112-2 工程數學(一) 期末報告，Solving Malthusian model with EM

Summary

The video script discusses the Malthusian model, which posits that population growth is exponential while resource growth is linear, leading to potential population crises. It introduces the logistic model as an improvement, considering resource limitations and carrying capacity, providing a more realistic population growth prediction. The script also addresses the impact of human overuse on the environment and the importance of sustainable resource management.

Takeaways

• 📚 The Malthusian Model, proposed by Thomas Robert Malthus in 1798, suggests that population growth is exponential while resource growth is linear.
• 🔍 The model predicts a population crisis if unchecked population growth surpasses resource capacity, leading to a population crash.
• 📈 The mathematical expression of the Malthusian Model is \( \frac{dP}{dt} = R \cdot P \), where \( P(t) \) is the population at time \( t \) and \( R \) is the constant growth rate.
• 🧩 The model can be used to estimate historical population numbers and predict future growth trends to assist in resource planning and crisis detection.
• 🌐 The application of the model can also help estimate if specific resources are sufficient to meet future population demands.
• 📊 The world population growth from 200,000 to 2010 was calculated using the Malthusian Model with an initial population of 6.14 billion and a growth rate of 1.4% per year.
• 🚫 The Malthusian Model is based on ideal conditions and does not account for real-world constraints such as resource limitations.
• 🔧 The Logistic Model was introduced by Pierre François Verhulst in 1838 to address the limitations of the Malthusian Model, considering limited resources and population stabilization.
• 📘 The Logistic Model's differential equation is \( \frac{dP}{dt} = R \cdot P \cdot (1 - \frac{P}{K}) \), where \( K \) is the carrying capacity.
• 🌟 The solution to the Logistic Model shows population growth slowing down and stabilizing near the carrying capacity \( K \).
• 🌍 As of May 2024, the global population has surpassed 8.1 billion, and real-world population growth lies between the predictions of the Malthusian and Logistic Models.
• 🌿 Human activities have caused significant environmental damage, leading to resource shortages, loss of biodiversity, and climate change, emphasizing the need for sustainable resource management.

Q & A

What is the Mausian model?

The Mausian model, proposed by British economist Thomas Robert Malthus in 1798, suggests that population growth follows an exponential pattern, while the resources necessary for survival grow linearly or arithmetically. It predicts that unchecked population growth will eventually surpass the capacity of available resources, leading to a population crash.

What is the mathematical expression of the Mausian model?

The Mausian model is represented by a simple differential equation: the derivative of P with respect to time T (dP/dT) equals to the constant growth rate R multiplied by the population at time T (R * P(T)).

How is the Mausian model solved mathematically?

The Mausian model is solved by separation of variables, integrating both sides to obtain the natural log of the population P equals to R times time T plus an integration constant C. The exponential form of the solution is P(T) equals to P0 (initial population) times e to the power of R times T.

What are the applications of the Mausian model?

The Mausian model can be used to estimate historical population numbers, predict future population growth trends to assist governments in planning resource allocations, and monitor population growth rates for early detection of potential population crises. It can also help estimate whether specific resources in certain areas are sufficient to meet future population demands.

How was the world population growth calculated from 200,000 to 2010 using the Mausian model?

The world population growth was calculated using the exponential growth formula with an initial population of 6.14 billion and an intrinsic growth rate of approximately 1.4% per year, resulting in a chart of population growth during 2000 to 2010.

What are the limitations of the Mausian model?

The Mausian model is overly idealistic as it assumes unlimited resources, which does not align with real-world conditions. In reality, there are resource constraints, and population growth is usually not exponential due to various factors such as medical advancements and policy influences.

What is the logistic model and how does it differ from the Mausian model?

The logistic model, introduced by Pierre-François Verhulst in 1838, assumes limited resources and predicts that population growth will slow down and eventually stabilize at a carrying capacity. It differs from the Mausian model by taking resource limitations into account, providing a more realistic prediction of population dynamics.

What is the mathematical expression of the logistic model?

The logistic model is represented by the differential equation dP/dT = R * P(T) * (1 - P(T)/K), where P(T) is the population at time T, R is the intrinsic growth rate, and K is the carrying capacity.

How is the logistic equation solved?

The logistic equation is solved to show that the population at time T (P(T)) equals K divided by 1 plus (K minus P0) over P0 times e to the power of negative R times T, where P0 is the initial population.

What is the carrying capacity in the context of the logistic model?

