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Non-standard discretization methods for some biological models. (English) Zbl 0989.65143
Mickens, Ronald E. (ed.), Applications of nonstandard finite difference schemes. Papers from the minisymposium on nonstandard finite difference schemes: theory and applications, SIAM annual meeting, Atlanta, GA, USA, 1999. Singapore: World Scientific. 155-180 (2000).
From the introduction: It has been observed for some time that the standard (classical) discretization methods of differential equations often produce difference equations that do not share their dynanics. An illustrative example is the logistic differential equations \[ {dx\over dt}= \beta x(1- x). \] Euler’s discretization scheme produces the logistic difference equation \[ x(n+ 1)= \mu x(n)(1- x(n)) \] which possesses a remarkably different dynamics such as periodic-doubling bifurcation route to chaos. A more popular discretization method is to modify the given differential equation to another with piecewise-constant arguments and then to integrate the modified equation. In some instances, this produces a difference equation whose dynamics is close to its original differential equation. However, oftentimes this is not the case. Nevertheless, many authors find it interesting to study the resulting difference equations. This is not a criticism of these authors’ research, since the study of nonlinear difference equations is of paramount importance regardness of whether or not they have connections with differential equations. But what we are actually saying is that from the point of view of numerical analysis such study is of less importance.
This paper concerns itself with those numerical schemes that produce difference equations whose dynamics resembles that of their continuous counterparts.
For the entire collection see [Zbl 0970.00024].

65P30 Numerical bifurcation problems
34A34 Nonlinear ordinary differential equations and systems, general theory
65L05 Numerical methods for initial value problems
65L12 Finite difference and finite volume methods for ordinary differential equations
37M20 Computational methods for bifurcation problems in dynamical systems
39A11 Stability of difference equations (MSC2000)
37G10 Bifurcations of singular points in dynamical systems
92D25 Population dynamics (general)