2- Introduction to Modeling.pdf

INTRODUCTION TO MODELING
MATH 3800
MATHEMATICAL MODELS AND NUMERICAL METHODS
© 2020 Gary Bazdell (excluding images)

WHAT IS A MATHEMATICAL MODEL?
1. Models are abstractions of reality!
2. Models are a representation of a particular thing, idea, or condition.
3. Mathematical Models are simplified representations of some real-world
entity
o Can be in equations or computer code
o Are intended to mimic essential features while leaving out
inessentials.
4. Mathematical Models are characterized by assumptions about:
o Variables (the things which change)
o Parameters (the things which do not change)
o Functional forms (the relationship between the two)
2
Introduction to Modeling - Introduction

MODELING PROCESS
3
Real World Data Model
Mathematical Conclusions
Predictions / Explanations
Simplifications
Interpretation
Reasonable?
yes – continue
no – go back to model
Verification – how good? Crunch numbers Solve equations
Introduction to Modeling - Introduction

MODELING CHANGE
Recall that the change can be modeled using the formula
change = future value − present value.
Change can take place over discrete or continuous time.
1. If the change occurs over discrete time periods (we know exactly what
comes next), then we get difference equations and we will actually be
observing the change.
2. If the change occurs continuously, then typically we will be observing the
rate of change and we will get a differential equation.
4
Introduction to Modeling - Modeling Change

FIRST DIFFERENCE EQUATIONS
DEFINITION
For a sequence of numbers 𝐴 = 𝑎0, 𝑎1, 𝑎2, … the first differences are
∆𝑎0 = 𝑎1 − 𝑎0
∆𝑎1 = 𝑎2 − 𝑎1
⋮
∆𝑎𝑛 = 𝑎𝑛+1 − 𝑎𝑛
Note: the first difference represents the rise or fall between
consecutive values of the sequence.
5

EXERCISE 1 – DIFFERENCE EQUATIONS
Write a difference equation to represent the change during the 𝑛th
interval as a function of the previous term in the following sequences:
1. 1, 3, 7, 15, 31, …
2. 1, 2, 5, 11, 23, …
6

DYNAMICAL SYSTEMS
Every difference equation
∆𝑎𝑛 = 𝑎𝑛+1 − 𝑎𝑛
determines a dynamical systems (relationship between terms of a sequence)
by solving for 𝑎𝑛+1 for a given initial value 𝑎0:
𝑎𝑛+1 = 𝑎𝑛 + ∆𝑎𝑛
A numerical solution is a table of values, 𝑎𝑗 for all 𝑗, satisfying the dynamical
system.
7

EXERCISE 2 – SIMPLE MODEL
Formulate a dynamical system that
models the following situation:
You owe $1000 on a credit card after
buying Christmas gifts for your family
that charges 1.5% interest each
month. You pay $50 each month and
you make no new charges.
8

SIMPLIFICATION OF OUR MODEL
One very powerful simplifying relationship is proportionality.
DEFINITION
Two variables 𝑦 and 𝑥 are proportional (to each other) if one is always
a constant multiple of the other. That is, if
𝑦 = 𝑘𝑥
for some nonzero constant 𝑘. We write 𝑦 ∝ 𝑥.
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Introduction to Modeling - Approximation of Change

EXERCISE 3 – POPULATION GROWTH
The following data was collected
from an experiment measuring
the growth of a yeast culture.
Formulate a dynamical system
that models the growth of the
yeast culture.
10
Time in hours 𝒏 Observed yeast biomass 𝒑𝒏
0 9.6
1 18.3
2 29.0
3 47.2
4 71.1
5 119.1
6 174.6
7 257.3
To simplify the model, we assume that the change in the population grows
proportionally with respect to the current size of the population.
∆𝑝𝑛 = 𝑘𝑝𝑛 for some growth constant 𝑘

SUMMARY CHART
11
We see that our model is not a great fit for the data. This may be due to the
limited amount of original data
0
50
100
150
200
250
300
0 1 2 3 4 5 6 7 8
Biomass
Time in hours
Comparison Chart - Original data vs Projected Data
Original Data Projected Data

ISSUES WITH CONSTANT GROWTH RATE
If we assume the change grows proportionally with the current
population size,
∆𝑎𝑛 = 𝑘𝑎𝑛
then 𝑎𝑛+1 would grow without bound.
This may, or may not, be a good simplification. To relax this
simplification, we could assume the population has a limiting value, say
𝑀. Then we could define the growth rate 𝑘 by a limiting factor.
𝑘 = 𝑟 𝑀 − 𝑎𝑛 → ∆𝑎𝑛 = 𝑟 𝑀 − 𝑎𝑛 𝑎𝑛
12

