Limiting Similarity

Limiting similarity is a foundational theoretical concept in community ecology and niche theory—formulated mathematically by Robert H. MacArthur and Richard Levins in 1967 and extended to stochastic environments by Robert M. May and MacArthur in 1972—that seeks to determine the maximum quantitative degree of ecological similarity (or minimum niche separation d / w) permissible between competing species before interspecific competition drives one or more species to local extinction via competitive exclusion (MacArthur & Levins, 1967; May & MacArthur, 1972; Abrams, 1983). Whereas Gause's Law states qualitatively that complete competitors cannot coexist, the theory of limiting similarity asks a quantitative question central to biodiversity and ecological assembly rules within an ecosystem: How different must two species be along a continuous resource continuum (such as seed size, prey body length, or rooting depth) to coexist stably? In trait-based community assembly, limiting similarity predicts that interspecific competition acts as a biotic filter that eliminates intermediate phenotypes, producing functional trait overdispersion (even spacing of morphological traits) and ecological character displacement among co-occurring guild members (Hutchinson, 1959; Dayan & Simberloff, 2005).

"Homage to Santa Rosalia" and the Empirical 1.28 Body-Size Rule

The empirical motivation for limiting similarity was introduced in 1959 by G. Evelyn Hutchinson in his celebrated address "Homage to Santa Rosalia, or Why Are There So Many Kinds of Animals?" Measuring the trophic appendages (skull length in mustelid carnivores and mandibular/culmen length in congeneric birds and aquatic Corixa water boatmen) of sympatric species pairs sharing the same foraging guild, Hutchinson observed that the ratio of the larger species' trophic linear dimension to the smaller species' dimension consistently averaged between 1.1 and 1.4, with a mean ratio of 1.28 (corresponding to an approximate 2 : 1 ratio in body mass, since mass scales as length cubed, 1.28³ ≈ 2.1; Hutchinson, 1959). Hutchinson proposed that a linear trophic size ratio of ~1.3 represents the empirical minimum morphological separation required for sympatric congeners to partition prey size classes and coexist in nature.

The MacArthur and Levins (1967) Gaussian Resource-Utilization Model

To derive a rigorous mathematical foundation for Hutchinson's observation, Robert H. MacArthur and Richard Levins (1967) modeled three species—two established resident species (Species 1 and Species 3) and a candidate intermediate species (Species 2) attempting to invade—competing along a continuous one-dimensional resource axis x (for example, log seed diameter or prey size):

1. Gaussian Resource Utilization Curves (U_i(x)): Each species i consumes resources along axis x according to a bell-shaped Gaussian utilization distribution with mean resource optimum μ_i and niche breadth (standard deviation) w, separated from its adjacent neighbor's optimum by distance d = |μ_i − μ_j|:

U_i(x) = [ 1 / (w √(2π)) ] · exp[ − (x − μ_i)² / (2w²) ]

1. Convolution Integral for the Competition Coefficient (α_ij): MacArthur and Levins defined the Lotka–Volterra interspecific competition coefficient α_ij as the normalized overlap integral (convolution) of the two species' resource-utilization curves:

α(d / w) = [ ∫ U_i(x) U_j(x) dx ] / [ ∫ U_i(x)² dx ] = exp[ − d² / (4w²) ]

1. Invasion Threshold for the Middle Species (K₁ = K₂ = K₃): Suppose Resident 1 (at μ₁ = −d) and Resident 3 (at μ₃ = +d) are at equilibrium with carrying capacities K₁ = K₃ = K, and an intermediate Species 2 (centered at μ₂ = 0, distance d from both neighbors) attempts to invade with carrying capacity K₂. Using the three-species Competitive Lotka-Volterra Equations, Species 2 achieves a positive invasion growth rate (dN₂/dt > 0 when N₂ → 0) if and only if:

K₂ / K > 2α / (1 + α²), where α = exp(−d² / 4w²) (assuming negligible overlap α⁴ between Species 1 and 3). When the middle species has a slightly lower or sensitive carrying capacity K₂, or when environmental variation perturbs K, MacArthur and Levins showed that stable species packing breaks down once 2α / (1 + α²) approaches K₂/K—yielding a typical minimum niche separation threshold of d / w ≥ 1.0 to 1.5 (meaning the distance d between adjacent species' resource optima must be at least 1 to 1.5 standard deviations of their niche width w; MacArthur & Levins, 1967).

A mathematical subtlety of the deterministic Lotka–Volterra equations is that if all S species along a resource continuum have specifically identical carrying capacities (K₁ = K₂ = ... = K_S) matching the resource supply curve, the competition matrix A (with entries α_ij = exp[−(i − j)²d² / 4w²]) is positive definite for any non-zero spacing d > 0—seemingly implying that infinitely many species could be packed arbitrarily close together in a perfectly constant world!

