R* Rule

The R* Rule (pronounced "R-star rule")—together with the resource-ratio hypothesis—is the core predictive theorem of mechanistic resource-competition theory in ecology, developed by Robert MacArthur (1972), R. A. Armstrong and Richard McGehee (1980), and synthesized experimentally and mathematically by David Tilman (1977, 1980, 1982). Whereas classical phenomenological Lotka–Volterra competition equations represent interspecific interactions via abstract competition coefficients (α_ij) without specifying what organisms compete for, Tilman's mechanistic framework explicitly models the coupled dynamics of consumer population densities (N_i) and the environmental concentrations of limiting resources (R_j, such as dissolved nitrate NO₃⁻, phosphate PO₄³⁻, silicate SiO₂, water, or light). For any species i growing on a single essential limiting resource R, R_i* denotes the break-even equilibrium resource concentration at which that species' reproductive growth rate balances its per-capita mortality or loss rate (dN_i/dt = 0). The R* rule proves that in a well-mixed, steady-state habitat with a single limiting resource, the competing species with the lowest R_i* draws the environmental resource concentration down to its own R_i* level, driving all competitors with higher resource requirements (R_j* > R_i*) to local competitive exclusion.

Monod Growth Kinetics and Chemostat Resource Balance

Consider n consumer species with population densities N₁, N₂, ..., N_n competing in a continuous-flow chemostat or natural mixed habitat for a single dissolved limiting resource of ambient concentration R(t). The per-capita reproductive growth rate f_i(R) of species i is a monotonically increasing, saturating function of resource availability, represented by the Monod (1949) rectangular hyperbola:

f_i(R) = ( r_i R ) / ( R + K_i )

where r_i (time⁻¹) is the maximum physiological specific growth rate of species i under resource saturation (R → ∞), and K_i (concentration units, e.g., μmol L⁻¹) is the half-saturation constant—the resource concentration at which species i achieves half of its maximum specific growth rate (f_i(K_i) = r_i / 2).

Simultaneously, each species experiences a density-independent per-capita loss rate m_i (comprising hydraulic washout rate D, sinking, senescence, or background mortality), while resource R is supplied from an external reservoir of concentration S at supply rate D(S − R) and depleted by the collective uptake of all consumers:

( 1 / N_i ) ( dN_i / dt ) = f_i(R) − m_i = [ ( r_i R ) / ( R + K_i ) ] − m_i

dR / dt = D ( S − R ) − ∑_i=1ⁿ q_i N_i f_i(R)

Here q_i (the cellular quota, inverse of yield Y_i = 1/q_i) is the quantity of resource R consumed to produce one unit of biomass of species i.

Algebraic Solution for R_i* and Competitive Exclusion

To find the equilibrium resource concentration required for species i to maintain a non-zero population (N_i* > 0), one sets its per-capita net growth rate to zero (dN_i/N_idt = 0):

( r_i R_i* ) / ( R_i* + K_i ) = m_i

Solving this equation explicitly for R_i* yields the fundamental R* formula:

R_i* = ( m_i K_i ) / ( r_i − m_i )

Provided r_i > m_i and S > R_i*, species i grown in monoculture increases until its uptake reduces the ambient resource concentration to R = R_i*, stabilizing at the monoculture equilibrium biomass N_i* = D(S − R_i*) / (q_i m_i).

Now suppose two species (1 and 2) compete for resource R with R₁* < R₂*. Because per-capita growth f_i(R) − m_i increases monotonically with R:

1. Whenever R > R₂*, both species experience positive net growth (dN₁/dt > 0, dN₂/dt > 0), collectively drawing down R(t).
2. Once R(t) enters the interval R₁* < R < R₂*, species 1 continues growing (dN₁/dt > 0) and depleting R, whereas species 2 suffers negative net growth (dN₂/dt < 0) and declines.
3. As t → ∞, species 1 drives the ambient resource concentration to R* = R₁*, where dN₁/dt = 0 and dN₂/(N₂dt) = f₂(R₁*) − m₂ < 0. Species 2 is driven to deterministic competitive exclusion regardless of initial population densities or cellular quotas q_i. Thus, R_i* measured in monoculture predicts a priori the outcome of interspecific competition.

