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2024年美赛35篇特等奖O奖论文-A-2405424.pdf
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大学生,数学建模,美国大学生数学建模竞赛,MCM/ICM,2024年美赛特等奖O奖论文
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Problem Chosen
A
2024
MCM/ICM
Summary Sheet
Team Control Number
2405424
Lamprey: a Vivid Example of the Doctrine of the Mean
The survival mechanism of the lamprey, one of the oldest organisms on earth, has
always been a topic of interest for researchers. When it comes to lampreys and resource
availability, the phenomenon that their populations can change their sex ratio has
become an inescapable topic. Studies have found that resource availability affects the
sex ratio of lamprey populations.
In order to study the ability of species to change their sex ratios according to resource
availability, we adopted the Lotka-Volterra predator-prey model and the Nicholson-
Bailey host-parasite model as modeling bases, and then constructed the Ideal Reference
Model, the Constant Sex Ratio Model, and the Models of Changing Sex Ratios, and
successfully solved the problem.
For Task 1,we first found that the total amount of resources in the ecosystem de-
clined with the introduction of lamprey or other parasites, while the number of species
surviving was highest in the model with lamprey. Therefore, we conclude that lamprey
with altered sex ratios are less stressful to other organisms than ordinary parasites,
which is conducive to increasing the diversity of the ecosystem.
For Task 2, we artificially worsened the environmental resources, and plotted the
change curves of the number of lamprey in Constant Sex Ratio Model and Models
of Changing Sex Ratios, respectively. Obviously, the population size of conventional
parasitic species is severely affected by the environment, while lamprey, which is able
to change its sex ratio, is very stable, which is the greatest population advantage
of lamprey.However, win some, lose some.With higher resilience to environmental
risk comes lower population size, which is the disadvantages to the population of
lampreys.
For Task 3, we collected a large amount of data and analyzed the final ecosystem
stability of the three models through the entropy weighting method, and concluded that
the introduction of parasites led to a decrease in ecosystem stability, but the property
of lamprey that can change the sex ratio significantly mitigated this phenomenon.
For Task 4,we then integrated these three models to explore the effects of introduc-
ing lamprey on other conventional parasites in the ecosystem. Under deteriorated
conditions, we found that the parasite was less able to resist the external environ-
ment in ecosystems lacking lamprey. However, after the introduction of lamprey, their
population size showed stable periodic fluctuations. This amplified the negative feed-
back regulation of the ecosystem, and also led to a significant strengthening of the
resistance of the original parasite to the adverse environment.
After that, in order to test the model, we firstly discussed the rationality of the
experimental results from various aspects. Then we analyzed the sensitivity of the model
from two perspectives: the disturbance of human activities and the change of the initial
value of the population, and found that the model is more resistant to the uncertainty
factors, and it is applicable to different initial values of the population, which
just shows the reliability and universality of the model. Finally, we analyzed the
advantages and disadvantages of the model and proposed a corresponding optimization
scheme.
Keywords:Changing Sex Ratios,Ideal Reference Model,Constant Sex Ratio Model
, Model of Changing Sex Ratios, Entropy Weighting Method.
![](https://csdnimg.cn/release/download_crawler_static/89271463/bg2.jpg)
Team # 2405424 Page 2 of 25
Contents
1 Introduction 3
1.1 Research Background[1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Model Preparation 5
2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Establishment of Models 6
3.1 Ideal Reference Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2
Constant Sex Ratio Model(Based on an ecosystem containing conven-
tional parasitic species with a constant sex ratio of 2 : 1) . . . . . . . . . . 8
3.3
Models of Changing Sex Ratios(Based on an ecosystem containing normal
lamprey) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.4 Evaluation Indicators and Stability Analysis of Ecosystems . . . . . . . . 14
4 Problem Solutions 14
4.1 Ecosystem Impacts of Changing Sex Ratios in Lamprey . . . . . . . . . . 15
4.2 Advantages and Disadvantages of Changing Sex Ratios for Lamprey . . 17
4.3 Effects of Changing Sex Ratios on Ecosystem Stability in Lamprey . . . . 18
4.4 Advantages of Altering Sex Ratio of Lamprey to Other Species . . . . . . 19
5 Rationalization Analysis and Sensitivity Testing of the Model 22
5.1 Rationalization Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2 Sensitivity Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6 Strengths and Weaknesses 24
6.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.2 Weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7 Conclusion 24
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Team # 2405424 Page 3 of 25
1 Introduction
1.1 Research Background[1]
1.The Living Habits of Lampreys.
The entire life cycle of the lamprey, from fertilized egg to adult, typically takes 5-8
years. Fertilized eggs hatch into worm-like larvae that reside in small burrows on the
bottom of sandy streams. The larvae feed on detritus and small plant algae, and after
three to six years become parasitic adults. Adults migrate to lakes and seas where they
feed on fish.As a typical migratory organism, the lamprey lives part of the time in the
sea. They enter large rivers in the fall, overwinter in the lower reaches of the rivers, and
return to the upper reaches to spawn when the water temperature reaches about 15
degrees Celsius in May to June of the following year. Both parents die after spawning.
