Introduction to Fisheries Economics

by the use of Mathematica programming

Arne Eide Copyright c 2017 Arne Eide

Published by Publisher book-website.com

Licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License (the “License”). You may not use this ﬁle except in compliance with the License. You may obtain a copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an “as is” basis, without warranties or conditions of any kind, either express or implied. See the License for the speciﬁc language governing permissions and limitations under the License.

First printing, March 2017 Contents

1 Introduction ...... 7

I Harvesting technology and population dynamics

2 Fishing effort ...... 11 2.1 Requirements of fishing 11 2.2 How to measure fishing effort? 11 2.3 The production of fishing effort 12 2.4 Introduction to elasticities 13 2.5 The Cobb-Douglas function 14 2.6 The CES function 15 2.7 Elasticity of scale 16

3 Fish catch production ...... 19 3.1 Catch production 19 3.2 Catch as a method of stock assessment 19 3.3 Cobb-Douglas catch production 23 3.4 Properties of ﬁshing gears 23 3.5 Exercises 23

4 Population dynamics ...... 25 4.1 Basic principles 25 4.2 Classical surplus growth models 25 4.3 Depensatory growth 31 4.4 Age structured model 34 4.5 Exercises 39

5 The concept of equilibrium harvest ...... 41 5.1 Equilibrium harvest 41 5.2 Surplus production models 42 5.3 Age structured models 46 5.3.1 Numbered List...... 47 5.4 Remarks 47 5.5 Corollaries 47 5.6 Propositions 47 5.6.1 Several equations...... 47 5.6.2 Single Line...... 47 5.7 Examples 47 5.7.1 Equation and Text...... 47 5.7.2 Paragraph of Text...... 48 5.8 Exercises 48 5.9 Problems 48 5.10 Vocabulary 48

II Fisheries economics

6 The economics of catch production ...... 51 6.1 The economics of eﬀort production 51 6.2 Exercises 54

7 Economic growth ...... 55 7.1 Labour creates capital 55 7.2 Neoclassical Economic Growth Theory 55 7.3 Exercises 56

III Fisheries management

8 Economic growth ...... 61 8.1 Labour creates capital 61 8.2 Table 61 8.3 Figure 61

Bibliography ...... 63 Articles 63 Index ...... 65

1. Introduction

Fishing, hunting and gathering food represent the oldest human forms of life, where people depended on wild food for subsistence. While the importance of hunting and gathering wild food decreased after the introduction of agriculture more than ten thousand years ago, ﬁshing has over the last hundred years been increasingly more important as a source of food and wealth for millions of people around the world.

Fishing is an economic activity to produce food for own consumption, but today also for commercial markets and for the purpose of recreation and pastime. These activities often coexist but they could also be regarded as stages in economic and cultural developments, starting with the subsistence ﬁshery, before developing a commercial industry and recreational use of the natural resource.

The aim of the earliest stage, the subsistence fishing, was to feed own household. Early barter economy allowed fish products to be traded but as a perishable product there were limits for the trade-ability of fish. Methods to preserve fish by drying, salting and smoking it, which contributed in developing fish markets where stored fish could be sold, were discovered and developed. Infrastructure then become the major obstacle, transporting the fish to the markets in the large cities demanded storing capacity, roads, vehicles for transport, but also organisations, agreements and security for those investing in the trade.

Increasing trade resulted in wealth creations, first in the business of trade and later also increasingly among fishers. A consequence of economic growth is that labour becomes relatively more expensive and capital more available, hence labour is substituted by capital as a consequence of economic growth, also in the fishing industry.

The aim of recreational fishing, the most recent utilisation of fish stock resources, is not primarily to produce catch, although the catch also often is utilised. The ultimate goals are however the adventure, utilising and enjoying the natural environment, sensing 8 Chapter 1. Introduction peaceful nature and experiencing relaxing activities. Recreational fishing has developed to become a major industry both in terms of accommodating the fishers and production of fishing tackle and equipment.

Fishing takes different forms in different societal and economic contexts. This book aims to discuss the economic activity of fishing through formal models, also including the dynamics of the utilised fish stocks. The analysis are based on basic microeconomic principles and standard theories of fisheries economics are presented.

The book includes however also some attempts of employing methodology which are more rarely seen in other bioeconomic textbooks. The book oﬀers an introduction to programming in the Wolfram Language, presenting a number of relevant examples of how to use the software Mathematica as a tool in ﬁsheries economics. Harvesting technology I and population dynamics

2 Fishing effort ...... 11 2.1 Requirements of fishing 2.2 How to measure fishing effort? 2.3 The production of fishing effort 2.4 Introduction to elasticities 2.5 The Cobb-Douglas function 2.6 The CES function 2.7 Elasticity of scale

3 Fish catch production ...... 19 3.1 Catch production 3.2 Catch as a method of stock assessment 3.3 Cobb-Douglas catch production 3.4 Properties of ﬁshing gears 3.5 Exercises

4 Population dynamics ...... 25 4.1 Basic principles 4.2 Classical surplus growth models 4.3 Depensatory growth 4.4 Age structured model 4.5 Exercises

5 The concept of equilibrium harvest .... 41 5.1 Equilibrium harvest 5.2 Surplus production models 5.3 Age structured models 5.4 Remarks 5.5 Corollaries 5.6 Propositions 5.7 Examples 5.8 Exercises 5.9 Problems 5.10 Vocabulary

2. Fishing eﬀort

2.1 Requirements of fishing Fishing is utilisation of a natural resource, the fish stock. In order to capture fish and other seafood products, proper catching methods have to be used. Some resources (shellfish, crabs, clams, etc.) could easily be picked on the beach or in shallow water when available, while other resources demand more elaborated catching techniques.

During the thousands of years humans have been utilising fish stocks for food, a large number of capturing methods have been developed. Some of them have proved to be efficient and are still in use, while others have been left behind. Today a large number of different catching technologies are in daily use, some of them known from ancient time while others only have been around for a few decades.

Angling is among the oldest fishing methods we know and is still used in commercial fisheries around the world. Hand line and long line fishing are the most important angling methods. Different types of fishing nets also have been in use since ancient times. Nets are used in modern fisheries as gill nets and in more sophisticated net constructions, as purse seines, Danish seines and trawling gears.

2.2 How to measure fishing effort? Fishing effort could be regarded as a commodity that could be produced in many different ways and in different quantities. As discussed above, it exists a vast number of different methods of catching fish. Their relative efficiency depends on a number of factors which we will discuss elsewhere in the book.

It is not obvious how to measure fishing effort, units that appear to be useful in some fisheries may be useless in others. In a fishery where the fishing fleet is fairly homogeneous, fishing effort may be measured in terms of 12 Chapter 2. Fishing effort

Gill net Purse seine

Trawl

Figure 2.1: Some standard ﬁshing gears used in modern ﬁsheries (Source: Wikimedia commons)

• Number of vessels per year • Number of ﬁshing hours, days or trips • Towing hours (for trawl ﬁsheries)

In the more common cases of heterogeneous fleets, useful fishing effort measures may be • Total numbers of engine horse power units • Total number of hooks (for hand line and long line gears) • Total number of nets (for gill net fleets) • Sum of vessel lengths • Sum of vessel tonnage

Each case must be evaluated on its own based on its own properties. The measurement method has high quality if all units are measured by the same scale, also over time. If performance data is available the best measure may be found by standardising an effort unit based on combinations of several of the measures shown in the bullet points above.

Summary 2.2.1 — Units to measure fishing effort. There is no standard unit for the measurement of fishing effort.

2.3 The production of fishing effort In general all fishing effort (E) is produced in production processes involving labour (L) and capital (K),

E(L,K) = f(L,K) (2.1)

where labour and capital may substitute each other, producing the same amount of effort. That means that the same fishing effort may be produced by a large quantity of labour and a small quantity of capital as by a small quantity of labour and a large quantity of capital. 2.4 Introduction to elasticities 13

The variables Labour (L) and capital (K) in Equation 2.1 are referred to as production input factors, while the ﬁshing eﬀort (E) is the output from the production process.

Figure 2.2 shows three possible shapes Equation 2.1 may have. Each of the three cases gives a constant production of fishing effort for different combinations of labour and capital. The red broken line represents a situation where labour and capital not are substitutable factors, increasing one factor does not contribute in increased fishing effort if not the other factor also is increased. Capital ( K )

Labour(L)

Figure 2.2: Three production processes with different substitution elasticities. The three curves indicates how a constant quantity of fishing effort may be produced by three different production technologies, where the substitution elasticity is zero (the red curve), between zero and infinity (the blue curve) or infinity (the green curve).

As discussed in Section 2.2 fishing effort may be measured in many ways. The labour involved could be measured in terms of number of hours the fishers spend on fishing. It may also be relevant to include the time spent on preparing for fishing and for finishing it. Of practical reasons, we prefer to measure labour and capital by the same units. The natural unit to use is value in monetary units per unit. The value of labour and capital used in the production then gives the total cost of the produced fishing effort.

