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

Before the QMS (Quota Management System) was introduced in 1986, Māori practiced their own sophisticated fisheries management through rahui — temporary restrictions on fishing in an area to allow stocks to recover. Mathematical models show that a rahui of just 18 months can allow snapper populations to increase by 40–70%.

1. If a fishing area has 12,000 snapper and a 20-month rahui allows 55% growth, how many snapper will there be after the rahui? If the fishing rate after the rahui is 15% per year, write a linear equation P(t) for the population over the following 5 years.
2. Exponential Growth: Under full protection, snapper numbers increase at approximately 18% per year. 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: The 1992 Treaty Fisheries Settlement gave Māori 50% of all NZ deepwater quota. Ngāti Porou now holds 4,200 tonnes of deepwater quota annually. At $8,200/tonne market valor, 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 from the perspective of Tangaroa — the guardian of the sea — about what a sustainable quota would be. Include at least one equation.