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\title{Making Room for Agriculture}
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Agricultural expansion often requires the transformation of forested landscapes and introduces chemical use on a drastic level, a process that dramatically alters ecosystems and imposes a new balance driven by human intervention. Our goal is to explore the transition from the natural forest to agricultural ecosystems, focusing on the interaction between natural processes and human decisions in the shaping of sustainable agricultural practices that would do less harm to the original landscape. We start by modeling the evolving food web in newly converted agricultural landscapes, giving mathematical expressions to each parameter of the ecosystem, then we would examine the role of producers, consumers, and decompose to better identify the importance of each role. Our analysis would emphasize critical aspects such as soil health, pest dynamics, and the impacts of herbicides and pesticides on biodiversity.

In the context of modern agriculture, where chemical dependence and monoculture dominate, our research highlighted the need for sustainable strategies that balance productivity with ecological stability. To address that problem, we use the incorporation of outside species into the food web, as pest controllers and pollinators, to take advantage of ecosystem services to reduce chemical dependency. To add a more realistic situation that a real agricultural ecosystem might be subjected to, we would also incorporate the reemergence of species over time into the agricultural space to better simulate the complex interactions between the transformed landscape and the surrounding landscape. In addition, our study will also investigate the implications of transitioning to organic farming practices, considering long-term sustainability, biodiversity restoration, and economic trade-offs, providing valuable insight into what the long-term effect of these measures will do to the ecosystem.

Through a combination of mathematical modeling and analysis in different scenarios, we aim to optimize the agricultural ecosystems for resilience and productivity. By understanding the effects of human interventions on ecological balance and exploring alternative options, we can provide statistically accurate math equations with actionable recommendations to promote harmonious co-existence between agriculture and nature.

\begin{keywords}
Agriculture, Ecological stability, Herbicide usage, Green, Reemergence of species
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\section{Introduction}

