Model Structures and Behavior Patterns
• The systems modeler believes that the behavior of the
system is a function of the system itself
• Translating that idea to the model realm, this means
that certain structures of elements should produce
certain types of behavior patterns
• We are going to look at five common behavior
patterns and their associated structures:
Linear
Growth
or Decay
Exponential
Growth
or Decay
Logistic
Growth
Overshoot
and
Collapse
Oscillation
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Linear Growth or Decay - Review
• When does linear growth or decay occur?
– The difference between all inflows and outflows is a constant
at all times
• What is the rate equation for linear growth or decay?
=
dR(t)
dt
n
=
n
Outflow
Inflow Σ
Σ
i=1
j=1
i
j
= constant = k
• What is the solution for the rate equation for linear
growth or decay?
– R(t) = R(0) + kt, growth when k > 0 and decay when k < 0
• Does a simple linear system contain any feedback?
– No, there are no closed loops
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Exponential Growth or Decay - Review
• When does exponential growth or decay occur?
– IFF the rate constant is a constant proportion of the size of
the reservoir
• What is the rate equation for exp. growth or decay?
=
dR(t)
dt
= kR(t), where k = Inflow Rate – Outflow Rate
• What is the solution for the rate equation for
exponential growth or decay?
– R(t) = R(0)ekt, growth when k > 0 and decay when k < 0
• Does an exponential growth/decay system contain any
feedback?
– Yes, there are closed loops in an exponential system
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
• Consider the following example of a system:
– A businessman in the far North decides there is a large,
untapped market for free range moose meat, and buys and
fences off a large tract of land for raising the moose
– The tract of land is capable of supporting a population of
100,000 moose based on the size of the tract
– A mating pair of moose are introduced
– How will the moose population progress through time?
• Once again, we will try to determine the structure of the
system in a series of steps
Step 1: Identify the reservoir(s)
– Since we are tracking the number of moose:
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
Step 2: Identify the process(es) that will change the
contents of the reservoir(s) over time:
– We obviously need to have moose being born, and moose
dying:
Step 3: Identify the converter(s) that determine the
rates of inflow and outflow:
– Here is where things differ from an exponential model
– We will need rates to regulate the Birth and Death processes,
but we also include a Carrying Capacity:
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
Step 4: Define relationships between system
elements with connectors:
– Once the cause and effect relationships in this system are
drawn in with the connectors, the structural difference
from an exponential system becomes apparent:
– In particular, Death is defined differently, with Carrying
Capacity also regulating this outflow process
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
• What is the difference equation for this system?
M(t+∆t) = M(t) + ({Birth - Death} * ∆t)
• We can find the expressions for the Birth and Death
processes, taking Carrying Capacity into account:
Birth = Birth Rate * M(t) = 1 * M(t) = M(t)
Death = Death Rate * M(t) * [M(t) / Carrying Capacity ]
• You can see that as M(t) Æ Carrying Capacity the
Death Rate is going to become quite large,
regulating the moose population’s growth, because
[M(t) / Carrying Capacity ] > 1
• However, early on [M(t) / Carrying Capacity ] will be
small, and the system will show exponential growth
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
• Using the following values for the converters:
– Birth Rate = 1 birth/capita/year
– Death Rate = 0.1 deaths/capita/year
– Carrying Capacity = 100,000
we can run the model and see the following output:
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
• We can interpret the shape of the curve as follows:
1
2
1. Initially, the logistic
system behaves like an
exponential system
2. Later, the growth levels
out to approach a steady
state value
• The logistic growth curve is often referred to as an
sigmoid curve or s-curve
• Logistic growth systems have closed loops that
provide reinforcing and counteracting feedback in
varying amounts at different times in the simulation
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth - Example
• QUESTION: Why does the growth in a logistic
system initially behave like an exponential growth
system?
• Using values from our example, initially
Moose at time (t)
is a very small value
Carrying Capacity
=
2
100,000
to begin with
thus the Outflow Rate is close to zero
• Logistic growth occurs whenever an exponential
system is constrained so that the reservoir achieves a
maximum level that is sustainable by the system (e.g.
