Logistic Growth - Northland Preparatory Academy
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Transcript Logistic Growth - Northland Preparatory Academy
Do Now: #1-8, p.346
Let f
x
50
1 5e
0.1 x
Check the graph first?
1. Continuous for all real numbers
2. lim f
x
x 50
lim f
x
x 0
3. H.A.: y = 0, y = 50
4. In both the first and second derivatives, the denominator
will be a power of 1 5 e
0.1 x
, which is never 0. Thus, the
domains of both are all real numbers.
Do Now: #1-8, p.346
Let f
x
50
1 5e
0.1 x
Check the graph first?
5. Graph f in [–30, 70] by [–10, 60]. f (x) has no zeros.
6. Graph the first derivative in [–30, 70] by [–0.5, 2].
Inc. interval: ,
Dec. interval: None
7. Graph the second derivative in [–30, 70] by [–0.08, 0.08].
Conc. up: ,1 6 .0 9 4 Conc. down: 1 6 .0 9 4,
8. Point of inflection: 1 6 .0 9 4, 2 5
LOGISTIC GROWTH
Section 6.5a
Review from last section…
Many populations grow at a rate proportional to the size of the
population. Thus, for some constant k,
dP
kP
dt
Notice that
dP dt
P
k
is constant,
and is called the relative growth rate.
Solution (from Sec. 6.4):
P P0 e
kt
Logistic Growth Models
In reality, most populations are limited in growth. The maximum
population (M) is the carrying capacity.
The relative growth rate is proportional to 1 – (P/M), with
positive proportionality constant k:
dP dt
P
dP
k
P
P M P
k 1
or
dt
M
M
The solution to this logistic differential equation is called
the logistic growth model.
(What happens when P exceeds M???)
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(a) Draw and describe a slope field for the differential equation.
Carrying capacity = M = 100
k = 0.1
Differential Equation:
dP
dt
k
M
P M P
0.1
100
P 100 P
0 .0 0 1 P 1 0 0 P
Use your calculator to get the slope field for this equation.
(Window: [0, 150] by [0, 150])
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
Differential Equation:
Initial Condition:
dP
P 0 10
dt
0.001 P 100 P
Rewrite
1
dP
P 100 P dt
0.001
1 1
1
dP
Partial Fractions
0.001
100 P 100 P dt
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
1 1
1
dP
0.001
100 P 100 P dt
1
1
Rewrite
dP 0.1dt
P 100 P
Integrate ln P ln 1 0 0 P 0 .1t C
Prop. of Logs ln
P
100 P
0.1t C
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
ln
P
100 P
Prop. of Logs ln
0.1t C
100 P
0.1t C
P
Exponentiate
100 P
e
0.1 t C
P
Rewrite
100 P
P
e
C
e
0.1 t
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(b) Find a logistic growth model P(t) for the population and draw
its graph.
100 P
e
C
P
0.1 t
–c 100
Let A = +– e
1 Ae
P
100
Solve for P P
0.1 t
1 Ae
Initial Condition 10
100
1 Ae
0
e
0.1 t
The Model:
P
100
1 9e
0.1 t
Graph this on top
of our slope field!
A9
A national park is known to be capable of supporting no more
than 100 grizzly bears. Ten bears are in the park at present. We
model the population with a logistic differential eq. with k = 0.1.
(c) When will the bear population reach 50?
Solve:
50
1 9e
1 9e
0.1 t
e
0.1 t
e
t
100
ln 9
0.1
0.1 t
0.1 t
2
1 9
9
21.972 yr
Note: As illustrated in this example,
the solution to the general logistic
differential equation
dP
dt
k
M
P M P
is always
P
M
1 Ae
kt
More Practice Problems
For the population described, (a) write a diff. eq. for the
population, (b) find a formula for the population in terms of t, and
(c) superimpose the graph of the population function on a slope
field for the differential equation.
1. The relative growth rate of Flagstaff is 0.83% and its
current population is 60,500.
dP
dt
0.0083 P
P 60, 500 e
How does the graph look???
0.0083 t
More Practice Problems
For the population described, (a) write a diff. eq. for the
population, (b) find a formula for the population in terms of t, and
(c) superimpose the graph of the population function on a slope
field for the differential equation.
2. A population of birds follows logistic growth with k = 0.04,
carrying capacity of 500, and initial population of 40.
dP
dt
P
k
M
P M P 0.00008 P 500 P
M
1 Ae
kt
500
1 11.5 e
0.04 t
How does the
graph look???
More Practice Problems
The number of students infected by measles in a certain school
is given by the formula
200
P t
1 e
5.3 t
where t is the number of days after students are first exposed
to an infected student.
(a) Show that the function is a solution of a logistic differential
equation. Identify k and the carrying capacity.
P t
200
1 e
5.3 t
200
1 e e
5.3
t
M
1 Ae
This is a logistic growth model
with k = 1 and M = 200.
kt
More Practice Problems
The number of students infected by measles in a certain school
is given by the formula
200
P t
1 e
5.3 t
where t is the number of days after students are first exposed
to an infected student.
(b) Estimate P(0). Explain its meaning in the context of the
problem.
P 0
200
1 e
5.3
0 .9 9 3
1
Initially (t = 0), 1 student has the measles.