Logistic Growth Calculator
Enter current population, intrinsic growth rate, and carrying capacity to compute the net growth for the next time step and understand the S-shaped (logistic) population curve.
Input Data
Results
At a glance:Logistic growth describes population growth under limited resources: near-exponential early, gradually slowing as the population approaches the carrying capacity K, finally saturating into an S-shaped (sigmoid) curve. Net growth is dN = r·N·(K − N)/K, where r is the intrinsic (maximum) growth rate, K the carrying capacity, and N the current population. The factor (K − N)/K is the fraction of unused environmental space: near 1 when N ≪ K (near exponential) and near 0 when N ≈ K (growth stalls). Net growth is fastest at N = K/2.
Formula
Net growth: dN = r × N × (K − N) ÷ K.
Next population: N(t+1) = N + dN.
Current per-capita rate: r × (K − N) ÷ K.
$$\Delta N = rN\left(\frac{K - N}{K}\right)$$$$N_{t+1} = N_t + rN_t\frac{K - N_t}{K}$$How to Use
- Enter current population N, intrinsic rate r, and carrying capacity K.
- The tool instantly shows this step's net growth dN, next population N(t+1), and current per-capita rate.
- Set N to K/2 to see net growth reach its maximum.
Net growth at different population sizes (r = 0.5, K = 1000)
| Current N | Unused space (K−N)/K | Net growth dN | Next N(t+1) |
|---|---|---|---|
| 100 | 0.90 | 45 | 145 |
| 300 | 0.70 | 105 | 405 |
| 500 | 0.50 | 125 | 625 |
| 700 | 0.30 | 105 | 805 |
| 900 | 0.10 | 45 | 945 |
Net growth peaks at N = K/2 = 500 (125) and is symmetric around it, forming the S-shaped curve.
Case Studies
Early rapid growth
A population with N = 100, r = 0.5, K = 1000.
dN = 0.5 × 100 × (1000 − 100) ÷ 1000 = 45, next N = 145.
Unused space is 0.90, growth is near-exponential, per-capita rate ≈ 0.45, close to max r.
Slowing as it nears capacity
The same population grows to N = 900.
dN = 0.5 × 900 × (1000 − 900) ÷ 1000 = 45, next N = 945.
Although more individuals, only 0.10 space remains, so net growth equals that at N = 100 and starts to fall — the population saturates.
FAQ
How does logistic differ from exponential growth?
Exponential growth assumes unlimited resources and accelerates continuously (J-curve); logistic growth adds carrying capacity K, slowing and leveling as N approaches K (S-curve). Real resources are always finite, so most natural populations resemble logistic growth.
What does carrying capacity K represent?
K is the maximum population the environment can sustain long-term, set by food, space, water, predators, and disease. Above K, death rises or birth falls, pulling the population back. K is not fixed and shifts with the environment.
Why is growth fastest at K/2?
Net growth dN = r·N·(K−N)/K is a downward quadratic in N, maximized at N = K/2, where 'enough individuals to contribute' and 'still plenty of resource space' balance — the basis of maximum sustainable yield (MSY).
Does this calculator use continuous or discrete growth?
Discrete one-step: from N it computes one time step's dN and N(t+1). The continuous form is dN/dt = rN(K−N)/K, whose solution is the S-curve; discrete iteration approximates the same trend when r is not too large.
How to estimate intrinsic rate r?
r is the maximum per-capita rate without resource limits, estimable from birth minus death rate, or fitted from early near-exponential growth. Larger r makes the population approach K faster; very large discrete r can cause oscillation or chaos.
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References
Content review: Calculatorism Science Team. Results are for reference only; please refer to the relevant authorities for the official figures.