Logistic Equation

1Introduction

This lecture explains how to analyze the logistic equation as an autonomous equation and determine its equilibrium solutions, monotonicity, and long-term behavior on the nonnegative state space.

Its standard form is

where $$r>0$$r>0 is the growth rate at low density and $$K>0$$K>0 is the carrying capacity.

When $$y$$y is small, $$1-y/K\approx 1$$1-y/K\approx1, so

and the behavior is close to exponential growth. As $$y$$y approaches $$K$$K, the factor $$1-y/K$$1-y/K becomes small and suppresses further growth.

2Why Analyze the Equilibria First?

The logistic equation is the autonomous equation

For an autonomous equation, the current value of $$y$$y alone determines whether a solution increases or decreases. The sign of $$F\left(y\right)$$F(y) therefore determines the direction of motion before an explicit solution is derived.

3Proposition 1: Equilibrium Solutions of the Logistic Equation

Assume that $$r>0$$r>0 and $$K>0$$K>0. The equilibrium solutions of

are

3.1Proof

An equilibrium is a constant solution and must satisfy $$y^{\prime }=0$$y'=0. Thus

Since $$r>0$$r>0, either $$y=0$$y=0 or $$1-y/K=0$$1-y/K=0. Hence the only possible equilibria are $$y=0$$y=0 and $$y=K$$y=K. Substitution shows that the right-hand side vanishes for both constants, so both are solutions.

4Proposition 2: Increase for $$0<y<K$$0<y<K and Decrease for $$y>K$$y>K

Assume that $$r>0$$r>0 and $$K>0$$K>0. If

then $$y^{\prime }>0$$y'>0. If

then $$y^{\prime }<0$$y'<0.

4.1Proof

For $$0<y<K$$0<y<K, both $$y$$y and $$1-y/K$$1-y/K are positive. Since $$r>0$$r>0,

so $$y^{\prime }>0$$y'>0 and the solution increases.

For $$y>K$$y>K, the factor $$y$$y is positive while $$1-y/K<0$$1-y/K<0. Therefore,

so $$y^{\prime }<0$$y'<0 and the solution decreases.

This sign analysis directly shows that, on the positive half-line, the arrows on the phase line point toward $$K$$K. Convergence to $$K$$K will be verified below from the explicit solution.

5Phase-Line Summary

When $$y$$y represents a population or another nonnegative quantity, the relevant range is usually $$y\text{[PARSE ERROR: Undefined("Command("ge")")]}0$$y\ge0. On this part of the phase line, the arrows point away from $$0$$0 and toward $$K$$K.

6Derivation of the Explicit Solution

Separate variables on a region where either $$0<y<K$$0<y<K or $$y>K$$y>K. In both cases $$y\ne 0$$y\ne0 and $$y\ne K$$y\ne K, so division by $$y(K-y)$$y(K-y) is valid. The equilibria $$y=0$$y=0 and $$y=K$$y=K, which were checked beforehand, are treated separately.

Since

we have

The partial-fraction decomposition is

Therefore,

Accounting for the derivative $$-1$$-1 of $$K-y$$K-y in the second term gives

After multiplication by $$K$$K, write

Solving for $$y$$y yields

For the initial condition $$y\left(0\right)=y_0$$y(0)=y_0 with $$y_0>0$$y_0>0 and $$y_0\ne K$$y_0\ne K,

and hence

This formula defines the solution on the maximal connected interval containing $$t=0$$t=0 on which its denominator is nonzero. For every $$y_0>0$$y_0>0 with $$y_0\ne K$$y_0\ne K, the formula also shows that $$y\left(t\right)\to K$$y(t)\to K as $$t\to \mathrm{\infty }$$t\to\infty. If $$y_0=K$$y_0=K, the solution is the equilibrium $$y\left(t\right)=K$$y(t)=K itself.

7Interpretation as a Bernoulli Equation

The logistic equation can be written as

or

This is the Bernoulli form

with $$n=2$$n=2.

Thus the logistic equation can be analyzed by a phase line as an autonomous equation or solved by linearization as a Bernoulli equation. A phase line is appropriate for qualitatively determining the direction of change on the positive half-line, whereas separation of variables or the Bernoulli substitution is appropriate when an explicit formula or a proof of convergence is required.

8Scope

The logistic equation assumes constant parameters $$r$$r and $$K$$K and assumes that the rate of change depends only on the current value of $$y$$y. Incorporating seasonal variation, age structure, or spatial movement leads respectively to time-dependent coefficients, systems of equations, or partial differential equations.

9Exercises

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