herring and mackerel
鲱鱼和鲭鱼

As the favorable food for Scotch, the herring and mackerel bring 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.
作为苏格兰威士忌的热门食物，鲱鱼和鲭鱼为酿酒公司带来了丰厚的利益。由于海洋变热，更多的鱼向北移动以寻找更好的栖息地，对鱼产业产生了负面影响。本报告的目的是建立一个迁移预测模型，以评估对捕鱼公司收入的影响。我们期望为能够在各种客观条件的约束下适应金融迁移并尽可能防止自身破产的金融公司提供一些策略。建立了三个模型：模型 I：海水温度预测模型;模型 II：鱼类洄游预测模型;模型 III：渔业公司收益评估模型。

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.
对于模型 I，首先收集了 1960 年至 2019 年每月的全球海洋温度日期。然后，基于对数据内在趋势的分析和平稳性的验证，证明了使用 ARIMA 模型预测温度的有效性。接下来，使用历史数据拟合 ARIMA 的参数，引入 k 折叠交叉验证以将最终预测模型标识为 ARIMA（1,1,0）。最后，根据 ARIMA（1,1,0），采用 bootstrap 方法模拟了 10000 种可能的预测情况，为预测 fish 的迁移奠定了良好的基础。

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.
对于模型 II，首先根据迁移速度和海洋温度数据确定温度梯度是影响迁移速度和方向的主要因素。并建立了相应的经验方程来确定温度对鱼洼迁移的影响。然后，基于模型 I 中 bootstrap 方法生成的 10000 个温度变化样本，模拟每个样本的迁移情况，以确定最可能的 FISH 位置。最终表明，50 年后，这些鱼主要分布在冰岛和法罗群岛之间的地区，结果如图 9 所示。

For Model III, the profit evaluation equation of fishing companies is determined 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 bankrupt 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%.
对于模型 III，根据经济原理确定渔业公司的资产评价方程式，通过引入实际管理数据估算所涉及的参数，结果见表 4;然后基于模型 II 的 10000 个渔业迁移样本，模拟每个样本的渔业公司资产变化，并得到 T 随时间变化的趋势如图 10 所示。最后，可以看出，最坏的情况是 2030 年，金融公司因金融迁移破产的概率为 0.02%，最好的情况是 50 年内不会破产，概率为 5.27%，最有可能的情况是 2039 年，金融企业因金融移民破产的概率为 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.
此外，本报告还讨论了小型渔业公司对渔业移民的有效应对措施，以及有效的应对策略。在不考虑领海带来的政策和法律问题的情况下，小型渔业公司应该将其港口转移到更靠近渔业的冰岛。最后，基于对这种策略效应的模拟，100.00% 的公司可以避免破产。至于考虑政策和法律问题，小型捕鱼公司应该升级他们的捕鱼容器，以延长捕鱼的保质期。经过仿真，62.68% 的公司可以避免破产。

Eventually, robustness and sensitivity analysis of the model are tested. When the initial distribution 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.
最终，测试了模型的稳健性和敏感性分析。当数据的初始分布是从均匀随机分布中随机生成的时，模型的最终收敛分布差异不大。对于影响模型的因素、社会转化率和渔船航行半径，发现这两个因素的增加将显著降低渔民企业破产的概率。

Keywords: ARIMA; Fish Migration; Earnings Evaluation; Computer Simulatio
关键词：ARIMA;鱼类洄游;收益评估;计算机模拟

Contents
内容

1 Introduction 3
1 介绍3

1.1 Problem Background 3
1.1 问题背景3

1.2 Restatement of the Problem 3
1.2 问题重述3

1.3 Our work 4
1.3 我们的工作4

2 Assumptions and Justifications 5
2 假设和理由5

3 Notations 5
3 符号5

4 Model Preparation 6
4 模型准备6

4.1 The Data 6
4.1 数据6

4.2 Geographic Coordinate System 7
4.2 地理坐标系7

5 Model I: Seawater Temperature Prediction Model 7
5 模型 I：海水温度预测模型7

5.1 Description of Temperature Field 7
5.1 温度场说明7

5.2 Autoregressive Prediction Model 8
5.2 自回归预测模型8

5.3 Results 9
5.3 结果9

6 Model II: Fish Migration Prediction Model 10
6 模型 II：鱼类洄游预测模型10

6.1 Kinematics of Migration 10
6.1 迁移的运动学10

6.2 Kinetics of Migration 11
6.2 迁移动力学11

6.3 Results 11
6.3 结果11

7 Model III: Fishing Company Earnings Evaluation Model 13
7 模型 III：渔业公司收益评估模型13

7.1 Fishing Company Operating Model 13
7.1 渔业公司经营模式13

7.2 Results 13
7.2 结果13

7.3 Discussion 16
7.3 讨论16

8 Test the Model 18
8 测试18 型

8.1 Sensitivity Analysis 18
8.1 敏感性分析18

8.2 Robustness Analysis 18
8.2 稳健性分析18

9 Conclusion 19
9 结论19

9.1 Summary of Results 19
9.1结果摘要 19

9.2 Strengths 21
9.2 优势21

9.3 Possible Improvements 21
9.3 可能的改进21

References 22
参考资料22

Appendices 24
附录24

Introduction
介绍

Problem Background
问题背景

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^[1]. The consortium wants to gain a better understanding of issues related to the potential migration 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 companies, who use fishing vessels without on-board refrigeration, to harvest and deliver fresh fish to markets in Scotland fishing ports.
全球海洋温度会影响某些海洋生物物种的栖息地质量。当温度变化太大而无法继续繁衍生息时，这些物种会迁移到更适合它们现在和未来生活和繁殖成功的其他栖息地^[1]。该联盟希望更好地了解如果全球海洋温度升高，苏格兰鲱鱼和鲭鱼可能从苏格兰附近的当前栖息地迁移的相关问题。这两种鱼种代表了对苏格兰渔业的重大经济贡献。鲱鱼和鲭鱼种群位置的变化可能会使苏格兰的小型渔业公司在经济上不切实际，这些公司使用没有船上冷藏的渔船来捕捞新鲜鱼并将其运送到苏格兰渔港的市场。

