Whales - Population Studies
Science Project 18

by
K. D. Stroyan
University of Iowa

Copyright 1993 by Academic Press, Inc. - All rights reserved.

NoteBook Overview

This project studeis a model of Sei Whale reproduction proposed by J. R. Beddington.  The model is similar to the discrete logistic equations studied in Chapter 12 of the main text, but involves a delay of 9 years corresponding to the time it takes a whale cow to birth a calf.  The NoteBook contains the technical details needed to simulate results of the model, so familiarity with Chapter 12 is not needed.  (You might take a brief look at the chapter just for comparison.)
	The written project explains the meaning of the model in more detail.
	Your job is to set the whaling quotas and to investigate what happens to the whales if some people cheat on the quotas.

Decline of the Sei Whale

The following list contains our estimates of the population of Sei Whales in the Indain Ocean.

WhaPopData = {{1900,26000},{1910,26000},{1920,25000},
				{1930,23900},{1940,22800},{1950,22000},
				{1960,21000},{1970,18000},{1980,4000}}

ListPlot[WhaPopData,
AxesOrigin->{1900,0},PlotJoined->True,
AxesLabel->{"year","population"}];

Problem Definition

The following equation was derived by J.R. Beddington in a article in the Report of the International Whaling Commission (1978) - it describes the population of Sei Whales at time  t  as a function of the carrying capacity k and populations at times (t-1) and (t - 9).  The equation accounts for deaths due to natural causes and to harvesting, as well as births (that occur only after 9 years).     
		  
p[t_] := p[t] = p[t-1] - 0.06 p[t-1] + p[t-9](0.06 + 0.0567(1 - (p[t-9]/k)^2.39)) - 0.94 h

The CHANGE in population from year  t - 1  to year  t is

		Natural change at t = - 0.06 p[t-1] + p[t-9](0.06 + 0.0567(1 - (p[t-9]/k)^2.39))

		Hunted change at t = -0.94 h

for a `harvest' of 94 % of  h  as a fraction of the carrying capacity.  This equation is described in detail in the Scientific Projects.

Interpreting the Formula

Your written project report should begin with a derivation of the above equation in your own words.  It would be helpful if you make a rough draft of that derivation before you do the mathematical experiments in this NoteBook.  SEE THE WRITTEN PROJECT 18 for help.

Equilibrium Values and  Sustainable Yields

When harvest levels are set so that the whale stock will tend to and reach some hunted equilibrium population, pH, then the formula for population will demonstrate that pH = p[t] = p[t-1] = p[t-9].  Explain why this is the case?  What does "equilibrium" mean?
	
	We want to understand the connection between a harvest level  h  and the resulting equilibrium values.  This means
	
	0.94 h = pH(0.0567(1 - (pH/k)^2.39))
	
Why?

Growth Below Carrying Capacity

In order to select a sustainable harvest level, we want to see how much natural net growth  fN[p]  the species could produce at a given population level.  If  the species is at the carrying capacity, k, the growth will be zero.  We know that hunting will supress the population somewhat and that the species will grow to try to recover.  The big question of this whole project is:  Will it grow enough to sustain a positive population?
	To simplify things, suppose that the population has been held at a constant level  p for a number of years, so that  p[t-1] = p[t-9] = p.  You should plot the natural growth for the next year as a function of  p.  This growth is given as part of the formula above and you should type the formula for  births[p]  and plot it now.
	
	Use  p  measured in thousands of whales to simplify scales.

Clear[p]
k = 26
births[p_] := - 0.06 p + p(0.06 + 0.0567(1 - (p/k)^2.39)) ;
Plot[births[p],{p,0,26},
			AxesLabel->{"Pop. X 1000","Growth X 1000"}];

A Key Question in Your Project

What does this graph mean?  For example, suppose we reduced the population to 80 % of the natural carrying capacity.  What would the net increase in the population be for the next year?

The Population Change with Hunting

The graph of the natural increase at populations below carrying capacity can help us determine harvest levels that the species can sustain.  Suppose that we decide to harvest one and one half percent of the carrying capacity of whales each year.  How does this effect the total change in next year's population fraction in the case where  p[t-1] = p[t-9] = p, a constant?
Explain why
	
	Next year's p = births[p] - hunted fraction who would not have died of natural causes
				= births[p] - 0.015 X 0.94
				
and if we hunt at  h  (as a fraction of natural carrying capacity),

	Next year's p = births[p] - 0.94 h
	
What does it mean if next year's total change function (fN[p] - 0.94 h) is positive?  Negative?

	Make a plot of  births[p] with  h = 0.015/0.94 superimposed on it.

