15 A primer on dynamic modelling

Dynamic modelling is about prediction over time, for which there are two approaches,

• Trend projections from past observations
• Prediction based on “rules for change” (how variables interact)

The dynamic modelling in EwE is based rules for change of two types,

• Tautologies that organize components of change, e.g., Next population = population new + births – deaths
• Functional relationships that propose testable hypotheses about how the components vary, e.g., Births = constant x population size.

The rules for change include two types of parameters,

• “State variables” that change over time, e.g, N_t the number in a population at time t.
• “Parameters” that are assumed constant over time, e.g., a birth rate.

There are two basic ways to represent rules for change,

• Difference equations, e.g., [latex]N_{t+1} = N_t + bN_t - mN_t \tag{1}[/latex]where b = birth rate and m = death rate
• Differential equations, e.g., [latex]dN/dt = bN_t - mN_t \tag{2}[/latex]

For complex continuous time models, such differential equations can be solved by simple numerical stepping approximations. For small time steps, we may use the one-step “Euler” method,

[latex]N_{t+dt}=N_t + X \cdot dt \tag{3}[/latex]

where X = dN/dt at time t. The Euler method thus predicts change using only the rate at time t.  It can give very poor results for rate equations where there are strong positive feedbacks, i.e. when X can increase rapidly as N increases.

For implementing dynamic model calculations in spreadsheets, it is better though to use the more precise, two-step “Adams-Bashforth” method,

[latex]N_{t+dt}=N_t+ [3X-Y]/2 \cdot dt \tag{4}[/latex]

where X is as in Eq. 3 and Y = dN/dt for the previous time step, time t-dt.  The Adams-Bashforth is used for the tutorial on predator-prey models (included in the web and pdf-versions of this book).

For numerical integration in computer languages like R and VB.Net, it is better though to use 4^th order Runge-Kutta integration, especially if the model includes state variables that change at very different rates (both “fast” and “slow” dynamic change).  The dynamic rate equations in the EwE software are solved using this more accurate integration method, which approximates the curve between two points by at 4^th degree polynomial.