Predator-Prey (Lotka-Volterra) Simulator
Solve the Lotka-Volterra predator-prey equations dPrey/dt = aPrey − bPrey·Predator and dPredator/dt = −cPredator + dPrey·Predator for the system's equilibrium point, and simulate how both populations cycle over time. Two 3D diagrams compare a prey-dominated moment in the cycle to a predator-dominated one, and charts trace the classic out-of-phase oscillation of both populations.
Reviewed by the ToolNestr Editorial Team — July 2026
Prey-dominated vs. predator-dominated moment
1. Prey-dominated moment
Abundant prey, few predators — the predator population is about to grow.
2. Predator-dominated moment
Many predators, scarce prey — the prey population is under heavy pressure.
Population cycle over time
How it works
The core idea in one line: predator and prey populations never settle down to a stable balance — instead, each one's boom feeds directly into the other's future bust, driving both populations around a perpetual, out-of-phase cycle centered on a single equilibrium point they never quite land on.
dPrey/dt = a·Prey − b·Prey·Predator
prey grow exponentially, minus losses to predation
dPredator/dt = −c·Predator + d·Prey·Predator
predators decline naturally, plus growth from consuming prey
Equilibrium: Prey* = c/d, Predator* = a/b
the one population combination where both rates of change are zero
Prey would grow exponentially on their own, but every encounter with a predator removes one — so prey growth slows as predator numbers rise. Predators, meanwhile, would starve and decline without food, but every prey they catch fuels new predator growth — so predator numbers rise only when prey is abundant. This feedback loop has exactly one equilibrium point, at Prey*=c/d and Predator*=a/b, where both effects perfectly cancel out — but because real populations rarely start exactly there, they instead orbit around it indefinitely: prey boom first, predators boom in response and overshoot, prey crash from overpredation, and predators then crash from starvation, resetting the cycle.
Worked example 1 — finding the equilibrium point
Given: Prey growth rate a = 0.5, predation rate b = 0.02, predator death rate c = 0.3, predator efficiency d = 0.01.
In practice, real (and simulated) predator-prey systems rarely sit exactly at this point — they orbit around it in a repeating cycle instead.
Worked example 2 — the paradox of enrichment
Given: Same parameters as example 1, but predation rate b doubles to 0.04 (predators become twice as effective hunters).
This is a genuinely counterintuitive result: making predators better hunters doesn't increase how many predators the system can support at equilibrium — it actually decreases it, since fewer predators are now needed to keep prey population in check.
How each parameter shifts the equilibrium point
Only two of the four parameters set each equilibrium value.
| Parameter | Affects | Direction |
|---|---|---|
| Prey growth rate (a) | Predator equilibrium ★ | Higher a → more predators supportable |
| Predation rate (b) | Predator equilibrium | Higher b → fewer predators needed at equilibrium |
| Predator death rate (c) | Prey equilibrium | Higher c → more prey needed to sustain predators |
| Predator efficiency (d) | Prey equilibrium | Higher d → fewer prey needed at equilibrium |
★ Prey growth rate (a) and predation rate (b) together set the predator equilibrium (a/b) — neither one alone determines it.
Where predator-prey modeling actually matters
🦌 Wildlife population management
Wildlife agencies use predator-prey models to predict how introducing or removing a predator species (like wolves) will affect prey populations (like deer or elk) over time.
🐟 Fisheries and marine ecosystem modeling
Fisheries scientists apply similar predator-prey models to understand how fishing pressure on one species ripples through the food web to affect its predators and prey.
🦗 Agricultural pest control
Biological pest control programs (introducing a natural predator to control a crop pest) are planned using predator-prey dynamics to predict whether the predator can actually establish a stable, self-sustaining population.
🧫 Epidemiology and immune system modeling
The same mathematical structure describes other consumer-resource interactions, including simplified models of immune cells "predating" on pathogens within the body.
Common misconceptions
"A stable ecosystem means predator and prey populations stay constant."
The Lotka-Volterra model shows that oscillation, not constancy, is the natural and stable long-term behavior of a simple predator-prey system — population cycles are a normal, expected pattern, not necessarily a sign of instability.
"More predators always means fewer prey, in direct proportion."
The relationship is cyclical and time-delayed, not simply direct — a predator population boom eventually crashes the prey population, which then causes the predator population itself to crash afterward, creating an oscillating rather than a simple inverse relationship.
"Removing predators is always good for the prey population long-term."
Prey populations without predator control can overshoot their environment's carrying capacity, leading to resource depletion and a population crash of their own — a dynamic not even captured by the basic Lotka-Volterra model, which assumes unlimited prey resources.
"This model perfectly describes real predator-prey populations."
The basic Lotka-Volterra model is a simplified idealization — it assumes unlimited prey food supply, no other predators or competitors, and no environmental variation, all of which real ecosystems violate to some degree.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, Biology 2e — Chapter 45, "Population and Community Ecology" (free, peer-reviewed). openstax.org
- • Lotka (1925) & Volterra (1926) — the original, independently derived predator-prey equations.
- • Odum, Fundamentals of Ecology — Predator-prey population dynamics chapter.
dPrey/dt=aPrey−bPrey·Predator; dPredator/dt=−cPredator+dPrey·Predator. Equilibrium: Prey*=c/d, Predator*=a/b. A simplified idealized model. Results are rounded for display.
How to use this calculator
Enter the four rate parameters
Prey growth, predation rate, predator death rate, and predator efficiency.
Enter starting populations
Initial prey and predator counts for the simulation.
Read the equilibrium and cycle
The equilibrium point solves instantly, and the chart shows the resulting population cycle.
Related tools
Frequently asked questions
What are the Lotka-Volterra equations?
They're a pair of differential equations modeling predator-prey population dynamics: prey grow exponentially but are consumed by predators (dPrey/dt = aPrey − bPrey·Predator), while predators decline without enough prey but grow by consuming it (dPredator/dt = −cPredator + dPrey·Predator).
What do a, b, c, and d represent?
a is the prey's natural growth rate, b is how effectively predators catch prey, c is the predator's natural death rate, and d is how efficiently consumed prey converts into new predators.
Why do predator and prey populations cycle instead of settling down?
As prey become abundant, predators have plenty of food and multiply, but the growing predator population then consumes prey faster than it can reproduce, crashing the prey population — which then starves the predators, letting prey recover, and the cycle repeats indefinitely.
What is the equilibrium point of the Lotka-Volterra system?
It occurs at Prey* = c/d and Predator* = a/b — the one population combination where both growth rates exactly cancel out, so neither population changes. In the idealized model, populations orbit around this point rather than settling directly onto it.
What is the 'paradox of enrichment'?
Counterintuitively, increasing the prey's resources or a predator's hunting efficiency (b) doesn't raise the predator equilibrium (which depends only on a/b) — instead it can destabilize the system into wider population swings, sometimes even risking extinction of one or both populations.