The carrying capacity (K) in the logistic model is the maximum population size that the environment can sustain indefinitely, given the food, habitat, water, and other resources available.

How does the logistic model account for real-world population growth?

The logistic model accounts for real-world population growth by incorporating the concept of carrying capacity, which reflects the environmental pressures and resource limitations that influence population growth rates and lead to stabilization.

What is the current global population as of May 2024 according to the script?

As of May 2024, the global population has surpassed 8.1 billion, indicating an acceleration in population growth rates after the rapid growth following the Black Death and the Great Famine in the 14th and 18th centuries.

How does human overuse of Earth's environment impact the planet?

Human overuse of Earth's environment leads to resource consumption exceeding the planet's regenerative capacity and ecosystem degradation. This results in visible damages such as deforestation, overfishing, and climate change, which can cause resource shortages, loss of biodiversity, and ultimately affect human development.

What is the conclusion of the engineering mathematics report on population growth models?

The conclusion is that while the Mausian model introduced the fundamental concept of exponential growth useful for analyzing population dynamics, it is overly idealistic. The logistic model improves upon the Mausian model by considering resource limitations and the importance of carrying capacity. However, real-world population growth has exceeded predictions, exerting even greater pressure on the environment due to human overdevelopment and exploitation.

Outlines

00:00

📊 Malthusian Model and Exponential Growth

The first paragraph introduces the Malthusian model, proposed by Thomas Robert Malthus in 1798. It discusses the model's assertion that population growth follows an exponential pattern while the growth of resources is arithmetic. The model predicts a population crash due to unchecked growth surpassing resource availability. The mathematical expression is given by the differential equation \( \frac{dP}{dT} = R \cdot P(T) \), where \( P(T) \) is the population at time \( T \) and \( R \) is the constant growth rate. The solution is derived using separation of variables, leading to an exponential form. Applications of the model include estimating historical population numbers, predicting future growth trends, and assessing resource sufficiency for future demands. An example calculation for world population growth from 2000 to 2010 is provided, using an initial population of 6.14 billion and a growth rate of 1.4% per year.

05:02

🔍 Logistic Model and Realistic Population Dynamics

The second paragraph delves into the logistic model, introduced by Pierre François Verhulst in 1838, as an improvement over the Malthusian model by incorporating resource limitations. The logistic model's differential equation is presented as \( \frac{dP}{dT} = R \cdot P(T) \cdot (1 - \frac{P(T)}{K}) \), where \( K \) is the carrying capacity. The solution to this equation is given, showing how population growth slows and stabilizes near \( K \). The paragraph also discusses applying the logistic model to the world population growth from 2000 to 2010, with an initial population of 6.14 billion, a growth rate of 1.4%, and a carrying capacity of 7 billion people. The actual global population growth is contrasted with the predictions of both the Malthusian and logistic models, noting that the real growth lies between the two. The paragraph concludes with a discussion on the environmental pressures caused by population growth and the need for sustainable resource management.

10:03

🌿 Addressing Environmental Impact and Resource Overuse

The third paragraph addresses the environmental impact of human overuse of Earth's resources, leading to resource shortages, loss of biodiversity, and climate change. It emphasizes the importance of cherishing resources and minimizing harm to the Earth. The paragraph concludes by summarizing the Malthusian and logistic models, highlighting the transition from an idealistic view of unlimited resources to a more realistic approach that considers resource limitations and carrying capacity. It also touches on the consequences of environmental degradation and the need for sustainable development to mitigate the pressures on the environment caused by human activities.

Keywords

💡Malthusian Model

The Malthusian Model, named after British economist Thomas Robert Malthus, is a theory that suggests population growth follows an exponential pattern while the resources necessary for survival grow linearly or arithmetically. In the video, this model is central to understanding the concept of population growth and its potential consequences when unchecked, leading to a population crisis. The model is used to estimate historical population numbers and predict future trends, which is vital for resource planning and crisis management.

💡Exponential Growth

Exponential growth is a pattern of growth where the rate of increase of a quantity is proportional to the quantity itself. In the context of the video, the Malthusian Model assumes that population growth follows this pattern, which can lead to rapid increases in population size. The script mentions that the mathematical expression of the Malthusian Model is represented by a differential equation that embodies this exponential growth.

💡Resources

Resources in the video refer to the materials and conditions essential for the survival and growth of a population. The Malthusian Model posits that these resources grow linearly, which contrasts with the exponential growth of the population. The script discusses how the finite nature of resources can lead to a population crash if growth remains unchecked, highlighting the importance of resource management in relation to population growth.