Zero Net Growth Isoclines (ZNGIs) for Essential Resources

Primary producers (such as phytoplankton and vascular plants) require multiple essential (non-substitutable) resources simultaneously—such as nitrogen, phosphorus, silicon, and light—in proportions related to the Redfield ratio. Following Tilman (1980, 1982) and Liebig's Law of the Minimum, the per-capita growth rate of species i on two essential resources R₁ and R₂ is governed by the resource in shortest relative supply:

( 1 / N_i ) ( dN_i / dt ) = min [ f_i1(R₁) , f_i2(R₂) ] − m_i

In the two-dimensional resource plane (R₁, R₂), the locus of resource pairs where dN_i/dt = 0 forms an L-shaped right-angle boundary called the Zero Net Growth Isocline (ZNGI) of species i:

• A vertical segment at R₁ = R_i1* (for R₂ ≥ R_i2*), where species i is limited by resource 1.
• A horizontal segment at R₂ = R_i2* (for R₁ ≥ R_i1*), where species i is limited by resource 2.
• A corner vertex at (R_i1*, R_i2*), where species i is co-limited by both resources.

Trade-Offs, Consumption Vectors, and Coexistence Criteria

For two species (A and B) competing for two essential resources (R₁ and R₂) to coexist at a stable equilibrium, three geometric conditions must hold in the (R₁, R₂) plane:

1. Interspecific Trade-Off (Crossing of ZNGIs): Species A must be the superior competitor for resource 1 (R_A1* < R_B1*), while species B must be the superior competitor for resource 2 (R_B2* < R_A2*). Their L-shaped ZNGIs then intersect at a two-species equilibrium point (R*) = (R_B1*, R_A2*), where species A is limited by resource 2 and species B is limited by resource 1.
2. Self-Limitation of Consumption Vectors (C_A and C_B): Let C_i = (c_i1, c_i2) denote the resource consumption vector of species i. For the intersection point (R_B1*, R_A2*) to be locally stable rather than an unstable saddle point, each species must consume relatively more of the resource that limits its own growth at equilibrium:

( c_A2 / c_A1 ) > ( c_B2 / c_B1 )

1. Intermediate Supply Ratio (S₁, S₂): The environmental resource supply point (S) = (S₁, S₂) must lie inside the conical coexistence sector bounded by the backward extensions of the consumption vectors C_A and C_B from (R_B1*, R_A2*).

Tilman's (1977) Asterionella vs. Cyclotella Chemostat Experiments

In his landmark 1977 study in Ecology, David Tilman tested the R* rule and the resource-ratio hypothesis using two freshwater planktonic diatoms from Lake Michigan: Asterionella formosa and Cyclotella meneghiniana, competing for dissolved orthophosphate (PO₄³⁻) and dissolved silicate (SiO₂, required to construct the diatom frustule).

1. Monoculture Parameterization: Single-species chemostat assays showed that Asterionella formosa had a much lower R* for phosphate (R_P* ≈ 0.004–0.02 μmol L⁻¹) than Cyclotella meneghiniana (R_P* ≈ 0.08–0.25 μmol L⁻¹), whereas C. meneghiniana had a much lower R* for silicate (R_Si* ≈ 0.15–0.6 μmol L⁻¹) than A. formosa (R_Si* ≈ 1.0–1.9 μmol L⁻¹).
2. Verification Across 76 Competition Chemostats: Tilman inoculated both species together into 76 continuous-flow chemostats spanning influent supply ratios from Si:P = 0.7 to Si:P = 1,020:
   • For Si:P > 90 (where both species were phosphate-limited), Asterionella formosa competitively excluded Cyclotella meneghiniana.
   • For Si:P < 6 (where both species were silicate-limited), Cyclotella meneghiniana competitively excluded Asterionella formosa.
   • For 6 ≤ Si:P ≤ 90, both diatom species coexisted stably because A. formosa was silicate-limited (with a high Si:P uptake ratio of ~90) while C. meneghiniana was phosphate-limited (with a low Si:P uptake ratio of ~6), matching natural distribution boundaries in Lake Michigan.

