Simple positive feedback Example: Bank balance Interest Payments = Bank Balance * Interest Rate Positive feedback = the stock will continue to increase (exponential growth, y = aXe bx ) Initial balance $10, interest rate 10% per year

Simple positive feedback Draw the flow diagram?

Simple positive feedback Positive feedback can not continue forever (there are always limits – food / space /…) “ Limits to Growth” Other positive feedback examples Cell division Spread of rumor Growth of a company Spread of forest fire

Simple positive feedback

Simple negative feedback Negative feedback exhibits goal - seeking behavior. Example: Coffee Cooling (non zero goal) Causal loop diagram (signs) Flow diagram

Simple negative feedback Solve using initial temp = 50 o C ; decline in temp = ? Common sense assumption - cooling stops at room temperature. decline in temp = (temperature - room temperature) / T Where: T is the cooling constant (function of coffee container; surface area; etc.). For room temperature = 20 o C and T = 5 min Decline in temp = (50-20)/5 = 6 o C/min

Simple negative feedback The above diagram has one problem: much of the detail involved in the rate (decline in temperature) equation is hidden in single arrow. A better representation of this is to define a new variable named ‘difference’. This is called an auxiliary variable (very useful in formulating complex rate equations).

Simple negative feedback

A conserved quantity has the property that it is never created or destroyed (within its system); it is only moved around.

S-shaped growth Known as “logistic” or “sigmoid” growth. It is a common type of behavior which combines both exponential (positive) and asymptotic (negative) growth. Population trends of various plant and animals typify S-shaped patterns.

S-shaped growth

The goal persists indefinitely and is called ‘stable equilibrium’. Prior to the growth (at level value of zero) the system rests at ‘unstable equilibrium’. The slightest disturbance produces a rate value other than zero (the system level increases).

S-shaped growth Draw a flow diagram for the problem in Assignment 1.

Transferability of structures Up to now you might have noticed some similarities amongst models. For example the Bathtub and Population models. This phenomenon occurs all the time in system dynamics and it is known as “ Transferability of Structures”. Structures common to several systems are called ‘generic structures’.

First order positive feedback Generic structure

First order positive feedback Generic equations stock(t) = stock(t-dt) + flow * dt [units] flow = stock * compounding_factor [units/time] compounding_factor = a constant [units/unit/time] OR flow = stock / time_constant [units/time] time_constant = a constant [time]

First order positive feedback Generic model behavior

First order positive feedback

First order negative feedback Generic structure

First order negative feedback Generic equations stock(t) = stock(t-dt) + (-flow) * dt [units] flow = adjustment_time * draining_fraction [units/time] adjustment_gap = stock - goal_for_stock [units] draining_fraction = a constant [1/time] goal_for_stock = a constant [units] OR flow = adjustment_gap / time_constant [units/time] time_constant = a constant [time]

First order negative feedback Generic behavior

First order negative feedback