Relative Nonlinearity of Competition

Relative nonlinearity of competition is one of the two canonical fluctuation-dependent coexistence mechanisms in Peter L. Chesson's Modern Coexistence Theory (alongside the storage effect). First demonstrated in consumer–resource differential equations by Robert A. Armstrong and Richard McGehee in 1976 and 1980, and unified into the general invasion-growth-rate partitioning framework by Peter L. Chesson (1994, 2000), relative nonlinearity explains how two or more competing species can stably coexist on a single shared limiting resource or competitive factor (C) when resource levels fluctuate through time or space. Coexistence occurs when the competing species differ in the second derivative (concavity or convexity) of their per-capita growth rate curves as a function of resource availability—classically manifest as a physiological trade-off between a gleaner (which has a lower equilibrium resource requirement R^* in constant environments) and an opportunist (which has a higher maximum growth rate during transient resource pulses)—and when each species, while dominant as a resident, modifies the variance of resource fluctuations in a way that favors its rare competitor.

Why Equilibrium Resource Competition Excludes All But One Species

In classical equilibrium consumer–resource theory—formalized by David Tilman's Resource-Ratio Hypothesis (1982) and the Competitive Exclusion Principle—if multiple consumer species compete for a single homogeneously mixed limiting resource R in a constant environment, the species with the lowest break-even resource concentration R^* (the level of R at which its per-capita birth rate balances mortality, dN_i/(N_i dt) = 0) draws the equilibrium resource down to R₁^*. At R = R₁^*, all other species (R_j^* > R₁^*) experience negative net per-capita growth rates and are driven to extinction.

In 1976 and 1980, however, Robert A. Armstrong and Richard McGehee proved mathematically that the rule "no more than k species can coexist on k limiting resources" applies only to fixed-point equilibria. When they replaced a linear Lotka–Volterra functional response with a saturating Monod or Holling Type II functional response, the consumer–resource system generated endogenous limit-cycle oscillations in resource abundance R(t). Within this fluctuating resource regime, a second competitor with a higher R^* but a less strongly saturating (more linear) growth curve invaded and stably coexisted with the low-R^* gleaner on a single biotic resource.

Concavity, Convexity, and the Variance of Competition

Suppose the per-capita growth rate r_i(R) = (1/N_i) dN_i/dt of species i depends on a single fluctuating resource R(t) (or, equivalently in Chesson's notation, a single shared competitive factor C(t)):

r_i(R(t)) = f_i(R(t)) − d_i

where f_i(R) is the resource uptake and conversion function (e.g., the Monod equation f_i(R) = μ_max,i R / (K_i + R)) and d_i is the per-capita mortality or chemostat dilution rate. Over a long time horizon T (or across a complete limit cycle of period T), the long-term average per-capita growth rate is the time average ⟨r_i(R(t))⟩.

Expanding f_i(R(t)) in a second-order Taylor series around the mean resource concentration R̄ = ⟨R(t)⟩ yields:

⟨r_i(R(t))⟩ ≈ f_i(R̄) − d_i + (1/2) · f_i″(R̄) · Var(R(t))

where f_i″(R̄) = d²f_i / dR² measures the nonlinearity (curvature) of species i's per-capita growth response, and Var(R(t)) = ⟨(R(t) − R̄)²⟩ is the temporal variance of the resource. By Jensen's inequality:

1. Concave Growth Response (f_i″(R) < 0, Saturating Type II Curve): Because a concave-downward curve bends below its tangent line, resource fluctuations depress the long-term average growth rate below the growth rate at the mean resource level: ⟨f_i(R)⟩ < f_i(R̄). High variance Var(R) penalizes a strongly saturating species because resource spikes above R̄ yield diminishing marginal gains, whereas resource troughs below R̄ inflict steep growth losses.
2. Linear or Weakly Concave / Convex Growth Response (f_i″(R) ≈ 0 or > 0): A species whose growth rate increases linearly with R (f_i″ = 0) is unaffected by resource variance (⟨f_i(R)⟩ = f_i(R̄)), while a species with a convex (accelerating or Type III sigmoid) response over the relevant range (f_i″ > 0) benefits from resource variance: ⟨f_i(R)⟩ > f_i(R̄).

In Modern Coexistence Theory (Chesson, 1994, 2000), the stabilizing contribution of relative nonlinearity (ΔN_i) to the invasion growth rate of rare species i invading resident species r is proportional to the product of their difference in nonlinearity and the difference in competitive variance generated when each species is resident:

ΔN_i ≈ −(1/2) · ( f_i″ − f_r″ ) · [ Var_r(R) − Var_i(R) ]

Why Fluctuations Must Be Endogenously Regulated for Mutual Invasibility

A difference in curvature (f₁″ ≠ f₂″) and a gleaner–opportunist crossover alone are necessary but not sufficient for stable coexistence if resource variance Var(R) is fixed purely by external weather. If Var(R) were identical regardless of which species dominated the community (Var₁(R) = Var₂(R)), then ΔN_i would equal zero: the variance term would merely shift average fitness differences, and one species would still exclude the other (Chesson, 1994; Letten et al., 2018).

