The Lotka-Volterra competition model describes how two species sharing a limited resource grow logistically but inhibit each other's growth through interspecific competition. For bacteria, the governing equations are dN₁/dt = r₁N₁(K₁ − N₁ − α₁₂N₂)/K₁ and dN₂/dt = r₂N₂(K₂ − N₂ − α₂₁N₁)/K₂, where K is the carrying capacity (maximum population a nutrient supply can support), r is the intrinsic growth rate, and α is the competition coefficient representing how much one strain suppresses the other. The competitive exclusion principle (Gause, 1934) states that two species competing for a single limiting resource cannot coexist indefinitely — one will always drive the other to extinction.
This simulation allows you to tune growth rates (r1, r2), carrying capacities (K1, K2), and competition coefficients (α₁₂, α₂₁) for two bacterial strains. The phase plane displays nullclines — the lines along which each population's growth rate is zero — whose arrangement predicts whether the outcome will be coexistence, competitive exclusion by either strain, or an unstable founder-effect equilibrium where the initial population sizes determine the winner.
What is the competitive exclusion principle?
Gause's competitive exclusion principle (1934) states that two species competing for exactly the same limiting resource in the same niche cannot stably coexist — one will be driven to local extinction. This was confirmed experimentally with Paramecium species competing for bacteria in controlled flasks. The principle implies that stable coexistence requires some degree of resource partitioning or niche differentiation.
What conditions allow stable coexistence?
Stable coexistence occurs when each species inhibits itself more than it inhibits its competitor: K₁/α₁₂ > K₂ AND K₂/α₂₁ > K₁. Geometrically, the nullclines cross in the interior of the phase plane and the intersection is a stable equilibrium point. Biologically, this means interspecific competition is weaker than intraspecific competition — each species "gets in its own way" more than it impedes the other.
What are nullclines and what do they show?
Nullclines are curves along which one population's growth rate is zero. The cyan line (N₁ nullcline) shows where dN₁/dt = 0: K₁ − N₁ − α₁₂N₂ = 0, or N₂ = (K₁ − N₁)/α₁₂. The red line (N₂ nullcline) similarly shows where dN₂/dt = 0. The intersection of the nullclines, if it exists in the positive quadrant, is the potential coexistence equilibrium, and its stability depends on the angle of crossing.
When neither coexistence condition holds — that is, both species can competitively exclude the other — the system reaches an unstable equilibrium. The actual winner then depends entirely on initial population sizes: whichever strain starts with a larger population relative to the unstable equilibrium point will drive the other to extinction. This is called the founder effect or priority effect, and is relevant to gut microbiome colonisation and hospital infection control.
The coefficient α₁₂ measures how strongly Species 2 suppresses Species 1, expressed in units of "Species 1 equivalents". If α₁₂ = 0.8, each individual of Species 2 suppresses Species 1 as much as 0.8 individuals of Species 1 would. Values greater than 1 indicate that Species 2 is a stronger competitor for the shared resource than Species 1 itself.
In a two-strain system, a pulse of antibiotic that kills the dominant competitor can cause "ecological release" — the previously suppressed strain undergoes rapid population growth into the newly available niche. This is clinically important: broad-spectrum antibiotics that eliminate commensal bacteria can allow resistant pathogens like Clostridioides difficile to expand, causing secondary infection.
K represents the maximum population size supported by the nutrient supply. A higher K₁ shifts the N₁ nullcline outward, making it harder for Species 2 to displace Species 1. When K₁/K₂ > α₁₂ (Species 1 has a much larger carrying capacity), Species 1 will dominate regardless of the competition coefficient. This is why fast-growing bacteria with access to abundant nutrients tend to competitively exclude slower-growing specialists.
Yes. With three or more species, coexistence is possible through rock-paper-scissors dynamics: Species A beats B, B beats C, but C beats A. This has been observed in E. coli strains competing via toxin production (colicins) in laboratory experiments. Non-transitive competition allows stable species-rich communities and is one explanation for the biodiversity of the human gut microbiome.
The coexistence equilibrium (N₁*, N₂*) is found by solving the two nullcline equations simultaneously: N₁* = (K₁ − α₁₂K₂)/(1 − α₁₂α₂₁) and N₂* = (K₂ − α₂₁K₁)/(1 − α₁₂α₂₁). The simulation marks this point with a yellow circle on the phase plane when it falls in the biologically valid region (both populations positive). When the denominator is near zero the equilibrium shifts to extreme values or becomes biologically meaningless.
Exponential growth (dN/dt = rN) assumes unlimited resources and leads to infinite populations, which is unrealistic. Logistic growth adds a density-dependent term that slows growth as N approaches K, reflecting nutrient depletion, waste accumulation, and space limitation. For bacteria in a closed culture (such as a microtiter plate), the logistic model captures the sigmoidal growth curve from lag phase through exponential phase to stationary phase.