2.4 Introduction to elasticities Economic theory often makes use of diﬀerent marginal measures - average values, derivatives and elasticities - all of them reﬂecting per unit changes. For a function f(x), the per unit values are f(x) • the average value x • the derivative with respect to the variable x, f 0(x) 0 f(x) • the elasticity with respect to the variable x, f (x)/ x

As seen from this listing the elasticity is the ratio between the two other, the derivative divided by the average value. As a ratio the elasticity is unit less, which in particular is an advantage when f(x) can be measured in diﬀerent units. The interpretation of the elasticity of f(x) then is the percentage change in f(x) when x is changed by 1%. 14 Chapter 2. Fishing eﬀort

2.5 The Cobb-Douglas function Equation 2.1 may have diﬀerent mathematical expressions depending on the properties of the production processes. The Cobb-Douglas function is often used to describe a production process, assuming unit elasticity of substitution and an elasticity of scale equal 1. We will return to these expression later, ﬁrst we will introduce the Cobb-Douglas production function.

Let and A be positive constant terms and the Cobb-Douglas production is expressed by

E(L,K) = A · Lα · K1−α (2.2)

where A and α are non-negative constants, and 0 ≤ α ≤ 1. The Mathematica implementation of equation 2.2 is given by In[1]1 in Code box 2.5.1 below where also the output elasticities are found. Code box 2.5.1 — Output elasticities in the Cobb-Douglas function. Deﬁning the Cobb-Douglas function in Mathematica:a

In[1]:= cd[l_, k_] := A * l^α * k^(1 - α)

Now ﬁnd the output elasticity with respect of labour (l):

In[2]:= D[cd[l, k], l] * l / cd[l, k]

Out[2]= α

and the output elasticity with respect of capital (k):

In[3]:= D[cd[l, k], k] * k / cd[l, k]

Out[3]= 1 - α

aSince all internal Mathematica commands start with a capital letter we prefer to use lower case letters in our variables, to avoid confusion. In this code the ﬁshing eﬀort symbol (E) in equation 2.2 is replaced by cd, indicating that it is a Cobb-Douglas equation.

The cost originating from the capital used is twofold: The market value of the fishing gears and other materials used in the fishing operation (c), plus the value of the forgone benefits by spending the capital on materials for fishing (co) and not elsewhere. The latter is often referred to as the opportunity cost of capital.

The cost of labour could be calculated in different ways. If the labour is hired, the labour cost is the wages paid (w) and the opportunity cost of the capital spend on labour (lo). For an independent fisher, spending the time on fishing, the labour cost is the forgone income by utilising the labour achievement in the best alternative placement (co).

The three fishing technologies mentioned above (angling, long line and hand line fishing) use different mixes of labour and capital in their production of fishing effort.

1Mathematica inputs are numbered in this way and can be referred to in various ways, for example by the expression ’%1’. 2.6 The CES function 15

2.6 The CES function The CES function gives a more general formulation of equation 2.1. The abbreviation CES means Constant Elasticity of Substitution and ﬁgure 2.2 shows the two extremes of an inﬁnitely large elasticity of substitution (green) and zero elasticity of substitution (red).

The Cobb-Douglas function is a special case of the CES function, with a constant elasticity of substitution equal one (unit elasticity of substitution). The blue curve in ﬁgure 2.2 shows a Cobb-Douglas curve while all three curves are special cases of a CES function. Code box 2.6.1 — Output elasticities in the CES function. Deﬁning the CES function in Mathematica:

In[1]:= ces[l_, k_] := (α l^r + (1 - α) k^r)^(1/r))

where r is a parameter related to the elasticity of substitution. The elasticity of substitution is η = 1/(1 − r). Now ﬁnd the output elasticity with respect of labour (l):

In[2]:= D[ces[l, k], l] * l / ces[l, k]

lr α Out[2]= kr (1-α) + lr α and the output elasticity with respect of capital (k):

In[3]:= D[ces[l, k], k] * k / ces[l, k]

kr (1-α) Out[3]= kr (1-α) + lr α Let us make 3D plots of the three cases indicated in ﬁgure 2.2, when the elasticity of substitution (η) equals ∞, 1 and 0 respectively. In the plots the input factors, labour and capital, are represented by the two horizontal axes while the output is measured upwards.

For η = ∞ (perfect elasticity of substitution):

In[4]:= Plot3D[ces[l, k] /. {r -> 1, α -> 1/2}, {k, 0, 1}, {l, 0, 1}, MeshFunctions -> {#3&}, Mesh -> {Transpose[ {Range[5]/6,{#,#,Directive[Thickness[.02], Green],#,#}} ]& @Black}]

Out[4]=

For η = 1 (the Cobb-Douglas function): 16 Chapter 2. Fishing eﬀort

In[5]:= Plot3D[Limit[ces[l, k]/.{α->1/2},{r->0}], {k, 0, 1}, {l, 0, 1}, MeshFunctions -> {#3&}, Mesh -> {Transpose[{Range[5]/6, {#,#,Directive[Thickness[.02],Lighter@Lighter@Blue],#,#}} ]& @Black}]

Out[5]=

For η = 0 (the Leontief function):

In[6]:= Plot3D[ ces[l, k]/.{r->1000000000000, α->1/2}, {k, 0, 1}, {l, 0, 1}, MeshFunctions -> {#3&}, Mesh -> {Transpose[ {Range[5]/6,{#,#,Directive[Thickness[.02], Red],#,#}} ]& @Black}]

Out[6]=

In each of the three plots above a constant value of output is indicated by colour according to the colours used in ﬁgure 2.2, green (η = ∞), blue (η = 1) and red (η = 0).

2.7 Elasticity of scale The principle of elasticises is presented in section 2.4 and output elasticities are found for the Cobb-Douglas function (Mathematica code 2.5.1) and the CES function (Mathematica code 2.6.1). The interpretations of the output elasticities are straightforward: When one of the input factors, l or k, changes (increases or decreases) by one percent, the production output is changed accordingly by the percentage equal the output elasticity of the input factor.

According to the results in Mathematica code 2.5.1 the output elasticity of labour in the Cobb-Douglas function is α and the output elasticity of capital is 1 − α. If both input factors are changed simultaneously by one percent, the total change of output equals the sum of the two output elasticities. For the Cobb-Douglas equation it is easy to see that α + (1 − α) = 1. This sum is usually referred to as the elasticity of scale (or returns to scale), indicating the eﬀect up- and down-scaling have on the produced quantity. 2.7 Elasticity of scale 17

Also for the case of the CES function the elasticity of scale equals one in this case. Here it is convenient to employ Mathematica to ﬁnd the solution.

In[7]:= Simplify[D[ces[l, k], l]*l/ces[l, k] + D[ces[l, k], k]*k/ces[l, k]]

Out[7]= 1

The elasticity of scale is usually assumed to equal one, meaning that there not is economics of scale in the production. This may be considered as the normal case but is certainly always the case. Both the Cobb-Douglas function and the CES function may be adjusted to accommodate scale elasticities diﬀerent from one. A simple and general formulation of the ﬁrst function is

E(L,K) = A · Lα · Kβ (2.3) where the elasticity of scale is = α + β.

3. Fish catch production

3.1 Catch production In section 2.1 necessary requirements of producing fishing effort were discussed. It is however not sufficient to know how to catch fish, fish stock resources also need to be available for catch. In economic terms we would say that the fish stock is an essential input factor in the production of catch. The other essential factor in the production of catch is the fishers’ fishing aptitude which in the following referred to as the fishing effort.

Catch production is a production process following the principle of technological eﬃ- cient production discussed in chapter 2 where some well-known production functions were introduced. In this chapter we will discuss further how to implement the Cobb-Douglas function (equation 2.3) to describe ﬁsh catch production.

In chapter 2 we have already discussed how fishing effort (E) is produced by the input factors labour (L) and capital (K). Now we consider a production process where fishing effort as an input factor in the output is fish harvest. Fishing effort alone is not sufficient to produce harvest of fish, it is also necessary to have access to a fish stock resource. Therefore the available fish stock also is an input factor in the production of fish harvest

In chapter 4 we will discuss further the properties of the fish stock, here we only consider the fish stock in terms of available biomass as one of the two input factors to produce fish catches.

3.2 Catch as a method of stock assessment Stock assessment is the core foundation of modern ﬁsheries management. Time series of stock assessments provides information about growth potential, variability and possible exploitation levels. 20 Chapter 3. Fish catch production

The simplest stock assessment method is to compare the amount of catch per unit of fishing intensity (catch per unit of effort, CPUE) between different areas or periods. The highest CPUE is believed to reflect the highest abundance.

This is however not a method to estimate the real number of fish in an area at a point in time but merely a relative measure to rank different observations. However, in order to tell if the fish abundance is increasing or decreasing it is sufficient to measure the relative changes.

The indexes by which we measure relative changes (e.g. CPUE) are regarded as stock estimates. In fact, this is the case also for the more sophisticated stock assessment methods utilised in data rich single species fisheries as for example the Northeast Arctic cod fishery. In this fishery there exists high resolution catch data, annual survey data and a number of scientific studies contributing in enriching the stock assessment methodology and tuning processes used to evaluate the state of the stock at any point in time.

