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2023年美赛特等奖论文-A-2322687-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
2322687
Is There Space for More? A Spatiotemporal Partial Differential
Equations Model for Plant Growth During Drought Cycles
Summary
Droughts pose a significant risk to the survival of a community of plants. Although individual
plants may struggle to adapt to a lack of water caused by a drought, existing literature has suggested a
positive relationship between the biodiversity of a plant community and its ability to survive drought
cycles. We have been tasked to verify this claim and build a generalized biome-independent model
that predicts changes in a plant community through cycles of drought.
Our approach was to develop a parameterized partial differential equations model for the growth
of multiple plant species over time on the basis of the amount of water available. Crucially, we also
incorporate spatial dimensions and analysis into the model, which allows us to consider plant dispersal
and water diffusion across physical boundaries. To support this growth model, we also implement a
sub-model for rainfall that can simulate the effect of a variety of droughts by changing drought timing
and severity. Incorporating this into the plant growth model, we model plant growth in Mathematica
using numerical analysis of the partial differential equations. We conduct a sensitivity analysis of
the model’s parameters, simulating the effects of biodiversity by steadily increasing the number of
unique species starting from a control of just one species. We also analyze the impact of different types
of droughts, as well as different types of plants as measured by their resistance and resilience. We
conclude by studying the impacts of external factors, such as pollution or habitat reduction.
Ultimately, we find that although increasing the number of unique plants in a given community
generates inter-species competition, it cumulatively benefits the overall system by increasing the
total amount of long-term plant biomass. However, marginal benefit decreases as total number of
unique species increases: going from one to two unique species generates significant benefits in total
biomass, but changes are smaller when going from two to three, and the transition from three to four
actually decreases total biomass. Meanwhile, our sensitivity analysis finds that changes in resilience
and resistance correspondingly impact total plant biomass but do not affect the general behavior of the
model. In addition, we determine that habitat destruction can cause individual species to dominate a
plant community due to increased spatial competition. We find under certain extreme conditions, a
given species of plants can either survive and dominate in the long-term or go extinct depending on
their initial conditions.
The main advantage of our unique spatial approach over traditional, space-independent models
that only depend upon time, like the classic Lotka-Volterra model, is the the ability to account for
fundamental physical behaviors of plants and ground water—that is, dispersal and diffusion across
a 2D space. Without spatial considerations, one cannot accurately model the fact that generally,
plants and water can only spread to adjacent locations. This also allows us to model phenomenon like
competition for habitation or water at a given location. Finally, the spatial approach also provides the
ability to simulate random initial locations, mirroring the stochasticity of real-world environments, as
well as forecast the impacts of habitat reduction easily. Combined with a realistic model for rainfall
data and drought conditions, our model serves as a comprehensive general phenomenological model
for simulating plant growth in any given community under cycles of droughts.
Keywords: Partial Differential Equations, Spatiotemporal Analysis, Drought Modeling
Team # 2322687 Page 2 of 25
Contents
1 Introduction 3
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Problem Restatement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Existing Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Developing the Plant Community Model 4
2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Model Overview and Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Variables and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 Plant Biomass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Water and Rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.6 Species Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.7 Rainfall, Droughts, Species Types, and Pollution . . . . . . . . . . . . . . . . . . . . . 8
2.8 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.9 Model Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Rainfall and Drought Model 10
3.1 General Rain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Drought-Adjusted Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Results 11
4.1 Number of Unique Plant Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Different Types of Droughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 Sensitivity Analysis 17
5.1 Types of Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Resistance and Pollution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.3 Habitat Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
6 Strengths and Weaknesses 23
6.1 Strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.2 Weaknesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7 Conclusion 23
8 Future Exploration 24
References 25
更多数模资讯和学习资料,请关注b站/公众号:数学建模BOOM
b站主页:https://space.bilibili.com/350975620
Team # 2322687 Page 3 of 25
1 Introduction
1.1 Background
The study of how drought cycles impact plant communities is crucial for understanding the impacts of
climate change on ecosystems and agriculture. Droughts can significantly alter plant growth, distri-
bution, and productivity, which can have cascading effects on ecosystem functions and services, such
as nutrient cycling, soil stability, and carbon storage. Moreover, droughts can impact agricultural pro-
ductivity, as they reduce crop yields, increase irrigation demands, and exacerbate soil erosion and land
degradation [1]. By studying how droughts affect plant communities, scientists can develop effective
management strategies to mitigate the impacts of droughts on ecosystems. This includes developing
drought-resistant crop varieties, improving water use efficiency, and implementing sustainable land
management practices. Ultimately, understanding the effects of drought on plant communities is cru-
cial for maintaining ecosystem health and supporting sustainable agriculture in the face of climate
change.
Plant species are often geographically grouped together into communities to better understand
species interaction and relationships [2]. The relative composition of the species within communities
is not static, but rather it changes as a response responds to stimuli and stresses. In particular, one
common source of stress that can drastically change the composition of a community is a period of
drought [3].
As different organisms have different characteristics, they react and adapt differently to extreme
conditions. Such responses are well described by the notions of resilience (ability to recover after
extreme conditions) and resistance (ability to persist and grow despite extreme conditions) [4]. Past
research has suggested that localized biodiversity within these communities strengthens the overall
ability of the community to adapt to periods of drought, relative to a more homogeneous community.
