Population growth calculator and simulation

With unlimited resources a population grows exponentially: N = N₀e^(rt), where r is the growth rate per individual. Real populations level off at a carrying capacity, K, which the logistic model captures: dN/dt = rN(1 − N/K). A population of 10 growing at r = 0.6 per day reaches about 201 after 5 days if growth is exponential, but only 123 if K = 300. This population growth calculator is our Population Growth Simulator: it works out both models and runs a population of individuals beside the equations.

It opens on that logistic population. Open the full population growth simulator over this page.

Try it

Population Growth Simulator

Model exponential and logistic growth with a carrying capacity

Open the simulatorGo to its page

How to use the population growth calculator

1. Choose One species, then Logistic or Exponential.
2. Set Starting population N₀, Growth rate r (per day) and, for logistic, Carrying capacity K. The Death rate only affects the simulated individuals: r is already births minus deaths.
3. Set At time t to the time you want answers for.
4. Read the worked answers: N after that time, the growth rate then, the per capita rate, the doubling time, the rate from two readings, the fastest growth and K. Each is marked on the curve.
5. Press play to run the individual-based population and compare its count with the equation's.
6. For two species, choose Predator and prey for a Lotka-Volterra model with a phase plot. To practise, press Random problem.

Population growth formulas

In the logistic model, growth is fastest when N = K/2, at a rate of rK/4. The doubling time while a population grows exponentially is t_d = ln 2 ÷ r ≈ 0.693 ÷ r. Some biology courses write r as r_max, the maximum per capita growth rate.

Worked examples you can check in the calculator

Example 1: exponential growth

With N₀ = 10, r = 0.6 per day and t = 5 days, the population is about 201. Choose Exponential to check.

Example 2: logistic growth

With K = 300, the same population is about 123 after 5 days, because crowding is already slowing it. Its growth rate then is 0.6 × 122.8 × (1 − 122.8/300) = 43.5 per day.

Example 3: doubling time

With r = 0.6 per day, t_d = 0.693 ÷ 0.6 = 1.16 days. In exponential growth that holds at any size. In logistic growth it only holds while N is far below K, and the calculator notes how much longer a later doubling takes.

Example 4: fastest growth

For K = 300 and r = 0.6, growth is fastest at N = K/2 = 150, where dN/dt = rK/4 = 45 per day. That is the steepest point of the S-shaped curve.

Example 5: rate from two readings

A population is 50 on day 2 and 120 on day 4. Its average growth rate over those days is (120 − 50) ÷ (4 − 2) = 35 per day. This is how exam questions often give growth from a table or graph.

Try one

Why the simulation matters

The equations give one smooth curve. A real population is individuals who are born and die by chance, so its count wobbles around the curve, and a small population can die out even when the equation says it should grow. Run the simulation with a small N₀ a few times to see this.

Common mistakes

• Using exponential growth for a whole curve. Real populations level off, so use logistic once resources limit them.
• Mixing time units. r per day needs t in days.
• Thinking growth is fastest at K. It is fastest at K/2 and zero at K.
• Confusing r with the death rate. r is births minus deaths per individual.
• Applying a doubling time to logistic growth near K. Doubling slows down as N approaches K.

Read next

Maths · GraphsExponential graphs: y = aˣ, eˣ, growth and decayHow exponential graphs work: the shape of y = aˣ, the asymptote, the point (0, 1), growth and decay, y = eˣ, transformations and log graphs.