(a) Herring （a） 鲱鱼	(b) Mackerel （b） 鲭鱼
Figure 1: Target fish 图 1：目标鱼 (a) Scottish herring: Atlantic herring are widely distributed throughout the north-east Atlantic, ranging from the Arctic ocean in the north to the English Channel in the south; （a） 苏格兰鲱鱼：大西洋鲱鱼广泛分布在整个东北大西洋，从北部的北冰洋到南部的英吉利海峡; (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. （b） 苏格兰鲭鱼：每年，海中鲭鱼的数量取决于从产卵中存活下来并作为新鱼进入成年渔业的幼鱼数量。

Restatement of the Problem
重述问题

Considering the background information and restricted conditions identified in the problem statement, we need to solve the following problems
考虑到问题陈述中确定的背景信息和限制条件，我们需要解决以下问题:

Build a mathematical model to identify the most likely locations for these two fish species over the next 50 years.
构建一个数学模型，以确定未来 50 年这两种鱼类最可能的位置。

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 populations will be too far away for small fishing companies to harvest if the small fishing companies continue to operate out of their current locations.
根据海水温度变化的速度，使用您的模型来预测最佳情况、最坏情况和最可能经过的时间，如果小型渔业公司继续在其当前位置运营，这些种群将距离太远，小型渔业公司无法收获。

In light of your predictive analysis, should these small fishing companies make changes to their operations?
根据您的预测分析，这些小型渔业公司是否应该改变他们的运营方式？

Use your model to address how your proposal is affected if some proportion of the fisherymoves into the territorial waters (sea) of another country.
使用您的模型来解决如果一部分渔业进入另一个国家的领海（海），您的提案会受到怎样的影响。

Our work
O你的工作

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 migration of the fish. Our work mainly includes the following:
该主题要求我们预测未来 50 年两种鱼类的迁徙情况，并根据鱼类的迁徙情况讨论渔业公司的经营策略和前景。我们的工作主要包括以下内容：

Based on the historical data of ocean temperature, a prediction model of ocean temperature is established;
基于海洋温度历史数据，建立了海洋温度预测模型;

The probability distribution of fish migration is given and the influence of randomness on the model is considered;
给出了鱼类洄游的概率分布，并考虑了随机性对模型的影响;

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.
本文基于渔业企业的经济效益模型，评估了鱼类洄游背景下各种捕捞策略的效益，并给出了合理的改进建议。

Firstly, set the Seawater Temperature Prediction Model. We use historical data of seawater 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.
首先，设置 Seawater Temperature Prediction Model（海水温度预测模型）。我们使用海水温度的历史数据来预测未来 50 年目标海域的海水温度变化。其次，设置 Fish Migration Prediction Model。我们描述了海水温度变化与鱼类洄游方向之间的相关性，然后根据未来 50 年目标海域的海水温度变化模拟鱼类洄游方向。最后，设置 Fishing Company Earnings Evaluation Model（渔业公司收益评估模型）。我们根据未来 50 年鱼类的洄游评估渔业公司的盈利能力变化，并在一些客观条件下讨论应对此类变化的策略。

In summary, the whole modeling process can be shown as follows:
综上所述，整个建模过程可以显示如下：

Figure 2: Model Overview
图 2：模型概述

Assumptions and Justifications
假设和理由

To simplify the problem, we make the following basic assumptions, each of which is properly justified.
为了简化问题，我们做出以下基本假设，每个假设都有适当的理由。

Assumption 1: The migration direction of population is predictable.
假设 1：种群的迁移方向是可预测的。

Justification: Although the swimming direction of each individual does not necessarily 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.
理由：虽然每个个体的游动方向不一定遵循迁徙规律，但根据大数规律，群体的行为会排除存在不可预测的偶然因素，因此我们可以通过预测海洋温度的变化来预测鱼类的迁徙方向。

Assumption 2: The migration of fish is carried out at the same depth.
假设 2：鱼类的迁徙在同一深度进行。

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.
理由：我们假设鱼类洄游过程中忽略了海洋深度的变化，因为在较长的时间跨度内，鱼类的洄游范围远大于其深度变化范围，因此可以忽略洄游过程中的深度变化。

Assumption 3: No macro-economic indicators, trade environment and technological breakthroughs in the research time.
假设 3：研究时间内没有宏观经济指标、贸易环境和技术突破。

Justification: Because the model considers the impact of ocean temperature change on the migration direction of fish, and then compares and analyzes the fishing strategies adopted by fishing companies before and after the migration of fish^[2]. Only when the external conditions are consistent can such a comparison be meaningful.
理由：因为该模型考虑了海洋温度变化对鱼类洄游方向的影响，然后对比分析了渔业公司在鱼类洄游前后采取的捕捞策略^[2]。只有当外部条件一致时，这样的比较才有意义。

Assumption 4: Assume the research data is accurate.
假设 4：假设研究数据准确。

Justification: We assume that the historical ocean surface temperature data, fishing 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.
理由：我们假设历史海面温度数据、捕捞数据和渔业公司的财务数据没有显示出明显的测量偏差，并认为它们是假的，因此我们可以基于它建立一个更合理的量化模型。

Notations
符号

The key mathematical notations used in this paper are listed in Table 1
表 1 列出了本文中使用的关键数学符号.

