Rate of change of prey (x). α is the intrinsic growth rate (reproduction without threats), and the term βxy models mortality proportional to encounters with predators (β is the predation rate).
Predator Dynamics
dtdy=δxy−γy
Rate of change of predators (y). δ is the efficiency with which consumed prey biomass is converted into new predators. γ is the natural mortality rate of predators (starvation in the absence of prey).
Equilibrium Points
(x∗,y∗)=(δγ,βα)
Non-trivial steady state where both derivatives are zero. It represents the perfect balance of the ecosystem. There is another trivial equilibrium at (0,0) (total extinction).
Linearized Period
T≈αγ2π
Approximation of the time it takes to complete one biological cycle. It is mathematically valid only for tiny oscillations near the equilibrium. For large orbits, the system is highly non-linear and the actual period diverges significantly.
1. Phase Space and Closed Orbits
A fundamental mathematical property of the original Lotka-Volterra equations is that they form a conservative system (they possess a constant of motion).
▸Perpetual cycle: Regardless of the initial conditions (as long as x,y>0), the populations will trace a closed orbit on the x vs y graph. They will never reach equilibrium nor mathematically go extinct.
▸Natural phase lag: The populations are in phase quadrature. The predator peak always occurs with a delayed lag after the prey peak.
2. Numerical Resolution (RK4 Method)
Since the non-linear system lacks an explicit analytical solution over time, this calculator employs the 4th-order Runge-Kutta (RK4) method for numerical integration. The use of this algorithm ensures minimal truncation error, avoiding numerical drift and preserving the topology of closed orbits, which lower-order methods like Euler fail to sustain under extreme parameters.
3. Limitations of the Classical Continuous Model
The model assumes an idealized ecosystem. It is crucial to consider the following limitations when interpreting the results:
①The fractional numbers paradox: The model is continuous, treating the population as a fluid. You may notice that the results yield population minimums like 10−18. Mathematically the curve survives, but in discrete biological reality, any value below 1 individual means irreversible extinction (flagged by our interface warnings).
②Infinite prey growth: In the absence of predators (y=0), prey grows exponentially to infinity. The model omits the environmental carrying capacity (logistic growth).
③Predator satiation: The consumption rate is βxy, assuming predators have an infinite appetite (Type I Functional Response). In reality, there is a satiation limit and prey handling time (Holling Type II).
4. How to read the graphs
Time Evolution: Shows the population history. The waves demonstrate the biological chain reaction: prey abundance generates a subsequent predator peak, which in turn causes a massive prey collapse, inevitably followed by predator starvation.
Time Units: The temporal axis has no fixed universal unit (days, months, years). It is an arbitrary unit that inherits its scale from the input rates. If the growth (α) and mortality (γ) rates are measured in "per year" (e.g., lynxes and hares), the X-axis represents years. If measured in "per hour" (microorganisms), the X-axis will indicate hours.
Phase Space: Removes the explicit time axis to show pure interaction. Time is "hidden" and flows by traveling along the curve (usually counter-clockwise). Every point on the orbit represents a snapshot of the entire ecosystem at an exact instant. The further the orbit stretches from the central equilibrium dot, the more severe and dangerous the fluctuations are.