limited by a Carrying Capacity)
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth – System Features,
Diagrams, and Equations
• A system that has logistic behavior generally takes the
following form:
• It is not strictly necessary to have a rate controlling
each of the processes (as in the moose example), all that
is required is that some sort of Carrying Capacity be
used to define the behavior of the Outflow process in
terms of a proportion on the size of the Reservoir
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth – System Features,
Diagrams, and Equations
• For our generic example, the difference equation is:
R(t+∆t) = R(t) + [Inflows – Outflows] * ∆t
• Substituting in the expressions for the processes in a
logistic system (symbolizing the Growth Rate with G
and Carrying Capacity with C):
R(t+∆t) = R(t) + [{G * R(t)} – {G * R(t) *
R(t)
} * ∆t
C
R(t)
} * R(t) * ∆t
= R(t) + G{ 1 –
C
• Subtracting R(t) from both sides, and dividing by ∆t:
R(t+∆t) - R(t)
∆t
= G{ 1 –
R(t)
} * R(t)
C
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth – System Features,
Diagrams, and Equations
• We can now take the derivative with respect to time:
lim
∆t Æ 0
R(t+∆t) - R(t)
∆t
=
dR(t)
dt
= k(t)R(t)
• The difference between logistic and exponential
behavior is that the growth rate is no longer a constant
proportion of the size the reservoir, but changes with
time:
R(t)
k = G{ 1 –
R(t)
C
}
• When R(t) << C, C Æ 0 and k Æ G (a constant),
therefore exponential growth
R(t)
• However, when R(t) Æ C, C Æ 1 and k Æ 0,
meaning the system is approaching a steady state
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Logistic Growth – System Features,
Diagrams, and Equations
• The solution to the rate equation is:
C
R(t) =
1 + Ae-Gt
C – R0
where A =
R0
•The shape of the curve
is governed by the sizes
of G and C
•Larger values of G
reach a steady state
more rapidly than
smaller values of G
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot & Collapse – System
Features, Diagrams, and Equations
• A characteristic of the logistic growth system is
that the system settles around carrying capacity
• This in turn implicitly assumes that the supply
of resources that are required to maintain that
carrying capacity are renewable, and will
always be present in sufficient quantity to
support a particular population
• If we wish to simulate a system where the
resources are non-renewable, we can see a very
different behavior
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot & Collapse – System
Features, Diagrams, and Equations
– The population can grow very
rapidly at first, but once there is
a scarcity of resources, it can
collapse and the population will
die out
– As the population increases, the
resources are consumed faster
and faster until they dip to a
dangerously low level …
• Between the plot shown above, and the previous
discussion of the importance of resource abundance in
this sort of model, it should be obvious to model this
type of behavior, we necessarily need to keep track of
more than one kind of thing, thus we need more than
one reservoir for such a model
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot & Collapse – System
Features, Diagrams, and Equations
•2 reservoirs mean 2 rate
equations, interlinked
•The Resource reservoir
has only an outflow
process
•Consumption of Resource
is a function of the size of
the Population (C)
•The size of Resource affects the Death Rate inversely, according
to the expression Death Rate = 1 - R(t)
(although other expressions
R0
can be used here, as long as Resource is non-renewable
•There are three feedback loops in this system: A reinforcing loop
between Birth and Population, a counteractive loop between
Death and Population, and a large counteractive loop
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot & Collapse – System
Features, Diagrams, and Equations
• Overshoot and collapse behavior occurs whenever one
reservoir depends on another non-renewable
reservoir for survival. As the population increases in
size and overconsumes the resource, the resource
becomes depleted to the point where the population
cannot survive, leading to eventual collapse
• As we have done with previous behaviors, we can
understand the behavior of this system in terms of its
rate equations, but since we have two reservoirs this is
a little more complex …
• We need to describe the behavior using a coupled pair
of rate equations (i.e. each equation contains terms for
both reservoirs)
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot and Collapse – System
Features, Diagrams, and Equations
• For our generic example, the difference equation is:
R(t+∆t) = R(t) + [Inflows – Outflows] * ∆t
• Substituting in the expressions for the processes in a
logistic system for each of the two reservoirs:
P(t + ∆t) = P(t) + {B – [1 -
R(t)
R0
]} * P(t) * ∆t
R(t + ∆t) = R(t) - P(t) * C * ∆t
• Performing the usual reshuffling of R(t) and ∆t, and
taking the derivative of the resulting rate equations:
dP(t)
R(t)
=
{B
–
[1
]} * P(t)
R0
dt
dR(t)
dt = -C * P(t)
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot and Collapse – System
Features, Diagrams, and Equations
dP(t)
•From dt = {B – [1 -
R(t)
R0
]} * P(t) , we can see that the
rate at which Population changes is proportional to the
current size of the population, but the constant in this
]} changes with time
equation {B – [1 - R(t)
R0
•Initially, R(t) ~ R0, so the proportionality constant
will be close to B, and the system behaves like an
exponential system (i.e. P = P0eBt)
•As time passes, the Resource is consumed [R(t) begins
to shrink], and the population growth rate changes in a
manner similar to a logistic system
•When the Resource is exhausted [R(t) = 0], the equation
reduces to {B – 1} * P(t), thus the collapse
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot & Collapse – System
Features, Diagrams, and Equations
• The decline in the Population begins as soon as the
Birth Rate < 1 - R(t)
R0 , because from that point onwards,
the proportionality constant is negative and the
Population decreases
• For this sort of system to reach steady state, the rate of
change of BOTH reservoirs has to be near zero
•We can find when this condition occurs by setting the
left side of the instantaneous rate of change equations to
zero:
0 = {B – [1 - R(t) ]} * P(t)
R0
0 = -C * P(t)
•Either P(t) = 0 or the proportionality constant AND C = 0
will produce the right conditions
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005
Overshoot & Collapse – System
Features, Diagrams, and Equations
•However, setting C to zero is not realistic, because it
indicates that the Population consumes none of the
Resource
•The second option, where P(t) =0 corresponds to the
case where the Population has totally collapsed, which
happens only once the Resource is totally consumed,
and this only occurs asymptotically as we go further out
in time
•So, as was the case with the exponential decay system,
the overshoot and collapse system only truly reaches
steady state as t Æ infinity, but reaches an effective
steady state when P = 0 and R = 0
David Tenenbaum – GEOG 110 – UNC-CH Fall 2005