Table 1: Notations used in this paper
Table 1： 本文中使用的符号

Model Preparation
模型准备

The Data
数据

Since the amount of data is large and not intuitive, we directly visualize some of the data for display.
由于数据量大且不直观，我们直接将部分数据可视化进行展示。

Data Collection
数据采集

The data we used mainly include historical seawater temperature data, fishery fishing data, fish distribution data, and financial indicators of some fishing companies. The data sources are summarized in Table 2.
我们使用的数据主要包括历史海水温度数据、渔业捕捞数据、鱼类分布数据以及一些渔业公司的财务指标。表 2 总结了数据来源。

Table 2: Data source collation
表 2：数据源排序规则

Data Cleaning
数据清理

The data is divided into groups by years and calculate the average value of the key data from April to July in each group. For the missing value in the data, we try to skip it and only seek the effective mean value. For the complete missing group from April to July, the values were recorded as a missing one. Then, the missing values are interpolated linearly along the time axis. If four or more missing values are in one column, the data in this column is considered as invalid. Finally, the location of invalid data column is set to be unreachable, which is ignored in model calculation.
将数据按年份分为几组，并计算每组中 4 月至 7 月关键数据的平均值。对于数据中的缺失值，我们尝试跳过它，只寻找有效平均值。对于 4 月至 7 月的整个缺失组，这些值被记录为缺失值。然后，沿时间轴线性插值缺失值。如果一列中有四个或更多缺失值，则此列中的数据将被视为无效。最后，将无效数据列的位置设置为 unreachable，在模型计算中被忽略。

Figure 3: Data cleaning
图 3：数据清理

Geographic Coordinate System
地理坐标系

The spherical coordinate is applied on the dataset to represent points. In order to obtain the true distance relation in the map, we regard the observed map area as a plane quadrilateral approximately^[3]. After using the geodesic equation (GRS80 sphere) to solve the quadrilateral length, we fit a projection transformation to get the corresponding relationship between the spherical coordinates and the plane coordinates. In this way, the Euclidean distance between points is approximately the geodesic distance on the sphere.
球坐标应用于数据集以表示点。为了获得地图中真实的距离关系，我们将观测到的地图区域近似视为一个平面四边形^[3]。使用测地线方程（GRS80 球体）求解四边形长度后，我们拟合投影变换，得到球面坐标和平面坐标之间的对应关系。这样，点之间的欧几里得距离大约是球体上的测地线距离。

Figure 4: Spherical coordinate transformation
图 4：球面坐标变换

Model I: Seawater Temperature Prediction Model
模型 I：海水温度预测模型

The temperature change of ocean is determined by various factors, namely sun radiation, heat loss and heat exchange of marine organisms, they can cause a significant change of ocean temperature. Therefore, for such a complex dynamic system, a method of multiple time series vector autoregression is applied to solve it. Formally, vector autoregression algorithm can consider the spatial-temporal correlation of each variable at the same time, and mine the data information to the maximum without introducing exogenous factors. Thus, the prediction based on Autoregressive Integrated Moving Average model (ARIMA) is a good approximation to the temperature field.
海洋的温度变化是由多种因素决定的，即太阳辐射、热量损失和海洋生物的热交换，它们会导致海洋温度的显着变化。因此，对于这样一个复杂的动态系统，采用多时间序列向量自回归的方法来解决。向量自回归算法从形式上讲，可以同时考虑各个变量的时空相关性，在不引入外生因素的情况下，最大限度地挖掘数据信息。因此，基于自回归综合移动平均模型 （ARIMA） 的预测是温度场的良好近似值。

Description of Temperature Field
温度场描述

According to the Assumption 2, the change of ocean temperature in the vertical plane is not considered. Therefore, for the target ocean area, based on longitude and latitude, a coordinate system is established to describe the location of each point. Therefore, the temperature of any point at time t can be expressed as
根据假设 2，不考虑垂直平面上海洋温度的变化。因此，对于目标海域，根据经纬度，建立一个坐标系来描述每个点的位置。因此，时间 t 处任意点的温度 可以表示为

where is the coordinate of the point , the abscissa represents the longitude and the ordinate represents the latitude.
其中 是点的坐标 ，横坐标表示经度，纵坐标表示纬度。

Autoregressive Prediction Model
自回归预测模型

The temperature series data of the -th () marked fishing point in the target sea area is numbered as . Firstly, the temperature change of each series in the past 60 years is plotted as shown in the Figure 5
目标海域中第 -th （） 个标记的钓鱼点的温度序列数据编号为 。首先，绘制了过去 60 年每个序列的温度变化，如图 5 所示.

Figure 5: Sea surface temperature in past 60 years (Spring)
图 5：过去 60 年的海面温度（春季）

It can be seen that the overall temperature variation has no obvious trend. Therefore, a model is applied to ARIMA() model the temperature series data. For the -th temperature series, the general situation of the model is as following
由此可见，整体温度变化没有明显的趋势。因此，将模型应用于 ARIMA（） 对温度序列数据进行建模。 对于第 -th 个温度序列，模型的一般情况如下

Where, represents the difference operator of order. is the residual value of the model. Therefore, the first-order difference prediction value of the -th series after the next years can be obtained as
其中， 表示 order 的差值运算符。 是模型的残差值。因此， 未来几年之后的第 -th 序列的 first-order 差分预测值 可以得到为

Furthermore, on the basis of the assumption from ARIMA model and the practical experience, the prediction values meet the normal distribution as . Therefore, with the consideration of randomness, the initial temperature prediction formula (4) of time can be modified as
此外，根据 ARIMA 模型的假设和实践经验，预测值满足正态分布 as。因此，在考虑随机性的情况下，可以将时间的初始温度预测公式 （4） 修改为

Different predictions can be obtained, based on the accumulation of the randomness in the process of progressively prediction. These relevant results which are related to the temperature change of ocean can lead the migration of fish to change. Therefore, the location of fish can be in different areas after 50 years, which lay a great impact on the fishing companies.
根据渐进预测过程中随机性的积累，可以获得不同的预测。这些与海洋温度变化相关的结果可以导致鱼类的洄游发生变化。因此，50 年后鱼的位置可以位于不同的区域，这对渔业公司影响很大。

