Ricker Model

The Ricker Model is a foundational discrete-time, nonlinear mathematical model of population dynamics and fisheries stock-recruitment introduced in 1954 by Canadian fisheries biologist and entomologist William Edwin ("Bill") Ricker (1908–2001) in his classic paper "Stock and Recruitment" in the Journal of the Fisheries Research Board of Canada. Formulated to explain the dome-shaped stock-recruitment curves of anadromous Pacific salmon (Oncorhynchus spp.), the Ricker difference equation models overcompensatory density dependence: as adult spawning density increases from zero, juvenile recruitment rises to a maximum at an intermediate stock size, and then decays exponentially as exp(−βS) at high parental densities due to adult-on-juvenile cannibalism, redd superimposition, disease transmission, or scramble competition for finite resources.

Beyond its status alongside the Beverton-Holt Model as one of the two primary pillars of global fisheries stock assessment, the Ricker map played a central role in the 1970s discovery of ecological chaos. In 1974 and 1976, mathematical physicist and ecologist Robert M. May demonstrated that as the intrinsic per-generation growth rate parameter r increases past 2.0, the Ricker map undergoes a cascade of period-doubling Feigenbaum bifurcations (from a stable equilibrium point to 2-point cycles, 4-point cycles, 8-point cycles, and deterministic chaos above r ≈ 2.6924) while remaining strictly positive and bounded across all initial conditions.

1. Ecological Intrinsic Growth Rate and Carrying Capacity Form (r and K)

In theoretical population ecology (May, 1974, 1976; Turchin, 2003), let N_t denote the population density of a species with non-overlapping generations at generation t. The Ricker difference equation is:

N_t+1 = N_t · exp[ r · ( 1 − N_t / K ) ]

where r (r > 0) is the intrinsic per-generation logarithmic growth rate (such that R₀ = e^r is the maximum finite per-capita reproductive multiplier as N_t → 0), and K (K > 0) is the equilibrium carrying capacity at which N_t+1 = N_t = K. Taking the natural logarithm of the per-capita reproductive ratio (N_t+1 / N_t) reveals that the Ricker model assumes a strictly linear decline in logarithmic per-capita reproduction with density N_t:

ln( N_t+1 / N_t ) = r − (r / K) · N_t

2. Classical Fisheries Stock-Recruitment Parameterization (α and β)

In fisheries science (Ricker, 1954, 1975), let S denote the adult spawning stock biomass (or egg deposition) and R denote the resulting number of surviving recruits:

R(S) = α · S · exp( −β · S )

where α (α = e^r > 0) is the density-independent productivity parameter (the slope at the origin dR / dS|_S=0, measuring maximum recruits per spawner at low stock abundance), and β (β = r / K > 0) is the density-dependent overcompensation coefficient. Setting the first derivative dR / dS = α · (1 − βS) · exp(−βS) equal to zero yields the peak recruitment coordinates:

• Spawning stock producing maximum recruitment (S_max): S_max = 1 / β = K / r
• Maximum possible recruitment (R_max): R_max = α / (β · e) = (K / r) · exp(r − 1)

Unlike the Beverton-Holt Model, where maximum recruitment occurs only at infinite stock size (S → ∞), the Ricker curve attains its global peak R_max at the finite stock level S_max = 1 / β, and whenever r > 1, peak recruitment occurs at a stock size smaller than the unfished carrying capacity (S_max < K).

Bill Ricker (1954) demonstrated that the exponential survival filter exp(−βS) emerges naturally from three distinct biological mechanisms operating during egg incubation and early juvenile life:

1. Adult-on-Juvenile Cannibalism and Predator Aggregation

Suppose a fish or insect cohort begins with E = α₀ S newly spawned eggs or larvae, which must survive a vulnerable juvenile window of duration T. If each adult present in the habitat (proportional to S) encounters and cannibalizes larvae at instantaneous per-adult clearance rate c, then the total instantaneous mortality rate experienced by the juvenile cohort is Z = μ₀ + c · S:

dN / dt = − ( μ₀ + c · S ) · N(t)

Because the mortality rate throughout the vulnerable window depends on the initial adult or egg density S rather than the instantaneous surviving larval density N(t), integrating from t = 0 to t = T yields R = N(T) = α · S · exp(−β · S). This mechanism directly describes cannibalistic teleosts (such as walleye pollock Gadus chalcogrammus, hake Merluccius, and perch Perca) as well as flour beetles (Tribolium castaneum), in which mobile adults and older larvae consume immobile eggs and pupae at rates proportional to adult density.

