Simulate predator-prey population dynamics using the classic Lotka-Volterra equations
Natural growth rate without predators
Rate predators consume prey
Natural death rate without prey
Prey-to-predator conversion efficiency
| Time | 🐰 Prey | 🦊 Predator |
|---|---|---|
| 0.0 | 100 | 20 |
| 1.0 | 64 | 34 |
| 2.0 | 32 | 41 |
| 3.0 | 16 | 39 |
| 4.0 | 8 | 32 |
| 5.0 | 5 | 25 |
| 6.0 | 4 | 20 |
| 7.0 | 3 | 15 |
| 8.0 | 2 | 11 |
| 9.0 | 2 | 8 |
| 10.0 | 2 | 6 |
| 11.0 | 2 | 5 |
| 12.0 | 2 | 4 |
| 13.0 | 2 | 3 |
| 14.0 | 2 | 2 |
| 15.0 | 2 | 2 |
| 16.0 | 3 | 1 |
| 17.0 | 3 | 1 |
| 18.0 | 3 | 1 |
| 19.0 | 3 | 1 |
Equations: dx/dt = αx − βxy · dy/dt = δxy − γy
Simulation using Euler method with dt = 0.1
The Lotka-Volterra equations are a pair of first-order nonlinear differential equations that describe the dynamics of biological systems in which two species interact—one as a predator and the other as prey. Independently derived by Alfred Lotka (1925) and Vito Volterra (1926), these equations represent one of the earliest and most fundamental models in mathematical ecology.
The model consists of two coupled equations: dx/dt = αx − βxy (prey dynamics) and dy/dt = δxy − γy (predator dynamics), where x is the prey population, y is the predator population, α is the prey growth rate, β is the predation rate, γ is the predator death rate, and δ is the predator reproduction efficiency. These equations capture the essential feedback loop: more prey means more food for predators (increasing predator population), which leads to more predation (decreasing prey population), which then reduces predator food supply (decreasing predator population), allowing prey to recover and restart the cycle.
The Lotka-Volterra model predicts cyclic oscillations in both populations, with predator peaks lagging behind prey peaks. While simplified compared to real ecosystems, these equations have been foundational in ecology, helping scientists understand population dynamics, biological invasions, and ecosystem stability. The model has been extended and applied to various fields including epidemiology, economics, and chemical kinetics.
Let's model a simplified ecosystem with rabbits (prey) and foxes (predators):
Starting above equilibrium, populations oscillate cyclically:
Key Insight: Populations oscillate cyclically around equilibrium values, with fox peaks lagging behind rabbit peaks—a pattern observed in real predator-prey systems like lynx and hares!
In the idealized Lotka-Volterra model, yes—cycles continue indefinitely with constant amplitude. Real ecosystems have damped oscillations due to factors like carrying capacity, environmental variation, and more complex interactions not captured in this simple model.
Predator population peaks lag behind prey peaks because it takes time for increased food availability to translate into predator reproduction. When prey are abundant, predators eat well and reproduce, but by the time predator population peaks, they've already depleted the prey.
In the continuous mathematical model, populations can approach zero but never reach it. However, in real discrete populations, once numbers drop below a critical threshold, extinction becomes likely due to factors like inability to find mates or demographic stochasticity.
The model assumes exponential prey growth (no carrying capacity), perfect predator specialization (only one prey species), no age structure, homogeneous mixing, and instantaneous responses. Real ecosystems are far more complex with multiple species, environmental factors, and spatial structure.
Classic examples include Canadian lynx and snowshoe hares (documented from Hudson Bay Company fur records), wolves and moose on Isle Royale, and planktonic predator-prey systems. However, most show additional complexity beyond the basic model.
To prevent extreme oscillations, you can decrease the predation rate (β) or increase prey growth rate (α). In real management, this translates to policies like predator control, prey supplementation, or habitat improvement to increase carrying capacity.
The equilibrium occurs at x* = γ/δ prey and y* = α/β predators, where population changes balance out (dx/dt = dy/dt = 0). This is the center around which populations oscillate. Note that populations rarely reach this point—they continuously cycle around it.
Yes! The Lotka-Volterra framework extends to multiple competing prey species, multiple predators, or complex food webs. However, the mathematics becomes significantly more complicated with each additional species, requiring systems of equations for each population.
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