Lotka-Volterra (LV)

Overview

Three two-species dynamical systems integrated via RK4, displayed as a live time series and phase-plane trajectory. The three modes are:

Classic predator-prey: the standard Lotka-Volterra system with no density dependence.
Dynamic predator-prey: adds logistic prey growth and predator self-limitation.
Competition: two-species Lotka-Volterra competition with carrying capacities and interaction coefficients.

The visualisation shows both how populations change over time and how they trace paths through state space (the phase plane).

Glossary

Controls: all modes

Control	Description
Speed	Playback speed: Slow / Normal / Fast.
Preset	Named parameter combinations for the current mode. Applied immediately.
\(\Delta t\)	RK4 integration time step. Smaller = more accurate but slower; larger = faster but potentially unstable. Default: 0.03.
Unbounded time	Disables the auto-pause checkpoints (default every 50 time units). Simulation runs until manually paused.

Controls: predator-prey (classic) mode

Control	Description
\(\alpha\)	Prey intrinsic growth rate: rate of increase of the prey population in the absence of predators. Default: 1.1.
\(\beta\)	Predation rate: rate at which each predator removes prey per unit time. Default: 0.06.
\(\delta\)	Predator conversion efficiency: prey consumed per new predator produced. Default: 0.03.
\(\gamma\)	Predator death rate: intrinsic rate of decrease of predators in the absence of prey. Default: 0.7.
\(N_0\)	Initial prey population. Default: 40.
\(P_0\)	Initial predator population. Default: 9.

Controls: dynamic predator-prey mode (additional)

Adds logistic prey growth and predator self-limitation to the classic system.

Control	Description
Dynamic preset	Named presets: Damped oscillations, Limit cycle, Predator collapse. All three share the same α/β/δ/γ; only K varies, isolating the effect of prey carrying capacity.
K (prey)	Prey carrying capacity: logistic ceiling for prey growth in the absence of predators. Default: 60.
m (predator)	Predator self-limitation coefficient: intraspecific competition among predators. Default: 0.00.

Controls: competition mode

Replaces the predator-prey parameters with two-species competition parameters.

Control	Description
Competition preset	Named presets: Coexistence, Species 1 wins, Species 2 wins.
\(r_1\)	Intrinsic growth rate of species 1. Default: 0.9.
\(r_2\)	Intrinsic growth rate of species 2. Default: 0.8.
\(K_1\)	Carrying capacity of species 1. Default: 75.
\(K_2\)	Carrying capacity of species 2. Default: 65.
\(\alpha_{12}\)	Effect of species 2 on species 1: per-individual competitive impact. Default: 0.60.
\(\alpha_{21}\)	Effect of species 1 on species 2: per-individual competitive impact. Default: 0.50.

Key symbols

Symbol	Meaning
\(N\)	Prey (or species 1) population size.
\(P\)	Predator (or species 2) population size.
\(\alpha\)	Prey growth rate (classic / dynamic modes).
\(\beta\)	Predation rate (classic / dynamic modes).
\(\delta\)	Predator conversion efficiency (classic / dynamic modes).
\(\gamma\)	Predator death rate (classic / dynamic modes).
\(K\)	Carrying capacity.
\(m\)	Predator self-limitation (dynamic mode).
\(r_1, r_2\)	Species intrinsic growth rates (competition mode).
\(K_1, K_2\)	Species carrying capacities (competition mode).
\(\alpha_{12}, \alpha_{21}\)	Interspecific competition coefficients (competition mode).
\(\Delta t\)	RK4 integration time step.

Numerical integration

All three modes are integrated using RK4 (classical fourth-order Runge-Kutta). The integration is in src/lv-engine.js.