Results
结果

Parameter Estimation
参数估计

Referring to the OLS method in the linear regression model, write out the linear equations corresponding to equation (2) as follows
参考线性回归模型中的 OLS 方法，写出与方程 （2） 对应的线性方程，如下所示

That is , so the closed-form solution of the corresponding parameter is
即 ，因此相应参数的闭式解 为

Therefore, OLS method can be used to estimate the parameters of formula. It is noted that there are lag order and difference order in the model, so we need to perform k-fold cross validation on the estimation results of the model, and find the model with the best prediction effect in the given alternative models. The solution results are shown in Figure 6
因此，OLS 方法可以用于估计公式的参数。值得注意的是，模型中存在滞后阶和差分阶，因此我们需要对模型的估计结果进行 k-fold 交叉验证，并在给定的备选模型中找到预测效果最好的模型。解决方案结果如图 6 所示.

Figure 6: Choose ARIMA model by using RMSE with 6-fold cross validation
图 6：使用具有 6 倍交叉验证的 RMSE 选择 ARIMA 模型

It can be seen from the figure that when the lag and difference orders are both 1, the prediction ability of the model is the best. So, we choose the optimal overall performance ARIMA(1,1,0) model.
从图中可以看出，当滞后阶数和差阶数均为 1 时，模型的预测能力最好。因此，我们选择整体性能最优的 ARIMA（1,1,0） 模型。

Calaculation Results
计算结果

We can get the predicted temperature distribution of target sea area Q after 50 years according to the above parameter estimation and prediction formula, as shown in Figure 7.
根据上述参数估计和预测公式，我们可以得到 50 年后目标海域 Q 的预测温度分布，如图 7 所示。

Figure 7: Temperature forecast after 50 years
图 7：50 年后的温度预测

Model II: Fish Migration Prediction Model
模型 II：鱼类洄游预测模型

The migration effect of fish need to be considered from two aspects: one is to explore the motivation and speed of fish migration based on dynamics; the other is to explore the relationship between the position change and migration speed of fish migration based on kinematics^[4]. A dynamic fish migration model can be obtained by combining these two aspects. These two parts will be described separately below.
鱼的迁移效应需要从两个方面考虑：一是基于动力学的探索鱼的迁移动机和速度;二是基于运动学的探索鱼迁移的位置变化与迁移速度的关系^[4]。通过将这两个方面结合起来，可以得到动态的 fish 迁移模型。下面将分别介绍这两个部分。

Kinematics of Migration
迁移的运动学

For each point , the corresponding fish situation is shown in the Figure 8
对于每个点，相应的鱼情如图 8 所示.

Figure 8: Fishes migration mechanism
图 8：鱼类洄游机制

It can be seen that, for the area shown in Figure 8, the migration direction of fish is due to . From the view of kinematics, after determining the moving speed of fish, the position updating condition can be calculated. Therefore, the expression of the corresponding moving speed based on the position calculation is
可以看出，对于图 8 所示的区域，鱼类的洄游方向是由于 。从运动学的角度来看，在确定了鱼的运动速度后，就可以计算出位置更新条件。因此，基于位置计算的相应移动速度的表达式为

There, the updated formula for position is
在那里，更新的 position 公式是

Kinetics of Migration
迁移动力学

After the kinematic description of fish migration, it is necessary to describe and model fish migration from the dynamic point of view. According to the living habits description of the two kinds of fish in Section 1, the fish tend to transfer to the area with lower temperature, so we believe that the relationship between the migration speed and the temperature field is
在对 fish migration 进行运动学描述之后，有必要从动力学的角度描述和建模 fishmigration 。根据第1节中两种鳍鸠的生活习性描述，鳍鳍倾向于向温度较低的地区转移，因此我们认为洄游速度与温度域之间的关系是

Where is an undetermined relation equation. We will refer to the actual data in next section to determine the specific form of
其中 是一个未确定的关系方程。我们将在下一节中参考实际数据 来确定 .

Results
结果

Estimation of
估计

For the equation (9), the calculation method of should be determined first. Based on definition, the expression for should be
对于方程 （9），应首先确定 的计算方法。根据定义，表达式 应为

We have the following approximation for
我们有以下近似值 ,

Therefore, the temperature gradient at each location can be calculated by equation (11).
因此，每个位置的温度梯度可以通过方程 （11） 计算。

Estimation of
估计

According to the migration speed of Scottish herring and mackerel and the temperature change, it is and. Therefore, to simplify the model, based on the structure form of function logistic regression, the new formula can be get as follows in the strict approximations of equation (9)
根据苏格兰鲱鱼和鲭鱼的洄游速度和温度变化，它是 and。因此，为了简化模型，基于函数 logistic 回归的结构形式，新公式在方程 （9） 的严格近似中可以得到如下

where is the functional symbol; is the maximum migration speed of fish; is the temperature scaling coefficient. According to the data, the estimated results can be represented as table 3.
其中 是功能符号; 是 fish 的最大迁移速度;是温度标度系数ficient。根据数据，估计结果可以表示为表 3。

Table 3: Parameter estimation results
表 3：参数估计结果

Migration Simulation Algorithm
迁移模拟算法

On the basis of the estimation of and , the process of re-identification of the fishs location can be get as follows
根据 和 的估计，重新识别鱼的位置的过程可以得到如下:

Calaculation Results
计算结果

Based on the process of relocation, the initial distribution of fish and the distribution after 50 years can be obtained in Figure 9.
根据搬迁过程，可以得到 fish 的初始分布和 50 年后的分布，如图 9 所示。

