## Transcribed Text

5.3 Geomatric Analysis and
290
The model equations and analysis. The predator-prey assumptions yield the
system of equations
=as-bry
Factoring gives
=y(cx-d)
for = 0 are I = 0 and y = a/b. The nuliclines for
oullclines.) A solution. The (d/c,a/b) solution has trajectories cirding abrein -
(d/c,a/b). (The hand-drawn phase plane is shown in Figure 5.39. Naturally, we
y = 0 and x = d/c. points (d/c,0) and (0,a/b) are not points of intersection of itd
The nuliclines Graphing these in the phase plane reveals two fixed points (0.0
interested it is in not the clear (0,0) motion. Jacobian solutions analysis determines spiraling this. in
from this analysis if the are or out or if the
though in pure oscillatory
are Here circling f(z,y) = ex - by and g(z,y) = axy - dy, and the Jacobian matrix
is
= cy CZ -ba d
While we are at it, we compute
((10.0)-(:-a).
y
a/b
x
d/c
FIGURE 5.39
Phase Piane and Nulicline Diagram for Predator-Prey Model.
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5.6
308
6. Consider the system of equations
fixed points of this system (there are two). of
b. a. Find Use Jacobian the analysis to determine the nature the fixed points (i.e, source,
saddie, spiral, etc.)
the nullclines on a phase plane.
d. e. Use Graph phase plane analysis to mark the flow arrows. Do the directions of flow
the differential equation solver capable of drawing phase plane plots to
resalts of b?
these several trajectories compare with what you expected from the previous amalysis by haxd
Use trajectories a (at least one from each region formed by the nullclines Hos
do
7. Consider the system of equations
:=sin(x)-y
dy
y
2
a. Find the fixed points and determine their nature using Jacobian analysis.
b. Draw the nullclines and flow arrows on a phase plane and compare with the behavios
suggested by the Jacobian analysis.
& For a final project, a student of ours was attempting to build a model for an
with a number of interacting variables changing over time. He set the equations 25
a
system of first-order homogeneous linear equations with constant coefficients. Without
consulting us, he was arbitrarily assigning values to the coefficients trying to find a si
of coefficients that would result in a stable ecosystem.
Try to help him Set up a software model that solves (numerically or symbolically
a system of three first-order linear homogeneous differential equations with constate
coefficients. Arbitrarily (and without mathematical analysis) assign coefficients and initial
conditions to try to get a system with solution curves that level out and are not all zere
Keep trying It can be done.
it can be done, but probably not by arbitrarily picking numbers. Try again. this tim
carefully choosing numbers and, perhaps, referring to some of the discussions on thest
types of systems in the text.
If this student wanted a stable model, what advice would you give him? Also feel free
in an ecosystem.
to advise about randomly choosing parameters which are supposed to have some
Assume that the following two logistic differential equations for time t between 0 and al. 10
aumerically 9. Use Euler's method (either directly or with a solver that uses Euler's method) to wht
Compare z(0) = 100. Solve both using the h values: 1. 0.5. 0.25. and
the solutions of the first equation for following the different h values. Explain whal
if
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