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Problem Chosen

2020

MCM/ICM

Summary Sheet

Team Control Number

2001334

Forecasts for the Ecology and Fisheries Economy of Scottish

herring and mackerel

As the favorable food for Scotch, the herring and mackerel br ing generous proﬁts to ﬁshing

companies. Due to the hotter ocean, more ﬁsh move to the north to seek better habitats, laying a

negative impact on the ﬁshing industry. The aim of this report is to build a migratory prediction

model to evaluate the inﬂuences on the income of ﬁshing companies. We are expected to provide

some strategies for ﬁshing companies who can adapt to the migration of ﬁsh under the constraints of

various objective conditions and prevent themselves from going bankrupt as much as possible. Three

models are established: Model I: Seawater Temperature Prediction Model; Model II: Fish Migration

Prediction Model; Model III: Fishing Company Earnings Evaluation Model.

For Model I, global ocean temperature date monthly from 1960 to 2019 is ﬁrstly collected. Then,

based on the analysis of intrinsic trend of the data and the veriﬁcation of the stationarity, the validation

of using ARIMA model to predict temperature is proved. Next, historical data is used to ﬁt the

parameters of ARIMA, with introduction of k-fold cross validation to identify the ﬁnal prediction

model as ARIMA(1,1,0). Finally, according to ARIMA(1,1,0), bootstrap method is used to simulate

10000 possible prediction cases, which lays a great foundation to predict the migration of ﬁsh.

For Model II, ﬁrstly, according to the data of the migration speed and the ocean temperature, it is

determined that the temperature gradient is the main factor aﬀecting the migration speed and direction.

And the corresponding empirical equation is established to determine the impact of temperature on

ﬁsh migration. Then based on the 10000 temperature change samples generated by bootstrap method

in Model I, migration situation of each sample is simulated to identify the most likely locations of the

ﬁsh. It was ﬁnally shown that the ﬁsh are mainly distributed in the area between Iceland and the Faroe

Islands 50 years later and the results are shown in ﬁgure 9.

For Model III, the proﬁt evaluation equation of ﬁshing companies is deter mined by the economic

principle, and the parameters involved are estimated by introducing the actual management data, the

results are shown in table 4; then based on the 10000 samples of ﬁsh migration from Model II, the

proﬁt change of ﬁshing companies is simulated for each sample and the proﬁt trend over time is shown

in ﬁgure 10. Finally, it can be seen that the worst case is in 2030, ﬁshing companies will go bankrupt

due to ﬁsh migration with a probability of 0.02%, the best case is that they will not go bankr upt in 50

years with a probability of 5.27% and the most likely case is that in 2039, ﬁshing companies will go

bankrupt due to ﬁsh migration with a probability of 8.25%.

In addition, this report discusses the eﬀective response to the ﬁsh migration for small ﬁshing

companies, together with eﬀective response strategies. Without considering the policies and legal

issues brought by the territorial sea, small ﬁshing companies should transfer their ports to Iceland,

which is closer to the ﬁsh. Finally, based on simulation of this strategys eﬀect, 100.00% of companies

can avoid bankruptcy. As for considering the policies and legal issues, small ﬁshing companies should

upgrade their ﬁshing vessels to extend the shelf life of ﬁsh. After simulation, 62.68% of companies

can avoid bankruptcy.

Eventually, robustness and sensitivity analysis of the model are tested. When the initial distribu-

tion of the ﬁsh is randomly generated from the uniform random distribution, the ﬁnal convergence

distribution of the model has little diﬀerence. As for the factors that aﬀect the model, social proﬁt rate

and ﬁshing boat navigation radius, it is found that the increase of these two factors will signiﬁcantly

reduce the bankruptcy probability of ﬁshing companies.

Keywords: ARIMA; Fish Migration; Earnings Evaluation; Computer Simulation

Team # 2001334 Page 1 of 27Team # 2001334 Page 1 of 27Team # 2001334 Page 1 of 27

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Problem Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Our Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 General Assumptions and Model Overview . . . . . . . . . . . . . . . . . . . . . . 2

3 Model Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3.2.1 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.2.2 Data Cleaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.3 Geographic Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

4 Model I: Seawater Temperature Prediction Model . . . . . . . . . . . . . . . . . . . 5

4.1 Description of Temperature Field . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.2 Autoregressive Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . . 5

4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4.3.2 Calaculation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

5 Model II: Fish Migration Prediction Model . . . . . . . . . . . . . . . . . . . . . . . 8

5.1 Kinematics of Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5.2 Kinetics of Migration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.3.1 Estimation of ∇u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.3.2 Estimation of f(∇u, v) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

5.3.3 Migration Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . 10

5.3.4 Calaculation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

6 Model III: Fishing Company Earnings Evaluation Model . . . . . . . . . . . . . . . 11

6.1 Fishing Company Operating Model . . . . . . . . . . . . . . . . . . . . . . . . 11

6.1.1 Assessment of Fishing Costs . . . . . . . . . . . . . . . . . . . . . . . . 11

6.1.2 Assessment of Fishing Income . . . . . . . . . . . . . . . . . . . . . . . 11

6.1.3 Assessment of Fishing Proﬁt . . . . . . . . . . . . . . . . . . . . . . . . 11

6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.2.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

6.2.2 Migration Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . 12

6.2.3 Calaculation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

6.3.1 The Management Strategies without Consider of Territorial Sea . . . . 14

6.3.2 The Management Strategies with Consider of Territorial Sea . . . . . . 15

7 Test the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

7.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Team # 2001334 Page 1 of 27Team # 2001334 Page 1 of 27Team # 2001334 Page 1 of 27

1 Introduction

1.1 Problem Background

Global ocean temperatures affect the quality of habitats for certain ocean-dwelling species.