(a) The most likely location of fishes （a） 最有可能的 fishes 位置	(b) The location of fishes (std) （b） 数据的位置 （std）
Figure 9: Prediction location of fishes over the next 50 years 图 9：未来 50 年的预测位置

Model III: Fishing Company Earnings Evaluation Model
模型 III：渔业公司收益评估模型

For the fishing companies without considering the policy factors, whether the fishing activities can bring positive profits to the companies is the decisive factor to decide whether they are going to sea^[5]. Therefore, it is necessary to make an effective evaluation of the cost and benefit on fishing activities in order to determine the fishing strategy.
在不考虑政策因素的情况下，对于渔业公司来说，渔业活动是否能为公司带来积极利益是决定他们是否出海的决定性因素^[5]。因此，有必要对捕捞活动的成本和效益进行有效评估，以确定捕捞策略。

Fishing Company Operating Model
渔业公司运营模式

Assessment of Fishing Costs
捕鱼成本评估

The cost of fishing is mainly divided into two parts: fixed cost and variable cost. Fixed cost refers to the equipment and workers’ fixed wages that must be used in each fishing operation; variable cost refers to the materials consumed with the increase of navigation distance in the fishing process. So, the cost of fishing is expressed as
固定成本主要分为固定成本和可变成本两部分。固定成本是指 每次捕捞作业必须使用的设备和工人的固定工资;可变成本是指在捕捞过程中随着航行距离的增加而消耗的材料。因此，捕鱼成本表示为

where is the fixed cost of fishing; is the variable cost of fishing; is the distance between the fishing site and the port.
其中 是 fishing 的固定成本;是 fishing 的可变成本;是安装站点与端口之间的距离。

Assessment of Fishing Income
渔业收入评估

The income mainly comes from catching fish, in which the number of fish harvested is positively related to the density of fish at the target fishing site as. Considering that there is no refrigeration on the fishing vessels, as the fishing operation time increases, the price of the catching fish will decline along the negative index as . So, the income from fishing is expressed as
收入主要来自捕捞鱼，其中捕捞的鱼的数量与目标捕捞地点的鱼碱密度呈正相关。考虑到捕鱼船上没有冷藏装置，随着捕鱼作业时间的增加，捕鱼的价格将沿着负指数下降。 因此，渔业收入表示为

where is the price decay coefficient.
其中是价格衰减系数。

Assessment of Fishing Profit
渔业风险评估

Due to the defination of profits, the fishing profit of fishing vessels is
由于专业定义，捕鱼船 的捕鱼权是

Results
结果

Parameter Estimation
参数估计

Note that in equation (15) four parameters need to be estimated and the ranges of them are discussed as follows.
请注意，在等式 （15） 中，需要估计四个参数，它们的范围讨论如下。

For the attenuation coefficient , according to the navigation records of fishing boats, the maximum navigation radius of general fishing boats can be obtained, so the maximum navigation range can be expressed as
对于衰减系数，根据捕鱼船的航行记录，可以得到一般捕鱼船的最大航行半径，因此最大航行距离可以表示为:

Besides, according to the equation (15), which is corresponding to the negative index attenuation, we can get , which can be changed into the estimated equation of
此外，根据对应于负指数衰减的方程 （15），我们可以得到 ，它可以改成 :

For variable cost coefficient and fixed cost coefficient , based on the investigation of fishing boats oil consumption and fishery companies financial data conducted by the Food and Agriculture Organization of the United Nations, the values of and are estimated. For the fishing coefficient , assuming initially can satisfy , then
对于可变成本 coef ficient 和 fixed cost coef ficient ，根据联合国粮食及农业组织对船只油耗和渔业公司财务数据的调查，估计了 和 的值。对于初始coefficient ，假设最初 可以满足 ，则

where is average cost per unit fishing; is the average profit per unit fishing and is the average income per unit fishing.
其中 是每单位计算的平均成本; 是每单位销售额 的平均收入，也是每单位销售额的平均收入。

According to the data about the financial status of fishery companies investigated by the Food and Agriculture Organization of the United Nations, the estimated fishing coefficient can be obtained by substituting equation (18). In summary, the estimation results of each parameter are shown in Table 4
根据联合国粮食及农业组织调查的渔业公司财务状况数据，可以通过代入方程 （18） 来获得估计的捕捞系数。综上所述，各参数的估计结果如表 4 所示.

Table 4: Parameter estimation results
表 4：参数估计结果

Migration Simulation Algorithm
迁移模拟算法

Combined with the dynamic migration process of fish in the previous section, real-time profit simulation is added to the simulation process of fish migration, and the simulation process is as follows
结合上一节的 fish 动态迁移过程，在 fish 迁移的模拟过程中增加实时 profit simulation，模拟过程如下:

Calaculation Results
计算结果

With the help of matlab, the above modeling process can be achieved, to get the relationship between the yearly profit of fishing companies and the time
借助matlab，可以实现上述建模过程，得到投资公司年度业绩与时间的关系

(a) Profit trends （a） 保护趋势	(b) Bankruptcy Time Distribution Histogram （b） 破产时间分布直方图
Figure 10: Simulation results of fishing company operations over the next 50 years 图 10：未来 50 年公司运营的模拟结果

It can be seen that, under the condition that the initial distribution of fish is set as uniform distribution, with the simulation of fish gradually enriched, such enrichment effect has brought increasing profits for fishing companies. But as time went on, the fish gradually left the coastal area and swam to the north, which led to a gradual decline in the profits of fishing companies. In the worst case, there will be no fishing companies to continue fishing in 2030; in the best case, fishing activities of fishing companies can last until 2070. The empirical distribution of fishing companies unable to continue fishing time points obtained by bootstrap method is shown in Figure 10. It can be seen that the time point of continuous operation with the greatest probability is 2039.
由此可见，在渔业初始分布设置为均匀分布的条件下，随着渔业模拟的逐渐丰富，这种富集效应为渔业企业带来了越来越多的收益。但随着时间的推移，鱼尾鱼逐渐离开沿海地区向北游去，这导致鱼矿公司的产量逐渐下降。在最坏的情况下， 2030 年将没有销售公司继续销售;在最好的情况下，销售公司的销售活动可以持续到 2070 年。图 10 显示了通过 bootstrap 方法获得的无法继续加密时间点的加密公司的经验分布。由此可见，连续运行概率最大的时间点是 2039 年。

