Dr. Eric Marland

Fisheries Lab, Analysis of the Ricker Model

Introduction

As an introduction to fisheries, please read the following three articles:

Marine conservation requires a new ecosystem-based concept for fisheries management that looks beyond sustainable yield for individual fish species, by Zabel et al.  American Scientist online, Volume 91, Number 2, Page 150.

Also, look at the NOAA  http://www.noaa.gov/nmfs/vision/sustain_vision.html .

Equivalence in Yield from Marine Reserves and Traditional Fisheries Management, by Alan Hastings and Louis Botsford. Science, Volume 284, 28 May 1999.

These articles give a broad view of different ideas about fisheries and management strategies.  For this lab we will consider a simple model and discuss strategies for addressing the questions brought up by the articles listed above.  In addition to these articles, find one additional source and cite it in your write-up.  The introduction and conclusion to your lab report should be based on these readings.

Model Background

We can begin with a discussion of the basic models of growth. Simple exponential growth can be modeled by the following difference equation,

P_t = P_t-1 + l P_t-1         (Exponential Growth Model)

Simple exponential growth is one of the few models that can be solved explicitly. The solution is,

P_t = e^{(l+1) t}

Unfortunately, the simple exponential model is only valid in quite specific circumstances. To begin our search for more realistic models, we first move to solve one of the primary problems with the exponential model, that of uncontrolled growth. While many populations begin by growing in an exponential fashion, inevitably the growth slows until it reaches level where the environment just sustains the population. The simplest model that incorporates this slowing of the growth rate as the population increases is the logistic model. The logistic model assumes that as the population grows, the growth rate decreases in a linear fashion as a function of the population size.

P_t = P_t-1 +  lP_t-1 (1- P_t-1/K)      (Discrete Logistic Model)

Again a solution is possible, but not really very pleasant. A model that has been used extensively for simple fishery models is the Ricker model (and a variety of modifications). The Ricker model, as with the logistic model improving upon the simple exponential model, improves on the logistic model. The logistic model does a very poor job (non-biological) of predicting very large populations. It assumes a decreasing growth rate with population size, but doesn't ever let that growth rate cause the population to become negative.

P_t = P_t-1 exp( r(1-P_t-1/K) )        (Ricker Model)

Methods of controlling fishing

• limiting effort (short season)
• limiting catch (number limits)
• limiting size (size limits)
• limiting location (reserves)

What are the goals of regulation? There are many goals of regulation. The one of least concern to the government may be concern for the species, although this concern has been increasing in recent years.  The real driver in many regulations is money. How do you make money with fish? You catch them and sell them. Not only that, you catch them and sell them every year for many years.

You don't really want to catch all of them the first year. If you did that, there would be no fish to catch the next year. What we want to do is to catch the amount that provides the maximum sustainable yield. Sustainable means that we can hope to catch this same number of fish every year, not just the first year. This is the strategy that most intelligent politicians and environmentalists can agree on. It is a strategy that seeks to perpetuate the population at a high level that also allows the fisher to still make money.

The key of course is to accurately calculate the maximum sustainable yield. What happens if you miscalculate the value? Under-fishing is not so bad. The population will grow to higher levels than otherwise and the fisher will make slightly less money than they could have. Over-fishing, by even just a little however, can cause the population to crash, perhaps irrecoverably. Making a determination of the maximum sustainable yield then, is of the utmost importance. While the easiest thing would be to play it safe and make sure you underestimate the value, but the fishers would complain loudly (which the politicians hear).

How to add in the effects of regulation

We have several options for how to incorporate harvesting of the fish, according to the goals of the different types of regulation. The easiest to incorporate is a set harvest during each period. For a set harvest, we simply subtract the harvest number from the updating function. For the exponential equation, this would be,

P_t =  l P_t-1 - h

where h is the number of fish to be harvested. In regulating the effort, via a fishing season, the effect is that a set proportion of the fish are harvested. The proportion of the fish harvested is related, among other things, to the length of the fishing season and the difficulty in catching the fish. Again, for the

The modifications for simulating reserves and size limits are a bit more complex and we reserve those modifications for the questions at the end of this lab.

How to maximize sustainable yield

The yield for a particular harvest is the total number of fish harvested. A sustainable yield would be the total harvest when the population is in steady state. To find this yield, first you will need to find the steady state population for a generic value for the harvest. With this value for the population, calculate the total harvest taken in such a year. You now have an expression for the sustainable yield.

Well, now that you have an expression for the sustainable yield, it is time to maximize this yield. How do we maximize the yield? The same way we maximize anything. Take the derivative, set it equal to zero and solve. The question is, which derivative are we taking in order to maximize the sustainable yield? We will leave that for you to ponder for a bit.

If the equations do not turn out to be easy to analyze, we can estimate the maximum sustainable yield by numerical methods. Using a few minute of guessing and checking on Berkeley Madonna, we can narrow in on the maximum sustainable yield. For larger, more complicated models, a more automated solution is required. While we will leave this discussion for another time, it is important to know that these methods are not only possible, but commonly used.

Questions

1. Run the model with r=0.1 for 120 time steps starting with P₀=1.  Describe the shape of the curve and comment on how it differs from the exponential model.
2. For a range of values of r between 1.5 and 3, plot the steady state level of the populations as a function of r.  Describe the plot and document r values where qualitative changes to the solution take place.  These changes in qualitative characteristics are called bifurcations.  Describe the changes.
3. Suppose we add a random component to the model,

   P_t = P_t-1 exp( r(1-P_t-1/K) ) + x_t-1

   where x is a random variable with normal distribution, mean 0, and variance = v.  The Berkeley Madonna function for a normally distributed random number is normal( mean, variance).  Pick a test value for v and repeat question 2.  Describe any changes in the solution.
    

4. What is the maximum sustainable yield for regulating effort in the Ricker model?
5. If you limit by effort, how might you do so to make the regulation most effective?
6. What is the maximum sustainable yield for regulating catch size in the logistic model? Also find the maximum sustainable yield for the Ricker model with regulated catch size (this will be a bit of a challenge).
7. What needs to be added to the model to adequately model marine reserves; propose some equations.
8. What needs to be added to the model to adequately model regulation by size limits; propose some equations.
9. Discuss the idea of extinction and how that might be relevant to different harvesting strategies.
10. How might the type of fish/sea creature determine which method is most appropriate?

Follow-up Questions

1. What do you feel was the purpose of this lab?
2. What did you learn in this lab?
3. What changes would you make to this lab?
4. Please rate the difficulty of this lab on a scale of 1 to 10.