Lottery Competition Model

The Lottery Competition Model is a foundational theoretical framework in community ecology and biodiversity maintenance originally proposed as the Lottery Hypothesis by the Australian marine ecologist Peter F. Sale in 1977 and 1978 to explain the coexistence of territorial coral-reef fishes, and mathematically formalized in 1981 by Peter L. Chesson and Robert R. Warner in The American Naturalist. Designed for space-limited ecological guilds characterized by sedentary, site-attached adults and widely dispersing pelagic larvae or seeds—such as pomacentrid damselfishes on coral heads, intertidal barnacles on rocky shores, and canopy trees in tropical rainforest gaps—the lottery model posits that vacant microsites created by adult mortality (δ_i(t)) are allocated at random ("by lottery") among recruitment-ready larvae in proportion to larval pool abundance (β_i(t) N_i(t)). Although Sale initially suggested that random lottery allocation of space could by itself prevent competitive exclusion among species sharing identical resource requirements, Chesson and Warner (1981) proved that in a constant environment or when adult generations are non-overlapping (δ_i = 1), random lottery allocation alone cannot maintain stable coexistence, leading instead to random-walk drift or exclusion by the highest β_i/δ_i. Crucially, Chesson and Warner demonstrated that combining temporal environmental variability in recruitment rates (β₁(t)/β₂(t)) with overlapping generations of long-lived adults (0 < δ_i < 1, "buffered population growth") produces positive invasion growth rates E[ln λ_i] > 0 for rare species—forming the foundational prototype of the temporal storage effect.

The Pomacentrid Coexistence Paradox on Coral Reefs

Under the classical Lotka–Volterra competitive exclusion principle and Tilman's R* rule, n competing species cannot stably coexist at a deterministic equilibrium on fewer than n limiting resources unless interspecific resource partitioning (niche complementarity) causes intraspecific competition to exceed interspecific competition. During the 1970s, field studies on the Great Barrier Reef by Peter F. Sale (1974, 1977, 1978) revealed a notable empirical exception to classical niche-partitioning theory among guilds of herbivorous territorial damselfishes (family Pomacentridae, including Pomacentrus wardi, Pomacentrus flavicauda, and Stegastes apicalis).

Within contiguous rubble patches on reef crests and lagoonal patch reefs, three to eight territorial damselfish species co-occurred at saturation density while defending mutually exclusive territories of 0.5 to 2.0 m². Dietary, behavioral, and microhabitat surveys demonstrated that these coexisting species overlapped extensively in food resources (browsing filamentous turf algae on coral rock), diurnal activity schedules, and structural microhabitat requirements. Furthermore, once a juvenile damselfish settled from the plankton onto a vacant rubble site and reached adulthood, it defended that territory against intruding juveniles and adults of all guild species until its death.

Sale's (1977) Pelagic Larval Lottery Concept

To explain how multiple ecological equivalents could coexist without classical resource partitioning, Sale (1977) highlighted the bipartite life history shared by most coral-reef fishes and benthic marine invertebrates:

1. Sedentary, Territorial Adult Stage: Space on the reef substratum is saturated (K total territories). Resident adults hold fixed territories for several years and cannot be competitively displaced by newly arriving recruits ("prior-residence advantage" or preemptive competition).
2. Open Pelagic Larval Pool and Random Vacancy Allocation: Adults spawn pelagic larvae that disperse across an open marine metacommunity before becoming competent to settle. Whenever the death of a resident adult opens a vacant territory on the reef, the vacancy is claimed at random among recruitment-ready larvae in proportion to each species' representation in the local larval pool.

Discrete-Time Equations for Preemptive Space Competition

To test whether random lottery recruitment can maintain stable species coexistence, Peter L. Chesson and Robert R. Warner (1981) constructed a stochastic discrete-time mathematical model of n competing species competing for a fixed total number of territories K (with adult populations N_i(t) or fractional space occupancies P_i(t) = N_i(t)/K satisfying ∑_i=1ⁿ P_i(t) = 1).

During time step (t, t + 1), each species i experiences two demographic rates that fluctuate stochastically over time in response to environmental conditions:

• Adult Mortality Rate (δ_i(t)): The fraction of resident adults of species i that die during interval t (0 < δ_i(t) ≤ 1), leaving surviving resident fraction 1 − δ_i(t).
• Per-Capita Larval Recruitment Rate (β_i(t)): The effective number of recruitment-ready larvae produced per adult of species i weighted by per-larva competitive ability to claim a vacant microsite (β_i(t) > 0), so that the species' larval pool contribution is proportional to β_i(t) N_i(t).

