Fishing Mathematically

“Oh you are getting a PhD in mathematics?  Are you going to teach?”

Here is my first attempt to try to explain why mathematicians exist outside school and should further be involvement in social issues.  We, as a society, are moving further away from the process of peer review.  Authors no longer have editors because editors cost too much.  If a company bothers to hire a mathematician, they also want her to be computer scientists so she can do computer programming as well as model development (and other mathematical duties).  We want our blogs unedited and our music uncensored.  There are fewer chances for someone to say, “That is not correct” before a million readers see something false or eat something inappropriate.  Companies need mathematicians to be guarantee they are mathematically trustworthy.  We are helpful and necessary members of industry and life.

Let’s take an example of over fishing.  There is currently a highly managed fishing business in Alaskan Salmon.  There are strict policies about how many fish can be caught and how many must escape capture. (This is the phrase the Alaskans use)  But who sets those limits?  What amounts to a “sustainable yield”[1]?

It’s easy to say that a fishing boat is removing X fish from the system per load.  Then anyone can count the number of loads a fishing boat makes in a year (say 250) and  the number of boats that fish that type of fish (say 100).  From this we have an idea of total fish removed (100*250*X fish).  But to make a good case that we are over fishing we need to show using population dynamics that given current conditions, we will run out of fish because there are not enough fish in the ocean to repopulate each year.  We want to say that the fish population can’t produce 25000X fish each year to sustain the population.  But how do we show this dynamic relationship?  Maybe the first year there is enough fish, but what about the next when there are 25000X less fish?  Let’s do an example.

Say there are 100 Salmon in the ocean.  The fishing companies think we can capture 50% of the fish each year.  So the first year we catch 50 fish and let 50 fish escape.  The surviving fish will reproduce.  Salmon under the age of 5 years old will not reproduce and they rarely live past 10 years old.[2] So we can assume ½ the population is under 5 and half the population over 5 is female.  So we have 12.5 female Salmon laying eggs.  Females lay up to 7 redds with up to 5,000 eggs per redd[3].   This means 7 * 5,000 = 35,000 eggs.  We know on average only 10% of the salmon eggs survive to be 2 years old. [4] Now we have 35,000 * .1 = 3,500 fish which are added to the fishing population in 2 years per female which yields 12.5*3,500 = 43750 Salmon.  (Assuming nothing else kills them of course!) During these two years we have fished up 25 fish the first year and 12 fish the next.  So when our 43750 fish enter the system there will be only 12 fish left.   We are saved! At 50% fishing our calculations predict that the population will survive.

Only, we didn’t consider a lot of variables.  Perhaps when there are less Salmon the death toll from natural predators increases- so less Salmon survive to be fished by humans.  There are may complicated relationships in the model of Salmon & Fishing.   To what degree do we need to scale back our fishing to allow the population to rebuild?  These questions require some differential equations, a personal computer with some computing software and a mathematician to produce the Salmon Model.  Now you might title the mathematician “engineer” or “scientist.”  But if they are actively doing mathematical modeling, then they are doing the work of a mathematician.  We need to make a mathematical model that calculates all of these details for us so we can see what will happen in the future given various sustainability tactics.   My favorite sentence from the Alaska Salmon Program at the University of Washington[5] is:

The forecasts are based on scientific information derived from research on climate-related and density-dependent processes governing the dynamics of the salmon populations.

Do you see how they make you think it was solely the scientists doing research who made the suggestion?  Really there is also a mathematician behind the scenes making models and testing parameters for robustness.

Without this mathematician and the research of mathematicians about differential equations for the last hundred years, the data produced by Salmon & Fishing Model would be worthless.  But many mathematicians made sure that these equations are consistent (the equations must come up with the same answer even if you take two paths to get there) and robust (if you put in weird initial conditions you still get a realistic solution). Most people blindly believe statistics and the mathematics behind data. We trust that the environmentalists who say fish stock is getting too low without understanding their models.  But someone needs to double check the models; someone needs to make sure we aren’t lying with our numbers.  The mathematicians are sort of the M.I.B. of data.  We are the first last and only line of defense against the wrongful use of models and formulas.  This is why mathematicians must work in industry.  It’s our job to make sure everyone else didn’t cheat on their math reports.  Just like your fourth grade teacher.  This is one reason I don’t have to be a math professor.

[1] Alaska’s constitution, Article VIII, Section 4 states: “Fish, forests, wildlife, grasslands, and all other replenishable resources belonging to the State shall be utilized, developed, and maintained on the sustained yield principle, subject to preferences among beneficial uses.”

Applied Mathematician and writer of socialmathematics.net.
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4 Responses to Fishing Mathematically

1. naomi says:

Nice example! Good thing there’s always one around when I need one! 🙂

• yay!!!!!!

2. Andy says:

what equation was used?