ECE 547 Nonlinear Systems and Control
Homework 4
1. In class we saw the Lanchester-Gause model of warfare, repeated here (all parameters are assumed </br>strictly positive, and the states are confined to the first quadrant):
,
.
A. Let’s start with the case we studied in class, when and . Analytically show
that the equilibria are indeed of the types we saw in class. Next, pick suitable values for the
parameters and enough initial conditions to produce a phase portrait of the system that
(hopefully) matches the analysis we did in class. Feel free to use MATLAB to do this numerically,
if you prefer.
B. Now, consider the case where and . Analytically, find what are its equilibrium
points, and determine their type. Next, similar to the previous case, verify your analysis by
producing a phase portrait of the system. Provide an interpretation of your results! What do
they mean for the warfare system?
C. Repeat for the case and .
2. A very famous system used in ecology to represent predator-prey dynamics is the Lotka-Volterra
system
,
.
Here, represents the number of prey, and the number of predators in some closed
ecosystem, with the predator’s success rate at catching prey, the prey’s birth rate,
and represents the predator’s death rate.
A. Verbally compare the LV model to the epidemic model we saw in class. How are they similar?
Different? Could the LV model be used for epidemic modeling purposes?
B. With parameters of your choice, do a computer simulation of the LV system. What do you
observe? Explain and interpret your results.
Note: you are welcome to read about the LV system online (there are plenty of links from a Google
search), but I urge you to try answering the questions yourself, before you look it up. I believe you
will learn more this way.
·x1 = r x1(1 − x1K) − a x1x2
·x2 = sx2(1 − x2L ) − bx1x2
r /a < L s /b < K
r /a > L s /b > K
r /a < L s /b > K
·S = − βSI + μS·I = βSI − γI
S ≥ 0 I ≥ 0
β > 0 μ > 0
γ > 0

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