At time t, adult mortality across all n species liberates a fractional area of vacant space V(t) = ∑_j=1ⁿ δ_j(t)P_j(t). Under lottery allocation, species i captures a share of these vacant microsites equal to its proportion in the total larval pool (β_i(t)N_i(t) / ∑ β_j(t)N_j(t)), yielding the Chesson–Warner lottery recurrence equation:

P_i(t + 1) = [1 − δ_i(t)] P_i(t) + [∑_j=1ⁿ δ_j(t) P_j(t)] × [ β_i(t) P_i(t) / ∑_j=1ⁿ β_j(t) P_j(t) ]

For two competing species (n = 2, P₁(t) + P₂(t) = 1), the finite per-step multiplication rate λ₁(t) = P₁(t + 1)/P₁(t) of species 1 is:

λ₁(t) = 1 − δ₁(t) + [ δ₁(t)P₁(t) + δ₂(t)(1 − P₁(t)) ] × [ β₁(t) / (β₁(t)P₁(t) + β₂(t)(1 − P₁(t))) ]

The Stochastic Mutual Invasibility Criterion

In a fluctuating environment, stable coexistence is governed by the mutual invasibility criterion: each species i must achieve a positive expected logarithmic per-capita growth rate E[ln λ_i] > 0 when invading from rare density (P_i → 0 while resident competitor j occupies nearly all space, P_j → 1). Setting P₁(t) → 0 and P₂(t) → 1 gives the boundary growth rates:

Δ₁ = E[ ln λ₁ ] = E[ ln( 1 − δ₁(t) + δ₂(t) (β₁(t) / β₂(t)) ) ] > 0

Δ₂ = E[ ln λ₂ ] = E[ ln( 1 − δ₂(t) + δ₁(t) (β₂(t) / β₁(t)) ) ] > 0

Proof That Constant Environments or Semelparity (δ_i = 1) Preclude Coexistence

Using these boundary growth rates, Chesson and Warner proved two foundational negative results:

1. Constant Environment (β_i and δ_i Constant): If demographic parameters do not vary over time, Δ₁ > 0 requires β₁/δ₁ > β₂/δ₂, whereas Δ₂ > 0 requires β₂/δ₂ > β₁/δ₁. Both inequalities cannot hold simultaneously: the species with the higher lifetime reproductive ratio β_i/δ_i excludes the other, or if β₁/δ₁ = β₂/δ₂, neutral random-walk drift drives one species extinct.
2. Non-Overlapping Generations (Semelparous / Annual Species, δ₁ = δ₂ = 1): When all adults die at the end of each reproductive cycle (δ₁(t) = δ₂(t) = 1), even large environmental fluctuations in recruitment β_i(t) cannot stabilize coexistence because:

Δ₁ = E[ ln(β₁(t) / β₂(t)) ] = E[ln β₁] − E[ln β₂] = −Δ₂ Because Δ₁ + Δ₂ = 0 identically, both species cannot have positive invasion growth rates; the species with the larger geometric mean recruitment E[ln β_i] excludes the other.

Convexity Proof and the Chesson–Warner Coexistence Bandwidth

The central discovery of Chesson and Warner (1981) is that stable lottery coexistence emerges when overlapping generations (0 < δ_i < 1) are combined with temporal variability in relative recruitment rates (β₁(t)/β₂(t))—a mechanism linked to evolutionary bet-hedging in biology and generalized by Peter Chesson (1983, 1994, 2000) as the temporal storage effect.