Figure 1: Lamprey[2]
Parasitism is the main way of survival
for the lamprey. Adult lampreys attach
themselves to a host fish through their
toothed sucker mouths and feed on its
blood. This parasitic behavior not only
causes damage to the health of the host
fish and reduces its ability to survive and
reproduce, but also affects the number
and structure of fish populations, thus af-
fecting the balance of the aquatic ecosys-
tem.
2.Population Dynamics of Lampreys.
Since the lamprey is regarded as a delicacy in some cultures, heavy human fishing in
recent years has led to a decline in its population. While considering the predator, prey
and parasitic relationships of lampreys, changes in sex ratio may also have a significant
impact on their ecological roles. For example, if an increase in the proportion of males
leads to a decrease in reproduction rates, then the impact of lamprey on their host fish
may be reduced, thus affecting the associated food chain. In addition, changes in sex
ratios may also affect the genetic diversity and adaptive capacity of lampreys, which in
turn may affect their long-term viability.
1.2 Restatement of the Problem
•
For Task 1, we need to build a mathematical model based on the basic property
of lampreys that can change the sex ratio, which can reflect the impact of the
sex-changing property of lampreys on the ecosystem, and draw the corresponding
conclusions based on the built model.
•
For Task 2,we need to utilize and improve the above model and derive the
strengths and weaknesses of lampreys population from the model results.
•
For Task 3,e need to establish an evaluation criterion on ecosystem stability, and
determine and validate the effect of the characteristic of lampreys that can change
the sex ratio on the stability of the ecosystem by comparing the change of this
indicator before and after the introduction of lampreys.
•
For Task 4,we need to further refine the model to find out if there are species (e.g.,
![](https://csdnimg.cn/release/download_crawler_static/89271463/bg4.jpg)
Team # 2405424 Page 4 of 25
parasites) that can profit from ecosystems containing populations of lampreys,
and to elaborate on the mechanisms of their profitability.
1.3 Literature Review
Since lampreys have relationships with other species in the ecosystem such as
interspecific competition, parasitism, etc..In order to better describe the status and role
of lampreys in the ecosystem, we need to understand the existing mathematical models
on interspecific relationships. According to Literature [3], [4], the existing models that
can describe interspecific relationships are mainly the Lotka-Volterra predator-prey
model that describes predation relationships and the Nicholson-Bailey host-parasite
model that describes parasitism relationships. Based on this, our approach to building
the model is shown in the Figure 2:
Figure 2: Modeling thoughts
As can be seen from the Figure 2, since all species obtain energy from environ-
mental resources, the environmental resources can be regarded as the prey and each
species as the predator, and the Lotka-Volterra predator-prey model can be used to
describe the relationship between species and environmental resources.And because
multiple species prey on the same environmental resources, when mathematical models
are used to describe the relationship between species and environmental resources, the
relationship of interspecific competition between individual species is also indirectly
described.
![](https://csdnimg.cn/release/download_crawler_static/89271463/bg5.jpg)
Team # 2405424 Page 5 of 25
1.4 Our Work
1.
Based on the Nicholson-Bailey host-parasite model and the Lotka-Volterra predator-
prey model, we will build the Ideal Reference Model , Constant Sex Ratio Model ,
and Models of Changing Sex Ratios .
2.
By adjusting the ecosystem parameters in each model, we will simulate the sur-
vival of different species in different climates as well as changes in total resources.
3.
We will analyze the resulting data and compare the three models above to discuss
the advantages and disadvantages of the ability of lamprey to alter sex ratios on
lamprey’s own populations, as well as the effects on factors such as the number of
ecosystem species surviving, total resources, and ecosystem stability.
4.
Combining the three models, we will discuss the impact of lamprey on conven-
tional parasitizers with a sex ratio of 1:1 by making reasonable improvements to
Constant Sex Ratio Model and Models of Changing Sex Ratios.
5.
Finally, we will perform a sensitivity analysis of the model by varying the initial
value of the population size of each species, and by adding the disturbance of
anthropogenic fishing factors both.
2 Model Preparation
2.1 Assumptions
•
Assumption 1: Natural enemies of parasitic species other than humans are not
considered. That is, the relationship between species and parasitic organisms
in the community is mostly parasitism and interspecific competition.
Justification: According to references [5], parasitic species such as the lampreys
are invasive in the Great Lakes of North America, suggesting that they have very
few natural enemies and can be ignored.
•
Assumption 2: The survivability of male and female lampreys in the population
is approximately the same.
Justification: From the references [6], it can be seen that there is little difference in
the survival ability of male and female lampreys, and this assumption can avoid
the extreme situation of a larger death in this party due to the lack of survival abil-
ity of either females or males, which ensures the reliability of the model prediction
results.
•
Assumption 3: All species in an ecosystem can obtain energy directly from
environmental resources.
Justification: A simple analysis shows that when this hypothesis is valid, it is more
effective to use Model 1 to reflect the competitive relationship between different
species, and it also simplifies the model to some extent.
These are the main assumptions, and other assumptions will be given at appropriate
places in the article.
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