4.4 The Logistic Equation

The Logistic Equation

Logistic equations model population growth when resources are limited. Unlike pure exponential growth (which assumes unlimited resources), the logistic model captures how populations grow rapidly at first, then slow down as they approach a maximum sustainable size called the carrying capacity. Direction fields and explicit solutions give you two complementary ways to analyze these equations: one visual, one algebraic.

Carrying Capacity in Logistic Growth

The carrying capacity ($K$) is the maximum population size an environment can sustain indefinitely. It depends on available resources like food, water, and shelter, and is further limited by competition for space and mates.

As a population approaches $K$, growth slows because individuals compete more intensely for scarce resources. Density-dependent factors like disease and predation also kick in harder at higher population densities.

At exactly $P = K$, the population stabilizes: births and deaths balance out, and the population holds at a steady state. This is one of the equation's two equilibrium points (the other is $P = 0$).

Direction Fields for Logistic Equations

A direction field is a grid of small line segments that show the slope (rate of change) at many points in the $t$-$P$ plane. Each segment tells you: "If the population were at this value at this time, here's how fast it would be changing." Tracing a path that follows these segments gives you an approximate solution curve without solving anything algebraically.

The logistic differential equation is:

$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$

where $P$ is the population size, $r$ is the intrinsic growth rate, and $K$ is the carrying capacity.

Constructing a direction field:

1. Choose a grid of points $(t, P)$ in the $t$-$P$ plane.

2. At each point, plug $P$ into $rP(1 - P/K)$ to compute the slope. (Notice the slope depends only on $P$, not on $t$, so all segments at the same height have the same slope.)

3. Draw a short line segment with that slope at each grid point.

Reading the field:

• Horizontal segments (slope $= 0$) appear at $P = 0$ and $P = K$. These are the equilibrium solutions.
• Between $P = 0$ and $P = K$, slopes are positive, so the population is increasing.
• Above $P = K$, the factor $(1 - P/K)$ is negative, so slopes are negative and the population is decreasing.
• Segments near $P = K$ from both above and below point toward $K$, which tells you $K$ is a stable equilibrium. Meanwhile, $P = 0$ is an unstable equilibrium: any small positive population will grow away from zero.

Solutions of Logistic Equations

You can solve the logistic equation exactly using separation of variables.

Step-by-step process:

1. Separate variables: $\frac{dP}{P\left(1 - \frac{P}{K}\right)} = r\,dt$

2. Use partial fractions on the left side. Rewrite $\frac{1}{P(1 - P/K)}$ as $\frac{1}{P} + \frac{1/K}{1 - P/K}$ (or equivalently $\frac{K}{P(K - P)} = \frac{1}{P} + \frac{1}{K - P}$).

3. Integrate both sides. The left side yields $\ln|P| - \ln|K - P|$, and the right side yields $rt + C_1$.

4. Combine the logarithms, exponentiate, and solve for $P(t)$.

The general solution is:

$P(t) = \frac{K}{1 + Ce^{-rt}}$

where $C$ is a constant determined by the initial condition $P(0)$. Setting $t = 0$ gives $C = \frac{K - P(0)}{P(0)}$.

Interpreting the solution:

• As $t \to \infty$, the term $Ce^{-rt} \to 0$, so $P(t) \to K$. Every positive initial population eventually approaches the carrying capacity.
• When $C > 1$ (meaning $P(0) < K/2$), the curve starts with a concave-up, nearly exponential phase before the inflection point, then becomes concave down as it levels off toward $K$. This produces the classic S-shaped (sigmoidal) curve.
• When $0 < C < 1$ (meaning $P(0) > K/2$ but still below $K$), the curve is concave down from the start, approaching $K$ from below without an obvious exponential-looking phase.
• When $C < 0$ (meaning $P(0) > K$), the population decreases toward $K$ from above.
• A larger $r$ means the population reaches $K$ faster; a smaller $r$ means a slower approach.

The inflection point (where growth is fastest) occurs at $P = K/2$. This is worth remembering: the population grows at its maximum rate when it's at half the carrying capacity.

Characteristics of Logistic Growth

• The solution curve is S-shaped (sigmoidal) when the initial population starts well below $K$.
• Early growth looks nearly exponential because $P/K \approx 0$, making the equation approximate $dP/dt \approx rP$.
• Growth rate peaks at $P = K/2$, then declines as the population nears $K$.
• The logistic model applies broadly: bacterial growth in a petri dish, spread of invasive species (kudzu, zebra mussels), epidemic dynamics in a susceptible population, and wildlife population management in ecology.