Let us move back to the basic observation which we may label the CPUE methodology, where a large catch with a given eﬀort signalise a larger stock abundance than a small catch with the same eﬀort at another point in time. In its most rudimentary form the CPUE methodology assumes a linear catch function:

H(E,X) = q · E · X (3.1) where H is the harvest produced by the fishing effort, E, and the stock abundance (measured in biomass), X. q is often referred to as the catchability coefficient, a scaling parameter also reflecting the technological property of the fishing gear in use.

As seen in section 2.7 It is easy to see from equation 3.1 that catch per unit of eﬀort (CPUE) is linear to the stock index q · X:

CPUE = H(E,X)/E = q · X (3.2)

Code box 3.2.1 — Simple stock assessment by CPUE calculations. Assume that we have access to time series of catch data and standardised ﬁshing eﬀort over a period of ten years. The catch time series is:

In[1]:= catch = { 2384, 2361, 1586, 1889, 1766, 2456, 1068, 1905, 1425, 1957};

The effort was 100 the first year and increased by 10 every year after. The CPUE development over time is seen by plotting catch per effort for each year

In[2]:= ListLinePlot[catch/Table[100 + 10*t, {t, 0, 9}], Mesh -> All, Frame->True,PlotTheme->"Detailed",FrameLabel->{"Year","CPUE"}] 3.2 Catch as a method of stock assessment 21

Out[2]= CPUE 10

0 0 2 4 6 8 10

Year

We observe a down-sloping trend in the CPUE measures and assume a linear model

In[3]:= model1 = LinearModelFit[ Transpose[{#, catch/#} &@Table[100 + 10*i, {i, 0, 9}]], x, x]

Out[3]= FittedModel 35.4802- 0.149425 x 

We see that about 67% of the variation is explained by the linear model

In[4]:= model1["RSquared"]

Out[4]= 0.675255

and we retrieve the analysis of variance by

In[5]:= model1["ANOVATable"]

DF SS MS F-Statistic P-Value

Out[5]= x 1 184.204 184.204 16.6347 0.00354115 Error 8 88.5878 11.0735 Total 9 272.792

The results of the linear regression is plotted together with the CPUE observation versus ﬁshing eﬀort.

In[6]:= Show[{Plot[model1[e], {e, 0, 240}, PlotStyle -> Dashed], ListLinePlot[Transpose[{#, catch/#} &@Table[100+10*i, {i,0,9}]], Mesh -> All]}, Frame -> True, PlotRangePadding -> None, PlotRange -> {0, All}, FrameLabel -> {"Effort", "CPUE"}] 22 Chapter 3. Fish catch production

Out[6]= CPUE 15

0 0 50 100 150 200

Effort

Multiplying the CPUE values above with eﬀort gives the catch and the linear regression now describes a parabolic curve through the origin.

In[7]:= Show[{Plot[model1[e]*e, {e, 0, 240}, PlotStyle -> Dashed], ListLinePlot[Transpose[{Table[100 + 10*i, {i, 0, 9}], catch}], Mesh -> All]}, Frame -> True, PlotRangePadding -> None, PlotRange -> {0, 2600}, FrameLabel -> {"Effort", "Catch"}]

2500

2000

1500

Out[7]= Catch 1000

500

0 0 50 100 150 200 Effort

If we had an accurate estimate of the value of q, we could actually determine the stock biomass in nature, X. In most cases it is however sufficient to know if the stock is growing or declining to make non-critical management decisions. Then the CPUE-measure is sufficient; given that equation 3.1 holds and that we are able to measure catch (H) and effort (E) correctly. The latter is a major problem which is dealt with other places in this document. Here we will discuss further the methodology, given that accurate catch and effort observations exist.

Measurement of fishing effort in a consistent manner is a challenge which is not easily solved. There are no standard methods of standardising effort in a heterogeneous fleet at a point in time. In addition, fleet efficiency changes (typically increases) over time, creating a systematic error when comparing fishing effort measures in different time periods.

In principle catch quantities are easier to measure than ﬁshing eﬀort. It demands however a system of retrieving such information without hidden, manipulated or illegal catches 3.3 Cobb-Douglas catch production 23

disturbing the data samples.

3.3 Cobb-Douglas catch production Catch is produced by two input factors, ﬁshing eﬀort (E) and stock biomass (X) as indicated in equation 3.1. Now assume a more general catch equation following the pattern of a Cobb-Douglas production function (equation 2.3)

H(E,X) = q · Eα · Xβ (3.3)

As seen from equation 3.3 this is equivalent to equation 3.1 when α = β = 1. There may be reasons to expect α = 1 and 0 < β < 1 (Hannesson, 1983[5]; Eide et al., 2003[3]). The value of β depends both on biological properties of the ﬁsh stock in question (typically β will be lower for schooling species than for non-schooling species) and the properties of the harvesting technology (gillnetting has for example higher β-values than longlining).

On the basis of the reasoning above let us assume an alternative catch equation where β = 1/2: √ H(E,X) = q · E · X (3.4)

3.4 Properties of fishing gears In chapter 2 a number of different fishing gears are mentioned. Some gears are fishing randomly fish passing through the area (some traps and gill nets), some gears aim to attract fish to swim into traps or onto hooks where they are captured (fish pots, long line, hand line, etc.), while other fishing gears actively move towards the fish to capture it (trawl, purse seine, Danish seine, etc.).

3.5 Exercises Exercise 3.1 Why increases the production more rapidly in the lower-right ﬁgure in

Code box 6.1.2 than in the upper-right ﬁgure?

4. Population dynamics

4.1 Basic principles A group of individuals of the same species living in a speciﬁc geographically area is often referred to as a population. Populations may be separated into smaller sub-populations (often referred to as stocks) in minor geographical areas. Population dynamics describes how a population or stock grows due to the genetic properties of the population/stock and the environmental constraints within the distribution area. In the following we will not diﬀerentiate between populations and stocks.

Population growth is the net eﬀect of individual growth, recruitment and mortality. All these factors depend both of the biological properties of the species and of the physical and biological environment. The size of a population could be measured in numbers of individuals (the common measure in the human population) or in terms of weight (the common measure of ﬁsh stocks). The latter is often referred to as the stock’s biomass, which is the total weight of the stock in nature.

The science of population dynamics originates from the same public discussions lead- ing to the economic discipline and mathematical demographics at the end of the eighteenth century. Several of the demographic models we will discuss in this chapter originates from that period, but at that time the focus was on the growth of the human population.

4.2 Classical surplus growth models Of course people had been interested in and fascinated by population growth long time before the eighteenth century. One example is the Fibonacci numbers, described by an Italian mathematician (Leonardo of Pisa Bonacci, about 1170 – about 1250)) during the ﬁrst years of the thirteenth century (the series has however been known from ancient times).

The Fibonacci numbers describes biological growth starting with a newborn pair of rabbits 26 Chapter 4. Population dynamics

(1). At the end of the first month they mate and a new pair is born at the end of the second month. Then there are two pairs. The first pair gives birth to a new pair every month (mortality is not existing in this model), while the second pair starts mating as the first pair after their first month. The Fibonacci series then gives the monthly number of pairs of rabbits.

Code box 4.2.1 — Fibonacci numbers. In Mathematica you have an internal function listing the Fibonacci numbers:

In[1]:= Table[Fibonacci[n], n, 20]

Out[1]= {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765}

In[2]:= ListLinePlot[%1, Mesh -> All, PlotRange -> All, Filling -> Axis]

7000

6000

5000

4000 Out[2]= 3000

2000

1000

5 10 15 20

Now have a look at the ratio between two and two consecutive numbers

In[3]:= Fibonacci[# + 1]/Fibonacci[#] & /@ Range[20]

3 5 8 13 21 34 55 89 144 233 377 610 Out[3]= 1, 2, , , , , , , , , , , , , 2 3 5 8 13 21 34 55 89 144 233 377 987 1597 2584 4181 6765 10946o , , , , , 610 987 1597 2584 4181 6765 The numerical values reveal that the ratios soon approach the same value while moving upwards in the series.

In[4]:= N[%]

Out[4]= {1., 2., 1.5, 1.66667, 1.6, 1.625, 1.61538, 1.61905, 1.61765, 1.61818, 1.61798, 1.61806, 1.61803, 1.61804, 1.61803, 1.61803, 1.61803, 1.61803, 1.61803, 1.61803}

Let us look at the numerical value of the built in function GoldenRatio, and we obtain the same number.

In[5]:= GoldenRatio // N 4.2 Classical surplus growth models 27

Out[5]= 1.61803

In[6]:= Show[{ListLinePlot[%3, PlotRange -> All, Mesh -> All], Plot[GoldenRatio, {n, 0, 20}, PlotStyle -> Red]}]

2.0

1.5

Out[6]= 1.0

0.5

5 10 15 20

The blue curve (Out[3] falls together with the red curve (the golden ratio, Out[4]). The golden ratio therefore could be obtained directly from the Fibonacci numbers:

In[7]:= Limit[Fibonacci[n + 1]/Fibonacci[n], n -> Infinity]

1 √ Out[7]= 1 + 5 2 Let us complete this session with the Fibonacci numbers by looking at one of many examples of how these numbers produce beautiful patterns in the nature. Here we use the golden ratio to draw patterns found in sunﬂower heads

In[8]:= Graphics[{Blend[{Yellow, Orange}, 1/4], Disk[{0, 0}, 23], Brown, PointSize[.02], Point[Table[Sqrt[n] { Cos[2 Pi n GoldenRatio], Sin[2 Pi n GoldenRatio]}, {n, 500}]]}]

Out[8]=

An interesting feature of the series is that each number is equal the sum of the two previous numbers. At ﬁrst sight the series may appear to be rather randomly constructed but it is following a strict rule where the population of rabbits after only a few months reach 28 Chapter 4. Population dynamics astronomical values. The series is a simple recruitment model and does not include growth in biomass and mortality. It demonstrates however the exponential power of population growth which was also discovered in European countries during the eighteenth century.