Writing U(t) = ln[β₁(t)/β₂(t)] so that β₁(t)/β₂(t) = e^U(t) (with constant adult mortality δ ∈ (0, 1)), species 1's invasion growth rate is Δ₁ = E[g(U(t))] where g(u) = ln(1 − δ + δ e^u). Differentiating g(u) twice with respect to u yields:

g″(u) = [ δ(1 − δ) e^u ] / (1 − δ + δ e^u)² > 0 for all 0 < δ < 1

Because 0 < δ < 1, g(u) is a convex function of the log recruitment ratio U = ln(β₁/β₂). By Jensen's inequality, variance σ_U² = Var[ln(β₁(t)/β₂(t))] increases E[g(U)] above g(E[U]). A second-order Taylor expansion for general mortality rates δ₁, δ₂ yields the Chesson–Warner coexistence bandwidth:

− (1 / 2)(1 − δ₁) σ_U² < E[ln(β₁ / δ₁)] − E[ln(β₂ / δ₂)] < + (1 / 2)(1 − δ₂) σ_U²

Conversely, if recruitment is constant (β₁/β₂ = r) and only adult mortality δ(t) fluctuates, Δ₁ = E[ln(1 + δ(t)(r − 1))] is a concave function of δ(t), so mortality fluctuations alone reduce invasion growth rates and oppose coexistence.

The Three Pillars of the Temporal Storage Effect

The Chesson–Warner convexity relation identifies the three biological ingredients of the temporal storage effect across marine and terrestrial ecosystems:

1. Differential Ecological Responses to the Environment (σ_U² > 0): Species respond asynchronously to environmental fluctuations so that the recruitment ratio β₁(t)/β₂(t) varies through time.
2. Covariance Between Environment and Competition: When a species is common (P_i ≈ 1), a strong recruitment year produces a large larval pool (β_iP_i), so its larvae compete against conspecifics for limited vacant sites. When a species is rare (P_i → 0), its larvae contribute negligibly to total larval competition, enabling uncapped per-capita recruitment gains during favorable years.
3. Buffered Population Growth via Long-Lived Adults (0 < δ_i < 1): During unfavorable recruitment years (β_i(t) → 0), population decline is bounded by adult survival 1 − δ_i because established territorial adults are immune to larval competition. Long adult lifespan (δ_i → 0) widens the coexistence bandwidth (1/2)(1 − δ_i)σ_U², a mechanism observed across coral-reef fishes (Warner & Chesson, 1985), gap-dynamic forest trees (Usinowicz et al., 2012; compare stochastic niche theory and spatial insurance hypothesis), and desert annual plants with persistent seed banks (Angert et al., 2009).

Can a purely random lottery in a constant environment allow competing species to coexist?

No. Although Peter Sale (1977) initially hypothesized that random allocation of vacant territories among pelagic larvae might prevent competitive exclusion by itself, Chesson and Warner (1981) proved that in a constant environment, the species with the highest ratio of per-capita larval recruitment to adult mortality (β_i/δ_i) deterministically drives all other species to extinction (or, if β_i/δ_i are equal, species go extinct via neutral random-walk drift).

Why are overlapping generations (long-lived adults) essential for lottery coexistence?

If all adults die every generation (δ_i = 1, as in annual species without a seed bank), the expected logarithmic invasion growth rates of two competitors sum to zero (E[ln λ₁] = −E[ln λ₂]), making mutual invasibility impossible. When adults survive across multiple reproductive cycles (0 < δ_i < 1), resident adults buffer the population against steep declines during poor recruitment years while allowing rare species to achieve large geometric gains during favorable recruitment years.

How does the Lottery Competition Model relate to the storage effect?

The Chesson–Warner (1981) lottery model is the mathematical prototype of the temporal storage effect formalized by Peter Chesson (1983, 1994). The storage effect generalizes the three properties discovered in the lottery model: (1) species-specific responses to environmental fluctuations, (2) covariance between environment and competition (so rare species escape intraspecific larval crowding during their favorable years), and (3) buffered population growth (via long-lived adults or dormant seed banks) that protects rare species during unfavorable years.

1. Sale, P. F. (1977). "Maintenance of high diversity in coral reef fish communities." The American Naturalist, 111(978), 337–359. https://doi.org/10.1086/283164
2. Sale, P. F. (1978). "Coexistence of coral reef fishes—a lottery for living space." Environmental Biology of Fishes, 3(1), 85–102. https://doi.org/10.1007/BF00006310
3. Chesson, P. L., & Warner, R. R. (1981). "Environmental variability promotes coexistence in lottery competitive systems." The American Naturalist, 117(6), 923–943. https://doi.org/10.1086/283778
4. Warner, R. R., & Chesson, P. L. (1985). "Coexistence mediated by recruitment fluctuations: A field guide to the storage effect." The American Naturalist, 125(6), 769–787. https://doi.org/10.1086/284379
5. 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