\subsection{Background}

Problem Chosen
E
2025
MCM/ICM
Summary Sheet
Team Control Number
2521641
Making Room for Agriculture
Summary
Agricultural expansion often requires the transformation of forested landscapes and introduces
chemical use on a drastic level, a process that dramatically alters ecosystems and imposes a new
balance driven by human intervention. Our goal is to explore the transition from the natural
forest to agricultural ecosystems, focusing on the interaction between natural processes and human
decisions in the shaping of sustainable agricultural practices that would do less harm to the original
landscape. We start by modeling the evolving food web in newly converted agricultural landscapes,
giving mathematical expressions to each parameter of the ecosystem, then we would examine the
role of producers, consumers, and decompose to better identify the importance of each role. Our
analysis would emphasize critical aspects such as soil health, pest dynamics, and the impacts of
herbicides and pesticides on biodiversity.
In the context of modern agriculture, where chemical dependence and monoculture dominate,
our research highlighted the need for sustainable strategies that balance productivity with ecological
stability. To address that problem, we use the incorporation of outside species into the food web,
as pest controllers and pollinators, to take advantage of ecosystem services to reduce chemical
dependency. To add a more realistic situation that a real agricultural ecosystem might be subjected
to, we would also incorporate the reemergence of species over time into the agricultural space to
better simulate the complex interactions between the transformed landscape and the surrounding
landscape. In addition, our study will also investigate the implications of transitioning to organic
farming practices, considering long-term sustainability, biodiversity restoration, and economic
trade-offs, providing valuable insight into what the long-term effect of these measures will do to
the ecosystem.
Through a combination of mathematical modeling and analysis in different scenarios, we aim to
optimize the agricultural ecosystems for resilience and productivity. By understanding the effects
of human interventions on ecological balance and exploring alternative options, we can provide
statistically accurate math equations with actionable recommendations to promote harmonious
co-existence between agriculture and nature.
Keywords: Agriculture, Ecological stability, Herbicide usage, Green, Reemergence of species
Contents
1 Introduction 3
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Restatement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Model Preparation 4
2.1 Assumption and Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Establishment of Models and Problem Solutions 6
3.1 Model 1: Introduction of Consumers and Competitors into Ecosystem . . . . . . . 7
3.1.1 Periodic Relationship between Plants, and Insects . . . . . . . . . . . . . . 7
3.1.2 Rate of Change of Population . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1.3 Impact of Seasonality Changes to the Ecosystem . . . . . . . . . . . . . . 12
3.1.4 Data Illustration and Equation Solving . . . . . . . . . . . . . . . . . . . . 13
3.2 Model 2: Incorporation of Reemergence of Species into Ecosystem . . . . . . . . . 18
3.2.1 Rate of Change in Population . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Data Illustrations for Reemergence of Species . . . . . . . . . . . . . . . . 22
3.3 Model 3: Removal of Herbicides and Introduction of Bats into Ecosystem . . . . . 23
3.3.1 Model 3A: Removal of Herbicides from Ecosystem . . . . . . . . . . . . . 23
3.3.2 Model 3B: Introduction of Bats into Ecosystem . . . . . . . . . . . . . . . 24
3.4 Model 4: Organic Farming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 Differential Evolution Algorithm . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 Objective Function for Model Building . . . . . . . . . . . . . . . . . . . . 28
3.4.3 Data Illustration of Organic Farming . . . . . . . . . . . . . . . . . . . . . 28
4 A Letter to Farmers 29
5 Rationalization Analysis and Sensitivity Testing of the Model 30
5.1 Rationalization Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2 Sensitivity Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.2.1 Control Equations and Boundary Conditions . . . . . . . . . . . . . . . . . 30
5.2.2 Definition of the Mean Temperature . . . . . . . . . . . . . . . . . . . . . 31
5.2.3 Determination of Heat Transfer Capacity . . . . . . . . . . . . . . . . . . . 31
6 Sub-model II: Adding Water Discontinuously 32
6.1 Heating Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.1.1 Control Equations and Boundary Conditions . . . . . . . . . . . . . . . . . 32
6.1.2 Determination of Inflow Time and Amount . . . . . . . . . . . . . . . . . 32
6.2 Standby Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3.1 Determination of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.3.2 Calculating Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7 Correction and Contrast of Sub-Models 33
7.1 Correction with Evaporation Heat Transfer . . . . . . . . . . . . . . . . . . . . . . 33
1
7.2 Contrast of Two Sub-Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Model Analysis and Sensitivity Analysis 33
8.1 The Influence of Different Bathtubs . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8.1.1 Different Volumes of Bathtubs . . . . . . . . . . . . . . . . . . . . . . . . 34
9 Strength and Weakness 34
9.1 Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.2 Weakness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
10 Further Discussion 35
Appendices 39
Appendix A First appendix 39
Appendix B Second appendix 39
Team # 2521641 Page 3 of 40
1 Introduction
1.1 Background
The conversion of forests into agricultural landscape is a critical process that drives food production
and is what humans have revolutionize 12,000 years ago to keep up with our increasing population,
but it is often accompanied by disrupts in the ecosystems and natural land, leading to habitat loss,
soil degradation, and reduced biodiversity. Modern agriculture’s reliance on chemical, such as her-
bicides and pesticides, has significantly amplified these challenges, creating ecological imbalances
and long-term sustainability concerns.
As the demand for sustainable farming practices that can maintain high crop yield rate grows,
integrating ecological principles and mathematical models that can simulate the change in the
converted forest area over time into agricultural systems has become essential. Methods like
organic farming and promoting beneficial species, such as pollinators and natural pest controllers
offer opportunities to balance productivity with environmental health. Our goal is to to explore
