Environmental Engineering Reference
In-Depth Information
The calculation and visualization of such curves requires also numerical means,
which are in most cases more sophisticated than the presented solution method
using numerical MATLAB
solvers for ordinary differential equations.
In practical applications it has been observed that the classical Lotka-Volterra-
system has several drawbacks. The simplistic assumptions have already been men-
tioned. Nevertheless, it is a jumping-off place for more realistic models, which are
obtained by extensions of the original system (
19.12
). Such extensions have been
proposed in the literature. Murray (
2002
) provides an overview. The most nearby idea
is to use a logistic growth term, as in (
19.1
), instead of the linear 1.order term.
In the presented approach is quite easy to implement extensions of the Lotka-
Volterra equations. All that needs to be done is to extend the formulae in the
function
of the '
predprey.m'
M-file. If additional parameters are required, these
should be added in the input specifications part of the M-file and considered as
formal parameters in the function call. Using various starting positions in following
runs, it is possible to examine, whether an equilibrium is stable or unstable or if a
limit cycle exists.
For 2D problems, i.e. settings with two variables, the '
pplane.m
' model can be a
very useful tool for the MATLAB
®
user. Briefly, we present an example applica-
tion for the phase space M-file '
pplane.m
', which was already mentioned in Chap.
18. The M-file is available from the web (
http://math.rice.edu/~dfield/
).
Figure
19.6
depicts the set-up and the output for a predator pray problem, for which
we used the MATLAB
®
version 7 file: '
pplane7.m
'. Four windows are depicted. In
the 'Setup' window the differential equations are specified. There are edit-boxes for
the input and change of parameters as well as for the basic settings concerning outlook
and axes of the display window. For the example case we did not edit the system of
differential equations, but chose the 'predator prey' entry from the 'Gallery' menu.
The display window shows the phase space. A field of grey arrows as a first
visualization depicts the trajectories. Blue lines for trajectories appear by mouse
click within the displayed phase space where the cursor location is taken as starting
value for a trajectory. The red dot, indicating an equilibrium location, appears when
the corresponding sub-menu entry 'Find an equilibrium point' under 'Solutions' is
selected. The user has to click into the displayed phase space and to select a starting
point for the search.
When an equilibrium position has been found, some of its characteristics are
shown in a separate small window with text output, depicted on the right side of
Fig.
19.6
. The exact position, the corresponding Jacobi matrix, the eigenvalues and
eigenvectors can all be read from the window. There is an additional button reading
'Display the linearization'. After pressing that button, the linearized system is
shown in another window named 'Linearization', into which trajectories are plotted
after a mouse click (Fig.
19.6
). In the example we see circles around the origin. The
pattern of the trajectories of the linearized system corresponds with the last row of
Table 18.2, with non-zero entries only in the off-diagonal of the matrix and two
purely imaginary eigenvalues.
The user may further explore the '
pplane7.m
' program by her/himself. There are
numerous new entries in the menu, which are added to the MATLAB
®
figure
®
