2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, 
BC, Canada.
Electronic Journal of Differential Equations,
Conference 12, 2005, pp. 9-19. 

Title: $O(\ell)$ shift in Hopf bifurcations for a class of
 non-standard numerical schemes

Authors: Murray E. Alexander (National Research Council, Canada)
         Seyed M. Moghadas (National Research Council, Canada)
 
Abstract: 
 Quantitative aspects of models describing the dynamics of biological
 phenomena have been mostly restricted to results of numerical
 simulations, often by employing standard numerical methods. However,
 several studies have shown that these methods may fail to reproduce
 the actual dynamical behavior of the underlying continuous model
 when the integration time-step, model parameters, or initial
 conditions vary in their respective ranges. In this paper, a
 non-standard numerical scheme is constructed for a general class of
 positivity-preserving system of ordinary differential equations. A
 connection between the dynamics of the system and that of the scheme
 is established in terms of codimension-zero bifurcations. It is
 shown that when the continuous model undergoes a bifurcation with a
 simple eigenvalue passing through zero (pitchfork, transcritical or
 saddle-node bifurcation), the scheme exhibits a corresponding
 bifurcation at the same bifurcation parameter value. On the other
 hand, for a Hopf bifurcation there is in general an
 $\mathcal{O}(\ell)$ shift in the bifurcation parameter value for the
 numerical scheme, where $\ell$ is the time-step. Partial results for
 the bifurcations of codimension-1 and higher are also discussed.
 Finally, the results are detailed for two examples: predator-prey
 system of Gause-type and the Brusselator system representing an
 autocatalytic oscillating chemical reaction.

Published April 20, 2005.
Math Subject Classifications: 65P30, 65C20, 37M05.
Key Words: Finite-difference methods, Codimension-zero bifurcations, 
           Quantitative analysis, Scheme failure.