Population growth

Choose a model, initial populations and end time T, then calculate. N and P are continuous quantities, not integer or stochastic counts. Use one consistent time unit. r, α and γ have units 1/time; β has units 1/(predator × time), δ has units 1/(prey × time). These idealized models are not calibrated forecasts of real ecosystems.

Exponential: dN/dt = rN, hence N(t) = N₀ exp(rt). The doubling time ln(2)/r is shown only when N₀ > 0 and r > 0. A zero population stays zero; r = 0 gives a constant population.

Logistic: dN/dt = rN(1 − N/K), with capacity K > 0. The calculator uses the closed solution. N₀ may exceed K: for r > 0 the population then decreases toward K. N₀ = K is constant; N₀ = 0 remains zero. This tool restricts r to nonnegative values. For N₀ = 100, r = 0.2, K = 1000 and T = 10, the result is N(T) ≈ 450.853.

Lotka–Volterra: dN/dt = N(α − βP), dP/dt = P(δN − γ). N is prey and P is predators. Prey resources are unlimited in this model. For all four rates positive, the isolated positive equilibrium is N* = γ/δ, P* = α/β. It is a stationary solution if initialized exactly there; the model contains no damping that would make surrounding cycles settle there. With zero rates, no isolated positive coexistence equilibrium is reported. Zero populations stay exactly zero; the remaining population evolves exponentially.

Reading the plots: time and phase views share 401 output times from 0 through T, or one point when T = 0. The phase view plots (N, dN/dt) for logistic growth and (N, P) for predator–prey dynamics. ○ marks the initial point, ◆ the final point; coincident endpoints share one ◆. Time is not encoded by equal distance along a phase curve.

Numerics and scope: positive predator–prey states use adaptive RK4 step doubling in ln N and ln P, with local log-error tolerance 10⁻¹⁰ and a local rate limit. This is a local error target, not a guarantee of the same global error. At most 20,000 attempted steps are allowed. Positive populations outside 10⁻²⁵⁰ to 10²⁵⁰, unresolvable steps or nonfinite derived values produce an error; populations are not clipped and no incomplete curve is presented as a result. Parameter limits are computational limits: N₀/P₀ ≤ 10⁶, 10⁻⁶ ≤ K ≤ 10⁶, T ≤ 100, r/α/γ ≤ 5, β/δ ≤ 1. Valid inputs are retained across navigation; Reset removes the result and keeps the inputs.

Sources: Chasnov, Mathematical Biology, §§1.1, 1.2 and 1.4 (the predator–prey rate symbols are renamed here); MIT 1.020, predator–prey model assumptions.