A most general characteristic equation can be deduced from:
where x and y are the deviations from
equilibrium values and A,B,C,D the coefficents of the linear
expansion of the dynamical system.
The characteristic equation in 
is then:
and the discriminant:
Oscillations occur when the discriminant is negative. A necessary condition is that BC<0. An interpretation of the fact that couplings should be larger than diagonal terms is that the system should be "truly" 2-dimensional. Otherwise the "fast" variable simply follows the "slow" variable decay towards equilibrium.
subsection Analysis of Variable Prices
Here we present the analysis for the case described in Section 4, of prices with a lower bound.
Suppose that
prices time 
are given by equation 39 with 
, so that prices
decrease from a maximum of 
toward a minimum of 
. In order to
make comparisons with the results of the other sections, a reasonable assumption is that the minimum price is
lower than under the constant price regime of section 2, i.e. that
, where 
is the price of section 2. Using equation 39
implies that there is no limit on the income from selling
the harvest, as there is when a=0. However,as long as a is small
compared to 1, we expect that the results given for the a=0 case
will essentially hold: namely that a nonzero equilibrium only exists
when b is above a minimum value near 
,
and that (when 
) the
equilibrium resource level is larger,
the equilibrium is more strongly attracting,
and oscillations are less likely than in the constant price case. However,
as a moves away from zero and price supports become stronger, we would
like to know how this affects the a=0 results.
When a > 0 the nonzero equilibrium of the nondimensionalized system
with 
, 
(
, 
given by equation 5)
is 
, 
, where
and we define 
,
, and
.
As we found in section 4, adding prices does not change the equilibrium
capital but significantly affects the equilibrium resource.
However, unlike the a=0 case, N-p is positive for all positive
values of b. What is different now is that if b<1 then
as 
while if b>1 then 
approaches the a=0
value as 
.
Letting
and linearizing about the nontrivial equilibrium, we find that the eigenvalues of this system are:
These eigenvalues always have negative real part. This can be seen
via the following argument. Note that in the remaining discussion of this section,
we drop the tildes.
Since 
, these eigenvalues both have negative real part when
. Using the relationship between a, b, c and 
to
replace each 
, this
condition on 
can be rewritten as
or
and applying the equality again shows that this is always true. Thus this is always a stable equilibrium point. It is a stable spiral point when
In the limit as 
, these results approach those found in section 3.
We expect that increasing a will have the effect of annuling the impact of
decreasing prices, and, in fact, increasing
the lower bound on prices, 
,
decreases the equilibrium level of the resource, 
.
However, the dependence of the eigenvalues
on a is much more subtle, and depends on the values of b and c.
For very small a, the upper bound on 
for oscillations
decreases as a increases (when 
satisfies the constraint
), implying a stabilizing effect.
When the parameters are such that 
satisfies this
inequality and the value
under the square root is negative, then the real part of 
increases in magnitude with a
when 1 < b < 3, and otherwise decreases.
These results are found by the following calculations:
which is negative for all a>0.
and
