 # Dynamic Equilibrium

Using a water tank to model the available employee pool is one example of a dynamic equilibrium, which occurs in many different areas in science and technology. Consequently, it is a cross-cutting concept exemplifying one of the three pillars of the Next Generation Science Standards.

Mathematically, we can express a dynamic equilibrium simply by equating the incoming and outgoing flows:

Finput = Foutput

In a real water tank, the pressure driving the outgoing flow is proportional to the water height, and the conductance is proportional to the cross-sectional area of the outgoing pipe, so the outgoing flow is proportional to the product.

Foutput = height × conductance

But since Finput = Foutput, we find

Finput = height × conductance

So the water height is given by

Height = Finput/Conductance

For our employee pool example, height corresponds to the number of employees and conductance corresponds to attrition rate, so

Finput = Fhomegrown + Fout-of-area = Final number of employees × attrition rate

so the final employee pool reaches the following value:

Final number of employees = (Fhomegrown + Fout-of-area)/attrition rate

In this simple model, which is not a perfect representation of reality, the incoming flows are independent of the currently available pool. The outgoing flow is an attrition rate proportional to the currently available pool times a factor that represents the undesirability of living in the area, caused by either the jobs themselves or a lack of educational, cultural, and natural assets in the area.

The solution to these types of problems is often an exponential decay from some starting condition to the equilibrium (final) value. In this case, the number of people in the employee pool, N, as a function of time is

N(time) = Ninitial + ((Fhomegrown + Fout-of-area)/attrition rate − Ninitial) × (1-exp(-attrition rate × time))

Perhaps surprisingly, the final number of employees is independent of the starting number of employees. This is because the value of exp(-attrition rate × time) goes to zero at long time, so the Ninitial terms cancel each other. Here are two examples of the way the employee pool approaches an equilibrium value in this simplified model: 