This particular system we've been experimenting with returns a
steady state because it contains a negative feedback mechanism
in the connection between the drain flow rate and the amount of water
in the tub. A negative feedback mechanism is a controlling mechanism,
one that tends to counteract some kind of initial imbalance or
perturbation. A good example of another negative feedback mechanism
is a simple thermostat in a home that responds to changes from the
steady state, returning the home to a specified temperature. Note
that the word negative, as used here, does not mean that this is bad
feedback; it just means that this feedback mechanism acts to reverse
the change that set the feedback mechanism into operation. So if we
first increase the amount of water in the tub, knocking the
system out of its steady state, the drain responds by increasing its
flow rate in order to decrease the amount of water in the tub.
If we instead decrease the initial amount in the tub, the
negative feedback forces the amount of water in the tub to
increase until the steady state is returned. The important
thing to remember here is that negative feedback mechanisms tend to
have stabilizing effects on systems.
In contrast, a positive feedback mechanism is one that
exacerbates some initial change from the steady state, leading to a
runaway condition - it acts to promote an enhancement of the initial
change. There are a number of ways to modify the simple water tub
system in order to create a positive feedback system, but a very
simple change is shown in Figure 2.12 below.
Figure 2.12. Modified Bathtub
model to generate a positive feedback mechanism. The drain flow in
this model is defined just like the faucet flow in the previous
models; the faucet here is just like the drain in previous
models.
The link between the water tub and the drain flow rate has been
removed and the drain flow rate is set to be a constant value of 1
liter/second, with the added condition that if there is no water in
the tub, the drain flow rate is equal to zero (otherwise we would end
up with negative water in the tub, which would be difficult to
explain). This can most easily be done by double-clicking on the
water tub reservoir and clicking in the Non-negative box. The faucet
flow rate has been changed so that it is now a function of the amount
of water in the tub - the more water in the tub, the greater the
faucet flow rate and vice versa. It is essentially just like the
drain used to be defined, as being equal to a rate constant, k, times
the amount of water in the tub, W; as before, we'll set k=0.1. First,
try to predict what will happen when this system is started with 10
liters in the water tub. You may be getting pretty good at this by
now, but it may help to think about what the faucet flow rate will be
at the very beginning of the simulation. We defined the faucet as 0.1
times the amount in the tub, which is initially 10, so the faucet
flow rate is 1 liters/second at the start. How does that compare with
the drain flow rate? They are the same; the inflow matches the
outflow, and the system is in a steady state.
Now, what if we increase the initial value of the water tub,
starting with 15 liters instead of 10? The details of this are a
little bit tougher to predict, but you might be able to see that at
the very beginning, the faucet flow rate will be greater than the
drain, so the amount of water in the tub should increase. What will
this do to the difference between the faucet and flow in the next
time increment? Figure 2.13 shows the results of first increasing and
then decreasing the starting amount of water in the tub.
Figure 2.13. With a positive
feedback mechanism installed, the system exhibits runaway behavior
and does not seek a new steady state.
Both of these changes lead to what we might call runaway behavior,
where the rate of change increases over time. Positive feedback
mechanisms, like negative feedback mechanism are not necessarily good
or bad. Epidemics and infections have positive feedback mechanisms
associated with them, but so does the growth of money in a bank
account with compounded interest. The Earth contains a wide variety
of both positive feedbacks and negative feedbacks and depending on
the conditions, wither kind of feedback may dominate. But, and this
is very important, the mere fact that we exist, the fact that our
planet has water and an atmosphere is compelling evidence to suggest
that ultimately, our Earth system is dominated by negative feedback
mechanisms. In addition, it is equally important to realize that
human time scales are much shorter than the history of the Earth and
over periods of time that interest humans, positive feedback
mechanisms may be very important; they have the potential to produce
dramatic changes.
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