**Mentor** Andrew Edwards, Fisheries and Oceans Canada
- Xiaoying Deng, St. Francis Xavier University
- Brian Goddard, University of British Columbia
- Fang He, University of Western Ontario
- Nancy Hernandez Ceron, Purdue University
- Alejandra Herrera Reyes, University of British Columbia

The waters off the Pacific coast of Canada are home to numerous species of fish, many of which are commercially harvested. Scientists conduct stock assessments to provide advice to fisheries managers concerning the status of fish stocks (such as whether a population is currently healthy). The advice is used by managers to help set total allowable catches so that stocks can be harvested in a sustainable manner.

In many cases, stock assessments are conducted using mathematical population models. In particular, age-structured population models are used for long-lived rockfish species that can live for 100 years, as well as shorter-lived species such as Pacific Herring. A key input for age-structured models is a set of data comprising the ages of individual fish.

For rockfish, the age data are determined from otoliths – the ear bones of the fish. The otolith is a chronological recorder of the environmental conditions inhabited by the fish, the most influential of which is temperature. Changes in temperature create a stress that is recorded within the otolith microstructure. These changes usually correspond to seasonal changes. For example, the warmer, more-productive summer months allow fish to grow at a faster rate than in the colder, less-productive winter months. This leads to the otolith microstructure having larger summer growth zones and smaller winter growth zones (Figures 1 and 2). Two zones combined therefore represent one year of life. Experienced ‘age readers’ count the zones to determine the age of the fish. For Pacific Herring, ages are determined from scales rather than otoliths (Figure 3).

However, age determination is accompanied by several sources of error. Interpretation of zones can be difficult (see Figures). Subsequently, the age reader records a “most likely” age together with minimum and maximum ages.

In this project, we will aim to quantify this source of uncertainty. In a recent rockfish stock assessment a simple sensitivity test was performed, which assumed that a fish that was given an age of, say, 37, had an 80% probability of truly being age 37, a 10% probability of being age 36, and a 10% probability of being age 38. This example demonstrates aging imprecision (a random error). A second issue is aging bias, where there is a systematic under- or over-estimation of the true age of the fish. In most cases, the imprecision would be larger than that in the example above; therefore, an approach using probability distributions should be more realistic.

The focus of this project will be to determine probability distributions that characterize the aging uncertainty. Extensive data will be available, including the ranges estimated for each otolith by the readers. For quality control, a second reader independently ages some of the otoliths that were aged by the original reader, and such data may also be explored.

If time permits, the results will be used to re-run existing stock assessment code to determine the impact of including aging error on the model results, and on the consequent advice that is given to fisheries managers. Given the highly nonlinear nature of the age-structured population model, it is not possible to predict (without re-running the model) how the inclusion of aging error will affect the results.

This project will demonstrate to students the interesting mathematical problems that can arise when modelling ecological populations, and that solutions can have a direct influence on the setting of allowable catches, and thus on the health of fish stocks.

**Prerequisites:**

Necessary: background in probability distributions and matrix algebra. Desirable: familiarity with likelihood analysis and the R or C++ programming languages.

Keywords:

Fisheries, uncertainty, matrix algebra, mathematical ecology, population models.

Figure 1. Otolith from a Silvergray Rockfish estimated at 18 years old. All photographs from the Sclerochronology Laboratory, Pacific Biological Station, Fisheries and Oceans Canada.

Figure 2. Otolith from a Shortraker Rockfish estimated at 82 years old. The arrows show the first three years of growth.

Figure 3. Scale from a Pacific Herring estimated at 9 years old.