Basic Terminology

General population models can always be written in the form of a standard balance equation,

where, for this application, the quantity of interest relates to the number of members of a given population. Also, for many studies, the production and loss rates are simply the birth and death rates associated with a given population. With this terminology and P(t) representing the amount of the species of interest present at time t, eqn. (1) can be written as

where B and D are the normalized birth and death rates, respectively. These normalized rates are defined precisely as follows:

Note that B and D can be functions of time or directly related to the population, P, depending on the specific situation of interest. A few special cases are illustrated below:

Constant Birth and Death Rates

If both B and D are constant, then is a constant, and the population balance in eqn. (2) becomes

Note that k can be positive or negative depending upon which normalized rate term is dominant. This first-order ODE is easily solved as a separable equation, leading to a general solution of the form

where P₀ represents the initial population size.

Birth Rate Decreases with Increasing Population (D = constant)

Most populations are limited in some fashion, often by restricted food supplies or physical space. A simple mathematical model that takes this observation into account models the normalized birth rate as a linearly decreasing function of the population size, or

where and are positive constants. This is simply one approximation that allows for a decreasing birth rate as the population increases. If we incorporate this birth rate representation with a constant normalized death rate, D₀, the population balance in eqn. (2) becomes

To simplify this expression a little, we define two new constants,

With these definitions, eqn. (6) can be written as

Solution of the Logistic Equation as a Bernoulli Equation

The logistic equation is also a Bernoulli equation and we will solve it here using this technique -- as another example of this solution scheme. First we rewrite the nonlinear population balance in standard form

and then recognize this as a Bernoulli equation of the form,

Since letting converts the nonlinear Bernoulli equation into a 1^st order linear system, we let for the current problem (since a = 2 here). With this substitution we have

Before substitution, let's first multiply eqn. (9) by , giving

Now, making the indicated substitutions from eqn. (11), this last expression simplifies to

This first order linear ODE has an integrating factor given by

Now, multiplying by the integrating factor gives

and one final integration gives

or

Finally, since , we can write the general solution to the logistic equation as

This is the general solution to the original population model stated in eqn. (8). To obtain a unique solution we simply apply the initial condition, . Doing this systematically gives

or

Putting this result for c into the general solution gives

or

This is the final unique solution for this mathematical model.

As a final note, we see that as time becomes large , the population approaches a limiting value, or

The quantity, M, which was defined in terms of the coefficients in the birth and death rate expressions [see eqn. (7)], is referred to as the limiting population. This result was expected and it is consistent with the physical model for this system.