For relative nonlinearity to act as a stabilizing mechanism (ΔN_i > 0 for both species), each species—when abundant as the resident—must drive the resource regime into a variance state that favors its rare competitor (negative frequency dependence via variance modification):

• When the Opportunist (Linear / Weakly Saturating Species 2) Is Resident: Because the opportunist has a nearly linear functional response, its consumption dampens resource oscillations. With low resource variance (Var₂(R) ≈ 0) and a high mean resource level R̄₂ = R₂^* > R₁^*, the rare gleaner (Species 1, with low R₁^*) easily invades because the environment is nearly constant.
• When the Gleaner (Strongly Saturating Type II Species 1) Is Resident: Because the gleaner's saturating Type II uptake destabilizes the consumer–resource interaction into large-amplitude limit cycles (or fails to buffer external resource pulses), resource variance becomes large (Var₁(R) ≫ Var₂(R)). By Jensen's inequality, this high variance penalizes the concave gleaner, forcing the gleaner's time-averaged resource level ⟨R(t)⟩₁ upward until the rare opportunist (which suffers little or no variance penalty) achieves a positive invasion growth rate and invades.

Chemostat Demonstrations and Modern Partitioning Studies

Relative nonlinearity has been tested experimentally in laboratory microcosms and natural field communities:

1. Rotifer–Algal Predator–Prey Cycles (Fussmann et al., 2000; Hiltunen et al., 2014): In continuous-flow chemostats where planktonic rotifers (Brachionus calyciflorus) graze on unicellular green algae (Chlorella vulgaris or Monoraphidium minutum), saturating functional responses generate endogenous consumer–resource oscillations that allow competing consumer genotypes or species differing in functional response curvature to coexist.
2. Phytoplankton under Pulsed Nutrient Supply (Sommer, 1984; Litchman & Klausmeier, 2001): In freshwater and marine phytoplankton communities competing for pulsed dissolved silicate or phosphate (PO₄³⁻), small-celled diatoms (Synedra) act as low-R^* gleaners during stratified low-nutrient periods, whereas fast-growing green algae or cyanobacteria with high μ_max act as opportunists during mixing or nutrient pulses.
3. Nectar-Inhabiting Yeasts and Annual Plants (Letten et al., 2018; Hallett et al., 2019): By parameterizing mechanistic consumer–resource models for nectar-dwelling yeasts (Metschnikowia reukaufii and Candida rancensis) experiencing floral sugar pulses and annual grassland communities in California, researchers demonstrated that when resource consumption is explicitly tracked, relative nonlinearity of competition contributes substantially—and sometimes more strongly than the storage effect—to stabilizing coexistence.

How does relative nonlinearity differ from the storage effect?

Both are fluctuation-dependent coexistence mechanisms in Modern Coexistence Theory, but they rely on different mathematical terms in the Taylor expansion of growth rates. The storage effect requires a direct density-independent response to an environmental variable E(t), covariance between environment and competition Cov(E, C), and a buffered life stage (∂²ln λ / ∂E∂C < 0). Relative nonlinearity requires no separate environmental response E(t) and no dormant seed bank: it operates purely through the second derivative of growth with respect to competition itself (∂²r / ∂C²) combined with variance in competition Var(C).

What is Jensen's inequality and why is it central to relative nonlinearity?

Jensen's inequality is a mathematical theorem stating that for any concave function f(x) (where f″(x) < 0), the average of the function is strictly less than the function of the average: E[f(X)] < f(E[X]). For a convex function (f″(x) > 0), the reverse holds: E[f(X)] > f(E[X]). In ecology, this means that temporal fluctuations in a resource automatically lower the long-term growth rate of a species with a saturating (concave) uptake curve while boosting or leaving unchanged a species with a convex or linear uptake curve.

Can relative nonlinearity generate negative (destabilizing) effects on coexistence?

Yes. If the species with the more concave growth response (f_i″ < f_r″) actually reduces resource variance when it is the abundant resident (Var_i(R) < Var_r(R)), then the sign of ΔN_i becomes negative. In that case, relative nonlinearity acts as a destabilizing mechanism (promoting priority effects and competitive exclusion) rather than stabilizing coexistence.

1. Armstrong, R. A., & McGehee, R. (1980). "Competitive exclusion." The American Naturalist, 115(2), 151–170. https://doi.org/10.1086/283553
2. Chesson, P. (1994). "Multispecies competition in variable environments." Theoretical Population Biology, 45(3), 227–276. https://doi.org/10.1006/tpbi.1994.1013
3. Chesson, P. (2000). "Mechanisms of maintenance of species diversity." Annual Review of Ecology and Systematics, 31, 343–366. https://doi.org/10.1146/annurev.ecolsys.31.1.343
4. Litchman, E., & Klausmeier, C. A. (2001). "Competition of phytoplankton under fluctuating light." The American Naturalist, 157(2), 170–187. https://doi.org/10.1086/318628
5. Letten, A. D., Dhami, M. K., Ke, P.-J., & Fukami, T. (2018). "Species coexistence through simultaneous fluctuation-dependent mechanisms." Proceedings of the National Academy of Sciences, 115(26), 6745–6750. https://doi.org/10.1073/pnas.1801846115