💡Population Crisis

A population crisis, as mentioned in the video, is a situation where the population growth rate surpasses the capacity of available resources, leading to potential societal and environmental collapse. The Malthusian Model predicts this outcome if there are no constraints on population growth, emphasizing the need for sustainable resource use and population control measures.

💡Differential Equation

A differential equation in the video is used to mathematically express the Malthusian Model. The script describes the equation as dP/dt = R * P, where P represents the population at time T, and R is the constant growth rate. This equation is key to understanding how the model predicts population growth over time.

💡Logistic Model

The Logistic Model, introduced by Pierre François Verhulst in 1838, is an improvement upon the Malthusian Model that takes into account resource limitations. The video explains that this model assumes a carrying capacity, beyond which population growth slows and eventually stabilizes. The Logistic Model is presented as a more realistic approach to predicting population dynamics in the face of finite resources.

💡Carrying Capacity

Carrying capacity in the video refers to the maximum population size that an environment can sustain indefinitely, given the food, habitat, water, and other resources available. The Logistic Model incorporates this concept, suggesting that population growth will slow as it approaches this capacity, contrasting with the unbounded growth predicted by the Malthusian Model.

💡Environmental Pressure

Environmental pressure, as discussed in the video, is the strain placed on the environment due to human activities and population growth. The script mentions that an increasing population leads to more resource consumption, more waste generation, and greater environmental impact, which can result in ecosystem degradation and other visible damages.

💡Deforestation

Deforestation is the clearing of trees and forests, often to make way for agriculture, urban development, or logging. In the video, it is cited as a specific example of human overuse of the Earth's environment, leading to resource shortages, loss of biodiversity, and climate change, which are part of the broader consequences of unsustainable population growth.

💡Overfishing

Overfishing is the act of catching fish at a rate faster than they can reproduce, leading to a depletion of fish stocks. Although not explicitly defined in the script, overfishing is mentioned as one of the consequences of resource overuse, illustrating the impact of population growth on marine ecosystems and the importance of sustainable fishing practices.

💡Sustainability

Sustainability in the video is implied as the ability to maintain or improve the environment, economy, and society without depleting resources or causing long-term damage. The discussion on population growth, resource use, and environmental impact all revolve around the concept of achieving sustainable practices to ensure the long-term viability of our planet.

Highlights

The Malthusian model, proposed by Thomas Robert Malthus in 1798, suggests that population growth follows an exponential pattern while resources grow linearly.

The model predicts a population crisis if unchecked growth surpasses available resources.

Population growth rate is directly proportional to the current population size in the Malthusian model.

The mathematical expression of the Malthusian model is a differential equation: dP/dt = R * P(t).

The model's solution is an exponential function: P(t) = P0 * e^(R * t).

The Malthusian model can estimate historical population numbers and predict future trends.

Governments can use the model for resource allocation and early detection of population crises.

The model helps estimate if resources are sufficient for future population demands.

World population growth from 200,000 to 2010 is calculated using the Malthusian model.

The model assumes ideal conditions without constraints, which is not realistic.

The logistic model, introduced by Verhulst in 1838, addresses the limitations of the Malthusian model.

The logistic model assumes limited resources and population growth that stabilizes at carrying capacity.

The logistic model's differential equation is dP/dt = R * P(t) * (1 - P(t)/K).

The logistic model's solution shows initial exponential growth slowing down and stabilizing near the carrying capacity.

The logistic model is applied to estimate population growth with a carrying capacity of 7 billion people.

Actual world population growth as of May 2024 has surpassed 8.1 billion, showing a slowdown in growth rates post-World Wars.

Human activities have caused significant environmental damage and overuse of Earth's resources.

Environmental overuse leads to resource shortages, loss of biodiversity, and climate change.

The Malthusian model is overly idealistic, assuming unlimited resources, which does not align with real-world conditions.

The logistic model provides a more realistic prediction of population dynamics, considering resource limitations.

Human overdevelopment and exploitation have led to environmental degradation, increasing pressure on the Earth.

The report concludes the importance of cherishing resources and minimizing harm to the Earth.