RK4 step

For a state vector \((N, P)\) at time \(t\) with step size \(\Delta t\):

\[ k_1 = f(N, P) \]

\[ k_2 = f\!\left(N + \tfrac{\Delta t}{2} k_{1,N},\ P + \tfrac{\Delta t}{2} k_{1,P}\right) \]

\[ k_3 = f\!\left(N + \tfrac{\Delta t}{2} k_{2,N},\ P + \tfrac{\Delta t}{2} k_{2,P}\right) \]

\[ k_4 = f\!\left(N + \Delta t\, k_{3,N},\ P + \Delta t\, k_{3,P}\right) \]

\[ N_{t+\Delta t} = N + \frac{\Delta t}{6}(k_{1,N} + 2k_{2,N} + 2k_{3,N} + k_{4,N}) \]

\[ P_{t+\Delta t} = P + \frac{\Delta t}{6}(k_{1,P} + 2k_{2,P} + 2k_{3,P} + k_{4,P}) \]

After each step, both populations are clamped to zero from below:

return {
  prey: Math.max(0, nextPrey),
  predator: Math.max(0, nextPred),
};

This prevents negative populations from numerical overshoot. In classic LV with large \(\beta\) or small initial populations, trajectories can approach zero and this clamp becomes relevant.

Time step \(\Delta t\)

Slider range: 0.005 to 0.080, step 0.001, default 0.030. This is the integration step size in units of model time. Decreasing \(\Delta t\) improves numerical accuracy at the cost of more computation per second of real time. Changing \(\Delta t\) triggers a Reset.

Speed and steps per frame

The simulation advances by a fixed number of RK4 steps per animation frame (requestAnimationFrame):

Speed	Steps per frame
Slow	1
Normal	2
Fast	6

At Normal speed and default \(\Delta t = 0.030\), the simulation advances by 0.060 model-time units per frame, approximately 3.6 model-time units per second at 60 fps.

Mode equations

Classic predator-prey

\[ \frac{dN}{dt} = \alpha N - \beta N P \]

\[ \frac{dP}{dt} = \delta N P - \gamma P \]

Parameters:

Symbol	Role	Default	Range	Step
\(\alpha\)	Prey intrinsic growth rate	1.10	0.2–2.0	0.01
\(\beta\)	Predation rate coefficient	0.06	0.01–0.20	0.001
\(\delta\)	Predator conversion efficiency	0.03	0.005–0.10	0.001
\(\gamma\)	Predator mortality rate	0.70	0.1–2.0	0.01

The system has two equilibria. The trivial one is \((0, 0)\). The non-trivial one is:

\[ N^* = \frac{\gamma}{\delta}, \quad P^* = \frac{\alpha}{\beta} \]

In the ideal system (no numerical error, no clamping) trajectories are closed neutral cycles around \((N^*, P^*)\). In practice, RK4 at finite \(\Delta t\) introduces small orbital drift.

Classic presets

Name	\(\alpha\)	\(\beta\)	\(\delta\)	\(\gamma\)	\(N_0\)	\(P_0\)	\(\Delta t\)
Balanced cycles	1.10	0.060	0.030	0.70	40	9	0.030
Predator crash	0.30	0.010	0.003	1.30	55	22	0.020
Prey crash	0.45	0.160	0.060	0.40	22	26	0.020
High-amplitude cycles	1.30	0.080	0.028	0.75	55	7	0.020

Dynamic predator-prey

Adds logistic prey growth and predator self-limitation:

\[ \frac{dN}{dt} = \alpha N\!\left(1 - \frac{N}{K}\right) - \beta N P \]

\[ \frac{dP}{dt} = \delta N P - \gamma P - m P^2 \]

Additional parameters:

Symbol	Role	Default	Range	Step
\(K\)	Prey carrying capacity	60	20–250	1
\(m\)	Predator self-limitation	0.000	0.000–0.200	0.001

In the engine, \(K\) is guarded: kSafe = max(1e-6, Kdyn) and mSafe = max(0, mPred) to prevent division by zero or negative values.

