The Logistic Equation and Saturating Growth
Growth proportional to the population and to the room left produces an S-shaped curve with a carrying capacity. The model is where equilibrium solutions first appear.
What a learner can do afterwards
- Solve the logistic equation for a given carrying capacity
- Identify the equilibrium solutions and say which is stable
- Compare logistic and exponential predictions over a long horizon
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Exponential growth assumes endless room: dP/dt = rP climbs forever. Real populations crowd, since food, space, and predation bite as numbers rise. The logistic rule adds the brake: dP/dt = rP times (1 minus P over K). Growth runs fast while P is small and chokes as P nears the carrying capacity K. Near K the bracket shrinks toward zero, which is why growth slows before the limit is reached.
Take dP/dt = 0.4 P times (1 minus P over 100). Set the rate to zero: either P = 0 or the bracket is zero, so P = 0 or P = 100. Test each side: at P = 1 the rate is 0.4 times 1 times 0.99, positive, so small populations grow away from 0. Below 100 the rate pushes up and above 100 it pushes down, so 100 attracts. The S-curve P = 100 over (1 + e to the minus 0.4 t) starts at 100 over 2 = 50 and climbs toward 100.
Over a long horizon the two forecasts split completely. Exponential keeps climbing without bound while logistic levels at K, because only the logistic model carries the braking term. The same lake gets two different futures. Note K is set by the environment, not by the animals.
Find equilibria first: set the rate to zero and solve. Then test one point on each side to call each equilibrium stable or not. Never argue stability from the solution formula before doing the zero algebra.
Logistic growth brakes toward K, with 0 repelling and K attracting.
The rewritten readings were written for this dataset.
More to read
Growth with a carrying capacity
Exponential population models leave resources unrestricted. Logistic growth accounts for a carrying capacity K through dP/dt = rP(1 - P/K). With positive r and K, per-capita growth slows as P approaches K; total growth is also small when P is near zero.
The equilibrium at 0 is unstable, while K is stable. Starting between these values, a population increases toward K. An S-shaped trajectory occurs when the initial population is below K/2. The explicit solution describes this approach to the limit. Logistic models are useful candidates when growth eventually levels off.
Resources that limit populations
Bacteria growing in a dish, island deer and lake fish can show rapid growth when resources are plentiful, followed by slower growth as crowding increases. Such populations may follow an S-shaped curve.
Food shortages, limited space and density-dependent predation can restrict growth. These environmental conditions help determine carrying capacity K. Their effects can slow a population before it reaches that capacity.
Checking logistic calculations
Check your equilibrium algebra before assessing stability. After separating variables and integrating, verify that the resulting expression satisfies the logistic equation and agrees with its explicit solution.
Compare the models over long periods. With a positive growth rate and initial population, exponential growth continues without bound. A logistic population starting between 0 and K instead approaches K. This difference helps check whether your solution predicts the intended behaviour.
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Optional connections
13 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.