The British scholar Thomas Robert Malthus (1766 - 1834) claimed that the exponential increase in the human populations throughout Europe in the eighteenth century eventually would be repressed at a maximum level (the level of subsistence) where the human population would suﬀer at the edge of natures capacity level of sustaining the human population.

Malthus’ ideas became very inﬂuential but also criticised by many. The idea that there existed an upper limit for how large a human population could be, and that this limit was related to environmental constraints (e.g. food), was shared by most of the critics.

Code box 4.2.2 — Gompertz’ population growth model. The Gompertz equation 4.1 is solved by Mathematica while providing a value for the initial stock biomass (X0, implemented below as x0). The solution includes a message indicating that other solutions may exist in the general case of such

In[1]:= DSolve[x’[t] == -r x[t] Log[x[t]/k],x[0] == x0, x[t], t][[1, 1]]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

-r t x0e Out[1]= x[t] → k k

Plotting equation 4.1 (left below) and its solution (right below) for some given parameter values (r = 0.5, K = 1000 and X0 = 10)

In[2]:= GraphicsRow[{Plot[-.5 x Log[x/1000], {x, 0, 1000}], Plot[1000 (10/1000)^Exp[-.5 t], {t, 0, 12}]}]

1000 150 800

100 600 Out[2]= 400 50 200

200 400 600 800 1000 2 4 6 8 10 12

Benjamin Gompertz (1779 – 1865) published in 1825[4] a mathematical demographic model. When replacing the variable (numbers of peoples) with a stock’s biomass (X), the Gompertz’ growth equation gives the time derivative of the stock biomass

dX(t) X(t) X˙ (t) = = −r · X(t) · ln (4.1) dt K where r is a growth rate while K is the environmental carrying capacity level. Since the time derivative is the per unit of time increment in the stock biomass, it provides a straight 4.2 Classical surplus growth models 29 forward biological interpretation on the speed by which the stock grows towards its natural equilibrium K. As seen from the plot of equation 4.1 in Code box 4.2.2, the point of maximum growth is placed to the left of K/2.

Another demographic model, published only a few years after the Gompertz model, and with an even greater scientiﬁc impact, was suggested by Pierre François Verhulst (1804 - 1849) in 1838 [8]. It is commonly referred to as the logistic growth equation and in line with the expression above it may be written as:

X(t) X˙ (t) = r · X(t) 1 − (4.2) K

As seen from equation 4.2 (and illustrated in Code box 4.2.3) the time derivative of the logistic growth describes a parabolic curve. The inﬂection point of X(t) therefore is found for X(t) = K/2. After Raymond Pearls introduction of the logistic growth curve as a biomass surplus growth model it has become the most common surplus growth model used in mathematical biology and ﬁsheries economics.

Code box 4.2.3 — Verhults population growth model. The Verhulst equation 4.2 is solved by Mathematica while providing a value for the initial stock biomass (X0, implemented below as x0). The solution includes a message indicating that other solutions may exist in the general case of such

In[1]:= DSolve[x’[t] == r x[t](1 - x[t]/k), x[0] == x0, x[t], t][[1, 1]]

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

ert k x0 Out[1]= x[t] → k - x0 + ert x0 Plotting equation 4.2 (left below) and its solution (right below) for some given parameter values (r = 0.5, K = 1000 and X0 = 10)

In[2]:= GraphicsRow[{Plot[.5 x (1 - x/1000), {x, 0, 1000}], Plot[(Exp[.5t]1000*10)/(1000-10+Exp[.5t] 10), {t, 0, 20}]}]

120 1000 100 800 80 600 Out[2]= 60 400 40 20 200

200 400 600 800 1000 5 10 15 20

In 1959 F. J. Richards (1901 – 1965) published a model in which the two previously presented models (equations 4.1 and 4.2) are special cases, referring to it as "a ﬂexible growth function for empirical use"[6]. In line with the previous equations we express 30 Chapter 4. Population dynamics

Richards function in this way

X(t)m−1 X˙ (t) = r · X(t) 1 − , (4.3) K introducing a third parameter (m), which moves the inﬂection point of the growth upwards and downwards depending on its value. It is easy to see that equation 4.3 is equal equation 4.2 for m = 2. It is also possible to show that equation 4.3 actually approaches equation 4.1 when m approaches one. Then both the Gompertz growth and the Verhulst growth equations are special cases of the Richards growth in equation 4.3. The graph shown in Code box 4.2.4 illustrates how the m-parameter in the Richards growth equation determines if the growth curve should be squeezed to the left or the right.

Code box 4.2.4 — Surplus growth models. This session makes use of the package PopulationGrowth (freely available at http://www.maremacentre.com/econmult). If the package is found by Mathematica in its ﬁle system, it is loaded by the command:

In[1]:= Needs["EconMult‘PopulationGrowth‘"]

This lists four surplus production models available in the package:

In[2]:= Grid[Text /@ {#, Notation@SurplusProduction[UseMSY -> False, GrowthModel -> #], SimplifyNotation@SurplusProduction[GrowthModel -> #]} & /@ $SurplusProductionModels, Frame -> All, Alignment -> Left]

VerhulstSchaefer r X1- X  4 MSY X(K-X) K K2 GompertzFox-r X log X  ⅇ MSY X(log(K)-log(X)) K K

X m m-1 MSYX-K   Out[2]= RichardsPellaTomlinson r X1- X   K K 1 m K m 1-m -m 1-m

b b 1 X b+1 b  +1 (b+1) MSY K  -X X -1 b K QuasiBevertonHolt r X1-  b+1  K K

In[3]:= Show[(Plot[SurplusProduction[ GrowthModel -> RichardsPellaTomlinson, CurrentBiomass -> x, UseMSY -> True, MaximumSustainableYield -> 100, CatchabilityCoefficient -> 1, BiomassMaximum -> 1000, RichardsPellaTomlinsonParameter -> #], {x, 0, 1000}, PlotRange->{0, 100}, Frame->True, FrameLabel -> {"Stock size", "Natural growth per unit of time"}, PlotRangePadding -> None ] &) /@ Table[Exp[.001 + .1*i^3] - 1, {i, 0, 4, .5}]] 4.3 Depensatory growth 31

100

Out[3]= 40

Natural growth per unit of time 20

0 0 200 400 600 800 1000 Stock size

4.3 Depensatory growth The models introduced in section 4.2 above are all included in the class of compensatory growth models. The term compensatory refers to the fact that in these models the relative growth rate (X˙ (t)/X(t)) is increasing for decreasing stock biomass levels. Biomass losses leads to an increasing compensation per unit of biomass.

This kind of compensating behaviour in the stock may however not always be the case in real life. The relative growth rate may increase by decreasing stock biomass down to a certain level where the relative growth rate also decreases. It may even turn negative in some cases. These cases are referred to as critical depensation, deﬁning a critical biomass level below which the stock will go extinct.

A depensatory growth model may be speciﬁed on the basis of the logistic equation 4.2 by adding a term including a depensation parameter D:

X(t) X(t) − D X˙ (t) = r · X(t) 1 − · K K − D which gives

r · X(t) · K − X(t) · X(t) − D X˙ (t) = (4.4) K · K − D

If the depensation parameter D is negative the depensation is non-critical, while the interpretation of a positive depensation parameter is the critically low biomass level. By the use of Mathematica it is easy to show that D = −∞ restores equation 4.2:

In[1]:= Limit[(r x (k - x) (x - d))/(k (k - d)), d -> -Infinity]

r (k - x) x Out[1]= k 32 Chapter 4. Population dynamics

Code box 4.3.1 — Depensatory growth. Depensatory growth as expressed in equation 4.4 includes both critical and non-critical depensation levels. The plot of equation 4.4 (below) for diﬀerent D-values illustrates this. The chosen colours goes from non-critical (blue) to critical (red) depensation levels.

In[1]:= Show[Plot[r x(k-x)(x-d))/(k(k-d)) /. {r->.5, k->1000, d->#}, {x, 0, 1000}, PlotStyle -> Hue[.8-Abs[Log[2 + #/3000]]]] & /@ {-50000, -3000, -1000, -300, 0, 200, 400}, PlotRange -> All, AxesLabel -> {"X", "dX/dt"}]

dX/dt

100

Out[1]= 50

X 200 400 600 800 1000

Two of the D-values above are positive and hence deﬁning critical biomass levels, respectively for stock sizes of 200 and 400. The blue curve (D = −50,000) is approaching the logistic growth function (obtained when D = −∞).

The plot below shows the relative growth (or average growth; surplus production per unit of biomass) for the D-values included in the plot above.

