popmodels.pdf - Population Models Basic Assumption:...

Population Models
Basic Assumption: population dynamics of a group controlled by two functions of time
Birth Rate β ( t , P ) = average number of births per group member, per unit time
Death Rate δ
(
t , P
) =
average number of deaths per group member, per unit time
If P ( t ) = population at time t , then, between t and t + Δ t :
• Total births
≈
β P Δ t
• Total deaths ≈ δ P Δ t
Change in population is the difference:
Δ P
≈ (
β
-
δ
)
P Δ t
= ⇒
Δ P
Δ t
≈ (
β
-
δ
)
P
= ⇒
d P
d t
= (
β
-
δ
)
P (take limit Δ t
→
0)
Different models depend on choices/observations/predictions of birth and death rates
Natural Growth Suppose β and δ constant
Natural growth ( β - δ > 0) or decay ( β - δ < 0) equation:
d P
d t
= ( β - δ ) P
= ⇒ P ( t ) = P
0
e
(
β
-
δ
)
t
where P
0
is the population at time t = 0
P
0
t
β
>
δ
β
<
δ
β = δ
P
0
1

Growth rate proportional to population Suppose δ = 0 and
β = kP with k > 0 constant: no death, while birth rate increases
with population. Then
d P
d t
= ( β - δ ) P = kP
2
= ⇒
Z
P
P
0
˜
P
- 2
d
˜
P
=
Z
t
0
k d
˜
t
= ⇒ -
P
- 1
+
P
- 1
0
=
kt
= ⇒
P
(
t
) =
P
0
1
-
P
0
kt
P
(
t
) →
∞ as t
→
1
P
0
k
(in finite time!)
Ridiculous model for long term: typically large population re-
duces birth rate in real-world situations.
P
0
t
P
0
1
P
0
k
The Logistic equation Assume growth rate β
-
δ a linear function of P :
β - δ = k ( M - P ( t ))
where M , k are constant.
Definition. The Logistic Differential Equation is
d P
d t
= kP ( M - P ) .
The logistic equation is separable, hence
Z
t
0
k d
˜
t
=
Z
P
P
0
1
˜
P ( M -
˜
P )
d
˜
P
=
1
M
Z
P
P
0
1
˜
P
+
1
M
-
˜
P
d
˜
P (partial fractions)
= ⇒
Mkt
=
ln
(
M
-
P
0
)
P