Motivation

In order to model the future of the global fish population, we chose a differential equation modelling approach. Such an approach, however,does rely on precise parameter definition – and, as a result, we spent considerable time refining these parameters. In particular, this “tuning” was done by taking a set of observed data (in our case, fish biomass throughout the last 60 years) and changing these parameters until our model’s predictions resembled the data as closely as possible. Being able to precisely predict past biomass values, ensured that we had some grounding for making future estimates.

Unfortunately the global fish biomass data, the cornerstone of the tuning process, was not something which was readily available. A “total fish biomass” time series did not, to our knowledge, exist. We had to create it ourselves.

RAM database

After further investigation, we found that there were many cases in which biomass data was available for specific species in specific regions – this data being produced mostly for the sake of commercial stock assessment. RAM Legacy Stock Assessment Database is a “compilation of stock assessment results for commercially exploited marine populations from around the world”. We believe that it is the most complete compilation of Stock Assessment Results to this date. Another advantage of the RAM Database, compared to other databases (NOAA, ICES & etc.), is that it combines data from different regional agencies, thus ensuring good global coverage. Effectively, the RAM Database includes data from all known to us sources; therefore we decided to use it for our further work.

• 

  RAM Database coverage

  Ricard D, Minto C, Jensen OP, Baum JK (in press) . 2011. Examining the knowledge base and status of commercially exploited marine species with the RAM Legacy Stock Assessment Database. Fish and Fisheries doi: 10.1111/j.1467-2979.2011.00435.x.

Data manipulation

The data presented in RAM, in some cases, was not entirely homogeneous. For example, the Spawning Stock Biomass (total weight of those fish that have reached the breeding age) – the data figure we were interested in - was often presented in different measures. These measures ranged from weight in tonnes/kg/pounds to the biomass of the annually produced eggs and non-specified measures. We had to omit the datasets, which were not directly convertible to tonnes.

Calculating total fish biomass

Chart: All datasets prior to any manipulations

1/5 Choosing boundaries for the datasets

The next step was to “combine” the individual species-specific time series into one, which would become our biomass data.

The time period coverage offered by different datasets varied significantly. Some datasets included biomass data from 1874 to 2005; while others gave information for the years 1990 to 2006.

FAO provided fish catch data from 1950, hence it was possible for us to run our model only from this date onwards. However, it was important that we could compare model predictions with ‘real’ data for as many years as possible, before we could use our model to predict future results. We, thus, chose 1950 as the baseline of our equation. The nature of our model, as a delay differential equation, meant that we had to have biomass data for 1-4 years (the average time for a fish to reach maturity) before 1950.

Chart: Datasets spanning different time periods

2/5 Extrapolating datasets between 1932 and 2006

The number of datasets present at 1932 was approximately the same as at 1945. Since the number of datasets influences the quality of extrapolation (refer to “How does extrapolation of datasets work?”), we could extrapolate just as well back to 1932 as we could to 1945.

The upper boundary of extrapolation was set to 2006, since after 2006, the number of datasets with information decreased rapidly.

Chart: The "standard" biomass sets

3/5 How does extrapolation of datasets work?

We termed those sets that had data for at least all years between 1932 and 2006 the “Standard Sets”. For the datasets requiring extrapolation, we assumed that the population of the species described by these datasets varied between 1932 and the first year of the dataset in a proportional way to the way in which the sum of the biomass of the standard sets varied during that time period. It is for this reason that it was extremely important, when searching for datasets, that we included as many as possible in our “Standard Sets” and our Standard Sets described species from many different temperate zones.

Chart: Extrapolated datasets

4/5 Combining the datasets

Once the charts were extrapolated from 1932 to 2006, we had 234 individual data sets. In order to combine them, we summed up all dataset values at each year. Since the data we had is only a fraction of the total world fish population (although, a representative one), there was a need to upscale it. We referred to Villlie Christensen’s prediction of the 1950 biomass to carry out this task.

Chart: The final time-biomass evolution curve

5/5 Our final result

‹ ›

Delay Differential Equations and Numerical Solution Approximation Methods - is it all really necessary?

Why model?

In our project we sought to:

In order to answer such questions about the future and theoretical, never before encountered, scenarios, one has to make assumptions about the nature of our world and how it 'works'. Very often, these assumptions can be expressed in a mathematical format. The mathematical format is often referred to as a "mathematical model" of the physical situation. Hence, as we sought to answer our own questions, we produced a mathematical model that predicted the evolution of world fish biomass over time. Our model involved parameter values which could be changed to enable us to ask different questions of the same model.

Why wet biomass?

We chose to model fish population in terms of fish biomass present at any time instead of number of fish for various reasons:

• Most data, e.g. recruitment rate is in terms of biomass. Therefore, we avoided unnecessary conversions and errors.
• More importantly, we are modelling total fish populations, so to model fish numbers when definition sustainable numbers of fish vary so differently from species to species is silly.

Why adult fish?

Having chosen to measure fish population in terms of wet fish biomass, it became necessary to measure population in terms of adult fish biomass instead of all fish biomass.

We are examining fish biomass throughout time and to model all biomass would require us to take into account the growth of the fish, thus, resulting in the modelling of multiple weight classes of fish throughout their lifetime. Investigating adult (mature) fish biomass, we can assume to a first approximation that a fish biomass is constant throughout time (not a bad approximation according to Von Bertalanffy fish growth model) and we can, thus, simplify our first model.

The mathematics

Our model takes into account what we believe to be the most fundamental factors that will alter adult fish biomass measurements between two years: the recruitment of junior fish into the adult population, the natural death of adult fish and the catching of adult fish.