Lecture 5

Bsc 417/517

Outline

• More pointers on models in STELLA• Behavior patterns

• Linear models• Exponential models

For building/manipulating models in STELLA

• Keep the model as simple as possible – Add complexity where needed (Occam’s Razor)

• Make sure you understand the mathematical relationships between elements/variables– Use common sense relationships

• If a credible mathematical relationship isn’t available, define the relationship using a graph– MAKE SURE you document this graph

• Make sure all units are in sync and compatible• Define time units and match these up• Ensure that the only entities that affect the reservoir

‘level’ are those inflows and outflows associated with that reservoir

Five common “behavior patterns”

• Linear growth or decay• Exponential growth or decay• Logistic growth• “Overshoot and collapse”• Oscillation

• Page 31 in the text• These are the ‘modular units’ that are the

mathematical building blocks of our models

Five common “behavior patterns”

• Linear growth or decay• Exponential growth or decay• Logistic growth• “Overshoot and collapse”• Oscillation

• Page 31 in the text• These are the ‘modular units’ that are the

mathematical building blocks of our models

For each behavior pattern:

• Be able to describe the relationship in words• Be able to give examples• Know the rate equation• Know the solution to the rate equation• Be able to graph it• Know steady state solution, if any• Summary tables in text are excellent for this

Linear growth or decay

• Fixed rate of growth or consumption• Back account balance at time t

= bank account balance at time t0 + wages dt

No FEEDBACK here, either counteracting (negative) or reinforcing (positive, amplifying)

Derivation of linear model

Exponential model• Exponential growth or decay is common in many

types of systems• Great for modeling feedback applications• Populations without predation• Microbes• Not really that useful or common over large time

spans• Exponential growth exists if and ONLY IF the rate

of growth or decay is proportional to the size of the reservoir

Generic exponential model

IF birth rate > death rate: ?

IF birth rate < death rate: ?

Exponential decay

Derivation of exponential model

How to interpret k?

• In growth: larger the k, the more rapid the growth

• In decay: larger the k, the more rapid the decay

• K = inflow rate – outflow rate

Steady state behavior• A system is in “steady state” as the change in reservoir

levels approaches (is) zero• Most environmental systems operate near steady state• Just the right mix of positive and negative feedback – no

explosions of growth or decay over the long term• We’re interested in modeling changes to natural systems

– perturbations – that may change the delicate balance• Interchangeable with ‘stability’• Functional definition = when the graph of reservoir levels

over time remains flat (constant)• dR/dt = 0 • Remember that anything with dR/dt is called a rate

equation

Steady state and linear/exponential models

• Linear models do not exhibit steady state behavior if k ≠ 0

• In exponential systems, steady state is achieved only as t ∞ and R 0 and k < 0– That is, in exponential decay, after a “long” time– Remember, time frames are relative to the

process

For next time

• Ch2, questions 1-3, 5-8, 11• Read up to page 43

• EXTRA: association and causation in systems models

Associations and causation

• Models are often used to identify and/or posit associations between variables in a model