Appendix C: Quick Reference Sheets

The 8-Step Modeling Framework

Step	Action	Behavior-Science Translation
1	Get the physical picture clearly in mind	Understand the behavioral phenomenon thoroughly before formalizing
2	Define the physical processes and boundaries	Specify which behavioral processes are in scope and which are excluded
3	Write down the laws and transport functions	Identify the quantitative laws and functional relationships to use
4	State the restrictive assumptions	List every simplifying assumption explicitly
5	Perform the balance in words, then symbols	Write the model verbally first, then translate to equations
6	Check units	Verify that both sides of every equation have consistent dimensions
7	Write down initial and boundary conditions	Specify starting values, valid ranges, and constraints
8	Verify, validate, and solve	Check the math, test against data, and derive predictions

Distribution	Probability Function	Mean	Variance	When to Use
Bernoulli	$P(X=1) = p$	$p$	$p(1-p)$	Single binary trial (response / no response)
Binomial	$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$	$np$	$np(1-p)$	Count of successes in $n$ independent trials
Poisson	$P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}$	$\lambda$	$\lambda$	Count of events in a fixed time window at constant rate
Exponential	$f(t) = \lambda e^{-\lambda t}$	$1/\lambda$	$1/\lambda^2$	Time between successive events (inter-response times)
Normal	$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2 / 2\sigma^2}$	$\mu$	$\sigma^2$	Continuous measurement with symmetric variability

Concept	Definition	Behavioral Example
State variable	A quantity defining the system's current state	Response rate, associative strength
Phase space	The set of all possible states	All possible combinations of two competing response rates
Trajectory	The path a system follows through phase space	The acquisition curve from initial to steady-state responding
Equilibrium	A state where the system does not change	Steady-state response rate on a VI schedule
Stable equilibrium	System returns to this state after perturbation	Response rate recovers after a brief disruption
Unstable equilibrium	Any perturbation drives the system away	Zero responding: once a single response is reinforced, rate grows
Attractor	A state or set of states the system tends toward	The matching-law equilibrium in concurrent schedules
Limit cycle	A stable oscillation the system settles into	Cyclic patterns in adjunctive behavior
Bifurcation	A qualitative change in dynamics when a parameter crosses a threshold	Transition from stable responding to extinction when reinforcement is removed