Studies have suggested the possibility that the genetic diversity offered by a diverse community provides
a higher chance for species-diverse communities to effectively thrive even during extended periods of
little to no rainfall [5]. However, little research has been done to explore the quantitative extent to
which biodiversity strengthens a community’s ability to persist and grow amid drought, and how this
survivability changes depending on the specific number of species.
1.2 Problem Restatement
To address this question, our team has been tasked to develop a model to predict how the composition
of a plant community changes through drought cycles. The model and its analysis must:
1. Account for species interaction, both competition and mutual support, during drought
2. Evaluate how many species are required to benefit from the biodiversity phenomenon, as well as
assess the impact of an increase in species within a community.
3. Consider the impact of changes in the frequency and severity of droughts
4. Consider the impact of differing types of species within a community
5. Consider the impact of pollution and habitat destruction
Team # 2322687 Page 4 of 25
1.3 Existing Literature
Prior to developing our customized model, we researched and analyzed several existing models. Below
are two important examples.
First, we examined the classic Lotka-Volterra Model as a basis to consider competition between
plant species and impacts on growth [6]. An abstracted version of the predator-prey equations are:
𝑑𝑥
𝑑𝑡
= 𝛼𝑥 − 𝛽𝑥𝑦
𝑑𝑦
𝑑𝑡
= 𝛿𝑥𝑦 − 𝜙𝑦
where 𝛼 and 𝛿 are the growth rates of each of the prey predator species respectively, while 𝛽 and 𝛿
represent the competitive interaction between the two species.
Although the Lotka-Volterra Model provided a method to factor in competition, we believed its
specificity towards a predator-prey relationship made it difficult to accurately model plant interactions
since those interactions could potentially also be mutually beneficial. At the same time, perhaps the
most important issue regarding the model for our task was the assumption of a constant growth rate. It
does not take into consideration the relationship that growth rate and water scarcity.
To address this resource utilization aspect of plant modeling, we also considered Monod’s equation
for modeling microbial growth rate [7].
𝜇 = 𝜇
𝑚𝑎𝑥
𝑆
𝐾
𝑆
+ 𝑆
where 𝜇, the growth rate, is determined by the product of the maximum growth rate and a fraction
where S represents the concentration of a limiting substrate, or resource and K is a constant.
These two models provided a fundamental starting point for our work but were ultimately insufficient
for the problem at hand. In particular, we hoped to account for the following factors that were not
considered in these systems:
1. Multiple species interaction
2. Storage of groundwater
3. The spatial dimensions of the community, which influences plant and water dispersal
4. Different types of droughts
5. Resilience and resistance of plants
6. External factors like pollution and habitat reduction
2 Developing the Plant Community Model
2.1 Assumptions
To simplify the modeling, we establish a few fundamental assumptions, as well as provide justifications
for them:
Team # 2322687 Page 5 of 25
• Assumption: Plant biomass is increased solely through plant growth.
– Justification: The problem statement revolves around analyzing the growth plants relative
to weather conditions. Therefore, we find it necessary to only discuss how plant biomass
changes based on plant growth or death, and to ignore other external factors like human
development, predators, etc.
• Assumption: New species of plants do not emerge suddenly in the environment throughout a
given simulation.
– Justification: Similar to the above assumption, we do not consider external influences that
may cause a plant to suddenly arise in an ecosystem. Instead, we will focus our analysis on
the existing plants in an ecosystem and study how their biomasses change with time.
• Assumption: Plants are differentiated solely by their resistance (how easily they die) and resilience
(how easily they grow).
– In the context of analyzing plant growth under extreme weather conditions, the most
important factor to consider is how plants survive within these conditions. Therefore,
we don’t consider factors like plant structure, shape, or size, and instead assume that the
resistance and resilience of a given plant are able to encapsulate its other defining qualities.
2.2 Model Overview and Justification
To explore this problem, our model will attempt to predict how the amount of water in the ground, as
well the total biomass of each species of plant, will change—these factors will serve as the dependent
variables. Specifically, because we are interested in studying the transition rates of these quantities,
we will develop a model based upon differential equations that can naturally characterize changes over
time.
However, we argue that analyzing the system across time alone is insufficient, as the physical
dimensions of an ecological community are highly influential in its development. For one, unlike
models for animals or microorganisms, plants generally do not move, which therefore limits their
water consumption and interactions with other plants at non-adjacent locations. For instance, different
species of plants may not be able to occupy the same physical location in the habitat due water shortages
in that particular location, and the size of the habitat will limit plant growth.
In addition, water dispersal is highly motivated by its diffusion across space and concentration
gradients. Modeling the spatial movement of water within a given area also provides us the ability to
use a more accurate measurement of the amount of water in a given location, rather than total water in
the system, which is far less accurate to plant growth. For instance, the growth of a plant at a given
location is not dependent on the amount of water in the overall system, but rather at the particular
location that it occupies.
Therefore, our model will, in addition to time, consider spatial dimensions as independent variables,
pointing to a need for partial rather than ordinary differential equations; this will also allow us to study
the impact of habitat reduction in later sensitivity analysis.
Within the model, we will incorporate parameters that allow us to explore and capture the following
conditions:
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