Discussion
讨论

The Management Strategies without Consider of Territorial Sea
不考虑领海的管理策略

For the solution result of question 2, the result marked with migration direction is shown in Figure 11, It can be seen that in the next 50 years, the temperature of ocean water will gradually increase, which led to the migration of fish to the north, and the density of fish in the offshore area will gradually decrease, which led to the gradual profits decline of small fishing companies. Therefore, in order to deal with this situation, the fishing strategy of fishing companies needs to be changed.
对于问题 2 的解题结果，用迁移方向标记的结果如图 11 所示，可以看出，在未来 50 年里，海水温度会逐渐升高，从而导致鱼卵向北迁移，近海区域的鱼卵密度会逐渐降低， 这导致小型金融公司逐渐衰落。因此，为了应对这种情况，需要改变注册公司的注册策略。

Figure 11: Fishing company port transfers
图 11：渔业公司港口转运

Referring to the migration direction of fish, transfer the initial position of the port to Iceland, and repeat the simulation process of section 6.2.2 with other conditions unchanged. The change trend of annual profit of fishing companies over time is shown in Figure 12,
参考 fi sh 的迁移方向，将港口的初始位置转移到冰岛，并在其他条件不变的情况下重复 6.2.2 节的模拟过程。 房地产公司年度投资额随时间的变化趋势如图 12 所示。

Figure 12: Simulation results of fishing company operations over the next 50 years
图 12：未来 50 年公司运营的模拟结果

As the location of the new port is closer to the migration direction of the fish in the future, the profits of fishing companies will gradually increase and eventually stabilize with the migration of the fish. Therefore, for small fishing companies, to effectively improve their survival rate and profit margin, it is necessary to flexibly select the area closer to the distribution of fish as the port of departure.
由于新港口的位置更接近未来鱼类的洄游方向，渔业公司的利润将逐渐增加，并最终随着鱼类的洄游而稳定下来。因此，对于小型渔业公司来说，要想有效提高其存活率和利润率，就必须灵活选择更接近鱼类分布的区域作为出发港。

The Management Strategies with Consider of Territorial Sea
考虑领海的管理策略

Reconsider the strategy in the previous section, because the migration range of fish is beyond the British territorial sea and a large number of fish enter the territorial sea of Iceland, considering the relevant policy and law that Scotland’s fishing companies cannot enter this area for fishing activities.
重新考虑上一节中的策略，因为渔业的迁移范围超出了英国领海，并且大量渔业进入冰岛领海，考虑到苏格兰渔业公司不能进入该地区进行渔业活动的相关政策和法律。

Figure 13: Discretize territorial seas into blocks
图 13：将领海离散为多个块

Therefore, in view of this situation, we will reconsider the fishing situation of fishing companies with randomness after excluding the territorial waters of Iceland and Norway. It can provide on-board refrigeration equipment for fishing boats, which can greatly improve the fresh-keeping situation of fish and thus enhance the fishing income. Using such a strategy to simulate, finally, the annual profit trend of the fishing company over time is shown in Figure 14
因此，鉴于这种情况，我们将重新考虑排除冰岛和挪威领海后随机性的渔业公司的捕捞情况。可为渔船提供船上制冷设备，可大大改善鱼类的保鲜状况，从而提高捕鱼收入。最后，使用这样的策略来模拟渔业公司随时间推移的年度利润趋势，如图 14 所示.

(a) Profit trends （a） 保护趋势	(b) Bankruptcy Time Distribution Histogram （b） 破产时间分布直方图
Figure 14: Simulation results of fishing company operations over the next 50 years 图 14：未来 50 年公司运营的模拟结果

It can be seen that after the installation of refrigeration equipment, the declining trend of the profits of fishing companies has been well contained. In the worst case, in 2036, the fishing company is unable to continue fishing; in the best case, the fishing activities of the fishing company can last until 2070. See figure 14 for the empirical distribution of fishing companies that can adapt to the migration of fish by using bootstrap method. Most fishing companies can stick to the end, which shows that our improvement strategy is effective.
由此可见，安装制冷设备后，渔业企业业绩下降的趋势得到了很好的遏制。在最坏的情况下，到 2036 年，金融公司将无法继续金融活动;在最好的情况下，金融公司的企业经营活动可以持续到 2070 年。参见图14，了解可以使用 bootstrap 方法适应 FISH 迁移的 FISH公司的经验分布。大多数美容公司都能坚持到底，这表明我们的改进策略是有效的。

Test the Model
测试模型

Sensitivity Analysis
敏感性分析

In section 6.3.2, two exogenous factors are introduced to estimate the parameters of the profit evaluation model of fishing companies: social profit rate and average navigation dis- tance. Therefore, the relationship between the final survival rate of fishing companies and these two factors is approximated by first-order difference
在第 6.3.2 节中，引入了两个外生因素来估计渔业公司利润评估模型的参数：社会利润率和平均航行距离。因此，渔业公司的最终存活率与这两个因素之间的关系用一阶差近似:

Therefore, the calculation results are shown in Figure 15
因此，计算结果如图 15 所示.

