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].

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].

##### MSC:

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) |

##### Keywords:

Euler method; biological models; global stability; positive equilibrium point; predator-prey model; logistic differential equations; logistic difference equation; periodic-doubling bifurcation; chaos; nonlinear difference equations
PDF
BibTeX
XML
Cite

\textit{H. Al-Kahby} et al., in: 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; Zbl 0989.65143)