1. The mathematics behind STELLA Global population model reservoir converter flow connector This system represents a simple differential equation this is essentially an integral Here, P is Population

2. We can solve this the old fashioned way with calculus to get an equation that will tell us P(t) — Population as a function of time

3. So if we are given a rate, r, and an initial value for P —P 0 —, this tells us the value of P at any time You could use this to figure out the doubling time for the population:

4. The mathematics behind STELLA Global population model dPdt is the change in population over time r is the annual growth rate (birth rate- death rate) Now, let’s look at what STELLA does The initial value for Population is 6.7e9 Population(t) = Population(t - dt) + (dPdt) * dt INIT Population = 6.7e9 INFLOWS: dPdt = Population*r r =.0121 {constant annual growth rate = birth rate - death rate} This is a finite difference equation

5. Population(t) = Population(t - dt) + (dPdt) * dt INIT Population = 6.7e9 INFLOWS: dPdt = Population*r r =.0121 {constant annual growth rate = birth rate - death rate} Population at each time in the model simulation Population at previous time Annual rate of change in population The time step of the calculation STELLA uses this finite difference equation to calculate the population change and the size of the population at each time step for the duration of the simulation.

6. dt=0.25 dt=10

7. think of the blue curve as the true solution dt y time Euler’s method consists of projecting a tangent line forward in time in small steps of dt, but if dt is too large, the numerical solution strays from the true solution

8. here, red is the true solution Euler’s method with dt=1 A Runge-Kutta method with dt=1 Runge-Kutta methods do a better job by finding the tangent lines at fractions of dt. Euler’s method does in fact give fine results is dt is small enough

9. from Watson and Lovelock, 1983,Tellus 35B, 284-289 Going from mathematics to STELLA

10. To represent both differential equations, we need two flows

11. 11 dt=5.0 Time Step and Numerical Instabilities dt=2.0 dt=0.25

12. This is just a graphical representation of a differential equation. A reservoir is just an integral

13. We then solve the integrals above to give:

16. = 20