Analysis of Questions A in 2021 American College Students Mathematical Contest in Modeling

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Question A: Fungi The
carbon cycle describes the exchange process of carbon in the entire geochemical cycle and is an important part of life on earth.
Part of the carbon cycle involves the decomposition of compounds, allowing carbon to be renewed and used in other forms. A
key component of this part of the process is the decomposition of plant material and wood fibers.
The key factor in decomposing wood fiber is fungus. The author of a recent research article on the decomposition of fungal wood has identified the
fungal traits that determine the rate of decomposition and noticed the connection between certain traits [1]. In particular, slow-growing fungal strains
tend to survive and grow better under environmental changes,
moisture and temperature, while faster-growing strains tend to be less robust to the same changes. For a summary of this article, see
page 3 below.
These researchers studied a large number of traits associated with different fungi and their role in breaking down ground litter (dead plant material) and
wood fibers. For this MCM problem, you should only pay attention to two characteristics of fungi: fungi and fungi's growth
rate and
their tolerance to moisture. Your main goal is to simulate the decomposition of wood fiber on a given land and do so in the same area
where multiple types of fungi decompose wood fiber.
When you explore the relationship between the two traits of interest, growth rate and moisture resistance, as the decomposition rate increases,
several questions may arise: Using these two traits, how do different fungi fix themselves in different environments Interaction
and decomposition of ground litter in patches ? In these different environments, as conditions change, how decomposition will be affected by time?
ring? How do environmental changes and changes in environmental changes affect the long-term kinetics of decomposition and the competition between fungi in a given environment? The estimation of the decomposition rate, given the growth rate, is shown in Figure 1. Taking into account the relative moisture tolerance, the estimated decomposition rate is shown in Figure 2.
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Figure 1: The relationship between the mycelial elongation (mm/day) of various fungi and the wood decomposition rate (% mass loss within 122 days) at different temperatures. ([1] Figure 1C).
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Figure 2: The water tolerance of various fungi (the relationship between the difference in the competition ranking of each isolate and the width of their water niche, are all scaled to [0,1] and the resulting wood decomposition rate (Quality loss% within 122 days, log conversion). ([1] Figure 4A).
Requirements: Your paper should discuss and solve the following aspects.

