Competitive Exclusion Calculator

Calculate competitive exclusion outcome using Lotka-Volterra competition equations.

Species 1 growth rate r1

Species 2 growth rate r2

Species 1 carrying capacity K1

Species 2 carrying capacity K2

Interspecific competition α (sp2 effect on sp1)

Interspecific competition β (sp1 effect on sp2)

Please check your inputs and try again.

The Four Possible Outcomes

The Lotka–Volterra competition model produces four outcomes, determined by how interspecific competition compares with intraspecific competition.

Condition	Outcome
α < K₁/K₂ and β < K₂/K₁	Stable coexistence
α > K₁/K₂ and β < K₂/K₁	Species 2 wins
α < K₁/K₂ and β > K₂/K₁	Species 1 wins
α > K₁/K₂ and β > K₂/K₁	Unstable — either may win, depending on starting densities

The underlying principle is that coexistence requires each species to limit itself more than it limits the other. When both competition coefficients are small, intraspecific competition dominates and neither can exclude its rival.

The fourth case is the interesting one: when both species compete strongly against each other, the outcome depends on initial abundances. Whichever gets established first tends to win — a priority effect.

Gause's Principle and Its Escape Routes

Complete competitors cannot coexist indefinitely on a single limiting resource. Yet ecosystems are full of apparently similar species, and several mechanisms explain how.

Mechanism	How it permits coexistence
Niche partitioning	Species use different resources or microhabitats
Temporal separation	Peak activity at different times or seasons
Environmental fluctuation	Conditions change before exclusion completes
Predation	Predators suppress the dominant competitor
Spatial heterogeneity	Different species win in different patches

Darwin's finches are the textbook example of niche partitioning through character displacement — beak sizes diverge where species co-occur, reducing dietary overlap. The plankton paradox is the counterexample that still provokes debate: dozens of species coexist on a handful of limiting nutrients, apparently sustained by environmental variation that prevents equilibrium ever being reached.

Worked Examples

Example 1: Stable coexistence: α=0.7, β=0.8, K1=200, K2=150

K1/K2=1.33>α, K2/K1=0.75<β? No

Result: Check both conditions carefully

Intraspecific > interspecific competition → coexist

Example 2: Competitive exclusion: α=1.5

α > K1/K2 → sp2 wins

Result: Interspecific competition too strong

One species eliminated

Example 3: Priority effects

α = 1.5, β = 1.4, both exceeding their thresholds

Result: Unstable — whichever establishes first wins

Strong mutual competition creates alternative stable states. This is why invasion order matters in community assembly.

Example 4: The plankton paradox

Dozens of species on few limiting nutrients

Result: Coexistence despite theory

Environmental fluctuation prevents equilibrium being reached, so competitive exclusion never completes. It remains an active area of research.

Common Mistakes

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Checking only one competition coefficient

Both conditions must hold simultaneously for coexistence. Testing α alone cannot determine the outcome.

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Assuming strong competition always excludes

When both coefficients are large the outcome is unstable and depends on initial densities. Either species may win.

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Treating the model as predictive for real communities

It assumes constant environment, no age structure and no spatial variation. Real coexistence often depends on precisely the fluctuations it excludes.

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Concluding niches are identical from apparent overlap

Species that appear to share resources usually differ in ways not immediately obvious — timing, microhabitat or preferred prey size.

Frequently Asked Questions

Gause's principle?

Two species competing for exactly the same resources cannot coexist indefinitely. Whichever has even slight advantage will exclude the other. But 'same resources': in practice, species partition resources slightly → coexistence.

Niche partitioning?

Limiting similarity: how similar can competing species be and still coexist? Character displacement: competing species evolve to use resources differently. Darwin's finches: bill size divergence reducing competition.

What does the competitive exclusion principle state?

Two species competing completely for the same limiting resource cannot coexist indefinitely. One will eventually exclude the other.

What condition allows coexistence?

Each species must limit itself more than it limits the other, meaning both competition coefficients fall below their respective carrying capacity ratios.

What happens when both coefficients are large?

The outcome is unstable and depends on initial densities. Whichever species establishes first tends to exclude the other, a priority effect.

How do similar species coexist in nature?

Through niche partitioning, temporal separation, environmental fluctuation, predation on the dominant competitor, or spatial heterogeneity.