Interior equilibrium (when \(m > 0\)):

The analytics panel computes this numerically. For \(m > 0\):

\[ N^* = \frac{\gamma + m \alpha / \beta}{\delta + m \alpha / (\beta K)}, \quad P^* = \max\!\left(0,\ \frac{\delta N^* - \gamma}{m}\right) \]

This system can exhibit damped spirals, limit cycles, or collapse depending on parameter choice, unlike the neutral cycles of classic LV.

The key stability criterion is whether \(N^* > K/2\). When \(N^* > K/2\), the interior equilibrium is stable and trajectories spiral inward (damped oscillations). When \(N^* < K/2\), a limit cycle emerges (paradox of enrichment). When \(K < N^* = \gamma/\delta\) (no \(m\)), the prey carrying capacity is too low to sustain predators and they collapse.

Dynamic presets

All three presets share the same \(\alpha\), \(\beta\), \(\delta\), \(\gamma\), so \(N^* = \gamma/\delta = 35\) is fixed. Only \(K\) varies, making the comparison clean.

Name	\(\alpha\)	\(\beta\)	\(\delta\)	\(\gamma\)	\(K\)	\(N_0\)	\(P_0\)	\(\Delta t\)
Damped oscillations	0.80	0.050	0.020	0.70	60	10	2	0.030
Limit cycle	0.80	0.050	0.020	0.70	200	50	3	0.020
Predator collapse	0.80	0.050	0.020	0.70	25	10	15	0.030

Damped oscillations: \(N^* = 35 > K/2 = 30\), so the equilibrium is stable. Limit cycle: \(N^* = 35 \ll K/2 = 100\), so a limit cycle forms (paradox of enrichment). Predator collapse: \(K = 25 < N^* = 35\), so prey carrying capacity is too low to sustain predators.

Competition

\[ \frac{dN_1}{dt} = r_1 N_1\!\left(1 - \frac{N_1 + \alpha_{12} N_2}{K_1}\right) \]

\[ \frac{dN_2}{dt} = r_2 N_2\!\left(1 - \frac{N_2 + \alpha_{21} N_1}{K_2}\right) \]

Parameters:

Symbol	Role	Default	Range	Step
\(r_1\)	Species 1 growth rate	0.90	0.1–2.0	0.01
\(r_2\)	Species 2 growth rate	0.80	0.1–2.0	0.01
\(K_1\)	Species 1 carrying capacity	75	10–200	1
\(K_2\)	Species 2 carrying capacity	65	10–200	1
\(\alpha_{12}\)	Effect of species 2 on species 1	0.60	0.00–2.00	0.01
\(\alpha_{21}\)	Effect of species 1 on species 2	0.50	0.00–2.00	0.01

In the engine, both \(K\) values are guarded: max(1e-6, K1) and max(1e-6, K2).

Coexistence equilibrium (when \(1 - \alpha_{12}\alpha_{21} \neq 0\)):

\[ N_1^* = \frac{K_1 - \alpha_{12} K_2}{1 - \alpha_{12}\alpha_{21}}, \quad N_2^* = \frac{K_2 - \alpha_{21} K_1}{1 - \alpha_{12}\alpha_{21}} \]

The analytics panel computes and displays these values live. The condition for stable coexistence is \(\alpha_{12} < K_1/K_2\) and \(\alpha_{21} < K_2/K_1\).

Competition presets

Name	\(r_1\)	\(r_2\)	\(K_1\)	\(K_2\)	\(\alpha_{12}\)	\(\alpha_{21}\)	\(N_{1,0}\)	\(N_{2,0}\)	\(\Delta t\)
Coexistence	0.90	0.80	75	65	0.60	0.50	40	35	0.030
Species 1 wins	1.00	0.70	85	60	0.45	1.35	55	45	0.030
Species 2 wins	0.70	1.00	60	85	1.30	0.45	45	55	0.030

State and time series

The state singleton (lv/state.js) holds current population values, the full integration parameters, and a rolling series buffer.

series: array of { t, prey, predator } records, capped at 1600 points by state.series.shift() when the cap is exceeded. The oldest point is removed first (FIFO). At Normal speed and default \(\Delta t\), 1600 points corresponds to approximately 48 model-time units of history. Used by the time-series chart.
phaseSeries: separate array of { prey, predator } records for the phase-plane chart, capped at 200,000 points. Because the cap is much higher, the phase plane retains the full trajectory of long runs where the time-series chart would have already scrolled.
t: current simulation time. Cumulative across runs until a Reset.
running: boolean.

