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Population Growth Simulator

Simulate and compare population growth models. Explore exponential (unlimited) growth, logistic (carrying capacity) growth, and Lotka-Volterra predator-prey dynamics with interactive graphs, real-time calculations, and ecological presets.

Parameters

Initial Population (N₀)10
Growth Rate (r)0.30
Time Span20 generations

Results

N(t)=N0⋅ertN(t) = N_0 \cdot e^{rt}
N(20)
4034
Doubling Time
2.31 gen
Growth Rate (r)
0.300
Growth Factor (per gen)
1.3499

Exponential Growth Curve

Reference Guide

Exponential Growth

When resources are unlimited, populations grow at a constant rate proportional to their size. This produces a J-shaped curve that accelerates over time.

Growth equation
N(t)=N0⋅ertN(t) = N_0 \cdot e^{rt}
Doubling time
td=ln⁡2rt_d = \frac{\ln 2}{r}

In reality, no population grows exponentially forever. Resource limits, disease, and competition eventually slow growth.

Logistic Growth

Logistic growth accounts for limited resources by introducing a carrying capacity K. Growth slows as the population approaches K, producing an S-shaped (sigmoid) curve.

Differential form
dNdt=rN(1−NK)\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)
Solution
N(t)=K1+(K−N0N0)e−rtN(t) = \frac{K}{1 + \left(\frac{K - N_0}{N_0}\right) e^{-rt}}

Carrying Capacity and Inflection

The carrying capacity K is the maximum population an environment can sustain indefinitely. At K, birth rate equals death rate and population growth stops.

Fastest growth at K/2
dNdt∣max⁡=rK4at N=K2\left.\frac{dN}{dt}\right|_{\max} = \frac{rK}{4} \quad \text{at } N = \frac{K}{2}

The inflection point of the logistic curve occurs at N = K/2. Below K/2, growth is accelerating. Above K/2, growth is decelerating as the population approaches its limit.

Predator-Prey Dynamics

The Lotka-Volterra equations model how predator and prey populations interact. Prey grow naturally but are consumed by predators. Predators depend on prey for reproduction and die without food.

Lotka-Volterra equations
dNdt=αN−βNP\frac{dN}{dt} = \alpha N - \beta NP
dPdt=δNP−γP\frac{dP}{dt} = \delta NP - \gamma P

The result is oscillating populations. When prey are abundant, predators thrive and increase. As predators grow, prey decline. Fewer prey cause predator decline, allowing prey to recover.