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.