Sustainable Fishing & Equations

Part 1 — Fish Stock Data and Linear Models

The Ministry for Primary Industries monitors key fish stocks using Spawning Stock Biomass (SSB) — the mass of breeding adults. Here is data for two key NZ species:

Linear Model Questions

1. Plot the Snapper SSB data on a graph (Year on x-axis, SSB on y-axis). Does it appear roughly linear? Sketch a line of best fit.
2. Calculate the average rate of change (gradient) in Snapper SSB from 1990–2008:
   gradient = (SSB₂ − SSB₁) / (year₂ − year₁) = (24,000 − 38,000) / (2008 − 1990)
   What does this slope tell us — in tonnes per year — about the health of the fishery?
3. Write the equation of a linear model for Hoki SSB from 1990–2008 in the form SSB = mt + c. Use it to predict the Hoki SSB in 2012. How close was your prediction to the actual recovery? What does the difference tell you about the limitations of linear models?

Part 2 — Quadratic Models and Maximum Sustainable Yield

The Maximum Sustainable Yield (MSY) is the largest amount of fish that can be caught without reducing the stock below its recovery point. Fisheries scientists model this with a quadratic equation:

Where: Y = yield, r = intrinsic growth rate, N = current population, K = carrying capacity (maximum natural population)

Your Turn

1. Graph the Yield Function: Using r = 0.4 and K = 80,000, calculate Y for N = 0, 10,000, 20,000, 30,000, 40,000, 50,000, 60,000, 70,000, 80,000. Plot these points. What shape do you get? What does this tell you about the model?
2. Finding the Maximum: Expand Y = rN − rN²/K. This is a downward parabola. Use completing the square (or calculus: dY/dN = 0) to find the value of N that maximises Y. Confirm this matches K/2.
3. Quota Setting: The current hoki catch limit is 156,000 tonnes. Estimated K = 500,000 tonnes, r = 0.3. Calculate the current yield at N = 190,000 tonnes. Is the quota set above or below the MSY? What recommendation would you make to the Minister of Fisheries?

Part 3 — Kaitiakitanga in Numbers: Māori Fisheries Management

Long before the QMS (Quota Management System) was introduced in 1986, hapū and iwi placed rāhui — restrictions on taking from an area, lifted when the conditions that prompted them have changed. A rāhui carries the authority of the hapū or iwi who place it; it is tikanga, not a management technique the QMS can be said to have replaced. What it shares with a quota model is the starting assumption that restraint is a legitimate outcome rather than a cost to be minimised. The recovery rates below are model figures chosen so the algebra works — they are not measured results for any real fishery or rāhui.

1. If a fishing area has 12,000 snapper and a 20-month rāhui allows 55% growth (a model figure, not a measured one), how many snapper will there be after the rāhui? If the fishing rate after the rāhui is 15% per year, write a linear equation P(t) for the population over the following 5 years.
2. Exponential Growth: For this model, assume a protected snapper population grows 18% per year. (A modelling assumption, not a measured rate — real recovery depends on the stock, the species and the season.) Write an exponential equation P(t) = P₀ × (1.18)^t for a population starting at 5,000. How many years until it exceeds 20,000? Solve algebraically using logarithms.
3. System Analysis: Under the 1992 Treaty of Waitangi (Fisheries Claims) Settlement Act, Māori received a 50% shareholding in Sealord Products Ltd and 20% of the quota for any new species entering the Quota Management System — not 50% of the existing deepwater quota outright. Suppose an iwi fisheries entity holds 4,200 tonnes of deepwater quota annually, at a market value of $8,200/tonne — scenario figures, chosen for the calculation; they are not any real iwi's holdings. Using them, calculate the annual value of their quota. Over a 25-year period (assuming 2% annual increase in quota value), calculate the total value using the geometric series formula:
   S = a(rⁿ − 1) / (r − 1) where a = first year value, r = 1.02, n = 25
4. Ethical Reflection: The orange roughy lives for 150 years and doesn't reproduce until age 20. Current stocks are at 17% of original levels. Write a short argument for what a sustainable quota would be, and defend it with at least one equation. Argue it as a kaitiaki would — from obligation to the moana and to the people who come after — not from this year's catch value. (Do not write in the voice of Tangaroa; the atua are not a role for us to perform.)