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## Problem A: Fungi…

The carbon cycle is an important process of the chemical cycle. One part of the carbon cycle is the decomposition of compounds, which allows carbon to be reused. The key to this process is the decomposition of plant materials and wood fibers.

The key to decomposing wood fiber is fungus. The latest research has identified fungal traits related to the rate of wood fiber decomposition and pointed out the relationship between these traits. In particular, slow-growing fungi are more likely to survive and grow in an environment with changes in humidity and temperature; while faster-growing fungi are less resistant to the same environmental changes.

The researchers studied a large number of traits related to different fungi, and also studied the fungus's role in the decomposition of litter and wood fiber. For this problem, we only need to pay attention to two characteristics of fungi: fungal growth rate and moisture resistance. Our modeling goal is to model the decomposition of wood fibers on a given ground, and model the decomposition of wood fibers in the same area by multiple types of fungi.

When dealing with the relationship between growth rate and moisture resistance and decomposition rate, the problems that may be encountered are:

- Using growth rate and moisture tolerance, how do different fungi interact and decompose litter on the ground in different environments?
- In different environments, how will the decomposition effect be affected over time as conditions change?
- In a given environment, how do environmental changes and rate of change affect the dynamics of "decomposition" and "competition between fungal communities"?

For a given growth rate, the decomposition rate is estimated as shown in Figure 1; for a given relative humidity, the decomposition rate is estimated as shown in Figure 2.

Your article should research and include the following aspects:

- Establish a mathematical model to describe the decomposition of litter and wood fiber in the presence of multiple fungi.
- In your model, combine the interactions of different types of fungi with different growth rates and different moisture resistance.
- Provide a model to describe the interaction between different types of fungi. The dynamic description of the interaction should include short-term and long-term trends. Your analysis should also check the sensitivity to fluctuations in environmental changes, and determine what the constant change trend of the atmospheric environment is to assist in evaluating the impact of weather changes on the model.
- It should also include predictions of the relative advantages and disadvantages of "each species" or "combination of species that may last for a period of time", as well as predictions for different environments such as arid, semi-arid, temperate, arbor, and tropical rain forest.
- Describe how the diversity of fungal communities affects the overall decomposition efficiency of litter. When there are varying degrees of variability in the local environment, predict the importance and role of biodiversity.

## Overview

"In particular, slow-growing fungi are more likely to survive and grow in an environment with changes in humidity and temperature; while faster-growing fungi are less resistant to the same environmental changes."

This sentence explains: The "rate of change" of temperature and humidity is related to fungal resistance/vitality.

For this problem, we only need to pay attention to two characteristics of fungi: fungal growth rate and moisture resistance. Our modeling goal is to model the decomposition of wood fibers on a given ground, and model the decomposition of wood fibers in the same area by multiple types of fungi.

This sentence shows: this article only needs to focus on "fungus growth rate and moisture resistance"; modeling focuses on how "external conditions (humidity, competition between communities) and internal traits (growth rate, moisture resistance) affect fungi on wood The rate of decomposition of fiber and litter."

**Keywords** : fungal decomposition rate, fungal growth rate and moisture resistance, humidity and the influence of community competition on fungi;

## Question 1: Establish a mathematical model to describe the decomposition of litter and wood fiber in the presence of multiple fungi.

This question only needs to establish a "decomposition rate model". Although the question mentioned "the situation where multiple fungi exist", the main point of the original English sentence is "to establish a model to describe the breakdown of ground litter and woody fibers through fungal activity in the presence of multiple species of fungi)”, which refers to the overall use of a method to describe the influence of fungi on plants.

Therefore, the general result of this question is a linear model of "fungus-decomposition rate". With the help of the background of the topic, the independent variables can be set to "temperature, humidity, overlap between communities, time", etc., and the dependent variable can be set to "decomposition rate (or vegetation decay rate/loss rate)" and so on. For example, Y=a *X_1+b* X_2+...

The data needed in the process of fitting the equation can be provided according to the title, or you can find other data by yourself. Because it is the first question, some factors can be considered as much as possible, but the solution process can be simplified appropriately, just write a general idea.

## Question 2: In your model, combine the interactions of different types of fungi with different growth rates and different moisture resistance.

"In particular, slow-growing fungi are more likely to survive and grow in an environment with changes in humidity and temperature; while faster-growing fungi are less resistant to the same environmental changes."

This sentence explains: The "rate of change" of temperature and humidity is related to fungal resistance/vitality.

The previous part mentioned "the relationship between growth rate and moisture resistance", so this question is mainly modeled in two aspects: the expression of moisture resistance and growth rate, and the combined treatment of different fungi.