Transcripts

my name is j today our topic is the

mathian model these are the members of

our

groups what is the mausan model the

mauian model proposed by the British

Economist Thomas Robert mos in

1798 suggests that population growth

follows an exponential pattern while the

resources necessary for survival growth

linearly or

arithmetically according to this model

if population growth remains unchecked

it will eventually surpass the capacity

of available resources leading to a

population

crasis the methan model assumes that the

population growth rate is directly

proportional to the current population

size the mathematical expression of this

model is represented by a simple

differential ination the derivative of P

of T equals to R * P of T where P of T

represents the population at the time T

and R is the constant growth rate now

let's delve into the mathematical

expression of the methan

model let's solve it by separation of

variables first

let's integrate both sides

simultaneously then we can obtain

natural log P equals to R * t + C where

C is the integration

constant the exponential form of the

solution is p of T equals to p 0 *

exponential of R * T where p 0 is the

initial

population applications of the methan

model first we can use this model to rly

estimate the historical population

numbers second we can predict future

population growth Trends through the

model which can assist governments in

planning resource allocations and early

detection of potential population crisis

by monitoring population growth rates

third through the model we can estimate

whether specific resources in certain

areas such as water land and energy are

sufficient to meet future population

demands in this page we will calculate

the world population growth from 200000

to 2010 by using the original matusan

model using the exponential growth

formula mentioned earlier let's plug in

the initial population of 6.14 billion

and an intrinsic growth rate of

approximately 1.4% per

year with the calculation of previous

page we can get this chart of population

growth during 2000 to

2010 the models discussed earlier are

all based on ideal conditions without

any

constraints however in reality there are

many issues to consider firstly there

are resource constraints ideally

resources are infinite and can

accommodate unlimited population growth

but this is not the case in reality

secondly advancements in medical

technology or policy influences can

affect population growth in specific

regions therefore population growth in

reality is usually not

exponential so in this page we are going

to talking about the introduction to the

larest model to address the limitation

of the mausan model

and this

words

FR introduced the logistic model in

1838 assuma limit Resources with

population grow slowing down and

eventually stabilizing at a c varing

capacity and on the next page we are

going to talking

about mathematical expression of the

logistic model differential equation

from the derivative of P equals RP where

P of T is the population at time t r is

the intrinsic growth rate and K is the

carrying

capacity on this page is going to

showing you how to solving the logistic

equation solving the logistic equation P

of T = k/ 1 + K minus p 0 over P0

exponential minus r t where p Z is the

indidual population describes population

growth that is exp exponential initially

slowing down as it approaches the

carrying capacity eventually stabilizing

near

K and the population growth 20 to 2010

logistic model

let's apply the exponential growth

formula mentioned yearly by

substituting the initial population of

6.14 billion and and intrinsic growth

rates of

approximately

1.4% per year additionally let's

consider a carrying capacity of 7

billion

people population growth curve is

displayed on the next

SL so as you can see the table of it in

contrast under constrain conditions the

ideal population size is not as high as

previously predict with a significant

decrease in the weight of the population

increase so the real world population

growth as of May

2024 the global population has surpassed

8.1 billion after the rapid population

grow following the black death and the

great F famine in the 4 18 century

factors contributing to a Slowdown in

population

growth diminishes after the two World

Wars this has led to

an

accelerated global population growth

rate exceeding

1.8%

anually the growing population has

intensified resour

consumption necessi more energy and

generating more waste th increasing

environmental

pressure from this chart we can observe

that the actual population growth lies

between the Mion model and the logic

model it is slightly lower than the

predictions made by the ausan model but

somewhat higher than those predict by

the logic

model and on this this page are talking

about human overuse of Earth

environment it is well known that human

activities have caused tremendous damage

to the Earth

when the environment is

overuses it Leeds to

Resource consumption exceeding

Earth's

regenerative capacity

and E

ecosystem

degradation which are iro visible

damages specific cases include

DeForest

deforestation over fish and climate

change the consequence of overusing

resource resulting in resource

shortages loss of

biod diversity and

criminate an noral PE impact

ultimately affect human

development therefore we must learn to

cherish resources and minimize harm to

the Earth as much as

possible the last in conclusion the M

Fusion model initially introduced the

foundamental concept of exponential

growth which is useful for analyzing po

population growth however the Maan model

is overly idealistic as it assumes

unlimits resource which does not align

with FW conditions the logistic model

improves through the shortcomings of

the Matan model by taking resource

limitations into account it provides

more realistic prediction of

population Dynamics and highlights the

importance of carrying

capacity in the real real in the real

world population growth has ex

exceed X

ations cising even greater pressure on

the

environment human over

development and

expl exploitation has led to

environmental overuse the de

gradation and this is the division of

our

work this is the end of our engineering

mathematics report for this time thank

you for your listening