In[1]:= Show[Plot[r x(k-x)(x-d))/(k(k-d))/x /. {r->.5, k->1000, d->#}, {x, 0, 1000}, PlotStyle -> Hue[.8-Abs[Log[2 + #/3000]]]] & /@ {-50000, -3000, -1000, -300, 0, 200, 400}, PlotRange -> All, AxesLabel -> {"X", "(dX/dt)/X"}]

(dX/dt)/X

0.4

0.2 Out[1]=

X 200 400 600 800 1000

-0.2

We see that the two orange or red curves, representing the two cases of critical depensation, becomes negative at suﬃciently low D-values. Other cases also indicate depensation 4.3 Depensatory growth 33

but not all of them. The blue and magenta coloured curves obviously shows an increasing trend by decreasing D-values in the whole range of D-values. This also seems to be the case for the green curve.

The plot below gives the marginal values of the relative growth, with respect of stock biomass X. This plot conﬁrms that the blue, magenta and green curves indeed belong to the class of compensatory growth models, while the other cases belong to the class of depensatory growth models. We base this conclusion on the fact that the ﬁrst three cases do not enter the positive region of marginal relative growth. Among the depensatory growth cases the orange (red) curves represent critical depensatory growth, while the other two (the yellow curves) represent depensatory growth where no critical biomass values above zero are found.

In[1]:= Show[Plot[D[r x(1-x/k)(x-d)/(k-d)/x, x] /. {r->.5,k->1000,d->#, x->y}, {y, 0, 1000}, PlotStyle->Hue[.8-Abs[Log[2+#/3000]]]]& /@ {-50000, -3000, -1000, -300, 0, 200, 400}, PlotRange -> All, AxesOrigin -> {0, 0}, AxesLabel -> {"X", "-(dX/dt)/X2"}]

-(dX/dt)/X2

0.0010

Out[1]= 0.0005

X 200 400 600 800 1000

-0.0005

The curves intersections with the horizontal axis corresponds to the maximum values of the relative growth. While the green curve reach zero at D = 0, the magenta and blue curves do not reach positive values for D ≥ 0.

Analytically it is possible to prove that the range of D-values of compensatory growth is from −∞ to −K, non-critical depensatory growth goes from −K to 0, while the critical area lays in the interval of D-values between 0 and K, being the maximum biomass level.

Most all examples provided in this text book assume compensatory growth. Shifting to depensatory growth may severely alter the conclusions of the compensatory growth models. If the stock’s ability to strive for increased growth per biomass unit with declining stock size is weakened, consequences of overﬁshing may be much more severe.

In the next section age structure models are introduced. Walters et al. (2008)[9] discusses the relationship between depensatory growth in surplus production models versus age structured models and claims that depensation is more commonly included in age structured models. 34 Chapter 4. Population dynamics

4.4 Age structured model Age structured population models decompose the population biomass to diﬀerent age components, each of them being the product of average individual weight in the age group and number of individuals. Changes in age composition of the stock will aﬀect the stock biomass development in ways not necessarily grasped by the surplus production models presented in section 4.2.

Age structured models are often referred to as cohort models, a more general term which allows the stock to be structured in other ways that by year classes, which is the normal structure. For some species, however, structuring by other time intervals (for example by month) or more aggregated groups (for example mature and immature individuals) is preferable to structuring by year-classes.

In 1934 Karl Ludwig von Bertalanﬀy (1901 – 1972)[2] launched an individual length growth model for asymptotic growth towards a maximum length (L∞) by increasing age. Let L(t) be the individual length at age t, k the length growth rate and t0 the theoretical age of zero length (L(t0) = 0). The von Bertalanﬀy equation is then given by

−k(t−t0) L(t) = L∞ 1 − e (4.5) The weight growth is closely related to length growth. Let parameter b represent the weight/length relationship and d be a scaling factor. The individual weight at age t, W (t), then is given by W (t) = d · L(t)b (4.6)

b Maximum individual weight also is deﬁned by equation 4.6, W∞ = d · L∞. Inserting equation 4.5 in equation 4.6 then gives1

−k(t−t0)b W (t) = W∞ 1 − e (4.7)

Code box 4.4.1 — Parameter relations. Some will emphasise that the gain of moving from aggregated surplus production models to cohort models is clearer biological interpretations of the model parameters.

The growth rate k is the percentage increment in length in relation to the diﬀerence between maximum length and current length. This gives a constant percentage of a diminishing diﬀerence, hence the length growth is approaching zero as the length approaches L∞.

L˙ (t) k = L∞ − L(t) b is the ratio between the percentage growth in weight and the percentage growth of length at the same age. This ratio is assumed to be constant and usually is close to 3, representing the cubic expansion of length.

W˙ (t)L˙ (t) b = W (t) L(t)

1Taking the time derivative of this function shows that the function in fact a Richards equation (equation 4.3). 4.4 Age structured model 35

d is simply a scaling parameter which value depends on the units by which weight and length are measured.

W∞ d = b L∞ The mortality rate Z gives a constant percentage decline in the number of individuals by age. On basis of the relations above the mortality rate may also be expressed in terms of percentage increment in weight minus percentage increment in biomass in a cohort.

N˙ (t) W˙ (t) x˙(t) L˙ (t) x˙(t) Z = − = − = b · − · N(t) W (t) x(t) L(t) x(t)

While the individual weight increases by age, the number of individuals of a cohort decreases over time due to natural mortality (predation, age and diseases). The standard mortality model was ﬁrst proposed by Baranov in 1918[1]. If R is the initial number of recruits in a cohort (at the age of recruitment, tR) and the mortality rate is Z, then the number of individuals in the cohort at time t, N(t), is

N(t) = R · e−Z(t−tR) (4.8)

The product of equations 4.6 and 4.8 gives the total biomass of the cohort in question:

−k(t−t0)b −Z(t−tR) x(t) = N(t) · W (t) = R · W∞ · 1 − e · e (4.9)

Code box 4.4.2 — Biomass of one cohort and all cohorts. This session makes use of the package PopulationGrowth (freely available at http://www.maremacentre.com/econmult). If the package is found by Mathematica in its ﬁle system, it is loaded by the command:

In[1]:= Needs["EconMult‘PopulationGrowth‘"]

The von Bertalanﬀy equation (equation 4.7) is implemented in the package as:

In[2]:= IndividualWeight[t] // Notation

Out[2]//TraditionalForm= -k t-t  b W∞ 1- ⅇ 0  By default the package assumes the total mortality rate (Z, as in equation 4.8) to be the sum of F and M, respectively the ﬁshing and natural mortality rate (see equation 5.1). The Baranov equation (equation 4.8) therefore is implemented by

In[3]:= IndividNumbers[t] // Notation

Out[3]//TraditionalForm= Rⅇ -(F+M) (t-tR) The product of number of individuals (N) and individual weigh (W ) gives the biomass of the cohort (equation 4.9):

In[4]:= CohortBiomass[t] // Notation 36 Chapter 4. Population dynamics

Out[4]//TraditionalForm= -k t-t  b -(F+M) (t-t ) RW∞ 1- ⅇ 0  ⅇ R Maximum biomass of cohort is found by

In[5]:= Solve[CohortBiomass’[t] == 0, t] // SimplifyNotation

Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>

Out[5]//TraditionalForm= b k log +1 F+M t→ +t 0 k This t-value is implemented in the PopulationGrowth package as AgeOfMaxGrowth, the age at which the cohort has its maximum biomass.

In[6]:= AgeOfMaxGrowth[] // Notation

Out[6]//TraditionalForm= log b k +1  F+M +t 0 k Maximum biomass of cohort could then be found by

In[7]:= CohortBiomass[AgeOfMaxGrowth[]] // Notation

Out[7]//TraditionalForm= b log b k +1  1 F+M RW∞ 1- exp -(F+ M ) -t R +t 0 b k +1 k F+M The maximum biomass is implemented in the package by MaximumBiomassGrowth. This tests that MaximumBiomassGrowth actually is equivalent to CohortBiomass[AgeOfMaxGrowth[]]:

In[8]:= MaximumBiomassGrowth[] === CohortBiomass[AgeOfMaxGrowth[]]

Out[8]= True

We assume some numerical values to parametrise the model. The previously presented parameters are represented in the package by descriptive names: InitialAge (t0), WeightLengthRelation (b), MaxWeight (W∞), GrowthRate (k), MortalityRate (M, a component of Z), FishingMortalityRate (F , a component of Z), Recruits (R) and RecruitmentAge (tR). Some parameters are to be introduced later: OldestAge (age of oldest cohort in the stock, t∞) and CatchAge (age of ﬁrst catch, tc).

