Biology Reference
In-Depth Information
We shall model the functions R
r(R) next, beginning with
the function R(P). Assume the resource exists in two forms: free and
bound (or consumed) by the population. Let F be the maximum amount
of free resource available when the population size P
¼
R(P) and r
¼
0,
the amount of available resource will decrease as P increases. Assuming
a fixed per capita rate of consumption c
¼
0. When P
>
>
0, we can write:
R
¼
F
cP
:
(1-14)
To model the dependence r
r(R), notice that the net per capita growth
rate needs to satisfy the following conditions:
¼
1. The population should be declining when no free resource is
available; so when R
¼
0, the net per capita growth rate should be
negative: r(0) < 0.
2. The population should be growing when the free resource is avail-
able. More of the free resource will cause a higher per capita growth
rate, so the function r
¼
r(R) should be an increasing function of R.
The simplest mathematical dependency r
¼
r(R) that satisfies conditions
1 and 2 is the line
r
ð
R
Þ¼
mR
n
;
(1-15)
where m
0 represents the rate the free resource affects the per capita
net growth rate, and n
>
0 represents the per capita rate at which the
population size will decline when the resource is lacking.
>
Substituting R from Eq. (1-14) into Eq. (1-15) yields
r
ð
R
Þ¼
m
ð
F
cP
Þ
n
and a subsequent substitution into Eq. (1-9) gives the following resource-
based population growth model:
dP
dt ¼ð
m
ð
F
cP
Þ
n
Þ
P
:
(1-16)
Equation (1-16) can be rewritten as
P
h
i P
dP
dt ¼ð
mc
mF
P
K
mF
n
Þ
1
n P
¼
a 1
;
(1-17)
where
mF
n
a
¼
mF
n and K
¼
:
(1-18)
mc
Therefore, this model is the same as the logistic model from Eq. (1-12),
with inherent per capita growth rate and carrying capacity as given
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