MATH314, Modeling Realizable Phenomena
Practice Midterm Solutions
Gidon Eshel, Physics Department, Bard College
AnnandaleonHudson, NY 125045000,
x7232,
[email protected]
.
Remember: succinct and to the point and
very
clearly printed, no cursive ever.
(1) Explain clearly in a sentence the Principle of Parsimony.
If the simulation (model output) resembles reality well, and if the model cannot be
simplified – it wins!
(2) For the scalar linear model
dx
dt
=
αx
please
(a) point out the state vector,
It is the 1vector (a scalar)
x
.
(b) obtain the general solution and state what information you need for it to be com
plete,
dx
dt
=
αx
=
⇒
dx
x
=
αdt.
=
⇒
x
(
t
)
x
(0)
dx
x
=
x
(
t
)
x
(0)
d
ln
x
=
α
t
0
dt
ln
x
(
t
)

ln
x
(0)
≡
ln
x
t

ln
x
o
= ln
x
t
x
o
=
αt
=
⇒
x
t
x
o
= e
αt
=
⇒
x
t
= e
αt
x
o
.
So
x
o
is the piece of information needed.
(c) define, and characterize the condition(s) for, instability.
If
α >
0
(which means that e
αt
>
1
for
t >
0
), the model is unstable.
(3) For the model
df
dt
=
αf

α
10
fg
dg
dt
= 10
βfg

βg,
please
(a) Write down the state vector.
(1)
x
=
f
g
.
1
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(b) Identify the coefficients and describe briefly their biological roles.
α
is
f
’s intrinsic growth rate.
β
is
g
’s intrinsic death rate.
The interaction terms’ coefficients are given in terms of the above two.
(c) Point out sources of nonlinearities.
In
f
’s evolution equation, it is the 2nd term,

αfg/
10
, and in
g
’s evolution equa
tion, it is the 1st term,
10
βfg/
10
, because both contain mutually multiplied state
variables.
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 Spring '10
 MBELK
 Math, Linear Algebra, Trigraph, Fundamental physics concepts, Orthogonal matrix

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