When dealing with the relationship between moisture resistance and growth rate, we can find that the moisture resistance and growth rate of different fungi are roughly linear (known from the picture provided by the title), so we need to abstract it into a linear relationship, such as X_1= c*X_2.

(Q: The graph is not a direct linear relationship

A: Yes. The linear relationship here can be regarded as a generalized binary relationship, as long as the two variables are directly related, it is not necessary to use the pure linear relationship y=ax. )

It is known from Figure 1 that the growth rate is related to the decomposition rate, and

from Figure 2 that the moisture resistance is related to the decomposition rate.

Therefore, combining the action of fungi with "different growth rates and moisture resistance" refers to the combination of the above two factors while keeping the fungus' "decomposition effect (decomposition rate)" unchanged. Because their combined effect is the decomposition rate, it is enough to keep the decomposition rate constant, and then combine the two factors (at the same time, this requires the help of the "growth rate and moisture resistance relationship" model, X_1=c *X_2). For example, Y=a* (c *X_2)+b* X_2.

In this way, we can get the "fungus decomposition rate" model after the combined treatment.

## Question 3: Provide a model to describe the interaction between different types of fungi. The dynamic description of the interaction should include short-term and long-term trends. Your analysis should also check the sensitivity to fluctuations in environmental changes, and determine what the constant change trend of the atmospheric environment is to assist in evaluating the impact of weather changes on the model.

Because different fungi have different moisture resistance, their growth rates in the same environment are different, which leads to interactions between different types of fungi.

Through the question, we need to see another point, which is why the interaction between fungi occurs? In fact, I mentioned the influence of fungal communities in the previous article. Generally speaking, it is the competitive relationship between the populations, which is mainly reflected in the "with or without nutrient supply" and "with or without oxygen". This will make it easier to understand.

The model we can build is the "community evolution model": what type of community evolution will occur between different types of fungi in the same initial environment. Fungi that are more suitable for environmental temperature and humidity will grow faster. Although two or more fungi will not compete under the conditions of sufficient nutrient supply at the beginning, when the number of fungi that are more suitable for the environment is much greater than that of the disadvantaged party, Vulnerable fungi will lack nutrients and may cause the fungus to die.

If you want to further deepen the topic, you can further consider a relationship, that is, the influence of "with or without oxygen" on aerobic and anaerobic respiration. If there are too many strong fungi, will it affect the oxygen of the weak fungi? This point can be further verified by consulting the literature, so I won't go into it here. At the same time, assuming that the effect of oxygen is ignored, if both parties are performing aerobic or anaerobic respiration or mixed with anaerobic, will the products of these two respirations further affect the survival of the other's fungus? For example, will the alcohol produced inhibit the growth of the other side’s fungi, or inhibit the growth of one’s side?

Therefore, through the above analysis, we can roughly get the trend changes in the short-term and long-term interactions required by this question. As for the sensitivity to environmental fluctuations, this refers to whether small changes in initial conditions will affect the different trends of subsequent community evolution? Will it cause the original strong fungus to become a weak fungus? After considering the impact of the environment on the initial conditions, the initial environmental factors can be added to the model to further assist the impact of weather changes on the model.

## Question 4: It should also include the prediction of the relative advantages and disadvantages of "each species" or "species combination that may last for a period of time", and predictions of different environments such as arid, semi-arid, temperate, arbor, and tropical rain forest.

This question is similar to the thinking of question 3, and the main consideration is "dynamic", that is, a trend.

For "prediction of relative advantages and relative disadvantages", time can be used as a measurement standard, and then the relative advantages and disadvantages of fungi in changes can be judged;

And for the forecast of "drought and humid" and other environments, that is, the impact of initial environmental conditions on community evolution. Factors that can be considered are: the duration of temperature and humidity, and what types of fungi may exist in dry and humid areas. In other words, the fourth question can be used in real life to try and apply.

## Question 5: Describe how the diversity of the fungal community affects the overall decomposition efficiency of litter. When there are varying degrees of variability in the local environment, predict the importance and role of biodiversity.

This question is considered based on the number of fungi species as an independent variable. It is only necessary to design a model to analyze the effect of fungi species on decomposition efficiency. For example, you can use the "community evolution model" in question 3, set the initial conditions to the number of fungal species, and then see how long it takes to decompose the litter.

When considering that there may be different degrees of variability in the local environment—that is, the difference in initial conditions—the “anti-interference ability” and “resilience” of the diverse fungal community are stronger, which should be easier to think of . The influence of the previously analyzed fungi species on the decomposition efficiency can be added to the judgment index, and the conditions under which the fungal species diversity can be fully restored (as much as possible) can be fully restored to the original state after the loss of fungal species diversity.