the transition from forest to farm, modeling ecosystem dynamics and examining the impacts of
human decisions and natural processes on biodiversity and sustainability, with the aim of deriving
a mathematical model that can provide accurate outcomes as to what a converted forest area will
be in the future, giving valuable information that can contribute to a more sustainable agriculture.
1.2 Literature Review
Due to its significant ecological effects, the conversion of forests into agricultural land has been the
subject of numerous studies. Complex forest ecosystems are disrupted by changes in land use, which
results in habitat loss, a drop in biodiversity, and a reduction in ecosystem services. According to
studies by (Foley, 2005, p. 573), these shifts result in the replacement of complex, multilayered food
webs with monoculture-dominated, simpler agricultural systems. This oversimplification weakens
resistance to illnesses and pests, highlighting the necessity of long-term plans to lessen their effects.
The dynamics of the food web evolve in agricultural settings. According to research, these
systems are frequently destabilized by the introduction of chemical inputs like fertilizers and
pesticides since they impact non-target species and reduce soil health (Tilman, 2002, p. 674).
Integrating beneficial species like birds and bats, which serve as pollinators and pest controllers,
has been the main focus of efforts to restore balance. These species provide essential ecosystem
services while lowering the agricultural reliance on chemicals (Boyles, 2011, p. 41).
Human decisions also play a critical role in shaping agricultural ecosystems. Studies have
consistently demonstrated that reliance on synthetic inputs disrupts biodiversity and compromises
long-term soil health (Pretty, 2018, pp. 451–452). The complex psychological behavior of humans
in decision making as shown in (Ilbery, 1978, pp. 452–454), in particular towards agricultural
development. Farmers will mostly prioritize their profits before considering sustainability as an
option, marking the human nature as one of the significant forces when it comes to the future
development of converted forest area.
In the context of reemergence of species from edge habitats back into the converted area, it act
as buffers between agricultural fields and surrounding ecosystems, which is critical for maintaining
biodiversity and providing ecosystem services to agriculture (Tscharntke, 2012, p. 672). While it
Team # 2521641 Page 4 of 40
adds unpredictability into the model, it is entirely necessary to incorporate these parameters into
the system to get a valid output.
While existing research provides significant insights into the ecological and economic dimen-
sions of agriculture, gaps remain in understanding how natural processes and human decisions
interact over time in transitioning systems. We seeks to address these gaps by developing an in-
tegrative model that captures the dynamic interplay between ecosystem evolution and agricultural
practices, offering actionable insights for sustainable land management.
1.3 Restatement of the Problem
The conversion of forests into agricultural land disrupts ecosystems, reduces biodiversity, and
creates dependency on chemical inputs. We aim to model the ecological dynamics of a newly
converted agricultural ecosystem, focusing on food web interactions, the effects of pesticides, and
the potential for species reintroduction to stabilize the system. The goal is to develop a framework
for balancing agricultural productivity with ecological health. The tasks that we aim to accomplish
is as follows:
1) Construct an Initial Food Web Model: Develop a model for the newly converted agricultural
ecosystem, including producers, consumers, and decomposers, with each parameters having
different importance weighting.
2) Incorporate Species Reemergence: Examine how the return of specific species from the
edge habitats back into the agricultural ecosystem impacts the stability of the agricultural
ecosystem and its ability to recover balance over time.
3) Analyze the Removal of Herbicides: Evaluate the effects of reducing chemical use on the
ecosystem, consider how these changes introduced by human intervention will influence the
interactions between species in the food web.
4) Model Organic Farming Practices: Investigate the ecological and economic implications
of adopting organic farming, including biodiversity impacts, pest control, crop health, and
sustainability under various scenarios. Embracing the Go Green concept.
5) Provide Recommendations for Farmers: Develop strategies for farmers that balance eco-
nomical trade-offs and ecosystem sustainability, which can encourage the topic of sustain-
ability to be further discuss in the agricultural sense.
2 Model Preparation
We will start our analysis starting from the simplest model, which is a basic food web for a newly
converted forest area. Then more and more restrictions and conditions will be added into the
equation, accounting for reemergence of species, then progressively the removal of herbicides, and
finally framers going green by implementing organic farming methods into agriculture land.
Team # 2521641 Page 5 of 40
2.1 Assumption and Justification
1) Assumption 1: Plants are the sole producers. Plants can be divided into 2 types, the
crops, and the weeds. They both serve the purpose of being the producer to the agricultural
ecosystem.
2) Assumption 2: Insects are the sole primary consumers. All insects are indistinguishable
by species, and serve the purpose of being the sole primary consumer to the agricultural
ecosystem.
3) Assumption 3: There are only 2 species of secondary consumers. To which there are only
birds and bats, both indistinguishable by their kind and prey on insects as their only food
source.
4) Assumption 4: Herbicides only directly affect the weeds. It will not directly harm the
primary and secondary consumers nor will it do any direct effects on the crops.
2.2 Notations
Table 1: Symbols and Descriptions
Symbols Description
N1(t) Population of producers, crops
N2(t) Population of primary consumers, insects
N3(t) Population of secondary consumers, birds and bats
N4(t) Population of producers, weeds
N5(t) Population of secondary consumers, new birds
N6(t) Population of primary consumers, new pests
r1 Population growth rate of crops
r2 Population growth rate of insects
r3 Population growth rate of birds and bats
r4 Population growth rate of weeds
r5 Population growth rate of new birds
r6 Population growth rate of new pests
r7 Population growth rate of bats
K1 Environmental carrying capacity of crops
K2 Environmental carrying capacity of insects
K3 Environmental carrying capacity of birds and bats
K4 Environmental carrying capacity of weeds
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