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag London Limited
About this chapter
Cite this chapter
Shaikhet, L. (2011). Difference Equations as Difference Analogues of Differential Equations. In: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer, London. https://doi.org/10.1007/978-0-85729-685-6_10
Download citation
DOI: https://doi.org/10.1007/978-0-85729-685-6_10
Publisher Name: Springer, London
Print ISBN: 978-0-85729-684-9
Online ISBN: 978-0-85729-685-6
eBook Packages: EngineeringEngineering (R0)