26 .015

Clear[p]
k = 26
births[p_] := - 0.06 p + p(0.06 + 0.0567(1 - (p/k)^2.39)) ;
h = .67
Plot[{0.94 h,births[p]},{p,0,26},AxesLabel->{"Pop. X 1000"," "},
PlotLabel->{"0.94 h and annual growth vs p"}];

Interpreting the Graph with Hunting

For which values of  p (simply read from the graph) is the total hunted change for next year positive?  Negative?

	In the two intervals of  p  where total hunted change is negative (at h = 0.015), what happens to the change as  p  increases or decreases?

The Maximum Sustainable Yield

If you have answered the previous question, you can see that there is a level of the line  h  allows for maximal harvest.  What value of  p  makes the annual natural recovery  fN[p] the largest?  Mathematica can help:

births[p_] := ??;
df = D[births[p],p]
FindRoot[ df == 0 , {p,0.7}]

What is the amount of the maximal annual recovery?

fN[0.6???]/.94

If the harvest level is set at  2.55 % of carrying capacity, what population level  pE  is in equilibrium?  What happens if  p < pE?  What happens if  p > pE?

	Why is it that when  h  equals the maximum of  fN[p]/.94, we say  h  is at the maximal sustainable yield? (MSY)  Can the function  fN[p] which assumes that  p[t-1] = p[t-9] answer this completely?

The Dynamics of Hunting over the Maximum Sustainable Yield

You have seen that the maximal sustainable yield should be about  2.5 % of carrying capacity based on computations with a simplified model.  Try some experiments with the full model at  h  levels slightly over this prediction:

Clear[p];
h = .67*.85	(*Annual Harvest 300 = 0.3 X 1000*)
Do[ p[i] = 10 , {i,1,9}];
p[t_] := p[t] = p[t-1] - 0.06 p[t-1] + p[t-9](0.06 + 0.0567(1 - (p[t-9]/k)^2.39)) - 0.94 h
ListPlot[Table[p[n],{n,100}],PlotJoined -> True,
			PlotRange->{0,27}];

Safety in the Measurements

Biologists and mathematicians have found that modeling the population of marine organisms like whales is a very complicated, uncertain process.  We need accurate observations of the real natural carrying capacity in our model.  Why?  It is therefore hard to calculate the real maximum sustainable yield.  For this reason, it has been suggested that the harvest quota should be set at a fraction of the maximum sustainable yield derived from a mathematical model such as this one.

The Equilibrium Values

Consider a case in which the harvest quota has been set at .85 MSY (MSY = maximum sustainable yield), using the MSY value you determined earlier.  With this harvest quota, what are the possible equilibrium level(s)?  Explain the meaning of all of your anwswers.

Clear[p]
births[p_] := p(0.0567(1 - (p/k)^2.39));
h = .55 (*550 actual*)
Plot[{0.94 h,births[p]},{p,0,26},PlotLabel->{"0.9 MSY and annual growth vs p"}];

Use FindRoot[] to calculate the two equilibrium values shown in the graph above and explain why they are true equilibria for the dynamic model given at the beginning of the project.

Explore the Consequences

Our 90 % harvest quota is intended to save the whales as a species, but we must understand mathematically why there ate two equilibria and then describe the bilogical consequences.  Here are some experiments to get you started.

Case #1

We want to test your solution with some real numbers.  Data indicates that the Sei whale population remained at .85 of the environmental carrying capacity from 1950 to 1960.  For the ten years following this, the whales were harvested at .9 MSY each year.  Make a table showing the population as a fraction of the carrying capacity from the year 1961 to the year 1971.  What can you say is happening to the population?    What if this continues for a century?

Graphical Representation - case #1

Case #2

What if we changed the above population, making the initial population .4 of the environmental carrying capacity.  Make a table of these population fractions, and discuss the results, comparing them with those from Case #1.

Graphical Representation - Case #2

Look into the Crystal Ball

Now, let's consider what the situation will be a few hundred years in the future if whales are harvested at the annual rate of .9 MSY.  Using various initial populations, consider the effects of this harvest level on the whale population.   Using at least four different initial populations, do projections 100, 200, 300, and 400 years into the future.  The following command cell gives you a function with which you can make these projections.
	 While doing these calculations, you want to consider the significance of the graphs that you create.  Be sure that you understand what your results imply in terms of your equilibrium points.  From the results that you derive, what is the minimal whale population that can be sustained with the given amount of harvesting?  Can you explain why this population is the minimal sustainable one?
	 Things to consider in your scientific report include the possibility that someone might cheat and harvest more than the quota for several years or that a disease temproarily reduces the population.  What if pollution lowers (or raises) the natural carrying capacity?