In[9]:= values = {InitialAge -> 0, WeightLengthRelation -> 3, MaxWeight -> 10, GrowthRate -> .2, MortalityRate -> .2, FishingMortalityRate -> F, Recruits -> 1, RecruitmentAge -> 0, CatchAge -> tc, OldestAge -> Infinity};

Without ﬁshing the cohort has its maximum biomass at a age close to seven years: 4.4 Age structured model 37

In[10]:= AgeOfMaxGrowth[Sequence @@ values, Fishing -> False]

Out[10]= 6.93147

This gives a graphical illustration on how the biomass develops over the life span of the cohort, when assuming no ﬁshing:

In[11]:= Plot[CohortBiomass[t, Sequence @@ values, Fishing -> False], {t, 0, 20}, AxesLabel -> {"Biomass of cohort", "Age"}]

Biomass of cohort

1.0

0.8

Out[11]= 0.6

0.4

0.2

Age 5 10 15 20

Now assume that a similar cohort is recruited to the stock every year after the ﬁrst one:

In[12]:= Show[Plot[CohortBiomass[t - #, Sequence @@ values, Fishing -> False], {t, 0, 30}, AxesLabel -> {"Age", "Biomass of cohort"}, PlotRange -> {0, All}] & /@ Range[0, 30]]

Biomass of cohort

1.0

0.8

Out[12]= 0.6

0.4

0.2

Year 0 5 10 15 20 25 30

By adding the biomass of all cohorts for each year we ﬁnd how the stock biomass grows from the ﬁrst cohort to a fully recruited stock. Negative biomass values are not possible and are ignored by using the Max function, assuming zero to be the lowest possible biomass value:

In[13]:= ListLinePlot[Total /@ Table[Max[CohortBiomass[t - i, Sequence@@values, Fishing->False], 0], {t, 0, 30}, {i, 0, 30}], AxesLabel -> "Year", "Stock biomass"] 38 Chapter 4. Population dynamics

Stock biomass

8 Out[13]= 6

Year 5 10 15 20 25 30

Since this curve is the integral of the cohort biomass curve, the surplus production graph of the age structured model is found by merging the two graphs:

In[14]:= ParametricPlot[ {PopulationBiomass[t, Sequence @@ values, Fishing -> False], CohortBiomass[t, Sequence @@ values, Fishing -> False]}, {t, 0, 50}, AspectRatio -> 1/GoldenRatio, AxesLabel -> {"Stock biomass", "Surplus production"}]

Surplus production

1.0

0.8

Out[14]= 0.6

0.4

0.2

Stock biomass 2 4 6 8 10 12

Since the mortality rate and the growth rate in our case are identical, the curve above follows the growth of the "QuasiBevertonHolt"-model in the package (stock biomass is denoted X):

In[15]:= SurplusProduction[CurrentBiomass -> X, GrowthModel -> "QuasiBevertonHolt", Sequence @@ values, MaximumSustainableYield -> MaximumBiomassGrowth[Sequence @@ values, Fishing -> False], BiomassMaximum -> EquilibriumBiomass[Sequence @@ values, Fishing -> False]]

1.8803 Out[15]= -0.8 1 - X X1/4

When assuming a constant recruitment (R) and a cohort biomass growth as in equation 4.9, the equilibrium biomass of the total stock at time τ is

Z t∞ Z t∞ −k(t−t0)b −Z(t−tR) X(τ) = xτ (t)dt = R · W∞ 1 − e · e · dt (4.10) t=0 t=0 when xτ (t) is the biomass of cohort of age t at time τ, described by equation 4.9. Usually recruitment is considered being a discrete process in time, which should change the integral 4.5 Exercises 39

in equation 4.10 to expressing a sum corresponding to the last part of code box 4.4.2. It follows from equation 4.10 that

X˙ (τ) = x(τ) (4.11)

test MORE AFTER INTRODUCING FISHING

4.5 Exercises Exercise 4.1 Why increases the production more rapidly in the lower-right ﬁgure in

Mathematica code 6.1.2 than in the upper-right ﬁgure?

5. The concept of equilibrium harvest

5.1 Equilibrium harvest

The idea of equilibrium harvest is straight forward: Equilibrium harvest is obtained when the surplus biomass production in the stock during a period of time is harvested during the same period. The practical implications of this include however some problems.

The surplus production is the net production when taking into consideration recruitment (in a fishery perspective this is often considered being the young fish when passing the size or age by which they could be captured), individual growth which is the total increment in size obtained by all individuals, and natural mortality which is the biomass loss due to all the fish dying during the period.

Obviously the harvest production during a period could not be exactly the biomass constituting the surplus production during the same period. It is equally obvious that it does matter how the harvested biomass is put together, if it is newly recruited young ﬁsh or older ﬁsh.

These are problems which are omitted in the standard surplus production models discussed in section 4.2. These models do not take into account the age composition of the stock or how the recruitment dynamics links to this composition. A certain biomass at the beginning of a year gives a certain surplus production in that year. The problems discussed above therefore does not aﬀect the surplus production models. But what about the age structured models presented in section 4.4? Is it possible to stick to the idea of equilibrium harvest in the case of age structured models?

The way the equilibrium concept is interpreted in age structured models is through ﬁxed ﬁshing mortality rates for each cohort. The total mortality rate Z is decomposed into natural mortality rate M, the mortality that naturally occurs through predation, age 42 Chapter 5. The concept of equilibrium harvest

and diseases, and F , the mortality rate imposed through fishing: Z = M + F (5.1) The harvest obtained from each cohort is then given by the product of the fishing mortality rate for the cohort, Fc, and the biomass of the cohort, Xc, which is the equilibrium biomass of the cohort after keeping all fishing mortalities constant for a sufficiently long period of time. Hence, equilibrium harvest is also a well defined concept within the tradition of age structured models.

5.2 Surplus production models In order to identify equilibrium harvest in surplus production models the ﬁrst step is to include the impact harvest has on the stock development in the growth equations 4.1, 4.2 and 4.3 respectively: X(t) X˙ (t) = −r · X(t) · ln − HE(t),X(t) (5.2) K

X(t) X˙ (t) = r · X(t) 1 − − HE(t),X(t) (5.3) K

X(t)m−1 X˙ (t) = r · X(t) 1 − − HE(t),X(t) (5.4) K The last term, H(E,X), is the harvest obtained throughout the period of one time step (often assumed to be a year). Let us assume that H(E,X) is a bi-linear catch equation given by equation 3.1. While keeping a constant ﬁshing eﬀort, E(t) = E, equilibrium is obtained when X˙ (t) = 0 for all t. The stock biomass reaches the equilibrium value X, which in the case of equation 5.3 is found through the following steps: X r · X 1 − = q · E · X K If X 6= 0: X r 1 − = q · E, K which by reorganising the terms gives the equilibrium biomass q X = K 1 − E (5.5) r

We see that the equilibrium biomass, while assuming a logistic growth equation (the Verhulst equation) and a Schaeffer production function, is a linear function of any value the constant fishing effort E may have.

Code box 5.2.1 — Equilibrium harvest. The linear relationship in equilibrium, between constant ﬁshing eﬀort E and stock biomass X, expressed in equation 5.5, is illustrated below for some given parameter values (K = 1, r = 1/2 and q = 1/2):

In[1]:= Plot[x /. Solve[r x (1-x/k) == q ee x && x != 0, x] /. {k->1, q->1/2, r->1/2, ee->e}, {e, 0, 1}, 5.2 Surplus production models 43

PlotStyle -> Red, AxesLabel -> {"E", "X"}]

1.0

0.8

0.6 Out[1]=

0.4

0.2

E 0.2 0.4 0.6 0.8 1.0

Recall the Cobb-Douglas function plot in Code box 2.6.1 and join a contour plot of the short term catch equation 3.1 with the plot above:

In[2]:= Show[ContourPlot[(q e x /. {k->1, q->1/2, r->1/2}) == #, {x, 0, 1}, {e, 0, 1}] & /@ Range[0.01, .6, .04], Plot[x /. Solve[r x (1 - x/k) == q ee x && x != 0, x] /. {k->1, q->1/2, r->1/2, ee->e}, {e, 0, 1}, PlotStyle -> Red], PlotRangePadding -> None, FrameLabel -> {"E", "X"}]

1.0

0.8

0.6 X Out[2]=

0.4

0.2

0.0 0.0 0.2 0.4 0.6 0.8 1.0 E

Imagine the red line in the plot above projected down at the surface of the production equation. Then the red line will describe a curve through the three dimensions of E, X and H as in this plot:

In[3]:= ParametricPlot3D[{e, k (1 - q/r e), q e k (1 - q/r e)} /. {k->1, q->1/2, r->1/2}, {e, 0, 1}, BoxRatios -> {1, 1, 1}, PlotStyle -> Directive[Thickness[.015], Red], AxesLabel -> {"E", "X", "H"}] 44 Chapter 5. The concept of equilibrium harvest

Out[3]=

We can also use Mathematica to view how the long term relationship (the red curve) is placed into the short term catch equation (the surface below):

In[4]:= Show[{Plot3D[q e x /. {k->1, q->1/2, r->1/2}, {x,0,1}, {e,0,1}, MeshFunctions -> {#3 &}, PlotStyle -> Directive[ Opacity[0.7], LightBlue, Specularity[White, 50]]], ParametricPlot3D[{e, k (1 - q/r e), q e k (1 - q/r e)} /. {k->1, q->1/2, r->1/2}, {e, 0, 1}, PlotStyle -> Directive[Thickness[.015], Red]]}, BoxRatios -> {1, 1, 1}, AxesLabel -> {"E", "X", "H"}, ViewPoint -> {2.3, -.6, .8}]

Out[4]=

While viewing the red curve from a view point in front of the X-axis (ViewPoint -> {Infinity, 0, 0}) we see the left graph below. With a view point in front of the E-axis 5.2 Surplus production models 45

(ViewPoint -> {0, -Infinity, 0}) we see the graph to the right below.

