Competitive Lotka-Volterra Equations

The Competitive Lotka-Volterra Equations are a system of coupled nonlinear first-order ordinary differential equations that describe the population dynamics of two or more biological species competing for shared, limiting resources within an ecological niche. Formulated independently in the 1920s by American biophysicist Alfred J. Lotka (1925) and Italian mathematician Vito Volterra (1926), and tested empirically in laboratory microcosms by Soviet ecologist Georgii Frantsevich Gause (1932, 1934), the competitive system extends Pierre-François Verhulst's single-species logistic growth equation by incorporating linear per-capita reductions in population growth caused by both conspecific and heterospecific individuals. Alongside the predator-prey Lotka-Volterra equations, this formulation provides the mathematical foundation for the competitive exclusion principle, phase-plane isocline analysis, resource partitioning, and modern coexistence theory in community ecology.

Unlike trophic models in which one species consumes another directly, the competitive equations assume that interactions among competing populations are mutually inhibitory. Each individual added to either population diminishes the per-capita rate of increase of both its own species (intraspecific competition) and its competitors (interspecific competition) relative to each species' environmental carrying capacity.

Coupled Two-Species Differential Equations

For a community composed of two competing species with population densities (or biomasses) N₁(t) and N₂(t) at time t, the continuous-time Competitive Lotka-Volterra system is written in terms of intrinsic growth rates (r_i), carrying capacities (K_i), and dimensionless competition coefficients (α_ij):

dN₁ / dt = r₁ N₁ [ 1 − ( N₁ + α₁₂ N₂ ) / K₁ ]

dN₂ / dt = r₂ N₂ [ 1 − ( N₂ + α₂₁ N₁ ) / K₂ ]

Each parameter in this coupled system has a distinct demographic and ecological meaning:

• r₁ and r₂ (units: time⁻¹) represent the maximum per-capita intrinsic rates of natural increase of Species 1 and Species 2 when growing in the absence of density-dependent crowding (N₁, N₂ → 0).
• K₁ and K₂ (units: individuals or biomass per unit area or volume) denote the single-species carrying capacities—the equilibrium population densities that Species 1 and Species 2 attain when cultured alone (N₂ = 0 or N₁ = 0, respectively). Within the bracketed term, the quantity 1 / K_i measures the per-capita strength of intraspecific competition (α_ii = 1 / K_i).
• α₁₂ (dimensionless) is the interspecific competition coefficient measuring the per-capita competitive effect of an individual of Species 2 on the population growth of Species 1, expressed in units of Species 1 equivalents. Specifically, adding one individual of Species 2 depresses the growth rate of Species 1 by the same amount as adding α₁₂ individuals of Species 1, reducing the residual carrying capacity by α₁₂ / K₁.
• α₂₁ (dimensionless) symmetrically measures the per-capita competitive effect of an individual of Species 1 on the population growth of Species 2, where each individual of Species 1 substitutes for α₂₁ individuals of Species 2.

Generalization to Multispecies Competitive Communities

For an assemblage of S competing species (i = 1, 2, …, S), the system generalizes to the matrix differential equation:

dN_i / dt = r_i N_i [ 1 − ( ∑_j=1^S α_ij N_j ) / K_i ]

where the diagonal elements of the interaction matrix A = [α_ij] are normalized to unity (α_ii = 1), or alternatively written in the absolute interaction form dN_i / dt = N_i (r_i − ∑_j a_ij N_j) with a_ij = r_i α_ij / K_i. In Robert MacArthur and Richard Levins's (1967) consumer-resource derivation, when two consumers utilize a continuous spectrum of renewable resources with utilization curves u₁(R) and u₂(R), the competition coefficient α₁₂ equals the convolution integral of their dietary overlap normalized by the niche breadth of Species 1.

Linear Nullclines in (N₁, N₂) State Space

The qualitative behavior of the two-species system is determined by analyzing its zero-net-growth isoclines (nullclines) in the (N₁, N₂) phase plane. Setting dN₁ / dt = 0 for N₁ > 0 and dN₂ / dt = 0 for N₂ > 0 yields two straight lines along which each population neither grows nor declines:

• Species 1 Isocline (dN₁ / dt = 0): N₁ + α₁₂ N₂ = K₁, which intersects the horizontal N₁-axis at (K₁, 0) and the vertical N₂-axis at (0, K₁ / α₁₂). Inside this boundary (closer to the origin), dN₁ / dt > 0; outside it, dN₁ / dt < 0.
• Species 2 Isocline (dN₂ / dt = 0): N₂ + α₂₁ N₁ = K₂, which intersects the vertical N₂-axis at (0, K₂) and the horizontal N₁-axis at (K₂ / α₂₁, 0). Below and to the left of this line, dN₂ / dt > 0; above and to the right, dN₂ / dt < 0.

Depending on the relative magnitudes of the axis intercepts (K₁ versus K₂ / α₂₁ on the N₁-axis, and K₂ versus K₁ / α₁₂ on the N₂-axis), the two linear isoclines arrange into four dynamical regimes.

