From Limited Growth to the Logistic Equation

From Limited Growth to a Model

A population may grow rapidly when resources are plentiful, yet slow down as it approaches the amount its environment can support. To represent this idea with a differential equation, we must specify how the growth rate depends on both the current population and the remaining capacity.

The logistic equation is built from two assumptions: population growth is proportional to the population already present, and the per-capita growth rate is proportional to the fraction of capacity that remains available. Together, these assumptions produce a particular differential equation.

Define the Quantities and Assumptions

Let \(y(t)\) be the population at time \(t\). Let \(K\) be the environment’s carrying capacity: the population level used to represent the limit imposed by available resources. We assume \(K>0\). Population and capacity are measured in the same units, such as individuals.

When the population is below capacity, the amount of capacity not occupied is \(K-y\). But the phrase fraction of capacity still available refers to a ratio, not an amount. The available fraction is the unoccupied capacity divided by total capacity:

This fraction is dimensionless: the population units in the numerator cancel the population units in the denominator. For \(0\leq y\leq K\), it ranges from \(1\), when no capacity is occupied, to \(0\), when the population equals capacity.

We also assume that the per-capita growth rate decreases in direct proportion to this available fraction. Per-capita growth means the population’s rate of change divided by the current population, when \(y>0\). Let \(r>0\) be the maximum per-capita growth rate, in units of reciprocal time. Under the model, this is the per-capita rate approached when the population is small relative to capacity.

Deriving the Logistic Equation

The assumptions specify the population’s per-capita rate. To obtain the rate of change of the whole population, multiply that per-capita rate by the population \(y\). This gives the logistic differential equation.

The same derivation can be written as a sequence of modeling steps. Each step follows from one stated assumption or from the definition of per-capita growth.

This derivation explains why the model has both factors \(y\) and \(1-y/K\). The factor \(y\) represents the assumption that more individuals produce more potential growth. The second factor reduces the per-capita rate as the available fraction shrinks.

Units and Parameter Meaning

Suppose \(y\) and \(K\) are measured in individuals and \(t\) in years. Then \(dy/dt\) has units of individuals per year. The ratio \(y/K\) has no units, so \(1-y/K\) is dimensionless. Therefore, \(r\) must have units of inverse years, and the right side has the required units:

The parameter \(r\) is not the population’s actual per-capita growth rate at every population size. The model makes that rate depend on \(y\): it is \(r(1-y/K)\). The capacity \(K\) sets the scale against which the current population is compared. A rate rule that omits the division by \(K\) generally has different units and does not use the fraction of capacity remaining.

Worked Example: Modeling a Rabbit Population

Worked Example: Turning a Verbal Rule into an Equation

Worked Example: Converting an Unnormalized Coefficient

Why the Fraction Matters

A common mistake is to use the unoccupied amount \(K-y\) where the model calls for the unoccupied fraction \((K-y)/K\). For example, with \(K=500\), \(y=100\), and \(r=1\) per week, an incorrect calculation might be

This is wrong for two reasons. First, it uses the amount \(400\) rather than the fraction \(400/500=0.8\). Second, \(r\) has units of inverse weeks, so \(ry(K-y)\) has units of individuals squared per week, not individuals per week.

The correction is to divide the unoccupied amount by total capacity:

Using the fraction makes the model dimensionally consistent and makes the per-capita rate equal to \(r\) times a proportion. The alternative form \(a y(K-y)\) can be valid, but its coefficient \(a\) must have different units and is not the same parameter as \(r\).

What the Assumptions Do—and Do Not—Say

The logistic equation is a model, not a consequence of the word “limited” alone. The equation follows when we assume that the per-capita rate is proportional to the fraction of capacity remaining. Other assumptions about resource use, interactions, or changing environments could produce different equations.

The standard interpretation takes \(r>0\), \(K>0\), and a population between zero and capacity when describing a growing population. Within that range, the available fraction is between zero and one. The equation gives an instantaneous rate at a given population; it does not, by itself, guarantee that real data follow the model. The assumptions should be appropriate to the situation and the time period being studied.

One practical benefit of the formulation is that its components can be checked separately. Ask whether growth should scale with the number already present, whether the per-capita rate should decline with crowding, and whether the decline should be linear in the remaining fraction. If those are reasonable approximations, the logistic equation is a coherent translation of them.