Surplus-Production State-Space Models for Fisheries (Concept Note)

Summary

Concept note, no raw source; citations from memory and should be verified before formal use. A surplus-production model treats a fish stock like a savings account: next year’s biomass equals this year’s, plus surplus production $g(B_t)$ (growth/recruitment net of natural mortality), minus catch $C_t$. In its state-space form the biomass is a latent process observed through an abundance index $I_t=q{B}_t{ϵ}_t$. The data (catches and an index) weakly identify $r$, $K$, $q$ jointly, so informative priors from biology (resilience, life history) do real work; this is the point of example E114.

Overview

The deterministic skeleton is the logistic (Schaefer) model. Fitting only to the trajectory (“observation-error” fit) ignores that real stocks fluctuate; putting process noise on biomass gives the state-space model. Catch is treated as known.

Main Content

Schaefer state-space surplus-production model

$$B_{t+1}=[B_t+r{B}_t(1-\frac{B_t}{K})-C_t]e^{{\eta }_t},{\eta }_t\sim N(0,{\sigma }_p^2),I_t=q{B}_t{e}^{{\epsilon }_t},{\epsilon }_t\sim N(0,{\sigma }_o^2).$$

$r$ intrinsic growth rate; $K$ carrying capacity; $C_t$ catch; $q$ catchability; ${\sigma }_p,{\sigma }_o$ process and observation sd. Reference points: $\text{MSY}=rK/4$ at $B_{MSY}=K/2$ and $F_{MSY}=r/2$. Pella-Tomlinson generalises the surplus function with a shape parameter $m$ ($B_{MSY}/K$ becomes $m^{-1/(m-1)}$).

Identifiability and the role of priors

Issue	Mechanism	Remedy
$r$-$K$ ridge	Many $(r,K)$ combinations produce similar biomass trajectories when the series lacks contrast (“one-way trip”: steady decline)	Informative prior on $r$ (e.g., from FishBase resilience classes) and/or on depletion
$q$ and $K$ scale confounding	Index is relative; $qK$ is identified, $q$ and $K$ separately are not	Prior on $q$ or on initial depletion $B_1/K$; absolute abundance survey
Process vs observation noise	Only the ratio ${\sigma }_p^2/{\sigma }_o^2$ is weakly identified	Fix the ratio or put a prior on the variance ratio
Collapse invisible to catches alone	Catch can be flat or falling while biomass crashes; the index reveals the decline	Always model the index; plot catch/biomass ratio $C_t/B_t$ (harvest rate)

Hierarchical extension. Fit many stocks jointly with log $r_s\sim N({\mu }_r({\text{resilience}}_s),{\tau }^2)$ and shared structure: partial pooling shrinks poorly informed stocks toward group behaviour, the same logic as Hierarchical Models; this is how 18-22 stocks in the RAM Legacy database are treated in E114.

Decision output. Posterior of $B_T/B_{MSY}$ and $F_T/F_{MSY}$, probability the stock is overfished ($B/B_{MSY}<1$), and catch advice as a function of risk tolerance (Decision Analysis).

Examples

Simulate a collapse and watch the index (runs; verified).

import numpy as np
rng = np.random.default_rng(1)
r, K, q, sp, so = 0.4, 1000., 0.001, 0.05, 0.1
T = 40; B = np.zeros(T); B[0] = 0.8*K
C = np.concatenate([np.full(15, 40.), np.full(25, 160.)])     # catch quadruples after year 15
for t in range(T-1):
    B[t+1] = max((B[t] + r*B[t]*(1 - B[t]/K) - C[t])*np.exp(sp*rng.normal()), 1e-3)
I = q*B*np.exp(so*rng.normal(size=T))
print(np.round(B[::8]), np.round(I[::8], 3))      # biomass 800, 908, 770, 190, ~0 ; index follows
print("MSY", r*K/4, "B_MSY", K/2)                 # MSY = 100: catch of 160 exceeds MSY and the stock collapses

Interpretation. A constant catch above MSY (160 vs 100) is unsustainable: biomass falls from about 770 to about 190 and then to zero within 16 years, yet the catch series itself looks flat. Only the index (or a model linking catch to biomass) shows it.

PyMC sketch (untested).

with pm.Model() as spm:
    r = pm.LogNormal("r", np.log(0.4), 0.3)           # biology-informed prior
    K = pm.LogNormal("K", np.log(1000), 1.0)
    q = pm.LogNormal("q", np.log(1e-3), 1.0)
    # latent biomass via pm.Scan or a non-centered log-random-walk with recursion; observation via index

Relevance to marketing measurement

Latent stock with replenishment and depletion, observed through a noisy relative index with an unknown scaling constant ($q$), is formally the same as brand equity/awareness observed through surveys. The identifiability lessons (scale confounding, need for informative priors on one component) carry over; see Linear-Gaussian State-Space Models.

Connections

Hierarchical Models, Prior Predictive Checking, Marginal Likelihood via the Kalman Filter (exact for the linearised case), Chen 2018 - Latent ODE Time-Series Model.

See Also

Chaos, Rough Likelihoods and Data Assimilation (Concept Note), Dose-Response with Weather Covariates in Long-Term Field Trials (Concept Note).