Four Dynamical Regimes and Jacobian Stability Conditions

1. Case 1 — Competitive Exclusion of Species 2 by Species 1 (K₁ > K₂ / α₂₁ and K₁ / α₁₂ > K₂): The isocline of Species 1 lies entirely outside the isocline of Species 2 in the positive quadrant. Regardless of initial abundances (N₁(0) > 0, N₂(0) > 0), Species 1 continues to grow even after Species 2 has reached its maximum tolerable total density. Trajectories converge globally to the boundary equilibrium (N₁*, N₂*) = (K₁, 0), driving Species 2 to local extinction.
2. Case 2 — Competitive Exclusion of Species 1 by Species 2 (K₂ > K₁ / α₁₂ and K₂ / α₂₁ > K₁): The isocline of Species 2 lies entirely above and to the right of the isocline of Species 1. All interior trajectories converge globally to (N₁*, N₂*) = (0, K₂), eliminating Species 1.
3. Case 3 — Unstable Interior Equilibrium and Priority Effects (K₁ < K₂ / α₂₁ and K₂ < K₁ / α₁₂): Both species exert stronger interspecific competition than intraspecific competition (α₁₂ > K₁ / K₂ and α₂₁ > K₂ / K₁, requiring α₁₂ α₂₁ > 1). The two isoclines cross at an interior equilibrium (N₁*, N₂*), but Jacobian eigenvalue analysis shows that this interior point is an unstable saddle point (determinant det(J) < 0). A separatrix passing through the origin and the saddle point partitions the phase plane into two basins of attraction, producing founder-controlled priority effects and alternative stable states: whichever species starts with a numerical advantage excludes the other.
4. Case 4 — Globally Stable Interior Coexistence (K₁ < K₂ / α₂₁ and K₂ < K₁ / α₁₂ reversed, i.e., K₁ / α₁₂ > K₂ and K₂ / α₂₁ > K₁): Each species inhibits its own growth more strongly than it inhibits the growth of its competitor (α₁₂ < K₁ / K₂ and α₂₁ < K₂ / K₁, which requires the reciprocal invasibility condition α₁₂ α₂₁ < 1). Both boundary equilibria (K₁, 0) and (0, K₂) are unstable against invasion when the rare species enters at low density, and all interior trajectories converge to a globally asymptotically stable coexistence point:

N₁* = ( K₁ − α₁₂ K₂ ) / ( 1 − α₁₂ α₂₁ ) ,     N₂* = ( K₂ − α₂₁ K₁ ) / ( 1 − α₁₂ α₂₁ )

Gause's Paramecium Microcosms and Tribolium Priority Experiments

In his classic 1934 experiments in Moscow, G. F. Gause cultured bacterivorous ciliate protozoans in buffered oatmeal broth replenished daily with Bacillus subtilis. When Paramecium aurelia and Paramecium caudatum were grown separately, each followed a logistic growth curve to its respective carrying capacity K_i. When inoculated together in a homogeneous test tube, both species competed for the same suspended bacterial resource: P. aurelia exhibited a higher per-capita growth rate and lower maintenance threshold, corresponding to Case 1 (K₁ > K₂ / α₂₁), and drove P. caudatum to extinction within 16 days. Conversely, when P. caudatum was paired with Paramecium bursaria—which harbors endosymbiotic Chlorella algae and forages on settled yeast cells in the oxygen-poor bottom layer while P. caudatum feeds in the aerated upper column—their spatial resource partitioning yielded α₁₂ α₂₁ < 1 (Case 4), enabling persistent coexistence.

Case 3 (unstable saddle point with priority effects) was demonstrated experimentally by Thomas Park (1948, 1954) using confused flour beetles (Tribolium confusum) and red flour beetles (Tribolium castaneum), which engage in reciprocal cannibalism and interspecific predation on eggs and pupae. Under intermediate temperature and humidity regimes, α₁₂ α₂₁ > 1, and initial abundance ratios determined which beetle species eliminated the other.

Mapping Lotka-Volterra Parameters onto Chesson's Modern Coexistence Theory

In Peter Chesson's (2000) Modern Coexistence Theory, the Competitive Lotka-Volterra parameters decompose directly into two orthogonal coexistence metrics across the biosphere:

• Stabilizing Niche Overlap (ρ) and Niche Difference (1 − ρ): Defined as the geometric mean of the interspecific relative to intraspecific competition coefficients:

ρ = √( ( α₁₂ α₂₁ ) / ( α₁₁ α₂₂ ) ) = √( α₁₂ α₂₁ )     (when α₁₁ = α₂₂ = 1)

When two species occupy identical niches (ρ = 1), niche difference 1 − ρ vanishes and coexistence is impossible. As species partition resources or natural enemies, ρ → 0 and stabilizing niche differences 1 − ρ approach 1.

• Average Fitness Ratio (κ₂ / κ₁): Quantifies the intrinsic competitive asymmetry between Species 2 and Species 1 in the absence of niche differentiation:

κ₂ / κ₁ = ( K₂ / K₁ ) · √( α₁₂ / α₂₁ )

The Lotka-Volterra mutual invasibility criteria (K₁ / α₁₂ > K₂ and K₂ / α₂₁ > K₁) condense into the single compact inequality:

ρ < κ₂ / κ₁ < 1 / ρ

Thus, stable coexistence requires that stabilizing niche differences (1 − ρ) be sufficiently large to overcome the equalizing fitness disparity (κ₂ / κ₁) between competing taxa, linking continuous Lotka-Volterra differential equations directly to discrete annual plant competition models such as the multispecies Beverton-Holt model and spatial Levins model dynamics.

1. Gause, G. F. (1934). The Struggle for Existence. Williams & Wilkins, Baltimore. https://doi.org/10.5962/bhl.title.4489
2. MacArthur, R., & Levins, R. (1967). The limiting similarity, convergence, and divergence of coexisting species. The American Naturalist, 101(921), 377–385. https://doi.org/10.1086/282505
3. Chesson, P. (2000). Mechanisms of maintenance of species diversity. Annual Review of Ecology and Systematics, 31(1), 343–366. https://doi.org/10.1146/annurev.ecolsys.31.1.343
4. Letten, A. D., Ke, P. J., & Fukami, T. (2017). Linking modern coexistence theory and contemporary niche theory. Ecological Monographs, 87(2), 161–177. https://doi.org/10.1002/ecm.1242