Nicholson’s Blowfly Application

Summary

The flagship application of the synthetic likelihood: resolving a decades-old question about Nicholson’s classic sheep-blowfly experiments. A stochastic version of the Gurney–Nisbet delay-differential model is fit by synthetic-likelihood MCMC to four experimental replicates. The full model (environmental + demographic stochasticity) fits quantitatively (${\chi }^2$ $p>0.2$), while a demographic-stochasticity-only simplification is decisively rejected ($p\ll 0.002$; $\Delta$AIC $>1800$). Overlaying the fitted parameters on Gurney–Nisbet’s stability diagram shows the dynamics are intrinsically-driven limit cycles perturbed by noise, not stochastically-forced quasi-cycles — settling whether the fluctuations are noise-driven (they are not).

Overview

Nicholson’s blowfly populations exhibit large, irregular cycles. Gurney & Nisbet (1980) gave the first plausible mechanistic model, but for the food-limited replicates it was impossible to decide — with the ad hoc estimation methods then available — whether the fluctuations were externally-forced quasi-cycles or intrinsic limit cycles, because plausibly-parameterized models are chaotic or near-chaotic. The synthetic likelihood provides the missing quantitative inference.

Main Content

Gurney–Nisbet delay model and its stochastic discretization (Wood 2010, Eqs. 3-4)

The continuous model for adult population $N$:

$$\frac{dN}{dt}=PN(t-\tau )e^{-N(t-\tau )/N_0}-\delta N(t),$$

with parameters $P,N_0,\delta ,\tau$; depending on values, dynamics range from stable equilibrium to chaos. Discretized with a daily timestep and demographic stochasticity ($N_{t+1}=R_t+S_t$):

$$R_t\sim \text{Poi}\{P{N}_{t-\tau }\exp (-N_{t-\tau }/N_0)e_t\},S_t\sim \text{binom}\{\exp (-\delta {ϵ}_t),N_t\},$$

i.e. egg production is an independent Poisson process per female, and each adult survives a day with probability $\exp (-\delta {ϵ}_t)$. The environmental-stochasticity terms $e_t$, ${ϵ}_t$ are independent Gamma deviates with mean 1 and variances ${\sigma }_p^2$, ${\sigma }_d^2$. The simplified, demographic-only model sets $e_t={ϵ}_t=1$.

Summary statistics used

Blowfly summary statistics (Wood 2010, "Blowfly statistics")

Autocovariances to lag 11; the cubic-regression “difference distribution” summary (as in the Ricker example); $\text{mean}\{N_t\}$; $\text{mean}\{N_t\}-\text{median}\{N_t\}$; the number of turning points observed; and the estimated coefficients $\hat{\mathbf{\beta }}$ of the autoregression

$$N_i={\beta }_0{N}_{i-12}+{\beta }_1{N}_{i-12}^2+{\beta }_2{N}_{i-12}^3+{\beta }_3{N}_{i-2}+{\beta }_4{N}_{i-2}^2+{\epsilon }_i.$$

Results

Full vs. demographic-only model fit (Wood 2010, Fig. 3)

Both models were fit to each of four experimental replicates with MCMC chains of 50,000 iterations.

• Full model (4): good ${\chi }^2$ fit ($p>0.2$) in all cases; simulated replicates qualitatively reproduce the irregular cycles (Fig. 3e–h).
• Demographic-only model: very bad fit ($p\ll 0.002$) in all cases; produces insufficiently variable dynamics (Fig. 3i–l).
• Model comparison: AIC differences $>1800$ favoring the full model for all four replicates.

The comprehensive rejection of the demographic-only model is because demographic stochasticity alone cannot produce the irregularity of the real cycles. So the stochastic Nisbet–Gurney model is not just qualitatively plausible — it fits quantitatively.

Limit cycles, not noise-driven (Wood 2010, Fig. 4)

Uncontrolled experimental variability dwarfs demographic stochasticity, raising the question of whether that drove the fluctuations rather than merely perturbing them. Overlaying 1,500 values of the stability-controlling parameters $P\tau$ and $\delta \tau$ — sampled from the second half of each replicate’s MCMC chain — on Gurney & Nisbet’s stability diagram for model (3) shows the parameter clouds sit in the region where the deterministic skeleton has limit-cycle (not stable-equilibrium) dynamics.

Conclusion: there is extremely strong statistical evidence that Nicholson’s blowfly fluctuations are limit cycles perturbed by noise — intrinsically driven by the population dynamics — and would have occurred no matter how constant the conditions or how large the cultures. They are not the result of stochastic forcing or resonance excitation, despite decisive evidence for stochasticity well above demographic levels.

Connections

• A worked demonstration of the synthetic-likelihood algorithm ($l_s$, MCMC, ${\chi }^2$ diagnostic, AIC) on real data.
• The stochastic Gurney–Nisbet model is a delayed, demographically-and-environmentally noisy population model — the ecological analog of the structural dynamic models fit by simulated moments and calibrated by ABC.
• Model selection by AIC mirrors standard likelihood-based comparison, made possible here by the synthetic likelihood.

See Also

• Synthetic Likelihood - Overview — the method and its broader significance
• Synthetic Likelihood Construction — the estimator used to fit these models
• Chaos and Phase-Insensitive Statistics — why these particular statistics were chosen