2. Redd Superimposition and Scramble Competition

In anadromous Pacific salmon (Oncorhynchus gorbuscha, O. nerka) and Atlantic salmon (Salmo salar), returning females excavate gravel nests (redds) in stream riffles of finite area. When spawner density S is high, late-arriving females dig their redds directly on top of earlier nests (redd superimposition), mechanically unearthing earlier embryos, while high egg density depletes interstitial dissolved oxygen (O₂) in gravel pore water and spreads fungal infections (compare disease ecology). Similarly, under A. J. Nicholson's (1954) scramble competition (such as blowfly maggots Lucilia cuprina or forest defoliating caterpillars sharing a finite resource equally), when initial larval density N_t exceeds the threshold required for individuals to reach minimum pupation mass, the cohort starves prior to metamorphosis—causing surviving adult recruitment N_t+1 to crash at high N_t (in contrast to r- and K-selection contest territoriality).

Stability Analysis and Feigenbaum Bifurcations

In 1974 (Science) and 1976 (Nature), Robert M. May used the Ricker map to demonstrate that simple deterministic single-species population models with zero external environmental noise can generate erratic, aperiodic fluctuations (chaos). Unlike the discrete logistic map (x_t+1 = a x_t (1 − x_t)), which produces unbiological negative population sizes whenever x_t > 1, the Ricker map remains strictly positive across all N_t > 0 and all r > 0.

Linearizing the Ricker map f(N) = N exp[r(1 − N / K)] at the equilibrium carrying capacity N* = K gives the local stability eigenvalue λ = f′(K) = 1 − r. Because local stability requires |λ| < 1 (−1 < 1 − r < +1), increasing the intrinsic growth rate r drives the population through five distinct dynamical regimes:

Experimental Verification in Laboratory Tribolium Beetles

To test whether real biological populations exhibit the bifurcation transitions and deterministic chaos predicted by Ricker overcompensation, Robert F. Costantino, Robert A. Desharnais, James M. Cushing, and Brian Dennis (1995, 1997) conducted a multi-year laboratory experiment using the cannibalistic flour beetle Tribolium castaneum. Formulating a three-stage (Larva-Pupa-Adult, LPA) Ricker matrix model in which adults and larvae cannibalize pupae and eggs via exp(−c_pa A_t − c_el L_t) and experimentally tuning adult mortality and recruitment rates across the predicted bifurcation boundaries, Costantino and colleagues drove replicate beetle populations from a stable equilibrium into a 2-cycle, a quasi-periodic invariant loop, and deterministic chaos—providing a landmark empirical confirmation of nonlinear Ricker bifurcation dynamics.

1. Ricker, W. E. (1954). Stock and recruitment. Journal of the Fisheries Research Board of Canada, 11(5), 559–623. https://doi.org/10.1139/f54-039
2. May, R. M. (1974). Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science, 186(4164), 645–647. https://doi.org/10.1126/science.186.4164.645
3. May, R. M. (1976). Simple mathematical models with very complicated dynamics. Nature, 261(5560), 459–467. https://doi.org/10.1038/261459a0
4. Costantino, R. F., Desharnais, R. A., Cushing, J. M., & Dennis, B. (1997). Chaotic dynamics in an insect population. Science, 275(5298), 389–391. https://doi.org/10.1126/science.275.5298.389
5. Melbourne, B. A., & Hastings, A. (2008). Extinction risk depends strongly on factors contributing to stochasticity. Nature, 454(7200), 100–103. https://doi.org/10.1038/nature06922