# Lotka–Volterra Interspecific Competition Simulator RK4 simulation of two competing species with phase plane, isoclines, equilibrium, and automatic outcome classification. > Canonical page: https://elysiatools.com/en/tools/lotka-volterra-competition - **Category:** Health - **Keywords:** lotka volterra, interspecific competition, competitive exclusion, population dynamics, phase plane, zero growth isocline, niche partitioning, coexistence equilibrium, ecology simulator, population ecology, competition model, rk4 ## Overview Simulate two competing species with the Lotka–Volterra interspecific competition model. Enter growth rates, carrying capacities, competition coefficients, and starting populations; RK4 integration produces N(t) time series, a phase-plane trajectory with zero-growth isoclines, the coexistence equilibrium when it exists, and automatic classification of the outcome (stable coexistence, competitive exclusion, or bistable priority effect). ## Inputs - **Intrinsic growth rate r₁ (species 1)** (number): Per-capita growth rate of species 1 with no competition. - **Intrinsic growth rate r₂ (species 2)** (number): Per-capita growth rate of species 2 with no competition. - **Carrying capacity K₁ (species 1)** (number): Population species 1 reaches alone. - **Carrying capacity K₂ (species 2)** (number): Population species 2 reaches alone. - **Competition coefficient α₁₂ (effect of species 2 on 1)** (number): Competitive impact of one species-2 individual on species 1, in K₁ units. α₁₂ = 1 means individuals are interchangeable. - **Competition coefficient α₂₁ (effect of species 1 on 2)** (number): Competitive impact of one species-1 individual on species 2, in K₂ units. - **Initial population N₁(0)** (number): Starting population of species 1 (0 allowed to test invasion-free scenarios). - **Initial population N₂(0)** (number): Starting population of species 2 (0 allowed). - **Simulation duration (time units)** (number): How long to integrate — long enough for the system to settle on its attractor (exclusion can be slow near equilibria). ## When to use - When you need to see whether two species coexist or one excludes the other from given r, K, and α values. - When you want the phase plane, isoclines, and equilibrium point rather than only population curves. - When you are checking invasion from low density or a species starting at zero. ## How it works - Set r₁, r₂, K₁, K₂, α₁₂, α₂₁, N₁(0), N₂(0), and simulation duration tMax. - The tool integrates dN₁/dt = r₁N₁(1−(N₁+α₁₂N₂)/K₁) and dN₂/dt = r₂N₂(1−(N₂+α₂₁N₁)/K₂) with RK4 over 6000 steps. - It plots both N(t) series and the N₁–N₂ phase plane with zero-growth isoclines and the simulated trajectory. - Outcome is classified from isocline geometry: stable coexistence, species-1 or species-2 exclusion, or bistable priority effect, plus N₁* and N₂* when they exist. ## Use cases - Compare mild versus strong competition coefficients to illustrate niche partitioning versus competitive exclusion. - Map isocline geometry onto the four classical outcomes for teaching population ecology. - Test whether a rare invader can increase when the resident is at carrying capacity. ## Frequently asked questions ### What equations does the simulator solve? The two-species Lotka–Volterra competition system: dN₁/dt = r₁N₁(1−(N₁+α₁₂N₂)/K₁) and dN₂/dt = r₂N₂(1−(N₂+α₂₁N₁)/K₂). ### How is the integration performed? Fourth-order Runge–Kutta (RK4) with 6000 steps over the chosen tMax. ### What outcomes can it classify? Stable coexistence (interspecific competition weaker than intraspecific on both sides), competitive exclusion of species 1 or 2, and bistable priority effect. ### Can a species start at population zero? Yes. N₁(0) or N₂(0) may be 0 to test invasion-free or single-species cases. ### How is the coexistence equilibrium calculated? N₁* = (K₁ − α₁₂K₂)/(1 − α₁₂α₂₁) and N₂* = (K₂ − α₂₁K₁)/(1 − α₁₂α₂₁) when the isoclines cross in the positive quadrant. ## Related tools - [Lean Body Mass Calculator (Boer / Hume / James)](https://elysiatools.com/en/tools/lean-body-mass-calculator): Estimate your lean body mass (LBM) from sex, age, weight and height using three clinical formulas side by side, with derived body-fat percentage. - [Extended BMR Calculator (Mifflin / Schofield Multi-Formula Comparison)](https://elysiatools.com/en/tools/basal-metabolic-rate-ext): Compare Mifflin-St Jeor, original and revised Harris-Benedict, Schofield (FAO/WHO/UNU), and Katch-McArdle BMR equations side by side with mean and spread in kcal/day. 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