In 1972, Robert M. May and Robert H. MacArthur resolved this paradox in "Niche Overlap as a Function of Environmental Variability." They added stochastic environmental white noise of variance σ² to the resource supply / carrying capacities K_i(t) = K̄_i + γ_i(t) and analyzed the stochastic Fokker–Planck stability of an S-species guild:

• For all S species to avoid stochastic extinction, the smallest eigenvalue (λ_min) of the S × S Gaussian competition matrix A must remain larger than the environmental variance (λ_min > σ² / K̄).
• Because λ_min ≈ 2 √(π) (w / d) exp(−π² w² / d²) vanishes exponentially fast as d / w drops below 1, λ_min transitions from near 0 (for d / w < 0.8) to ~1 (for d / w > 1.5) over a remarkably narrow transition zone around d / w ≈ 1.0.
• Consequently, across a wide range of environmental noise levels (σ² spanning two orders of magnitude), niche separation cannot fall below approximately one standard deviation (d / w ≈ 1.0) without triggering rapid stochastic extinction of intermediate species (May & MacArthur, 1972).

Simberloff and Boecklen (1981), Abrams (1983), and Chesson (2000)

During the late 1970s and 1980s, the universality of a fixed "1.3 size ratio" or "d / w = 1" constant was rigorously refined on both statistical and mechanistic grounds:

1. Null-Model Testing of Size Ratios (Simberloff & Boecklen, 1981; Dayan & Simberloff, 2005): Daniel Simberloff and William Boecklen (1981) tested published claims of constant Hutchinsonian 1.3 size ratios against null models in which S species log-body-sizes are drawn uniformly at random across a bounded interval. While many casual whole-body-mass comparisons did not depart from random draws, Tamar Dayan and Daniel Simberloff (2005) showed that when measurements are restricted to the actual trophic killing or crushing apparatus—such as the cross-sectional diameter of the upper canine tooth (C¹) in sympatric felids and mustelids (Felis, Mustela, Martes), or beak depth in Galápagos Geospiza finches—coexisting guild members exhibit statistically significant minimum segment-length overdispersion (even spacing) that disappears in allopatry (character displacement).
2. Synthesis with Modern Coexistence Theory (Abrams, 1983; Chesson, 2000): Peter A. Abrams (1983) and Peter L. Chesson (2000) demonstrated that there is no single universal "magic number" for limiting similarity because coexistence depends on the ratio between stabilizing niche overlap (ρ = α) and average fitness inequality (κ_j / κ_i): ρ < κ₁/κ₂ < 1/ρ. When two competing species have nearly identical demographic fitness (κ₁ ≈ κ₂), they can coexist stably with very high niche overlap (d / w ≪ 1); conversely, if one species has a large fitness advantage (κ₂ ≫ κ₁), even a large niche separation (d / w > 2) may be insufficient to prevent exclusion.
3. Self-Organized Trait Clumping ("Emergent Neutrality" — Scheffer & van Nes, 2006): Analyzing continuous Lotka–Volterra competition along a trait axis with dozens of invading species, Marten Scheffer and Egbert H. van Nes (2006) discovered that before reaching a final evenly spaced equilibrium, communities spontaneously self-organize into discrete, evenly spaced clumps (clusters) of species along the body-size axis! Between adjacent clumps, species obey MacArthur–Levins limiting similarity (d / w ≈ 1.5); within each clump, several species share nearly identical traits and equal fitness, coexisting for thousands of generations via slow neutral drift—explaining why natural lake plankton and beetle communities display multimodal "lumpy" body-size spectra.

What does the ratio d / w ≈ 1 mean in the MacArthur–Levins and May–MacArthur models?

In the Gaussian resource-utilization model, d is the distance between the peak resource optima of two adjacent competing species (for example, the difference in mean seed size eaten by two finch species), and w is the standard deviation (niche width) of each species' diet around its mean. The rule d / w ≥ 1 states that for a chain of competing species to coexist stably in a fluctuating environment, the distance between neighboring species' peak diets must be at least as large as one standard deviation of their individual diet breadth.

How do ecologists test for limiting similarity in functional trait data?

Whereas environmental filtering reduces the total range of traits at a site, limiting similarity acts at the fine neighborhood scale to prevent co-occurring species from having identical traits. Ecologists test for limiting similarity by computing the Mean Nearest Taxon/Trait Distance (MNTD) or the Standard Deviation of Nearest-Neighbor Trait Distances (SDNND) along a sorted trait axis: when SDNND is significantly smaller than in randomized null communities, coexisting species are spaced more evenly (overdispersed) along the trait axis than expected by chance (Cornwell & Ackerly, 2009).

1. Hutchinson, G. E. (1959). "Homage to Santa Rosalia or why are there so many kinds of animals?" The American Naturalist, 93(870), 145–159. https://doi.org/10.1086/282070
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. May, R. M., & MacArthur, R. H. (1972). "Niche overlap as a function of environmental variability." Proceedings of the National Academy of Sciences, 69(5), 1109–1113. https://doi.org/10.1073/pnas.69.5.1109
4. Scheffer, M., & van Nes, E. H. (2006). "Self-organized similarity, the evolutionary emergence of groups of similar species." Proceedings of the National Academy of Sciences, 103(16), 6230–6235. https://doi.org/10.1073/pnas.0508024103