MR
May 16, 2026
How to model population growth using logistic differential equations?
I need to understand the logistic growth model:
where is the growth rate and is the carrying capacity.
I know this is a separable ODE, but I'd like help with:
- Step-by-step separation of variables and integration using partial fractions
- Finding the equilibrium solutions and their stability
- What happens when ? Does the population decrease to ?
- How to fit this model to real-world data
Can someone solve:
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1 Answer
PJ
Prof. James ChenMay 16, 2026 AcceptedLet's solve $\frac{dP}{dt} = 0.5P(1 - P/1000)$ with $P(0) = 50$.
Step 1: Separate variables:
$$\frac{dP}{P(1 - P/1000)} = 0.5 \, dt$$
Step 2: Partial fractions. Rewrite as:
$$\frac{1}{P(1 - P/1000)} = \frac{1}{P} + \frac{1/1000}{1 - P/1000}$$
Integrating:
$$\ln|P| - \ln|1 - P/1000| = 0.5t + C$$
$$\ln\left|\frac{P}{1 - P/1000}\right| = 0.5t + C$$
Step 3: Solve for $P$:
$$\frac{P}{1 - P/1000} = Ae^{0.5t}$$
Using $P(0) = 50$:
$$\frac{50}{1 - 0.05} = \frac{50}{0.95} = A \implies A \approx 52.63$$
$$P(t) = \frac{1000Ae^{0.5t}}{1000 + Ae^{0.5t}} = \frac{1000}{1 + 19e^{-0.5t}}$$
**Equilibrium analysis:**
- $P = 0$ is unstable (population grows away from 0)
- $P = K = 1000$ is stable (population approaches 1000 from either direction)
If $P_0 > 1000$, the growth rate becomes negative (the $1 - P/K$ term is negative), so the population decreases toward $K$. The logistic equation always approaches the carrying capacity in the long run.
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