  • Establish a mathematical model to describe the decomposition of ground litter and wood
    fiber through fungal activity in the presence of various fungi .
  • In your model, combining the interactions between different types of fungi, they have different growth rates and
    different moisture tolerances, as shown in Figures 1 and 2.
  • Provide model analysis and describe the interaction between different types of fungi. The dynamics of the interaction should be described and described
    , including short-term and long-term trends. Your analysis should examine the sensitivity to rapid fluctuations in the environment, and you should
    determine the overall impact of the changing atmosphere and assess the trend of changes in local weather patterns.
  • It includes the prediction of the relative advantages and disadvantages of each species and the combination of species that may continue to exist, and predictions
    of different environments including arid, semi-arid, temperate, arbor forest and tropical rain forest.
  • Describe how the diversity of a system's fungal community affects the overall efficiency of the system in decomposing ground waste.
    Predict the importance and role of biodiversity in the local environment with varying degrees of variability.
    Includes a two-page result article. Your article should be suitable for inclusion in an introduction to university-level biology textbooks, discussing the latest developments in our understanding of the role of fungi in ecosystems.
    Your PDF solution does not exceed 25 pages and should include:
  • One page summary
  • table of Contents.
  • Your solution.
  • Two-page article.
  • List of references.
    Note: MCM competitions now have a 25 page limit. All aspects of your submission are calculated to a limit of 25 pages (summary form, table of contents,
    reference list and any appendices).
    References:
    [1] Nicky Lustenhouwer, Daniel S. Maynard, Mark A. Bradford, Daniel L. Lindner,
    Brad Oberle, Amy E. Zanne, and Thomas W. Crowther, "A trait-based section of
    wood recography," Proceedings of the National Academy of Sciences of the
    United States, May13,
    2020 https://www.pnas.org/content/pnas/117/21/11551.full.pdfResearch
    article Synop sis
    We provide below the research of Lustenhouwer et al. A brief summary of the article [1]. The full text is available at
    https://www.pnas.org/content/pnas/117/21/11551.full.pdf. Please note that you do not need to read the original
    article to complete this MCM question.
    The decomposition of organic matter is a key component of the carbon cycle. Large-scale modelling of the carbon cycle and global climate models is becoming
    more refined and sophisticated,
    incorporating smaller-scale details. An important detail is the rate at which microbial and fungal communities are associated with the decay of organic matter.
    The focus of this article is the different decay rates of different types of fungi.
    This article explores several different characteristics of fungi to determine the impact of wood decomposition. They did this by measuring
    how much mass was lost after introducing different types of fungi into the wood block . Researchers have studied a large number of different
    traits associated with each fungus and tried to determine the role of these traits in the decomposition of wood blocks.
    For example, an important feature is the mycelial elongation rate. Hyphae are cells that stretch out and form the filaments and structures of
    fungi. Different types of hyphae play different roles in the life cycle of fungi. Mycelial elongation is essentially the growth rate
    of the fungus . Another characteristic is the density of hyphae in a given volume.
    These two characteristics are related to many characteristics of fungi. For example, it was found that if the mycelium elongation rate is greater (grow faster),
    fungus is easier to decompose wood faster. Similarly, if the filaments are denser, the decomposition of the wood is more likely to be slower. In addition, these
    two characteristics are also related to how the fungus reacts to different environmental conditions.
    In particular, the researchers found that fungi that can better adapt to different ranges of moisture conditions also tend to decompose wood more slowly.
    Fungi that grow faster and exceed other fungi tend to decompose wood faster. Figure 1 and Figure 2 in the MCM question A sentence show
    these relationships.
    Wood materials have gone through many stages, and the fungi investigated in the research articles are most
    related to the decay of wood materials in their decay cycle . For other decay stages, the results may be different. For the purpose of this modeling exercise, you can focus on
    the result of the intermediate stage and assume that it is consistent with the other decomposition stages. Another consideration is that the local environmental
    conditions in an area may vary greatly and affect the overall dynamics.
    Vocabulary:
    Biodiversity: Broadly speaking, the diversity of life in the world. On a smaller scale,
    the diversity of life in a particular habitat or ecosystem .
    Carbon cycle: A continuous process (or series of
    processes) that exchanges carbon between organisms and the environment, and then reuses it throughout the earth .
    Competitive ranking: Under similar conditions, one fungus surpasses other fungi in a series of paired tests ability to measure.
    biosphere: lithosphere (crust and upper mantle), the water ring (the water all the earth's surface) and atmosphere (around the earth
    the envelope gas.
    fungus (plural: fungi): eukaryotes group Any member (with a nucleus in a cell surrounded by a nuclear envelope)
    organisms. Such as yeast, mold and mushrooms.
    Geochemical cycle: the various pathways and steps of element exchange between the earth's biosphere and the earth's biosphere.
    Mycelium : Cells that form filaments in the fungal community.
    Mycelial elongation rate: the growth rate of the fungus.
    Water niche width: half the difference between the maximum and minimum water levels at which the fungal community can maintain its fastest growth rate
    .
    Water tolerance Sex: the difference between the competition ranking of fungi and the width of their water niche.

    Thinking analysis

    Question A Comprehension
    Modeling Center: The influence of the growth rate and moisture resistance of fungi (various) on the decomposition of lignocellulosic fibers, as well as the evolution mode of space (spatial heterogeneity) over time (changes in time).
    Subject requirements:
    (1) Establish fungus-wood Decompose a single model. This question needs to simplify the classification of fungi based on the characteristics of growth rate and moisture resistance, descriptive modeling of different fungal decomposition activities, and discuss the change patterns of indicators including but not limited to attenuation, decomposition rate, CO2 emissions and other indicators.
    (2) Establish an integrated fungus-wood decomposition model (fungus decomposition interactive model). This question needs to consider all types of fungi as a whole, and the first question constitutes a sublimation or advanced relationship. There is no need to consider the activity interactions between too many different bacterial species, and further analyze the overall law of fungal activities, as shown in the reference diagram.
    (3) From a short-term and long-term perspective, analyze the interaction patterns between different fungi. (This work can be placed in the second question) Analyze (judge), determine (quantitative, relationship), and evaluate (qualitative) the influence of fungal activities on weather changes. This question should make assumptions about the characteristics of fungi. Consider establishing a fungus-environmental impact model.
    (4) According to the relationship expressed by the interactive model in Question 2-3, discuss the situation in different environments, use predictive models such as exponential smoothing, linear regression and other simple models to make predictions for different environments, and determine what dependent variable to represent For species sustainability, is it necessary to set indicators for species sustainability? Consider whether the data is easily accessible.
    (5) Open questions. The impact of biodiversity on garbage classification can be written into conclusions and suggestions, and of course, modeling can also be used. The variability of fungi in different environments is also a discussion direction. The focus of this question is to find out what kind of fungus combination can promote or slow down the garbage classification.
    The highlight of the problem lies in the early model construction

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