In[5]:= GraphicsRow[ParametricPlot3D[ {e, k (1 - q/r e), q e k (1 - q/r e)} /. {k->1, q->1/2, r->1/2}, {e, 0, 1}, BoxRatios -> {1, 1, 1/GoldenRatio}, PlotStyle -> Red, ViewPoint -> #, AxesLabel -> {"E", "X", "H"}] & /@ {{Infinity, 0, 0}, {0, -Infinity, 0}}, Spacings -> 50]

0.10 0.10

H H Out[5]= 0.05 0.05

0.00E0.00.51.0 0.00X0.00.51.0 0.0 0.5 1.0 0.0 0.5 1.0 X E

Equation 5.5 is the isocline for value zero of diﬀerential equation 5.3 (X˙ (t) = 0). The isocline separates the E − X-plane into two sections, one section where the stock biomass increases (below the isocline) and one section where the stock biomass is reduced (above the isocline), see Figure 5.1.

K  X =0

 X <0

 X >0

r E q

Figure 5.1: The isocline of equation 5.3 is a down-sloping straight line separating the plane into two sections: Below the isocline where the stock biomass increases (X˙ (t) > 0) and above the line where the stock biomass declines (X˙ (t) < 0).

Figure 5.1 shows all possible equilibriums as a line while all other combinations of X and E causes increase or decline in the stock biomass X. Note that X = K (for E = 0) and r X = 0 (for E = q , see equation 5.5) also are equilibriums, deﬁning the outer boundaries of the isocline. 46 Chapter 5. The concept of equilibrium harvest

5.3 Age structured models Code box 5.3.1 — Age structured model (Beverton and Holt). Continuing from code box 4.4.2

Parameter values are loaded in input number 9. TotalCatch is the yield function in the PopulationGrowth package, including two variables defining an equilibrium fishery: The fishing mortality rate F and the age of the cohort first recruited to the fishable stock, tc. The surface of the yield function in the F − tc−plane could be presented by the Plot3D function in Mathematica:

In[15]:= Plot3D[TotalCatch[Sequence @@ values], {F, 0, 1}, {tc, 0, 20}, PlotPoints -> 35, PlotRange -> All, MeshFunctions -> {#3 &}, Mesh -> 10, PlotLabel -> "Sustainable Yield per recruit (kg)", AxesLabel -> {"Fishing mortality rate (F)", "Selection age tc", ""}]

Out[15]=

n Notation 5.1. Given an open subset G of R , the set of functions ϕ are: 1. Bounded support G; 2. Inﬁnitely diﬀerentiable; a vector space is denoted by D(G).

This statement requires citation. 1 ConditionalExpression − ,<(a)=(r) 6= =(a)<(r) ∨ ((a + r > 0 ∨ a + r∈ / ) ∧ ((<(a) < r ∧ =(a) = 0) ∨ a − r∈ / )) ∨ r∈ / 2(a2 − r2) R R R (5.6) 1 Out[16]= 2

In[17]:= Integrate[{y^(-3)*(1-(a/y)^2)^(-2)},{y,r,Infinity}] 5.4 Remarks 47 1 Out[17]= 2 (a2 - r2)

5.3.1 Numbered List 1. The ﬁrst item 2. The second item 3. The third item

5.4 Remarks This is an example of a remark.

Learn The concepts presented here are now in conventional employment in mathematics. Vector spaces are taken over the ﬁeld K = R, however, established properties are easily extended to K = C.

5.5 Corollaries This is an example of a corollary.

Corollary 5.5.1 — Corollary name. The concepts presented here are now in conventional employment in mathematics. Vector spaces are taken over the ﬁeld K = R, however, established properties are easily extended to K = C.

5.6 Propositions This is an example of propositions.

5.6.1 Several equations Proposition 5.6.1 — Proposition name. It has the properties:

||x|| − ||y|| ≤ ||x − y|| (5.7) n n X X || xi|| ≤ ||xi|| where n is a ﬁnite integer (5.8) i=1 i=1

5.6.2 Single Line 2 Proposition 5.6.2 Let f,g ∈ L (G); if ∀ϕ ∈ D(G), (f,ϕ)0 = (g,ϕ)0 then f = g.

5.7 Examples This is an example of examples.

5.7.1 Equation and Text 2 0 Example 5.1 Let G = {x ∈ R : |x| < 3} and denoted by: x = (1,1); consider the function:

(e|x| si |x − x0| ≤ 1/2 f(x) = (5.9) 0 si |x − x0| > 1/2

2 0 The function√f has bounded support, we can take A = {x ∈ R : |x − x | ≤ 1/2 + } for all ∈ ]0;5/2 − 2[. 48 Chapter 5. The concept of equilibrium harvest

5.7.2 Paragraph of Text

Example 5.2 — Example name. Nam dui ligula, fringilla a, euismod sodales, sollicitudin vel, wisi. Morbi auctor lorem non justo. Nam lacus libero, pretium at, lobortis vitae, ultricies et, tellus. Donec aliquet, tortor sed accumsan bibendum, erat ligula aliquet magna, vitae ornare odio metus a mi. Morbi ac orci et nisl hendrerit mollis. Suspendisse ut massa. Cras nec ante. Pellentesque a nulla. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Aliquam tincidunt urna. Nulla ullamcorper vestibulum turpis. Pellentesque cursus luctus mauris.

5.8 Exercises This is an example of an exercise.

Exercise 5.1 This is a good place to ask a question to test learning progress or further

cement ideas into students’ minds.

5.9 Problems Problem 5.1 What is the average airspeed velocity of an unladen swallow?

5.10 Vocabulary Deﬁne a word to improve a students’ vocabulary. Vocabulary 5.1 — Word. Deﬁnition of word. II Fisheries economics

6 The economics of catch production .... 51 6.1 The economics of eﬀort production 6.2 Exercises

7 Economic growth ...... 55 7.1 Labour creates capital 7.2 Neoclassical Economic Growth Theory 7.3 Exercises

6. The economics of catch production

6.1 The economics of effort production While technological efficiency is assumed in the formulation of the production functions above, we will now introduce the concept of economic efficiency. There are two approaches to economic efficiency in production, minimising cost of a given production or maximising production within a given budget. In the box below we provide the Mathematica code following the latter principle.

In order to study economic eﬃciency in production we need to have an expression for the cost of production. We assume the production function in equation 2.3 and assume constant market prices for the two input factors, labour (l) and capital (k). Let the price (wage) of labour be (w) and the price (interest rate) of capital be (i). The total cost of a consuming labour and capital in the production process then is

C(L,K) = w · L + i · K (6.1)

Code box 6.1.1 — Economic eﬃciency in production. The Cobb-Douglas function with constant elasticity of scale equal α + β:

In[1]:= cd[l_, k_] := A * l^α * k^β

The cost of production,w being the cost of labour and i the cost of capital:

In[2]:= c[l_, k_] := w*l + i*k

The problem is to maximise the production within a given budget restriction R. We formulate the lagrange equation of the problem of constrained maximisation:

In[3]:= lagrange[l_, k_] := cd[l, k] - λ (R - c[l, k]) 52 Chapter 6. The economics of catch production

In[4]:= Sequence@@Solve[ ((λ /. Solve[D[lagrange[l, k], l] == 0, λ][[1]]) == (λ /. Solve[D[lagrange[l, k], k] == 0, λ][[1]])) /. {l -> k * x}, x][[1]] /. {x -> l/k} l i α Out[4]= → - k w β

From the calculation above we see that technological and economically eﬃcient Cobb- Douglas production is obtained when the following condition is met L i · α = − (6.2) K w · β We see that the ratio between the two input factors changes when one of the prices changes. When the wages (w) increases, the use of labour (L) is reduced and substituted by capital (K) if the previous production level should be maintained. We also see that the output elasticities of labour and capital aﬀect how labour is substituted by capital.

From expression 6.2 we can see that the cost efficient mix of input factors is fixed for given prices (i and w) and given output elasticities(α and β). The expansion path described by the optimal mix when regarding different production levels therefore is linear, as shown in Code box 6.1.2 below.