Figure 15: Sensitivity analysis of and
图 15：和 的敏感性分析

It is indicated that the ultimate survival rate of fishing companies increased with the social profit margin, which reflects the feedback effect of social development on fishing companies; correspondingly, the ultimate survival rate of fishing companies is also increase with the average navigation distance, which shows the importance of long-distance navigation ability in promoting the profitability of fishing companies. The trend of the model obtained by sensitivity test is consistent with the actual situation, which also proves the rationality and robustness of the profit evaluation model of fishing companies.
结果表明，渔业企业的最终生存率随着社会效益的扩大而提高，这反响了社会发展对渔业企业的反馈效应;相应地，渔业企业的最终生存率也随着平均航行距离的增加而增加，这表明了远程导航能力对促进渔业生产的重要性 金融公司的能力。敏感性检验得到的模型趋势与实际情况相符，也证明了渔业企业产权评价模型的合理性和稳健性。

Robustness Analysis
稳健性分析

For model 2, it is significant to consider whether the final migration states can be stable under different initial distribution samples. Therefore, the migration of fish based on model 2 under different initial distribution conditions is simulated, and the final distribution for -th is expressed as . Then the degree of fit between any two final distributions is calculated by
对于模型 2，考虑最终迁移状态在不同初始分布样本下是否能保持稳定具有重要意义。因此，模拟了不同初始分布条件下基于模型 2 的鱼类迁移，-th 的最终 分布表示为 。然后，任意两个最终分布之间的拟合度计算公式为:

And is combined into pair series to draw a Q-Q plot as
并组合成对级数以绘制 Q-Q 图，如

Figure 16: QQ Plot of RMSE v.s. Standard Normal
图16：RMSE与标准正态的 QQ图

It can be seen that the scattered points of the figure are distributed on the straight line, which shows that the overall fit situation is stable. It means that the similar final fish distribution can be obtained on the premise of different initial fish distribution, which verifies the decisive role of temperature change on fish migration and reflects the stability of the model.
可以看出，图的散点分布在直线上，说明整体拟合情况是稳定的。这意味着在不同初始数据分布的前提下可以获得相似的最终数据分布，验证了温度变化对数据迁移的决定性作用，反映了模型的稳定性。

Conclusion
结论

Summary of Results
结果摘要

Result of Problem 1
问题1 的结果

According to the calculation results from Model I and II, the distribution of two kinds of fishes after 50 years can be determined as follows
根据模型 I 和 II 的计算结果，可以确定两种鱼类在 50 年后的分布情况如下:

Figure 17: The most likely location of fishes over the next 50 years
图 17：未来 50 年最有可能的 fishe 位置

Result of Problem 2
问题 2 的结果

According to the prediction for the migration of fish in the next 50 years, take use of the Model III to evaluate the management of the companies to achieve the histogram:
根据对未来 50 年渔业迁移的预测，利用模型 III 来评估公司的管理，以实现直方图：

(a) The most likely location of fishes （a） 最有可能的 fishes 位置	(b) The location of fishes (std) （b） 数据的位置 （std）
Figure 18: Prediction location of fishes over the next 50 years 图 18：未来 50 年的预测位置

(a) Profit trends （a） 保护趋势	(b) Bankruptcy Time Distribution Histogram （b） 破产时间分布直方图
Figure 19: Simulation results of fishing company operations over the next 50 years 图 19：未来 50 年公司运营的模拟结果

Result of Problem 3
问题 3 的结果

Taking account that the fish has trend to move north, the strategy given is changing the fishing port to Iceland. The later management trend curve can be get using Model III:
考虑到渔民有向北移动的趋势，给出的策略是将渔民港改为冰岛。后面的管理趋势曲线可以使用模型 III 获得：

Figure 20: Simulation results of fishing company operations over the next 50 years
图 20：未来 50 年公司运营的模拟结果

It can be concluded that after changing the initial seaport, the management situation can be greatly improved, and by 2070, 100% of fishing operations can be maintained.
可以得出结论，在改变初始海港后，管理状况可以大大改善，到 2070 年，可以保持 100% 的渔业运营。

Result of Problem 4
问题 4 的结果

Taking the related policies and laws about territorial water, fishing companies cannot change their departure ports to Iceland. The corresponding strategy given by us is the improvement of the ships technology, which can improve the profit at the price of higher cost. For this strategy, the profit trend curve can be determined with the help of Model III. It can be seen in Figure 21, if the departure ports are changed, 62.68% companies can keep their fishing business 50 years later thanks to the better business situation.
根据领海相关政策和法律，捕捞公司不能将其出发港更改为冰岛。我们给出的相应策略是船舶技术的改进，可以以更高的成本为代价提高性能。对于此策略，可以在 Model III 的帮助下确定趋势曲线。从图 21 中可以看出，如果更改出发港，62.68% 的公司可以保留 50 年后的业务，这要归功于更好的业务状况。

(a) Profit trends （a） 保护趋势	(b) Bankruptcy Time Distribution Histogram （b） 破产时间分布直方图
Figure 21: Simulation results of fishing company operations over the next 50 years 图 21：未来 50 年公司运营的模拟结果

Strengths
强度s

The sea temperature prediction model based on time series is scientific and reasonable, and can pass various statistical tests. The predictions obtained have a reliable statistical description;
基于时间序列的海水温度预测模型科学合理，能够通过各种统计检验。获得的预测具有可靠的统计描述;

The sensitivity analysis of the model demonstrates the effectiveness of the model under different parameter combinations and prove the robustness of the model;
模型的敏感性分析证明了模型在不同参数组合下的有效性，证明了模型的鲁棒性;

The business strategies are evaluated scientifically based on large sample data, which is convenient for the managers of the companies to make rational decisions according to the actual situation.
根据大样本数据对经营策略进行科学评估，便于公司管理者根据实际情况做出理性决策。

Possible Improvements
可能的改进

The analysis of fish migration can be more accurate if we have more complete data;
如果我们有更完整的数据，对鱼苗迁移的分析会更准确;

Some approximate analysis methods are applied to model the management of fishing companies, which may lead to the situation contrary to the actual in extreme cases.
采用一些近似分析方法对金融公司的管理进行建模，在极端情况下可能会导致与实际相反的情况。

References
引用

Corten, A. (2001). Northern distribution of North Sea herring as a response to high water temperatures and/or low food abundance. FisheriesResearch, 50(1-2), 189-204.
Corten， A. （2001 年）。北海鲱鱼的北部分布是对高水温和/或低食物丰度的回应。渔业研究， 50（1-2）， 189-204.

Jansen, T., Campbell, A., Kelly, C., Hatun, H., & Payne, M. R. (2012). Migration and fisheries of North East Atlantic mackerel (Scomber scombrus) in autumn and winter. PLoSOne, 7(12).
詹森，T.，坎贝尔，A.，凯利，C.，哈顿，H.和佩恩，M. R.（2012）。东北大西洋鲭鱼 （Scomber scombrus） 在秋冬季的迁徙和捕鱼。公共科学图书馆， 7（12）。

Nøttestad, L., Misund, O. A., Melle, W., Hoddevik Ulvestad, B. K., & Orvik, K. A. (2007). Herring at the Arctic front: influence of temperature and prey on their spatiotemporal distribution and migration. MarineEcology, 28, 123-133.
Nøttestad， L.， Misund， O. A.， Melle， W.， Hoddevik Ulvestad， B. K.， & Orvik， K. A. （2007）.北极前沿的鲱鱼：在温度变化中，捕食它们的时空分布和迁徙。 海洋生态学， 28， 123-133。

Misund, O. A., Vilhjálmsson, H., Jákupsstovu, S. H. Í., Røttingen, I., Belikov, S., Asthorsson, O., ... & Sveinbjórnsson, S. (1998). Distribution, migration and abundance of Norwegian spring spawning herring in relation to the temperature and zooplankton biomass in the Norwegian Sea as recorded by coordinated surveys in spring and summer 1996. Sarsia, 83(2), 117-127.
Misund， O. A.， Vilhjálmsson， H.， Jákupsstovu， S. H. Í.， Røttingen， I.， Belikov， S.， Asthorsson， O.， ...& Sveinbjórnsson， S. （1998）。1996 年春季和夏季的协调调查记录的挪威春季产卵鲱鱼的分布、迁移和丰度与挪威海温度和浮游动物生物量的关系。萨西亚，83（2），117-127。

Peer, A. C., & Miller, T. J. (2014). Climate change, migration phenology, and fisheries management interact with unanticipated consequences. North American Journal of Fisheries Management, 34(1), 94-110.
Peer， A. C.， & Miller， T. J. （2014年）。气候变化、迁徙物候学和渔业管理会带来意想不到的后果。 北美渔业管理杂志， 34（1）， 94-110.

Watch for fish migration
观察鱼类洄游

However, the fifth evaluation report concerning global warming from Intergovernmental Panel on Climate Change indicates that, since 1870, the observational data shows the continuing and accelerating rises of global ocean temperature, which rises more than one degree. Herring and mackerel have to leave current habitats due to the continued rises of ocean temperature to seek better habitat northly, which lay a burden on the Scottish fisherman. Because the distant fish increase the fishing expenses, while the amount of capture will decrease inevitably.
然而，政府间气候变化专门委员会关于全球变暖的第五份评估报告表明，自 1870 年以来，观测数据显示全球海洋温度持续加速上升，上升超过 1 度。由于海洋温度的持续上升，鲱鱼和鲭鱼不得不离开目前的栖息地，向北寻找更好的栖息地，这给苏格兰渔民带来了负担。因为远处的鱼增加了捕捞费用，同时捕获量必然会减少。

Based on our research, under the current technical level and business strategy, facing the possible large-scale migration of fish in the next 50 years, the survival probability of small fishing companies, which is the way most people to make a living, is worrying. Following figure shows our estimated survival rate distribution.
根据我们的研究，在目前的技术水平和经营策略下，面对未来 50 年可能出现的鱼类大规模迁徙，作为大多数人谋生方式的小型渔业公司的生存概率令人担忧。下图显示了我们估计的存活率分布。

After 10000 times simulation of fish migration and the corresponding income of small fishing companies, the worst case is that in 2030, the probability of fishing companies going bankrupt due to fish.migration is 0.02%; as for the most likely case, the probability is 8.25% in 2039. The best situation is that it has not been bankrupt for 50 years, with a probability of 5.27%.
经过 10000 次鱼类洄游和小型渔业公司相应的收入模拟后，最坏的情况是 2030 年，渔业公司因 fish.migration 而破产的概率为 0.02%;至于最有可能的情况，2039 年的概率为 8.25%。最好的情况是它 50 年没有破产，概率为 5.27%。

Finally, our research of the relationship between the social return rate and the maximum navigation radius of fishing boats indicates that the increase of the social return rate and the maximum navigation radius can both significantly reduce the probability of bankruptcy. Based on this, our advice is to actively participate in the industry with better overall development and improve the technology of fishing boat, expand the navigation radius, and always achieve low-carbon travel, energy conservation and environmental protection, contribute to slowing down global warming, so that you can enjoy the fun brought from fishing in a longer period of time while maintaining due benefits.
最后，我们对渔船社会回报率与最大航行半径之间关系的研究表明，社会回报率和最大航行半径的增加都能显著降低破产的概率。基于此，我们的建议是积极参与整体发展较好的行业，提高渔船的技术，扩大航行半径，始终实现低碳出行、节能环保，为减缓全球变暖做出贡献，让您在保持应有利益的同时，在更长的时间内享受钓鱼带来的乐趣。

MCM
MCM 公司

17^th Feb 2020
17th二月 2020

Appendices