The Logistic Observation
The following theorem helps us understand the logistic equation in a more
useful and realistic way. Let P be the density of a fixed
population, v the population's birth rate at small population
(or rather the fecundity), u the population's death rate, and K
the population's carrying capacity (the population maximal density for a
sustainable population). Recall
one way to justify the use of the logistic equation
dP/dt = (v-u)(1-P/K)P
is to note that if we know the birth rate and death rate of a population
then the population would exponential increase and be determined by dP/dt
= (v-u)P but as we approach the carrying capacity of that population
we need the derivative to go to zero. This equation does the trick of
capturing this statement, but if we try to decouple the birth and death
rate from each other we might be led to
dP/dt = v(1-P/K)P-u(1-P/K)P
But this makes less sense. The birth rate going to zero as we approach
the carrying capacity makes a certain amount of sense but clearly the
magnitude of the death rate should probably not be getting
smaller! We might say this as "this
is not the right way to
decouple
the birth and death-rate". Another
possibility
is to assume a more sensible form to this equation, something like
dP/dt = v(1-P/C)P-uP
Here we are making the some more reasonable assumption that
the death-rate is independent of population
size. We would like a stable equilibrium at the carrying capacity to we
solve
0 = v(1-K/C)K-uK
for C and find
C = vK/(v-u)
In fact as you might expect...
Theorem:
Letting C= vK/(v-u) we have
dP/dt = v(1-P/C)P-uP=(v-u)(1-P/K)P.
This isn't much of theorem mathematically but biologically it is rather
important since it shows us how to correctly decouple the birth and
death rates. It tells us what the correct birth rate in population
with a given carrying capacity should be v(1-T/C) where
K is the carrying capacity, v the birthrate at small
population, u the natural death rate, and C= vK/(u-v).
This will be particularly useful when the a population
is split into categories and birth is viewed as a transition from
one
category to another. For
example someone infected by a given disease might have offspring which are
not infected and hence should be modeled by
dS/dt= v(1-T/C)(S) - u S - beta SI
dI/dt= beta SI - u I - alpha I
where T is the total population contributing to the utilization of
resources, in this case T=S+I. Recall beta is
measuring how catchy the disease is,
and alpha is the increase in the death rate do to the disease.