Checkpoint auto-pause

By default, the simulation pauses automatically every 50 model-time units. This gives users a chance to observe the state before it evolves further.

pauseInterval = 50 (fixed).
nextPauseAt is advanced by pauseInterval each time a pause triggers.
When paused by checkpoint, autoPausedAt is set and the play button changes to “▶ Resume”.
Unbounded time checkbox disables the checkpoint: setUnboundedTime(true) clears autoPausedAt and prevents the pause check.
Toggling unbounded time off during a run advances nextPauseAt to the next 50-boundary above the current t.

The exact pause logic in advanceSimulation:

if (!state.unboundedTime && state.t >= state.nextPauseAt) {
  state.running = false;
  state.autoPausedAt = state.nextPauseAt;
  state.nextPauseAt += state.pauseInterval;
  break;  // stop the step loop for this frame
}

Controls

Parameter sliders: behaviour on input vs change

All parameter sliders use a two-event pattern:

input event: updates the parameter value and display label immediately (live preview of the number).
change event (fires when slider drag ends): sets running = false and calls resetState().

This means moving a slider always resets the simulation when you release it. There is no live-update mode for parameters in LV.

Model type

Switching model type stops the simulation, applies the default preset for the new mode, and resets. The Complications panel shows only the section relevant to the active mode.

Reset vs Reset Defaults

Reset: resets simulation time and populations to current initial conditions (prey0, predator0) without changing any parameters.
Reset Defaults: restores all parameters to the default preset for the active mode, then resets. For classic PP this is the “Balanced” preset; for dynamic PP the “Damped oscillations” preset; for competition the “Coexistence” preset.

Speed buttons

Speed change does not reset the simulation. It changes SPEED_TO_STEPS selection for the next animation frame.

Analytics panels

lv/analytics.js subscribes to state and re-renders both charts on every notification (which fires after each advanceSimulation call, after resets, and after parameter changes).

Time series chart

Blue line: prey (species 1 in competition mode). Red line: predators (species 2). X-axis: time window over the visible series buffer (scrolling). Y-axis: maximum population across both series, with 5% headroom. The chart is rebuilt from scratch each frame (no incremental D3 update).

The series scrolls naturally because old points are dropped from series when the cap is hit: the x-axis domain always spans [min(t), max(t)] of the current buffer.

Phase-plane chart

X-axis: prey (or species 1). Y-axis: predators (or species 2). The trajectory is drawn as a single path through all points in the phaseSeries buffer (up to 200,000 points). A filled black dot marks the current position (last point). The chart is also rebuilt from scratch each frame. Axes expand to fit the current trajectory extent with 5% headroom.

Estimates strip

Displays current \(t\), current \(N\) (prey), current \(P\) (predators), and analytical equilibrium values (\(N^*\), \(P^*\)). The equilibria are recomputed on every state notification from the current parameter values using the closed-form expressions above.

Assumptions and limitations

The model is deterministic; there is no demographic stochasticity. Populations can reach fractional values.
Non-negativity is enforced by clamping, not by the ODEs. In particular, classic LV has no built-in carrying capacity on prey; prey grows without bound if predator density is very low.
In classic LV, the interior equilibrium is a centre (neutral stability). Integration with finite \(\Delta t\) can produce slow orbital drift. Smaller \(\Delta t\) reduces but does not eliminate this.
The 1600-point series cap means long runs lose early history in the time-series chart. The phase plane uses a separate phaseSeries buffer (200,000 points) and retains the full trajectory of typical runs.
Parameters update live in the display but always trigger a full reset when the slider is released.