EXERCISE 4 – CARRYING CAPACITY
The following data was
collected from an
experiment measuring the
growth of a yeast culture.
Formulate a dynamical
system that models the
growth of the yeast culture.
13
Time in
hours 𝒏
Observed yeast
biomass 𝒑𝒏
Time in
hours 𝒏
Observed yeast
biomass 𝒑𝒏
0 9.6 10 513.3
1 18.3 11 559.7
2 29.0 12 594.8
3 47.2 13 629.4
4 71.1 14 640.8
5 119.1 15 651.1
6 174.6 16 655.9
7 257.3 17 659.6
8 350.7 18 661.8
9 441.0
The population appears to
be approaching a limiting
value, or carrying capacity

SUMMARY CHART
14
With the addition of the limiting value, we see that our model is a better fit to
the original data Introduction to Modeling - Approximation of Change
0
100
200
300
400
500
600
700
0 2 4 6 8 10 12 14 16 18 20
Biomass
Time in hours
Comparison Chart - Original data vs Projected Data
Original data Projected Data

SOLUTIONS TO DYNAMICAL SYSTEMS
When we develop a mathematical model, we would like to be able to
find the solution; we would like an easy way to express
𝑎𝑛+1.
An obvious method for finding the solution of a simple model is to
hypothesize the form of a solution and then test if it is correct. This
method is known as the Method of Conjecture.
15
Introduction to Modeling - Analytical Solutions

THE METHOD OF CONJECTURE
The following summarize the procedure for applying this method.
1. Observe a pattern.
2. Conjecture a form of the solution to the dynamical system.
3. Test the conjecture by substitution.
4. Accept or reject the conjecture depending on whether it does or
does not satisfy the system after the substitution and algebraic
manipulation.
16

LINEAR DYNAMICAL SYSTEMS
What if ∆𝑎𝑛 = 𝑐𝑎𝑛, where 𝑐 is some nonzero constant?
THEOREM
The solution of the linear dynamical system 𝑎𝑛+1 = 𝑟𝑎𝑛 for 𝑟 any
nonzero constant is
𝑎𝑘 = 𝑟𝑘
𝑎0
where 𝑎0 is a given initial value.
17

EXERCISE 5 – SOLUTIONS
Find the solution to the difference equations in the following problems:
1. 𝑎𝑛+1 = 5𝑎𝑛, 𝑎0 = 10
2. 𝑎𝑛+1 =
3𝑎𝑛
4
, 𝑎0 = 64
18

EQUILIBRIUM VALUE
DEFINITION
A number 𝑎 is called an equilibrium value or fixed point of a dynamical
system
𝑎𝑛+1 = 𝑓 𝑎𝑛
if 𝑓(𝑎𝑛) = 𝑎 for all 𝑛 ≥ 𝑁.
If 𝑓 𝑎𝑛 = 𝑎 for all 𝑛 ≥ 0, 𝑎 is a constant solution to the dynamical
system.
19

LONG-TERM BEHAVIOR
Let’s consider some significant values of 𝑟.
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𝑟 = 0 Constant solution and equilibrium value at 0
𝑟 = 1 All initial values are constant solutions and equilibrium
values.
𝑟 < 0 Oscillation
𝑟 < 1 Decay to limiting value of 0
𝑟 > 1 Growth without bound

What if ∆𝑎𝑛 = 𝑐𝑎𝑛 + 𝑏, where 𝑐 is some nonzero constant?
THEOREM
The equilibrium value for the dynamical system
𝑎𝑛+1 = 𝑟𝑎𝑛 + 𝑏, 𝑟 ≠ 1
is
𝑎 =
𝑏
1 − 𝑟
If 𝑟 = 1 and 𝑏 = 0, every number is an equilibrium value. If 𝑟 = 1 and
𝑏 ≠ 0, no equilibrium value exists.
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TYPES OF EQUILIBRIUMS
There are two kinds of equilibrium, called stable and unstable.
1. If the equilibrium is stable, then 𝑎𝑛 → 𝑎 as 𝑛 → ∞ regardless of 𝑎0.
2. If the equilibrium is unstable, then the equilibrium is achieved only
if 𝑎0 = 𝑎.
For the dynamical system 𝑎𝑛+1 = 𝑟𝑎𝑛 + 𝑏,
1. If 𝑟 < 1, the equilibrium is stable.
2. If 𝑟 > 1, the equilibrium is unstable.
22

EXERCISE 6 - EQUILIBRIUMS
Suppose we give a patient a daily dosage of 0.2mg of a drug at the end
of the day. 40% of the drug remains in the patient’s system.
1. How much drug is in the patient’s system at time 𝑛 + 1?
2. What is the equilibrium value?
3. Examine what happens for 𝑎0 = 0.25, 𝑎0 = 0.30, and 𝑎0 = 0.35.
23

