--- title: "The Logistic Equation and Saturating Growth" description: "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." canonical: https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L source: https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Mathematics · Area: Differential Equations · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L ## Ready when they can - 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 ## Lesson: Growth with a ceiling 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. **Example.** 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. **Tip.** 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. **Recap.** Logistic growth brakes toward K, with 0 repelling and K attracting. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [First-Order Linear Equations and the Integrating Factor](https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8) - [Compound Growth and Decay](https://lightmysky.com/learn/mathematics/compound-growth-and-decay-mt_xf5FQ2sugf) ## Opens up - [Second-Order Linear Equations with Constant Coefficients](https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ)