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2020 Mathematical Contest in Modeling (MCM) Summary Sheet

Title:Analysis of Fish Migration in the North Atlantic

Summary

The thesis analyzes the impact of global ocean temperature rise on the habitats of two species of fish in the North Atlantic, we mainly researched on the migration location of two kinds of fish in the next 50 years, the situation beyond the fishing range of the small-scale fishery company and the operation mode and strategy of the company, etc. Our team established the corresponding mathematical models and use software tools such as Excel and MATLAB to solve the problem.

For the problem 1, firstly, collect and extract data, and then use interpolation method to add the data. Based on the sea surface temperature and fishing data in the past 20 years, our team formulated the model of ‘fish group spatial position representation’. Then we analysed the spatial distribution of fish groups, and formulated the ‘fish group spatial autocorrelation’ model . Finally, after our comprehensive consideration , we formulated the ‘fish swarm prediction’ model. From the models we can describe the specific location of the two fishes in the next 50 years including the specific longitude and latitude in detail.

We adopted particle swarm optimization algorithm to solve the problem 2 .The basic core of particle swarm optimization algorithm is to make use of the information sharing of individuals in the group, so that the movement of the whole group in the problem solving space will produce the evolution process from disorder to order, so as to obtain the optimal solution of the problem. This algorithm is a kind of probabilistic global optimization algorithm, which has more chances to solve the global optimization. The algorithm has uncertainty, which embodies the biological mechanism of natural organisms, and is superior to the deterministic algorithm in solving some specific problems. It is

appropriate to find out the best as well as the worst situation in the problem 2.

For the problem 3, as far as we are considered, the company needs to change its business strategy. However, different business strategy correspond to different economic benefits, hence，it is necessary to evaluate the economic income of different strategies, in another words, to establish an income model to evaluate the income and cost situation of different strategies, pursue the maximization of revenue by comparing the cost and revenue models of different strategies. Our team adopted the Stackelberg model to solve this problem.

we should consider the impact of cross territorial fisheries on the company's income, as companies are unable to fish across territorial waters. If fishes cross domestic territorial waters with rising temperatures，the actual number of fishes available to companies is lower than predicted. After reevaluated our earnings by considering the company's geographic location and other information , we modified the model and verified the rationality of it.

Finally, our team evaluated the advantages and disadvantages of the model, and proposed the further improvement direction and application in other aspects.

Keywords:(1) Resource density center(2)Particle swarm optimization algorithm

(3)Income assessment(4)Game theory .

content

Overview (3)

Basic assumption (4)

Analysis of problem 1 (5)

Models of problem1 (6)

Analysis of problem 2 (8)

Models of problem2 (8)

Location update formula: (9)

Analysis of problem 3 (10)

Models of problem 3 (10)

Analysis of problem 4 (11)

Model of question 4 (12)

Suggestions for fishermen (14)

Appendix (15)

Overview

As is known to all, movements of the habitats for ocean-dwelling species greatly affects the economy of small fishery. It is urgent to predict the location of migration of fishes with the change of water temperature and help small-scale fishery companies to design the corresponding operation mode. After

mathematically analyzing these problems , our modeling group would like to present our conclusions, strategies, and recommendations to the company.

Basic assumption

1.The extent of fishing in the fishery can cover the partitioned area.

2.The data collected are reliable and consistent with the facts.

3.The herring and mackerel migrate independently of topography, water quality, and salinity.

4.ignore the economic impact of other kinds of fish on the fishery.

Symbols Definition

n Number of fish capture stations involved in the analysis

fish resource density of station I

X

i

X

fish resource density of station J

j

W

matrix value of weight in n * n space

ij

x average resource density of all captured fish in stations

white noise sequence

a

t

p non negative integer

q non negative integer

N the total number of particles in this grou

Rand() Rand () is used to generate random numbers between (0,1)

learning factors.

C

1

learning factors.

C

2

Pbest the local optimal positions of particle swarm, respectively

Gbest the global optimal positions of particle swarm, respectively the potential cost

Q

1

α the cost elasticity coefficient

τ the unit time value

R’ p rofit of the company

C total operating cost of the company

Uj the utility of the quantity (or quality) level of the fishery ecosystem in the sea area of country j

Gj the quantity (or quality) function of the fishery ecosystem in the sea area of country j

Ej the quantity (or quality) of the catch in the sea area of country j

t the diffusion function of the quantity (or quality) of the catch

T(Ei) the quantity (or quality) of the catch in the sea area of country i

Ti the diffusion function of the quantity (or quality) of catches in the sea area of country I

Tj the diffusion function of the quantity (or quality) of catches in the sea area of country J.

D the damage cost function of fishery ecosystem

the initial quantity of catches in country 1

S1

1 the quantity of catches reduced in country 1 to protect fishery

S

r

ecosystem

1) the quantity of catches increased in country 2 after the quantity

T(S

r

of catches decreased in country 1

D1 the cost of damage suffered by country 1

1

C1 the cost of country 1 which is to reduce the amount of catch S

r

S the quantity of catches

the initial number of domestic catches

S2

S2

the number of catches decreased for the protection of fishery r

ecosystem

Analysis of problem 1

According to the requirements of the topic，first of all, we should determine an index to describe the

spatial distribution information of the fish, and we planned to use the resource center of the fishery to describe it. Then we analyzed the spatial distribution and characteristics of fishes, which can be analyzed by spatial autocorrelation method.

Finally, we established a time prediction model to predict the most likely spatial position of the fish affected by the change of water. The resource center of a fishery is an important parameter to formulate the position of the fishery, which is usually used to describe the change of spatial position of fishery. It is the basis of fishery resources and fish ecology. Also，it can be used to judge the migration route of fish, determine the central fishery and describe the distribution characteristics and change rules of fish stocks in fishery, and so on. Fish stocks have migratory characteristics，which are essentially different from fixed objects, and the resourse center of fishery is related to environmental elements and biological characteristics. It is of great ecological significance to study the resource center of fishery.

Models of problem1

Spatial autocorrelation is an important index to test whether the attribute value of an element is significantly related to the attribute value of its adjacent spatial points. The content of statistics includes the spatial location and attribute of spatial variables, that is, the spatial location relationship between each variable and its adjacent statistical analysis variables and the attribute value characteristics. Positive correlation indicates that the attribute value change of a unit is significantly related to its adjacent units of space . However, the negative correlation is the opposite. Spatial autocorrelation can be divided into global spatial autocorrelation and local spatial autocorrelation.

Global Spatial Autocorrelation is as follows:

(2)

In fact, it is almost impossible to choose the best spatial weight matrix, it select mainly by evaluate the sensitivity of results to weight selection. The nearest neighbor number method can be used to calculate the spatial weight matrix.

Local spatial autocorrelation is usually expressed by Lisa (local indicators of spatial association) index. Lisa index can reveal the spatial autocorrelation properties of local and even each spatial unit, which is essentially decomposed to each regional unit. For a spatial unit I, its calculation formula is as follows:

(3)

ARMA spatiotemporal series model is an important method to study time series, which is composed of autoregressive model (AR model) and moving average model (MA model)

The form of the model is:

(4)

Using backward operator B, the model above can be written as:

(5)

At this time, we seek RT's expectation to get the equation (5):

(6)

It is as like as two peas in our AR model. So it has the same characteristic equation:

(7)

Analysis of problem 2

This topic requires us to analyse the rate of temperature change of sea water and establish models to predict the best, the worst, and the most likely time for fishes to exceed the range of fishing by small fisheries companies. That is, on the basis of considering the fishing range of small fisheries companies, the optimal solution and the worst solution (corresponding to the best case and the worst case beyond the fishing range) are considered for the models of problem 1, and find out when the two situations above are most likely to occur.

We planned to use Particle swarm optimization algorithm. The basic core of particle swarm optimization algorithm is to make use of the sharing of information by individuals in the group, Thus, make the movement of the whole group generate the evolution process from disorder to order in the problem solving space, so as to obtain the optimal solution of the problem. This algorithm is a kind of probabilistic global optimization algorithm, which can have more opportunities to solve the global optimal solution. The algorithm has uncertainty, which reflects the biological mechanism of natural organisms and is superior to deterministic algorithm in solving some specific problems. It is suitable for solving the worst and best case of the model in problem 1.

Models of problem2

PS0 is initiated into a group of random particles (random solutions). Then find the best solution by iteration. In each iteration, particles update themselves by tracking these two extremums (pbest, gbest). After finding these two optimal values, the particle updates its speed and position by using the following formula.

Location update formula:

(8)

Speed update formula:

(9)

In the update formula, I = 1,2…N, n is the total number of particles in this group. Rand () is used to generate a random number between (0,1). C1 and C2 are learning factors.

Pbest and gbest represent the local and global optimal positions of particle swarm, respectively. When C1=0, the particle has no cognitive ability and becomes a social model (social-only):

It is called the global PS0 algorithm. Particles have the ability to expand and search space and have fast convergence rate, but due to the lack of local search, it is easier to fall into local optimal position for complex problems than standard PSO.

When C2=0, there is no social information between particles, and the model becomes only cognitive (cognition-only) model:

(10)

It is called the local PS0 algorithm. Because there is no exchange of information between individuals, the whole group is equivalent to multiple particles to carry out blind random search, the convergence rate is slow, therefore, the possibility of obtaining the optimal solution is small.

Algorithm flow:

1) Initialize a group of particles (the population size is N), including their random position and speed.

2)Evaluate the fitness of each particle.

3)Compare each particle’s fitness value with the best position(pbest) it has passed by , and if so, it would be regarded as the current best position pbest.

4)Compare each particle’s fitness value with the best position(gbest) it has passed by , and if so, it would be regarded as the current best position pbest.

5)Adjust the particle speed and position according to formula (2), (3);

6)If the end condition is not met, go to step 2).

According to the specific problem, the iteration termination condition is generally selected as the

maximum number of iterations GK or (and) the current optimal position of particle swarm optimized meets the predetermined minimum adaptive threshold.

Analysis of problem 3

This topic requires that according to the prediction results of the model, it is necessary to analyze whether these small fishing companies should change their way of doing business. That is, analyze the result of the second problem. For example, if the worst-case solution occurs in a very short period of time, the business model needs to be changed.

As far as we are considered, the company needs to change its business strategy, however, different business strategy correspond to different economic benefits, hence ，it is necessary to evaluate the economic income of different strategies, in another words, to establish an income model to evaluate the income and cost situation of different strategies, pursue the maximization of revenue by comparing the cost and revenue models of different strategies. Our team intended to adopt the Stackelberg model.

Models of problem 3

Relevant literature points out that the company's operating cost includes fixed cost and variable cost, and its total cost is shown as follows:

()0C cN c vx N λ==+ (11)

Combined with the actual situation, we can get:

12k a k e N Q A E +=+ (12)

The income obtained after the company's investment is modeled in different ways:

According to the change of income, we get a more reasonable model, as shown in the following formula:

()1exp d Q Q P t kt ατ=-++???? (13)

Q 1represents the potential cost, α is the cost elasticity coefficient, represents the percentage change in a certain period of time; τ represents the unit time value.

Assume that the total income only comes from the income funds of two kinds of fish stocks, and at the same time, it needs to subtract its own adaptability cost for the early stage. The profits of the companies are as follows:

'R PQ aN η=-

Then solve it to maximize the benefits:

'L: max 0 ..00.73R PQ aN

a A s t e E

N Q η=-≤≤??≤≤??>?(14)

Finally, evaluate the strategies according to the maximum benefit of different strategies,

Analysis of problem 4

In this topic, we should consider the impact of cross territorial fisheries on the company's income, as companies are unable to fish across territorial waters. If fishes cross domestic territorial waters with rising temperatures ，the actual number of fishes available to companies is lower than predicted.

Also, we need to reevaluate our earnings by considering the company's geographic location and other information.

Marine fishery resources are a kind of international public resources or regional public resources. The political boundaries delineated by human society do not make any sense to fish. The fishery

resources of coastal countries are directly linked to each other through seawater media, which we call straddling fisheries resources. In international fishery industries, the use and activities of many

fisheries resources (such as migratory fish) occur within the boundaries of the jurisdiction of more than two regulatory bodies or in areas not subject the jurisdiction of any regulatory body, such as the use of fisheries resources on the high seas.

This paper will analyze the game results of cross-border marine fishery under cooperative game and noncooperative game, and analyze the corresponding economic benefits. The measurement of costs and benefits of straddling fisheries may include certain special considerations, in order to develop fishery cooperation among countries, leading to special costs and benefits.

Model of question 4

There is a diffusion function of catches in the management of cross-border marine fisheries. That is to say, the quantity and quality of fishery ecosystem in the sea area of a country (country j) are not only affected by its own catches, but also by the increase of catches by the country (I) reflected by the catch diffusion function t.

We use utility function to express the quantity (or quality) of fishery ecosystem

(15)

In the formula, U j represents the utility of the quantity (or quality) level of the fishery ecosystem in the sea area of country j; G j represents the quantity (or quality) function of the fishery ecosystem in the sea area of country j; E j represents the quantity (or quality) of the catch in the sea area of country j; t represents the diffusion function of the quantity (or quality) of the catch; T(E i) represents the quantity (or quality) of the catch in the sea area of country I, and E i increases, thus reducing the quantity (or quality) of the catch in the sea area of country J.

The migration of fishes can be one-way or two-way, which leads to the increase and decrease of catches in different sea areas. In the formula（1）, it represents a one-way migration, that is, the migration of catch from country I to country J. If it is two-way, the quantity (or quality) of the fishery ecosystem of the two countries is shown as follows with the utility function:

(16)

In this function, T i represents the diffusion function of the quantity (or quality) of catches in the sea area of country I; Tj represents the diffusion function of the quantity (or quality) of catches in the sea area of country J.

Assume that each country minimises its costs, and the cost should be equal to the sum of the cost of ecosystem damage to fisheries and the cost of catch reduction. We introduce the damage cost function of fishery ecosystem D, d = D (s), S represents the quantity of catches; D′>0 shows that the damage cost of fishery ecosystem increases with the increase of catches. D1>0 indicates that the damage cost of fishery ecosystem increase progressively. The initial catch levels of S01 and S02 are established in both countries. Minimise country1’s total cost and represent it as the following target function:

In the function, S01 represents the initial quantity of catches in country 1; S r1represents the quantity of catches reduced in country 1 to protect fishery ecosystem; T (S r1) represents the quantity of catches increased in country 2 after the quantity of catches decreased in country 1; D1 represents the cost of damage suffered by country 1 due to the degradation of fishery ecosystem, which is affected by the quantity of catches in domestic fishery ecosystem.C1 is the cost of country 1 which is to reduce the amount of catch S r1.For the optimal catch reduction S1r, the optimal condition is as follows:

(17)

As the increase of catches in fishery ecosystem will lead to the degradation of fishery ecosystem, and the reduction of catches will reduce the degradation of fishery ecosystem, there are:

(18)

By the same token, minimise country 1’s total cost and represen t it as the following target function:

It can be seen that the number of catches in the fishery ecosystem of country 2 is affected by the migration of catches from country 1 to country 2, which is equal to the initial number of domestic catches S02minus the number of catches decreased for the protection of fishery ecosystem S2r, then

plus the number of catches migrated from country 1. For the optimal catch reduction S2r, the optimal conditions are as follows:

(19)

The optimization conditions (2) and (3) mean that Marginal cost of reduced catches which is to protect fishery ecosystems (on the right side of the equation) is equal to the marginal damage which is to avoid the degradation of the fishery ecosystem (on the left side of the equation).

Suggestions for fishermen

As we all know, with the increasing greenhouse effect, the ocean temperature is continuously rising. First of all, fishermen should not despise this matter, which may bring you heavy losses. Just as we move to a warm place when we are cold, marine organisms will choose to move because of the change of living environment temperature, and they are more sensitive to the change of environment temperature. We all know that when we fish, we have to choose the place where fish groups gather to catch more fishes. Experienced fishermen often find the fish group through their own experience. The change of water temperature makes it difficult to find the right place. It is reported by Taiwan's https://www.360docs.net/doc/ee142859.html,. that tropical fish appear thousands of kilometers away in New Zealand due to the warming of the ocean, and more and more "wandering species" of fish are emerging in the waters of New Zealand. It can be seen that experience can no longer accurately lead us to find where the most fish are, and there are fewer and fewer fish caught. According to the report of the Department of marine fishery statistics of Scotland, in 2015, the catch value and weight of fishing vessels registered in Scotland decreased by 15% and 8% respectively compared with the previous year. Mackerel is still the most valuable fish in Scotland, accounting for 30% of the total fishing value. In 2015, the catch value of mackerel decreased by 33% and 17%, respectively.

Based on such a heavy blow to the fishermen caused by the change of ocean temperature, our team has established several mathematical models on the migration of mackerel and herring by studying the

data of previous years. These mathematical models can provide scientific guidance and make more accurate judgments for the fishermen.

First of all, we predict the change of ocean temperature in the next 50 years based on the change trend of marine temperature. We find that the rise trend of temperature will continue. In other words, mackerel and herring will be more affected.According to the model, we can predict the most likely areas of mackerel and herring in the next 50 years. Therefore, our suggestion is that the fishermen who mainly catch mackerel and herring should change their main fishing areas in the next 50 years according to the model guidance to ensure their income.

Secondly, we consider the impact of the cross territorial sea issue on the company's earnings, because the company cannot fish across the territorial sea. If the temperature changes enough to make the fish group cross the territorial sea of the country, the earnings of the fishery company will decline. This needs a comprehensive consideration of the company's geographic location and other relevant information, The political boundaries delineated by human society do not make any sense to fish. The fishery resources of coastal countries are directly linked to each other through seawater media, which we call straddling fisheries resources. In international fishery industries, the use and activities of many fisheries resources (such as migratory fish) occur within the boundaries of the jurisdiction of more than two regulatory bodies or in areas not subject the jurisdiction of any regulatory body, such as the use of fisheries resources on the high seas.

In general, fishermen should follow the changes and use scientific methods to ensure their steady income and the harmony of environment.

Appendix

[1]Mingxiang Niu, Spatiotemporal Dynamics of Anchovy Population and its Response to Environmental Factors in the Yellow Sea Based on Ocean RS and GIS [D],2012,Shang Dong Agricultural University.

[2]Shuang Cheng,Xinjun Cheng , Study on Changes of Fish Community Structure Grade in North Sea Fishery of Northeast Atlantic[R],2019,Shang Hai Ocean University.

[3]Huayong Xiao, Guide to Mathematical Modeling Competition for College

Students[M],Beijing,Publishing House of Electronics Industry,2015.

[4] https://www.360docs.net/doc/ee142859.html,/news/201906/1817608.html,2020/2/14

[5]https://https://www.360docs.net/doc/ee142859.html,/view/73ddbeac88eb172ded630b1c59eef8c75ebf9532.html?from=search, 2020/2/15

[6]Audun H. Rikardsen，J. Brian Dempson,Dietary Life-Support:The Food and Feeding of Atlantic Salmon at Sea[J],2010.

[7] https://https://www.360docs.net/doc/ee142859.html,/view/7924b2a20b4c2e3f572763ca?pcf=2,2020/2/15

[8]Johnson,Andrew,Chillon Kelly,Mackerel Migration and Fisheries in the Northeast Atlantic in Autumn ang Winter[J],2012.

[9]G.Huse,BR MacKenzie,H.Arrizabalaga,Comparative Ecology of the Wide Distribution of Pelagic Fish in the North Atlantic[J],2014,French Institute of Oceanography.

[10]Teunis Jansen,Henrik Gislasion,Temperature Affects the Timing of Spawning and Migration of North Sea Mackerel[J],2011.