Lotka–Volterra Equations Calculator — Predator–Prey Model
Explore the classic Lotka–Volterra predator–prey model: enter the four ecological rate parameters plus the starting populations and instantly compute the coexistence equilibrium, the approximate cycle period, and an animated phase-plane portrait of the population orbit.
Time for one complete predator–prey oscillation (2π / √(αγ))
- 1
α × γ (prey rate × predator death rate)
0.5 × 0.3 = 0.15 - 2
√(αγ) — angular frequency
√(0.15) = 0.3873 - 3
Period T = 2π ÷ √(αγ)
2π ÷ 0.3873 = 16.22Approximate period from linearisation of the Lotka–Volterra system around the equilibrium.
How does this calculator work?
The Lotka–Volterra model: dx/dt = αx − βxy (prey); dy/dt = δxy − γy (predators). Coexistence equilibrium: x* = γ/δ, y* = α/β. Approximate oscillation period: T = 2π/√(αγ). The system produces closed population cycles — the phase portrait shows these as closed orbits around the equilibrium point.
Formula
How this is calculated
The Lotka–Volterra equations, formulated independently by Alfred Lotka (1925) and Vito Volterra (1926), are a pair of first-order nonlinear differential equations describing the dynamics of two species — prey (x) and predators (y). Prey grow exponentially at rate α in the absence of predators, and are consumed at rate β per predator–prey pair. Predators reproduce at rate δ per prey consumed and die at rate γ in the absence of prey. The system produces perpetual oscillations: prey rise → predators rise → prey fall → predators fall → repeat.
The coexistence equilibrium (x* = γ/δ, y* = α/β) is the fixed point where both populations remain constant. Any departure from equilibrium produces closed orbits in phase space (prey on the x-axis, predators on the y-axis) — these are the curves shown in the plot. The period of oscillation around the equilibrium is approximately T = 2π/√(αγ), derived by linearising the system around (x*, y*).
The calculator integrates the full nonlinear system numerically using the 4th-order Runge–Kutta (RK4) method with dt = 0.05, which gives accurate trajectories for typical parameter values. Limitations: the model assumes unlimited prey food, no spatial structure, no age structure, and perfectly efficient mixing. Real ecological populations are also subject to stochastic events, carrying-capacity constraints (add a logistic prey term), and multi-species interactions not captured here.
Frequently asked questions
α is the prey birth rate (per unit time per prey, in the absence of predators). β is the predation rate — how often a single predator–prey encounter leads to a kill. δ is the conversion efficiency — what fraction of consumed prey becomes new predator biomass. γ is the predator death rate in the absence of prey.
The equilibrium point (x*, y*) is a centre, not an attractor — trajectories neither spiral inward nor outward (for the pure Lotka–Volterra model). Any perturbation produces a closed orbit that never settles. In practice, real populations often do spiral in toward a stable equilibrium because of additional damping effects such as prey carrying capacity or predator saturation.
The phase portrait plots prey population (x) on the horizontal axis against predator population (y) on the vertical axis, tracing the trajectory over time. A closed loop means the system repeats its cycle. The marked dot is the equilibrium (x*, y*) — the populations rotate around it. A large loop means strong oscillations; a small loop near the equilibrium means gentle fluctuations.
Also known as
TG we-Calculate Editorial Team. (2026). Lotka–Volterra Equations Calculator — Predator–Prey Model [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/lotka-volterra-equations-calculator
TG we-Calculate Editorial Team. "Lotka–Volterra Equations Calculator — Predator–Prey Model." TG we-Calculate. 2026. https://we-calculate.com/calculator/lotka-volterra-equations-calculator.
TG we-Calculate Editorial Team, "Lotka–Volterra Equations Calculator — Predator–Prey Model," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/lotka-volterra-equations-calculator
@misc{wecalculate_lotka_volterra_equations_calculator, title = {Lotka–Volterra Equations Calculator — Predator–Prey Model}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/lotka-volterra-equations-calculator}}, year = {2026}, note = {TG we-Calculate} }
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