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2023年美赛特等奖论文-A-2309229-解密.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,2023年美赛特等奖O奖论文
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Problem Chosen
A
2023
MCM/ICM
Summary Sheet
Team Control Number
2309229
The Warriors against Drought: Plant Communities
Summary
As the severe drought increases since the global warming, many plant communities die
out for lack of water, while others can recover after drought. To explore this phenomenon,
we develop GIID model to describe the response of communities to drought and analyse
whether species diversity can help communities withstand drought.
Before all the models are established, we select five typical plant species and classifiy them
into three categories, according to their inherent growth rate and optimum soil moisture. In
addition, we estimate the parameters used in our models.
For Task 1: To simulate plant growth and interspecific competition, we first developed
the Lotka-Volterra competition model. Then, considering optimum soil moisture of each
species, we set the environmental capacity of them as a function of soil moisture. Finally, we
modeled the soil moisture variation to describe the interaction between plants and the en-
vironment. We consider the water absorption by plants, irregular rainfall, and other factors.
We summarize these analysis and devise our GIID model. Then, we verify the correctness of
model. The results are shown in Figure 7.
For Task 2: First, we do the qualitative analysis and find that the Five Species Community is
more stable in drought. To quantify how the community benefits from the species diversity,
we calculate total biomass, coefficient of variation of total biomass, and S-W index to evaluate
the vigor, stability, and evenness of the ecosystem respectively. The experimental results
show that as the number of species increases, the total biomass of the community increases
and then decreases, while the stability and evenness keep increasing. Besides, two species
are required for the community to benefit. The results are shown in Figure 10 to 13.
For Task 3: To explore the effect of species type, we choose three typical species com-
binations and each combination is simulated separately. We find that combinations with
moderate differences in optimum soil moisture increase the total biomass and stability of the
community. And if the differences in optimum soil moisture of species are too small or too
large, the community would not benefit.
For task 4: We adjust the magnitude and frequency of droughts, and the experiment
results show that: when droughts become stronger and more frequent, the complementary
effect among species also becomes stronger. And the biomass of the community increases
as the number of species increases. When droughts are less frequent, the competition among
species is stronger, and the increase in the number of species lead to decrease in total biomass.
The increase in the number of species always enhances community stability.
For Tasks 5 and 6: We attributed the effects of habitat loss and environmental pollu-
tion to the decrease in environmental capacity. So we set up a step function to simulate this
process. We find that habitat loss and environmental pollution will lead to decrease in total
community biomass, while multi species community will still have high stability in this pro-
cess. Next, we illustrate the measures to ensure the long-term viability of the community. We
also describe the impact of the community on the environment based on above findings.
Finally, we change the plant competition intensity, water absorption rate and climate pat-
terns in order to perform sensitivity analysis on these parameters. We find that our model
is sensitive to all of them. We also analyze the strengths and weaknesses of the model and
discuss how to improve it.
Keywords: Lotka-Volterra Model, Biodiversity, Irregular Drought, Complementary Effect
Team # 2309229
Page 2 of
25
Contents
1 Introduction 3
1.1 Problem Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Our Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Model Preparation 4
2.1 Assumptions and Justifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Classification of Plants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Data Collection and Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 6
3 Task1 : The GIID Model 7
3.1 The Growth and Compete of Plants . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 The Interaction between Plants and Water . . . . . . . . . . . . . . . . . . . . . . 9
3.2.1 Relationship Between Water and Environmental Capacity . . . . . . . . 9
3.2.2 Variation of Soil Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 The Simulation of Irregular Weather . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4.1 Single Species Community . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4.2 Five Species Community . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Task 2 and 3 : Effect of Species Diversity 12
4.1 Task 2 : Effect of the Number of Species . . . . . . . . . . . . . . . . . . . . . . . 12
4.1.1 Qualitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.1.2 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1.3 Model Implementation and Validation . . . . . . . . . . . . . . . . . . . . 14
4.2 Task 3 : Effect of the Type of Species . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.1 Three Representative Species Combinations . . . . . . . . . . . . . . . . 16
4.2.2 Our Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Task 4,5 and 6 : Impacts of Environmental Change on Species 18
5.1 Task 4 : Changes in the Magnitude and Frequency of Drought . . . . . . . . . . 18
5.2 Task 5 : Impacts from Habitat Reduction and Environmental Pollution . . . . . 20
5.3 Task 6 : Further Discussion on Community Viability . . . . . . . . . . . . . . . . 21
5.3.1 Ensuring Community Viability . . . . . . . . . . . . . . . . . . . . . . . . 21
5.3.2 The Impact of Communities on the Larger Environment . . . . . . . . . 22
6 Sensitivity Analysis 22
6.1 Plant Water Absorption Rate β . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Plant Competition Intensity α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.3 Climate Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7 Model Evaluation 24
7.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
7.2 Weaknesses and Possible Improvements . . . . . . . . . . . . . . . . . . . . . . . 25
References 25
Team # 2309229
Page 3 of
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1 Introduction
1.1 Problem Background
In recent years, the frequency of extreme weather events has increased dramatically around
the world because of the global warming. Drought is one of the extreme weather events that
will have a huge impact on our lives. The image shown below is collected from the U.S.
drought monitoring website Drought. gov. It reflects the drought conditions in the U.S. be-
tween 8 Feb. 2023 and 14 Feb. 2023.
Figure 1: Distribution of drought in the United States
Source :
https://www.drought.gov/current-conditions
As we can see from the graph,
34
.
55%
of U.S. territory is exposed to drought, which affects
252.9 million acres of crops and 74.3 million people. During the extreme droughts, we can
easily discover that different plant communities behave differently. Plants in grassland and
farmland are almost dead, while multi-species communities are less affected and thrive after
drought. Based on the above analysis, it is essential and urgent to study how plants respond
to drought and how the diversity of species in a plant community helps the plants withstand
drought.
1.2 Restatement of the Problem
• According to the requirements, we should firstly develop a mathematical model to pre-
dict how a plant community changes over time as it is exposed to various irregular
weather cycles. And we are required to consider the interaction between different plant
species during the cycles of drought.
• Secondly, we should extend our model to a long-term time and a larger environment.
• Thirdly, we need to do a further discussion on the relationships between the number of
plant species and the benefits of the plant community gets, and to explore what happens
when the number of species increases. Besides, we should pay attention to the impact
of the type of species on the results we obtained previously.
• We are also required to study whether the number of species will have the same effect
on the total plant population under the drought with higher or lower frequency and
wider variation. Then we should take other factors like pollution and habitat reduction,
etc. into account.
• Finally, based on all results above, we should figure out what need to be done to ensure
the viability of the community and what the impacts of the plant community on the
larger environment are.
Team # 2309229
Page 4 of
25
1.3 Literature Review
In order to study the plant communities in drought, we need to know the existing meth-
ods to simulate plant communities and the corresponding ecological principals. According
to the literature [1, 2, 3, 4], for the simulation, there are mainly three methods and for the
principles, there are mainly two principles we need to use. As the following figure shows:
Figure 2: Literature Review
1.4 Our Work
The problem requires us to explore interactions between plants and enviroments, and
measure the effects of species diversity. We mainly carry out the following work.
• Based on the analysis of several factors such as plant growth, interspecific competition,
and the competition for water resources, a GIID model was developed.
• Explore the effects of number of species and type of species
• Explore the effects of other factors such as changes in drought frequency, habitat reduc-
tion, and environmental pollution.
In order to avoid complex textual descriptions, we use flowcharts to show our work,
which is shown in Figure
3.
2 Model Preparation
2.1 Assumptions and Justifications
To simplify the given problems, we make the following basic assumptions:
Assumption 1: The water absorption rate of each plant is proportional to its biomass.
Justification: According to literature[
7], When intense competition between
plants occurs, each resource obtained by a plant is proportional to its biomass.
Assumption 2: The main factor affecting plant growth is the soil moisture. And there’s
different optimum soil moisture for different plants to grow.
Team # 2309229
Page 5 of
25
Figure 3: Our Work
Justification: The factors affecting plant growth such as temperature, mois-
ture, and sunshine are coupled in the real world. However, we just focus on
the growth of plant community under extreme drought. Therefore, to sim-
plify our models, we ignore other factors, and only take the soil moisture
into consideration. When a plant grows under moderate soil moisture, it will
grow fast, but it can not grow when it is lack of water or flooded. So there’s a
optimum soil moisture.
Assumption 3: Under the appropriate conditions, all plants share a common environmen-
tal capacity.
Justification: Most of plants are producers and obtain energy in the same way
(all from photosynthesis). Under the appropriate conditions, they all have
the opportunity to absorb all solar energy in the habitat. So they can reach
the same maximum biomass, which means they share a same environmental
capacity.
Assumption 4: During our study, the effects of accidental events such as plant invasion
and sudden changes in climate patterns are not considered.
Justification: Although the plants grow in a natural environment, to simplify
our model, we assume that the plant community is in a relatively closed en-
vironment. So there is no invasive species and no large, rapid changes in
climate patterns.
Assumption 5: In plant communities, the only source of water is precipitation.
Justification: There may be various ways for communities to get water, such
as dew, fog and precipitation. Since dew and fog carry little water, we ignore
their effects, and only consider the precipitation.
2.2 Notations
The key notations are shown in Table 1.
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