THEOREM
The solution of the dynamical system
𝑎𝑛+1 = 𝑟𝑎𝑛 + 𝑏, 𝑟 ≠ 1
is
𝑎𝑘 = 𝑟𝑘
𝑐 +
𝑏
1 − 𝑟
for some constant 𝑐 (which depends on the initial condition).
24

SYSTEMS OF DIFFERENCE EQUATIONS
For the systems considered here, we find the equilibrium values and
then explore starting values in their vicinity. If we start close to an
equilibrium value, we want to know whether the system will:
1. Remain close
2. Approach the equilibrium value
3. Not remain close.
Does the long-term behavior described by the numerical solution
appear to be sensitive to the initial condition?
25
Introduction to Modeling - Systems

MODELS FOR INTERACTING POPULATIONS
When species interact the population dynamics of each species is
affected. We consider here two-species systems.
There are three main types of interaction.
1. If the growth rate of one population is decreased and the other
increased the populations are in a predator-prey situation.
2. If the growth rate of each population is decreased then it is
competition.
3. If each population’s growth rate is enhanced then it is called
symbiosis.
26

PREDATOR-PREY MODEL
If 𝑁𝑛 is the prey population and 𝑃𝑛 that of the predator at time 𝑛 then our
model is
∆𝑁𝑛 = 𝑁𝑛 𝑎 − 𝑏𝑃𝑛
∆𝑃𝑛 = 𝑃𝑛 𝑐𝑁𝑛 − 𝑑
where 𝑎, 𝑏, 𝑐 and 𝑑 are positive constants.
1. The prey in the absence of any predation grows unboundedly.
2. The effect of the predation is to reduce the prey’s per capita growth rate
by a term proportional to the prey and predators population.
3. In the absence of any prey for sustenance the predator’s death rate
results in exponential decay.
4. The prey’s contribution to the predator’s growth is proportional to the
available prey as well as to the size of the predator population.
27

EQUILIBRIUM VALUES
The equilibrium values for the system are those values for which no
change in the system takes place.
Let 𝑁 and 𝑃 be those values. Substitution in our model yields the
following requirements for the equilibrium values:
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∆𝑁𝑛 = 0 = 𝑁 𝑎 − 𝑏𝑃
∆𝑃𝑛 = 0 = 𝑃 𝑐𝑁 − 𝑑
𝑁 = 0 or 𝑃 =
𝑎
𝑏
𝑃 = 0 or 𝑁 =
𝑑
𝑐
𝑃, 𝑁 → 0,0 𝑎𝑛𝑑
𝑎
𝑏
,
𝑑
𝑐

EXERCISE 7
Suppose we have Owls and Mice in a park and the Owls feed on the
Mice. Let 𝑀𝑛 and 𝑂𝑛 be the population at time 𝑛 and let’s say
𝑀𝑛+1 = 1.5𝑀𝑛 − 0.005𝑂𝑛𝑀𝑛
𝑂𝑛+1 = 0.5𝑂𝑛 + 0.001𝑂𝑛𝑀𝑛
1. What do these equations represent?
2. What are the equilibrium values?
3. What happens if 𝑀0 = 200 and 𝑂0 = 100?
29

SUMMARY
After the first 5 iterations it appears the mice population is growing while the
owl population increases.
After the 5th iteration, the owl population begins to grow but so does the
mice population.
At the 11th iteration, we see that the mice population has grown large
enough to allow the owl population to make large growth jumps which
allows the owls to take over.
Since the owls no longer have a food source, they begin to die off.
30

COMPETITIVE EXCLUSION
Here two or more species compete for the same limited food source or
in some way inhibit each other’s growth.
Let 𝐴𝑛 and 𝐵𝑛 be the populations of species 𝐴 and 𝐵 respectively at
time 𝑛 and suppose that in the absence of the other species, each
individual species exhibits unconstrained growth. Let 𝑘1 and 𝑘2 be the
positive growth rates.
∆𝐴𝑛 = 𝐴𝑛 𝑘1 − 𝑘3𝐵𝑛
∆𝐵𝑛 = 𝐵𝑛 𝑘2 − 𝑘4𝐴𝑛
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COMPETITIVE EXCLUSION
Solving each equation for the 𝑛 + 1 term gives
𝐴𝑛+1 = 1 + 𝑘1 𝐴𝑛 − 𝑘3𝐵𝑛𝐴𝑛
𝐵𝑛+1 = 1 + 𝑘2 𝐵𝑛 − 𝑘4𝐴𝑛𝐵𝑛
When there is no change we have equilibrium values at
∆𝐴𝑛 = 0 = 𝐴𝑛 𝑘1 − 𝑘3𝐵𝑛
∆𝐵𝑛 = 0 = 𝐵𝑛 𝑘2 − 𝑘4𝐴𝑛
𝐴, 𝐵 = (0,0) and 𝐴, 𝐵 =
𝑘2
𝑘4
,
𝑘1
𝑘3
32

EXERCISE 8
Suppose a species of spotted owls competes for survival in a habitat that also
supports hawks. Let 𝑂𝑛 and 𝐻𝑛 be the population at time 𝑛 and let’s say
𝑂𝑛+1 = 1.2𝑂𝑛 − 0.001𝑂𝑛𝐻𝑛
𝐻𝑛+1 = 1.3𝐻𝑛 − 0.002𝑂𝑛𝐻𝑛
1. What do these equations represent?
2. What are the equilibrium values?
3. What happens if 𝑂0 = 151 and 𝐻0 = 199? What happens if 𝑂0 = 149
and 𝐻0 = 201? What happens if 𝑂0 = 10 and 𝐻0 = 10?
33

SYMBIOSIS
There are many examples where the interaction of two or more species
is to the advantage of all. Symbiosis often plays the crucial role in
promoting and even maintaining such species: plant and seed dispersal
is one example.
Let 𝐴𝑛 and 𝐵𝑛 be the populations of species 𝐴 and 𝐵 respectively at
time 𝑛. Let 𝑘1 and 𝑘2 be the positive growth rates.
∆𝐴𝑛 = 𝐴𝑛 𝑘1 + 𝑘3𝐵𝑛
∆𝐵𝑛 = 𝐵𝑛 𝑘2 + 𝑘4𝐴𝑛
34

THE MODELING PROCESS
Suppose we want to understand some behavior or phenomenon in the
real world. We may wish to make predictions about that behavior in
the future and analyze the effects that various situations have on it.
35
Mathematical World
1. Models
2. Equations/Formulas
3. Operations/Rules
4. Conclusions
Real World Systems
1. Observed Behavior
2. Phenomenon
Introduction to Modeling - Modeling Process

SYSTEM
A system is a collection of objects, typically represented by the
variables in the model, that interact or are interdependent.
We are interested in understanding how a particular systems works,
what causes changes in the system, and how sensitive the system is to
certain changes to help us predict what changes might occur and when
they occur.
EXAMPLE:
In our previous examples, what happened then we saw too many mice
in our system?
36

REAL-WORLD VS MATH WORLD
37
Real-World System Mathematical Model
Mathematical
Conclusions
Real-World
Conclusions
Observation
Simplification
Analysis
Trials/Experiments
Interpretation

WHY MODEL?
Why not just test things in the real world?
There are many situations in which we would not want to follow such a
course of action (even with the risk of loss of fidelity).
1. Level of concentration of a drug (lethal).
2. Radiation effects of a failure in a nuclear power plant.
3. Atmospheric changes
May not even be possible to produce a trial, not just for moral reasons.
38

MATHEMATICAL MODEL
A mathematical model is defined as a mathematical construct designed
to study a particular real-world system or phenomenon. It can include
graphical, symbolic, simulation, and experimental constructs.
When developing our model, we should keep a few things in mind:
1. Feasibility: is the real-world system just to much?
2. Fidelity: how close does the model match reality?
3. Costs: do we have the resources, etc.?
4. Flexibility: can the model be expanded easily?
39

CONSTRUCTION OF MODELS
The process of constructing a model is similar to the scientific method:
1. Identify the problem. This is not as straightforward as it sounds. Be
as clear as you can.
2. Make assumptions. Simplify relationships, reduce the number of
variables, classify the variables (independent, dependent, neither),
determine the relationships between the variables.
3. Solve or interpret the model. Solve the equations (possible
numerically).
40

CONSTRUCTION OF MODELS
4. Verify the model. Did we answer the question, is it feasible, does it
make sense, compare with observations.
5. Implement the model. Make it useful/practical.
6. Maintain the model. Update as necessary.
NOTE:
This is like the scientific method for testing hypothesis and we do try to
be as scientific as possible, but we must remember that this is a
creative/artistic/intuitive part of modelling and that for the most part,
our models can never duplicate reality exactly.
41

SIMPLIFYING OR REFINING
Model Simplification
1. Restrict problem identification.
2. Neglect variables.
3. Group effects of several
variables.
4. Set some variables to be
constant.
5. Assume simple (linear)
relationships.
6. Incorporate more assumptions.
Model Refinement
1. Expand the problem.
2. Consider additional variables.
3. Consider each variable in detail.
4. Allow variation in the variables.
5. Consider nonlinear
relationships.
6. Reduce the number of
assumptions.
42

PROPERTIES OF A MODEL
The following are terms that are useful in describing models:
1. A model is said to be robust when its conclusions do not depend on
the precise satisfaction of the assumptions.
2. A model is fragile if its conclusions do depend on the precise
satisfaction of some sort of conditions.
3. Sensitivity refers to the degree of change in a model’s conclusions
as some condition on which they depend is varied; the greater the
change the more sensitive is the model to that condition.
43

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