Abstract
We consider certain types of discretization schemes for differential equations with quadratic nonlinearities, which were introduced by Kahan, and considered in a broader setting by Mickens. These methods have the property that they preserve important structural features of the original systems, such as the behaviour of solutions near to fixed points, and also, where appropriate (e.g. for certain mechanical systems), the property of being volume-preserving, or preserving a symplectic/Poisson structure. Here we focus on the application of Kahan’s method to models of biological systems, in particular to reaction kinetics governed by the Law of Mass Action, and present a general approach to birational discretization, which is applied to population dynamics of Lotka–Volterra type.
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Acknowledgments
KT’s studentship was funded by the EPSRC and the School of Mathematics, Statistics & Actuarial Science, University of Kent. We are grateful to the referees for their comments.
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Hone, A., Towler, K. Non-standard discretization of biological models. Nat Comput 14, 39–48 (2015). https://doi.org/10.1007/s11047-014-9463-4
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DOI: https://doi.org/10.1007/s11047-014-9463-4