Equation 6.2 can be modiﬁed to w · β K(L) = L (6.3) i · α

Code box 6.1.2 — Economic eﬃcient expansion paths in production. We start as previously by deﬁning the Cobb-Douglas function:

In[1]:= cd[l_, k_] := A * l^α * k^β

The cost equation (equation 6.1):

In[2]:= c[l_, k_] := w*l + i*k

Cost eﬃcient input of capital as a function of labour input is found from expression 6.3:

In[3]:= k[l_] := l * w * β / (i * α)

Now we plot the directions of the expansion paths for diﬀerent values of α and β. The elasticity of scale ( = α + β)in ﬁrst three plots equal one, while the last plot shows a case where = α + β = 1 + 1 = 2:

In[4]:= GraphicsGrid[Partition[Show[{ ContourPlot[cd[l, k] /. {A -> 1, α -> #, β -> If[ # < 1, 1 - #, 1]}, {l, 0, 1}, {k, 0, 1}, ContourShading -> None, Contours -> {.02, .1, .2, .4, .6, .8}], Plot[k[l] /. {α -> #, β -> If[# < 1,1 - #, 1], w -> 1, i -> 1}, 6.1 The economics of eﬀort production 53

{l, 0, 1}, PlotStyle -> Red]}, PlotLabel -> "α = " <> ToString[#] <> ", β = " <> ToString[ If[# < 1,1 - #, 1]],FrameLabel -> {"Labour (L)", "Capital (K)"}, FrameTicks-> None]&/@ {.3, .5, .7, 1}, 2], Spacings -> {0, 20}]

α= 0.3,β= 0.7 α= 0.5,β= 0.5 Capital ( K ) Capital ( K )

Labour(L) Labour(L) Out[4]= α= 0.7,β= 0.3 α= 1,β=1 Capital ( K ) Capital ( K )

Labour(L) Labour(L)

Note that the red expansion path in the case of α = β = 0.5 is similar to the case when α = β = 1. From expression 6.2 it is easy to see that this indeed must be true. The production level of the latter case increases, however, at the rate 2:1 compared with the ﬁrst case.

As described in Section 2.6 the elasticity of substitution of the Cobb-Douglas function is one. In a normal production process also the elasticity of scale is expected to equal one, as in equation 2.2. The plots in Code box 6.1.2 also display the case of > 1, representing cases where there are economics of scale.

Code box 6.1.3 — Unit cost of production. Let us continue from Code box 6.1.2. Let us further assume that β = 1 − α, as in equation 2.2. The unit cost of production is equation 6.1 divided by equation 2.2 when including equation 6.3. When specifying basic assumptions we ﬁnd

In[5]:= Simplify[c[l, k[l]] / cd[l, k[l]] /. {β -> 1 - α}, Assumptions -> {Element[{l, w, i, A, α}, Reals], w > 0, i > 0, l > 0, A > 0, 0 < α < 1}]

i (i α)-α (w - w α)α Out[5]= - A (- 1 + α) 54 Chapter 6. The economics of catch production

From this result we can conclude that for a Cobb-Douglas production process with elasticity of scale equal one, the unit cost of output is constant in an economically eﬃcient production.

As seen below this is not necessarily the case when the elasticity of scale is diﬀer- ent from one:

In[6]:= Simplify[c[l, k[l]] / cd[l, k[l]], Assumptions -> {Element[{l, w, i, A, α, β}, Reals], w > 0, i > 0, l > 0, A > 0, 0 < α < 1, 0 < β < 1}]

l1-α w (i α)β (l w β)-β (α + β) Out[6]= A α In the special case of α = β = 1/2 the unit price of production is found to be

In[7]:= Simplify[c[l, k[l]] / cd[l, k[l]] /. {α -> 1/2, β -> 1/2}, Assumptions -> {Element[{l, w, i, A}, Reals], w > 0, i > 0, l > 0, A > 0}] √ 2 i w Out[7]= A

6.2 Exercises Exercise 6.1 Why increases the production more rapidly in the lower-right ﬁgure in

Code box 6.1.2 than in the upper-right ﬁgure?

Exercise 6.2 What is the unit cost of a product following a Cobb-Douglas production process and economic eﬃcient use of input factors when the output elasticities equal

one? 7. Economic growth

7.1 Labour creates capital In chapter 2 we discussed how commodities are produced by the input factors labour and capital. This chapter presents the neoclassical economic theory of how capital is produced by the production of commodities.

When labour and capital becomes embedded in commodities through the production processes described in chapter 2, some of the commodities are consumed. This includes both ﬁnal consumption and the use of the produced commodities in the production of other products. Outputs from the lumber industry are for example input factors in the production of houses, furnitures, paper and many other products.

But not everything is consumed immediately, commodities could also be saved for later consumption or invested. These fractions of the produced commodities then eﬃciently become parts of the capital stock.

7.2 Neoclassical Economic Growth Theory The Solow-Swan model describes economic growth model within the framework of neoclassical economics. The following refers the model set-up by Solow[7].

Assume there is only one commodity, Y , which is the output of a production process where labour (L) and capital (K) are inputs:

Y (L,K) = f(L,K) (7.1)

The commodity is partly consumed and the rest is saved and invested. If the saved fraction is constant and equal s, then s · Y is added to the capital stock K. In continuous time this gives

K˙ = s · Y (7.2) 56 Chapter 7. Economic growth

The labour population is assumed to grow exponentially with the growth rate n, thus the labour at time t will be

n·t L(t) = L0 · e (7.3)

Since both capital and labour grow it is of interest to investigate the ratio between the two. Let the variable r represent this ratio:

K(t) r(t) = (7.4) L(t)

Solving this for K(t) and inserting from equation 7.3 gives

n·t K(t) = r(t) · L(t) = r(t) · L0 · e (7.5)

which after diﬀerentiating with time gives

n·t n·t K˙ (t) =r ˙(t) · L0 · e + n · r(t) · L0 · e (7.6)

Setting this equal equation 7.2, we obtain

n·t r˙(t) + n · r(t) L0 · e = s · f(L,K) (7.7)

Assuming constant returns to scale we have

r˙(t) + n · r(t) = s · f(L,1) (7.8)

and further

r˙(t) = s · f(L,1) − n · r(t) (7.9)

By deﬁnition n = L/L˙ and since r = K/L we have

r˙(t) K˙ (t) L˙ (t) K˙ (t) = − = − n (7.10) r(t) K(t) L(t) K(t)

K˙ (t) s · f(L,K) r˙(t) = r − n · r = r − n (7.11) K(t) K

corresponding to equation 7.6. When this expression equals zero, the capital/labour ratio (r) is constant and the capital grows at the same pace as the population (n):

s · f(L,K) n = (7.12) K

Summary 7.2.1 — Deﬁnition name. Given a vector space E, a norm on E is an application, + denoted || · ||, E in R = [0,+∞[ such that:

7.3 Exercises 7.3 Exercises 57

Exercise 7.1 Why increases the production more rapidly in the lower-right ﬁgure in

Mathematica code 6.1.2 than in the upper-right ﬁgure?

III Fisheries management

8 Economic growth ...... 61 8.1 Labour creates capital 8.2 Table 8.3 Figure

Bibliography ...... 63 Articles

Index ...... 65

8. Economic growth

8.1 Labour creates capital 8.2 Table

Treatments Response 1 Response 2 Treatment 1 0.0003262 0.562 Treatment 2 0.0015681 0.910 Treatment 3 0.0009271 0.296

Table 8.1: Table caption

8.3 Figure

Figure 8.1: Figure caption

References

[1] F. I. Baranov. “On the question of the biological basis of fisheries.” In: 1.1 (1918), pages 81–128 (cited on page 35). [2] L. von Bertalanffy. “Untersuchungen über die Gesetzlichkeit des Wachstums.” In: 131 (1934), pages 613–652 (cited on page 34). [3] A. Eide et al. “Harvest Functions: the Norwegian Bottom Trawl Cod Fisheries.” In: 18 (2003), pages 81–93 (cited on page 23). [4] B. Gompertz. “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies.” In: 123 (1825), pages 513–585 (cited on page 28). [5] R. Hannesson. “Bioeconomic production function in fisheries: Theoretical and empirical analysis.” In: 40.7 (1983), pages 968–982 (cited on page 23). [6] F. J. Richards. “A flexible growth function for empirical use.” In: 10 (1959), pages 290– 300 (cited on page 29). [7] R. M. Solow. “A contribution to the theory of economic growth.” In: 70.1 (1956), pages 65–94 (cited on page 55). [8] P. F. Verhulst. “Notice sur la loi que la population pursuit dans son accroissement.” In: 10 (1838), pages 113–121 (cited on page 29). [9] C.J. Walters, R. Hilborn, and V. Christensen. “Surplus production dynamics in declining and recovering fish populations.” In: 65 (2008), pages 2536–2551 (cited on page 33).

Index

A Production ...... 12 Fishing gears Age structured models Angling...... 11 Equilibrium harvest ...... 46 Danish seine ...... 11 Gill net ...... 11 C Hand line ...... 11 Long line ...... 11 Catch production ...... 19 Purse seine ...... 11 Corollaries ...... 47 Trawl...... 11 E L Economic Growth Theory...... 55 Elasticity ...... 13 Lists of output ...... 14 Numbered List ...... 47 of scale ...... 16 of substitution ...... 15 P Equilibrium harvest...... 41 Examples ...... 47 Problems ...... 48 Equation and Text ...... 47 Production functions ...... 12 Paragraph of Text ...... 48 CES ...... 15 Exercises ...... 48 Cobb-Douglas ...... 14 Chapter 2 ...... 54 Cost of production...... 51 Chapter 3 ...... 23 Propositions ...... 47 Chapter 4 ...... 56 Several Equations ...... 47 Chapter 5 ...... 39 Single Line ...... 47 F R Fishing eﬀort ...... 11 Measurement ...... 11 Remarks...... 47 66 INDEX

Stock assessment by CPUE...... 20 Surplus production Depensatory growth ...... 31 Equilibrium harvest ...... 42 Gompertz model...... 28 Richards model ...... 29 Verhulst model ...... 29

Table ...... 61

Vocabulary ...... 48