Terrestrial Grasslands, Biodiversity, and Stochastic Extensions

Tilman extended the R* rule to terrestrial plant communities at the Cedar Creek Ecosystem Science Reserve in Minnesota (Tilman, 1982, 1988; Tilman & Wedin, 1991). Field monocultures of perennial prairie grasses (such as Schizachyrium scoparium, Poa pratensis, and Agrostis scabra) demonstrated that species investing heavily in fine root biomass achieved the lowest soil extractable nitrate R_N* (<0.02 mg kg⁻¹) and displaced faster-growing grasses on nitrogen-poor sandy soils. When chronic nitrogen fertilization shifted competition toward a single spatially uniform resource—light at the soil surface (I*)—tall, leafy species shaded out low-stature root specialists, explaining why heavy nutrient enrichment reduces terrestrial plant biodiversity. In contemporary ecology, Tilman's deterministic R* framework is complemented by the niche complementarity hypothesis and stochastic niche theory.

Why does the species with the lowest R*, rather than the highest maximum growth rate r, win competition?

A high maximum growth rate (r_i) provides an advantage only transiently when resources are abundant (R ≫ K_i), such as shortly after a disturbance. As consumer populations grow and deplete the limiting resource toward equilibrium, ambient concentrations fall to low levels where survival depends on combining high resource uptake affinity (low K_i) with low loss rate (m_i). The species with the lowest R_i* = m_iK_i / (r_i − m_i) continues growing at a resource concentration where faster-growing opportunist species starve.

How can dozens of plankton species coexist on only a few limiting nutrients ("The Paradox of the Plankton")?

Under equilibrium resource-ratio theory, at most k consumer species can stably coexist on k homogeneously mixed limiting resources. In natural lakes and oceans, G. Evelyn Hutchinson's (1961) "Paradox of the Plankton" is resolved through spatial and vertical heterogeneity (light vs. deep nutrient gradients), temporal fluctuations and endogenous oscillations in resource supply (Armstrong & McGehee, 1980; Huisman & Weissing, 1999), specialist viral and zooplankton grazers, and recruitment stochasticity.

What happens if two species have crossing ZNGIs, but each species consumes more of the resource that limits its competitor?

If species A is limited by resource 2 and species B is limited by resource 1 at their ZNGI intersection, but species A consumes a higher ratio of resource 1 to resource 2 than species B (c_A2/c_A1 < c_B2/c_B1), the two-species equilibrium is an unstable saddle point. Instead of coexisting at intermediate supply ratios, the system exhibits priority effects (bistability): whichever species begins at a higher initial density drives the resource trajectory into its favored domain and excludes the other.

1. Tilman, D. (1977). "Resource competition between planktonic algae: An experimental and theoretical approach." Ecology, 58(2), 338–348. https://doi.org/10.2307/1935608
2. Tilman, D. (1980). "Resources: A graphical-mechanistic approach to competition and predation." The American Naturalist, 116(3), 362–393. https://doi.org/10.1086/283633
3. Tilman, D., & Wedin, D. (1991). "Plant traits and resource reduction for five grasses growing on a nitrogen gradient." Ecology, 72(2), 685–700. https://doi.org/10.2307/2937208
4. Huisman, J., & Weissing, F. J. (1999). "Biodiversity of plankton by species oscillations and chaos." Nature, 402(6760), 407–410. https://doi.org/10.1038/46540
5. Wilson, J. B., Spijkerman, E., & Huisman, J. (2007). "Is there really insufficient support for Tilman's R* concept? A comment on Miller et al." The American Naturalist, 169(5), 700–706. https://doi.org/10.1086/513113