Calculator Inputs
Formula Used
The classic Lotka Volterra predator prey system is defined by two coupled differential equations:
x is prey population and y is predator population.
α is prey growth, β is predation strength, δ converts encounters into predator growth, and γ is predator decline.
The nonzero equilibrium is:
How to Use This Calculator
- Enter the initial prey and predator populations.
- Set α, β, δ, and γ to match your scenario.
- Choose the simulation duration and time step.
- Select RK4 for better accuracy or Euler for simplicity.
- Choose how many rows you want visible in the table.
- Press Calculate to view summary metrics, charts, and downloads.
Example Data Table
Illustrative example using x(0)=40, y(0)=9, α=0.60, β=0.03, δ=0.02, γ=0.80, duration=30, and RK4.
| Time | Prey | Predator | Invariant |
|---|---|---|---|
| 0.00 | 40.0000 | 9.0000 | -3.19944 |
| 5.00 | 26.0842 | 32.6455 | -3.19944 |
| 10.00 | 48.6171 | 9.4647 | -3.19944 |
| 15.00 | 21.8772 | 26.8544 | -3.19944 |
| 20.00 | 58.0224 | 11.1023 | -3.19944 |
| 25.00 | 20.3434 | 21.3655 | -3.19944 |
| 30.00 | 66.2474 | 14.5158 | -3.19944 |
Frequently Asked Questions
1. What does the Lotka Volterra model measure?
It models how two interacting populations change over time. In the classic version, prey grows naturally, predators decline naturally, and encounters transfer energy from prey to predators.
2. Why can both curves oscillate?
Predators increase after prey becomes abundant. Then prey drops because predation rises. Later predators fall from food shortage, allowing prey to recover. The repeated feedback creates cycles.
3. Which method should I choose?
RK4 is usually better for smooth, stable estimates because it reduces numerical error. Euler is simpler and faster, but larger time steps can distort amplitudes or even destabilize the simulation.
4. What do alpha, beta, delta, and gamma mean?
Alpha is prey growth rate. Beta is predation pressure. Delta converts prey encounters into predator growth. Gamma is predator decline rate without enough prey.
5. Why does the invariant drift matter?
The Lotka Volterra system has a conserved quantity in the ideal continuous model. Drift shows how much the numerical method moved away from that conservation, helping you judge solver accuracy.
6. Can I model extinction with this calculator?
Yes, but remember this is a simplified deterministic system. Zero or near zero populations can appear from chosen parameters or coarse steps, though real ecosystems include randomness and additional biological constraints.
7. How should I choose the step size?
Use a smaller time step when curves change rapidly or when you need better accuracy. If outputs change a lot after halving the step, your original step was probably too large.
8. Is this only for ecology?
No. The same equations also appear in chemistry, economics, epidemiology, and nonlinear systems courses whenever two coupled quantities reinforce and limit each other through interaction.