## Feedback Mechanisms

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.