When temperature changes are too great for their continued thriving, these species move to

seek other habitats better suited to their present and future living and reproductive success.

The consortium wants to gain a better understanding of issues related to the potential migra-

tion of Scottish herring and mackerel from their current habitats near Scotland if and when

global ocean temperatures increase. These two ﬁsh species represent a signﬁcant economic

contribution to the Scottish ﬁshing industry. Changes in population locations of herring and

mackerel could make it economically impractical for smaller Scotland-based ﬁshing com-

panies, who use ﬁshing vessels without on-board refrigeration, to harvest and deliver fresh

ﬁsh to markets in Scotland ﬁshing ports.

(a) Herring (b) Mackerel

Figure 1: Target ﬁsh: (a) Scottish herring: Atlantic herring are widely distributed through-

out the north-east Atlantic, ranging from the Arctic ocean in the north to the English Channel

in the south; (b) Scottish mackerel: Each year, the number of mackerel in the sea depends on

the number of young ﬁsh which survive from spawning to enter the adult ﬁshery as recruits.

1.2 Restatement of the Problem

• Build a mathematical model to identify the most likely locations for these two ﬁsh

species over the next 50 years.

• Based upon how rapidly the ocean water temperature change occurs, use your model

to predict best case, worst case, and most likely elapsed time(s) until these popula-

tions will be too far away for small ﬁshing companies to harvest if the small ﬁshing

companies continue to operate out of their current locations.

• In light of your predictive analysis, should these small ﬁshing companies make changes

to their operations?

• Use your model to address how your proposal is affected if some proportion of the

ﬁsherymoves into the territorial waters (sea) of another country.

1.3 Our Approach

The topic requires us to predict the migration of two kinds of ﬁsh in the next 50 years

and discuss the business strategies and prospects of ﬁshing companies according to the mi-

gration of the ﬁsh. Our work mainly includes the following:

Team # 2001334 Page 2 of 27Team # 2001334 Page 2 of 27Team # 2001334 Page 2 of 27

• Based on the historical data of ocean temperature, a prediction model of ocean tem-

perature is established;

• The probability distribution of ﬁsh migration is given and the inﬂuence of randomness

on the model is considered;

• Based on the economic beneﬁt model of ﬁshing companies, this article evaluates the

beneﬁts of various ﬁshing strategies under the background of ﬁsh migration and gives

reasonable suggestions for the improvement of them.

2 General Assumptions and Model Overview

To simplify the problem, we make the following basic assumptions, each of which is

properly justiﬁed.

• Assumption 1: The migration direction of population is predictable.

→ Justiﬁcation: Although the swimming direction of each individual does not neces-

sarily follow the law of migration, according to the law of large numbers, the behavior

of the group will exclude the existence of unpredictable accidental factors, so we can

predict the migration direction of ﬁsh by predicting the change of ocean temperature.

• Assumption 2: The migration of ﬁsh is carried out at the same depth.

→ Justiﬁcation: We assume that the change of ocean depth is ignored in the process

of ﬁsh migration, because in a relatively long time span, the migration range of ﬁsh is

far larger than its depth change range, so the depth change in the process of migration

can be ignored.

• Assumption 3: No macro-economic indicators, trade environment and technological

breakthroughs in the research time.

→ Justiﬁcation: Because the model considers the impact of ocean temperature change

on the migration direction of ﬁsh, and then compares and analyzes the ﬁshing strate-

gies adopted by ﬁshing companies before and after the migration of ﬁsh. Only when

the external conditions are consistent can such a comparison be meaningful.

• Assumption 4:Assume the research data is accurate.

→ Justiﬁcation:We assume that the historical ocean surface temperature data, ﬁsh-

ing data and ﬁnancial data of ﬁshing companies do not show obvious measurement

deviation and are believed that they are fake, so we can establish a more reasonable

quantitative model based on it.

Firstly, set the Seawater Temperature Prediction Model. We use historical data of seawa-

ter temperature to predict seawater temperature changes in the target sea area in the next

50 years. Secondly, set the Fish Migration Prediction Model. We describe the correlation

between seawater temperature changes and ﬁsh migration directions, and then simulate

ﬁsh migration directions based on seawater temperature changes in the target sea area over

the next 50 years. Finally, set the Fishing Company Earnings Evaluation Model. We assess

changes in the proﬁtability of ﬁshing companies based on the migration of ﬁsh in the next 50

years, and discuss strategies to deal with such changes subject to some objective conditions.

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