Abstract
Particular cases of nonlinear systems of delay Volterra integro-differential equations (denoted by DVIDEs) with constant delay τ > 0, arise in mathematical modelling of ‘predator–prey’ dynamics in Ecology. In this paper, we give an analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for systems of this type. Then, from the perspective of applied mathematics, we consider the Volterra’s integro-differential system of ‘predator–prey’ dynamics arising in Ecology. We analyze the numerical issues of the introduced collocation method applied to the ‘predator–prey’ system and confirm that we can achieve the expected theoretical orders of convergence.
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References
Volterra V (1927) Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie del R. Comitato talassografico italiano, Mem. CXXXI
Cushing JM (1977) Integrodifferential equations and delay models in population dynamics. Lecture notes in biomathematics, vol 20. Springer, Berlin
Volterra V (1939) The general equations of biological strife in the case of historical actions. Proc Edinburgh Math Soc 2: 4–10
Bocharov GA, Rihan FA (2000) Numerical modelling in biosciences using delay differential equations. J Comput Appl Math 125: 183–199
Gopalsamy K (1992) Stability and oscillation in delay differential equations of population dynamics. Kluwer, Boston
Kuang Y (1993) Delay differential equations with applications in population dynamic. Academic Press, San Diego, CA
Baker CTH, Ford NJ (1990) Asymptotic error expansions for linear multistep methods for a class of delay integro-differential equations. Bull Greek Math Soc 31: 5–10
Enright WH, Hu M (1997) Continuous Runge–Kutta methods for neutral Volterra integro-differential equations with delay. Appl Numer Math 24: 175–190
Brunner H (1988) The approximate solution of initial- value problems for general Volterra integro-differential equations. Computing 40: 125–137
Makroglou A (1983) A bloc-by-block method for the numerical solution of Volterra delay integro-differential equation. Computing 30: 49–62
Brunner H (1989) Collocation methods for nonlinear Volterra Integro-differential equations with infinite delay. Math Comp 53: 571–587
Brunner H (1994) The numerical solution of neutral Volterra integro-differential equations with delay arguments. Ann Numer Math 1: 309–322
Brunner H, Zhang W (1999) Primary discontinuities in solutions for delay integro-differential equations. Methods Appl Anal 6: 525–534
Baker CTH, Willé D (2000) On the propagation of derivative discontinuities in Volterra retarded integro-differential equations. N Z J Math 29: 103–113
Brunner H (1998) The use of splines in the numerical solution of Volterra integral and integro-differential equations. In: Dubuc S (ed) Splines and the theory of wavelets I, II, CRM Proceedings and Lecture Notes. American Mathematical Society, Providence, RI
Brunner H (1984) Implicit Runge–Kutta methods of optimal order for volterra integro-differential equations. Math Comp 42: 95–109
Atkinson KE (1989) Introduction to numerical analysis, 2nd edn. Wiley, New York
McKee S (1982) Generalized discrete Gronwall lemmas. Z Angew Math Mech 62: 429–434
Brunner H (2004) Collocation methods for Volterra integral and related functional equations. Cambridge University Press, Cambridge
Enright WH (2005) Tools for verification of approximate solutions to differential equations. In: Einarsson B(eds) Handbook for scientific computing. SIAM Press, Philadelphia, pp 109–119
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Shakourifar, M., Dehghan, M. On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments. Computing 82, 241–260 (2008). https://doi.org/10.1007/s00607-008-0009-4
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DOI: https://doi.org/10.1007/s00607-008-0009-4
Keywords
- Piecewise polynomial collocation
- Delay Volterra integro-differential equations
- Global convergence
- Optimal order of superconvergence