I have used the bifurcation packages AUTO [1] and LOCBIF [2] to investigate the dynamical behaviour of an oceanic plankton ecosystem model. The model consists of three autonomous coupled ordinary differential equations, involving variables which measure concentrations of nutrient, phytoplankton and zooplankton. There are 14 parameters in the model, many of which are extremely difficult to measure accurately in the open ocean. It is therefore important to understand how robust the model is with respect to these parameters. Analytically it can be shown that a positive steady state exists, and may undergo Hopf bifurcations, but a numerical approach is required to elucidate the behaviour of the subsequent limit cycles.
AUTO, written in FORTRAN, runs on the department Sun workstations, whereas LOCBIF consists of a set of executable files which run on a PC. LOCBIF actually exists as four programs of the same format, each program being used to examine a different type of problem; continuation and bifurcation analysis of equilibrium points of autonomous ODEs or iterated maps, of periodic solutions of periodically forced ODEs, and of limit cycles of autonomous ODEs. Up to 10 variables and 10 parameters can be used in LOCBIF; AUTO does not specify such upper limits.
In addition to investigating autonomous ODEs AUTO can additionally be used to explicitly calculate bifurcations of periodic waves in diffusive systems. When investigating autonomous ODEs AUTO is restricted to tracking bifurcation curves of codimension one or two, whereas LOCBIF can investigate bifurcations of up to codimension four.
I started my investigation using AUTO, but subsequently found LOCBIF to be more user-friendly and less time consuming. I used AUTO in conjunction with DSTOOL [3] and TRAX [4], which graphically display time evolutions of trajectories. LOCBIF will also iterate trajectories, but only one window can be displayed at once, whereas DSTOOL and TRAX allow for multiple windows (for example to simultaneously display 2-d projections of the trajectories and time series plots of the variables). Graphical display from AUTO is implemented by the built in package PLAUT, a rather strange creature designed to give the user the problem of figuring out its many quirks. In particular, the 3-d option does not seem to work, but otherwise the output is adequate. Other users at Leeds have adapted the output files to enable the use of more sophisticated graphical displays.
For each run of AUTO the file containing the computational constants has to be edited and then the program re-run. Although a graphical user interface has been implemented in AUTO94 I found that it did not work effectively, as it appeared to require a large amount of memory (and would only work on Thursday evenings on one particular computer!). The output is in the form of three data files which can be inspected to check the accuracy of computations. These files can be appended to previous results, enabling complete bifurcation diagrams of steady states and limit cycles to be produced. To compute, say, a branch of fixed points of a system of ODEs, an initial fixed point and value of the bifurcation parameter have to be specified. AUTO will locate bifurcation points along the branch, and a Hopf bifurcation can be used as the initial point for the subsequent branch of limit cycles. I have found that when AUTO reaches the end of such a branch, it often doubles back along itself, not seeming to realise that it has found the endpoint. Spurious results were sometimes obtained when trying to trace branches of limit cycles after a secondary doubling bifurcation. It is not too clear from the manual how best to change the computational parameters in order to overcome such difficulties. The AUTO86 manual is notoriously incomprehensible, the AUTO94 manual is far more concise and has a range of helpful examples.
LOCBIF is much more user-friendly in its presentation, with pull-down menus giving simple control of the output, branches to be followed and computational accuracy. The equations for the system are specified in a simple editor. One large graphics window is always on display, the axes of which are easily changed to display any of the variables, parameters or even eigenvalues of fixed points and Floquet multipliers of Poincaré maps of limit cycles. This information is also displayed at each step of computation along a branch, which I have found to be especially useful for following how the Floquet multipliers move along a branch of limit cycles. The LOCBIF computational parameters have much more meaningful abbreviations than those in AUTO, and hints in the manual on when they should be changed to improve computational accuracy are forthcoming. The LOCBIF manual is written mainly for the version which analyses equilibrium points of autonomous ODEs, with details of the other 3 programs placed in the appendix. I have mainly used the limit cycle program, and some hints on choice of Poincaré section would have been helpful. The separate programs concept means that a branch of steady states and a branch of limit cycles of the same system cannot be plotted at the same time. Thus I have continued to use AUTO to illustrate the behaviour of a limit cycle emanating from a Hopf bifurcation of a steady state.
I have used LOCBIF on a dan 486, and computations are generally fairly quick, although computing branches of limit cycles resulting from a series of period doubling bifurcations can become time consuming, due to the accuracy needed. Tracking two parameter bifurcation curves of limit cycles is, however, quite quick. AUTO computations are rapid but can take up a lot of memory. LOCBIF is quicker to use in the sense of switching branches and changing computational parameters as there are no files to edit and the program does not need to be recompiled. LOCBIF plots the output as it computes it, so even if a computation takes a long time, the user can see how far it has advanced along the branch. Watching a branch slowly be calculated can also help to actually understand how the system is behaving. I shall continue to use LOCBIF, but revert to AUTO to verify some calculations. LOCBIF and AUTO have both been used in the recent literature to examine the dynamical behaviour of biological systems and are currently being incorporated into the latest version of DSTOOL, which should result in a comprehensive user-friendly package for the numerical analysis of dynamical systems.
Andrew M. Edwards ( andy@amsta.leeds.ac.uk).