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Numerical modelling in biosciences using delay differential equations. (English) Zbl 0969.65124
Summary: Our principal purposes here are (i) to consider, from the perspective of applied mathematics, models of phenomena in the biosciences that are based on delay differential equations and for which numerical approaches are a major tool in understanding their dynamics, (ii) to review the application of numerical techniques to investigate these models. We show that there are prima facie reasons for using such models: (i) they have a richer mathematical framework (compared with ordinary differential equations) for the analysis of biosystem dynamics, (ii) they display better consistency with the nature of certain biological processes and predictive results. We analyze both the qualitative and quantitative role that delays play in basic time-lag models proposed in population dynamics, epidemiology, physiology, immunology, neural networks and cell kinetics. We then indicate suitable computational techniques for the numerical treatment of mathematical problems emerging in the biosciences, comparing them with those implemented by the bio-modellers.
65R20Integral equations (numerical methods)
92B20General theory of neural networks (mathematical biology)
92C45Kinetics in biochemical problems
92-08Computational methods (appl. to natural sciences)
92D25Population dynamics (general)
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations