Announcements
Topics:
- roughly sections 3.3, 4.1, and 4.2
* Read these sections and study solved examples in your
textbook!
Work On:
- Practice problems from the textbook and assignments
from the coursepack as assigned on the course web
page (under the link “SCHEDULE + HOMEWORK”)
MODELLING WITH DTDSs
Bacterial Population Growth:
bt +1 = rbt
The parameter r is called per capita production.
It represents the number of new bacteria
produced per bacterium.
MODELLING WITH DTDSs
Bacterial Population Growth:
r=2 Þ each bacterium divides into two daughter bacteria
and each daughter has a 2/2=1=100% chance of survival
r=1.5 Þeach bacterium divides into two daughter
bacteria and each daughter has a 1.5/2=3/4=75% chance
of survival
r=0.5 Þ each bacterium divides into two daughter
bacteria and each daughter has a 0.5/2=1/4=25% chance
of survival
Bacterial Population Growth in General
Solution:
bt = b0r
t
(Unrealistic) Assumption: r is constant
Reality: r depends on the size of the population
(resources are limited)
small populations
large populations
Þ less competition Þhigher r
Þ more competition Þ lower r
Bacterial Population Growth in General
Solution:
bt = b0r
t
(Unrealistic) Assumption: r is constant
Reality: r depends on the size of the population
(resources are limited)
small populations
large populations
Þ less competition Þhigher r
Þ more competition Þ lower r
Bacterial Population Growth in General
Solution:
bt = b0r
t
(Unrealistic) Assumption: r is constant
Reality: r will depend on the size of the population
(resources are limited)
small populations
large populations
Þ less competition Þhigher r
Þ more competition Þ lower r
Bacterial Population Growth in General
Solution:
bt = b0r
t
(Unrealistic) Assumption: r is constant
Reality: r will depend on the size of the population
(resources are limited)
small populations
large populations
Þ less competition Þhigher r
Þ more competition Þ lower r
MODELLING WITH DTDSs
Model for Limited Bacterial Population Growth:
bt +1 = r(bt )× bt
Replace the constant r by a function which matches
natural observations:
.
MODELLING WITH DTDSs
Model for Limited Bacterial Population Growth:
bt +1 = r(bt )× bt
Replace the constant r by a function which matches
natural observations:
1
1
ra
Þ r(bt ) = k ×
bt
bt
.
MODELLING WITH DTDSs
Model for Limited Bacterial Population Growth:
2
Example: r(bt ) =
1+ 0.001bt
r(bt )
bt
MODELLING WITH DTDSs
Model for Limited Bacterial Population Growth:
Example:
æ
ö
2
bt +1 = ç
÷ × bt
è1+ 0.001bt ø
Determine equilibria and behaviour of nearby
solutions by cobwebbing.
elimination of chemicals
*** filtration by kidneys (kidneys break down
constant amount per hour … caffeine)
*** breaking down the chemicals using
enzymes from the liver (amount of chemical
broken down depends on the amount present …
alcohol)
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Absorption of Caffeine:
Our bodies eliminate caffeine at a constant rate
of 13% per hour.
DTDS:
c t +1 = 0.87c t + d
amount of caffeine
(mg) 1 hour later
amount of
caffeine now
amount of
“new” caffeine
consumed at
time t+1
* Similar to “methadone” example
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Absorption of Caffeine:
Our bodies eliminate caffeine at a constant rate
of 13% per hour.
DTDS:
c t +1 = 0.87c t + d
amount of caffeine
(mg) 1 hour later
amount of
caffeine now
amount of
“new” caffeine
consumed at
time t+1
* Similar to “methadone” example
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Absorption of Caffeine:
Our bodies eliminate caffeine at a constant rate
of 13% per hour.
DTDS:
c t +1 = 0.87c t + d
amount of caffeine
(mg) 1 hour later
amount of
caffeine now
amount of
“new” caffeine
consumed at
time t+1
* Similar to “methadone” example
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Absorption of Caffeine:
Our bodies eliminate caffeine at a constant rate
of 13% per hour.
DTDS:
c t +1 = 0.87c t + d
amount of caffeine
(mg) 1 hour later
amount of
caffeine now
amount of
“new” caffeine
consumed at
time t+1
* Similar to “methadone” example
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
The amount of alcohol that is broken down by
the liver depends on the amount of alcohol
present in the body.
The larger the amount, the smaller the
proportion of alcohol being eliminated.
*Similar to the limited growth population model
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
The amount of alcohol that is broken down by
the liver depends on the amount of alcohol
present in the body.
The larger the amount, the smaller the
proportion of alcohol being eliminated.
*Similar to the limited growth population model
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
The amount of alcohol that is broken down by
the liver depends on the amount of alcohol
present in the body.
The larger the amount, the
smaller the proportion of
alcohol being eliminated.
*Similar to the limited growth population model
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
The amount of alcohol that is broken down by
the liver depends on the amount of alcohol
present in the body.
The larger the amount, the
smaller the proportion of
alcohol being eliminated.
*Similar to the limited growth population model
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
The amount of alcohol that is broken down by
the liver depends on the amount of alcohol
present in the body.
The larger the amount, the
smaller the proportion of
alcohol being eliminated.
*Similar to the limited growth population model
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
DTDS:
rate of elimination
at +1 = at - r(at )at + d
amount of alcohol
(g) 1 hour later
amount of
alcohol now
amount of
“new” alcohol
consumed at
time t+1
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
Example:
Rate of Elimination:
10.1
r(at ) =
4.2 + at
æ 10.1 ö
DTDS: at +1 = at - ç
÷at + d
è 4.2 + at ø
r(at )
at
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
Example:
r(at )
Rate of Elimination:
10.1
r(at ) =
4.2 + at
at
æ 10.1 ö
DTDS: at +1 = at - ç
÷at + d
è 4.2 + at ø
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
Example:
A standard drink contains 14g of alcohol.
Compare what happens over time for the following
situations:
(a) You consume two drinks right away and
continue to have half of a drink every hour
(a) You consume one drink every hour
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
Example:
A standard drink contains 14g of alcohol.
Compare what happens over time for the following
situations:
(a) You consume two drinks right away and
continue to have half of a drink every hour
(a) You consume one drink every hour
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
Example:
A standard drink contains 14g of alcohol.
Compare what happens over time for the following
situations:
(a) You consume two drinks right away and
continue to have half of a drink every hour
(a) You consume one drink every hour
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
Example:
A standard drink contains 14g of alcohol.
Compare what happens over time for the following
situations:
(a) You consume two drinks right away and
continue to have half of a drink every hour
(a) You consume one drink every hour
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
(a) You consume two drinks right away and
continue to have half of a drink every hour
æ 10.1 ö
f (at ) = at - ç
÷at + 7, a0 = 28
è 4.2 + at ø
Substance Absorption (Elimination) and
Replacement (Consumption) Models
Elimination of Alcohol:
(b) You consume one drink every hour
æ 10.1 ö
f (at ) = at - ç
÷at + 14, a0 = 0
è 4.2 + at ø
Calculus On Continuous Functions
In order to start studying continuous-time
dynamical systems, we need to develop the
usual tools of calculus on continuous functions:




Limits
Continuity
Derivatives
Integrals
}
these are all defined
in terms of limits
The Limit of a Function
Notations:
f (2) = 5
means that the y-value of
the function AT x=2 is 5
ì x2 - 4
ï
if x ¹ 2
f (x) = í x - 2
ïî5
if x = 2
means that the y-values of
the function APPROACH 4
as x APPROACHES 2
The Limit of a Function
Definition:
“the limit of f(x), as x approaches a, equals L”
means that the values of f(x) (y-values)
approach the number L more and more closely
as x approaches a more and more closely (from
either side of a), but x¹a.
Limit of a Function
Some examples:
Note: f may or may not be defined at x=a. Limits are only asking how f is defined NEAR a.
Limit of a Function
Some examples:
Note: f may or may not be defined at x=a. Limits are only asking how f is defined NEAR a.
Limit of a Function
Some examples:
Note: f may or may not be defined at x=a. Limits are only asking how f is defined NEAR a.
Limit of a Function
Some examples:
Note: f may or may not be defined at x=a. Limits are only asking how f is defined NEAR a.
Left-Hand and Right-Hand Limits
means
as
from the left (x < a).
means
as
from the right (x > a).
** The full limit exists if and only if the left and right
limits both exist (equal a real number) and are the
same value.
Left-Hand and Right-Hand Limits
For each function below, determine the value of
the limit or state that it does not exist.
Left-Hand and Right-Hand Limits
For each function below, determine the value of
the limit or state that it does not exist.
Left-Hand and Right-Hand Limits
For each function below, determine the value of
the limit or state that it does not exist.
Left-Hand and Right-Hand Limits
For each function below, determine the value of
the limit or state that it does not exist.
Evaluating Limits
We can evaluate the limit of a function in 3
ways:
1. Graphically
2. Numerically
3. Algebraically
Evaluating Limits
Example:
Use a table of values to
estimate the value of
x
f(x)
3.5
3.9
3.99
4
4.01
4.1
4.5
undefined
Evaluating Limits
Example:
Use a table of values to
estimate the value of
x
f(x)
3.5
7.5
3.9
7.9
3.99
7.99
4
undefined
4.01
4.1
4.5
Evaluating Limits
Example:
Use a table of values to
estimate the value of
x
f(x)
3.5
7.5
3.9
7.9
3.99
7.99
4
undefined
4.01
4.1
4.5
Evaluating Limits
Example:
Use a table of values to
estimate the value of
x
f(x)
3.5
7.5
3.9
7.9
3.99
7.99
4
undefined
4.01
8.01
4.1
8.1
4.5
8.5
Evaluating Limits
Example:
Use a table of values to
estimate the value of
x
f(x)
3.5
7.5
3.9
7.9
3.99
7.99
4
undefined
4.01
8.01
4.1
8.1
4.5
8.5
Evaluating Limits
Example:
Use a table of values to
estimate the value of
x
f(x)
3.5
7.5
3.9
7.9
3.99
7.99
4
undefined
4.01
8.01
4.1
8.1
4.5
8.5
Evaluating Limits Algebraically
BASIC LIMITS
Limit of a Constant
Function
Limit of the Identity
Function
Example:
Example:
LIMIT LAWS
[used to evaluate limits algebraically]
Suppose that c is a constant and the limits
exist. Then
1.
2.
3.
LIMIT LAWS
[used to evaluate limits algebraically]
Continued…
4.
5.
Evaluating Limits Algebraically
Example:
Evaluate the limit and justify each step by
indicating the appropriate Limit Laws.
Evaluating Limits Algebraically
Example:
Evaluate the limit and justify each step by
indicating the appropriate Limit Laws.
Evaluating Limits Algebraically
Example:
Evaluate the limit and justify each step by
indicating the appropriate Limit Laws.
Evaluating Limits Algebraically
Example:
Evaluate the limit and justify each step by
indicating the appropriate Limit Laws.
Direct Substitution Property
From the previous slide, we have
f (1)
f (x)
Notice that we could have simply found the
value of the limit by plugging in x=1 into the
function.
Direct Substitution Property
Direct Substitution Property:
If f(x) is an algebraic, exponential, logarithmic,
trigonometric, or inverse trigonometric
function, and a is in the domain of f(x), then
Equal Limits Property
Consider the functions:
x2 - 4
f (x) =
x -2
g(x) = x + 2.
* Note: f(x)=g(x) everywhere except at x=2
Equal Limits Property
Example:
Calculate
Note: direct substitution does not work
FACT:
If f (x) = g(x) when x ¹ a , then
provided the limits exist.
Strategy for Evaluating Limits
#
= ±¥
0
real #
0
0
Evaluating Limits Algebraically
Evaluate each limit or state that it does not exist.
(a)
(b)
(c)
(d)