Logistic Differential Equation (general solution)

Summary

This video tutorial explains the logistic differential equation, a model for population growth with a carrying capacity. The presenter demonstrates how to solve it by separating variables, integrating, and using partial fractions. The solution leads to the logistic growth equation, which describes how populations approach their carrying capacity. The video concludes with finding the constant 'C' using the initial population, making it a practical tool for predicting population dynamics.

Takeaways

• 📐 The logistic differential equation models population growth with a carrying capacity, represented as 'm'.
• 🔄 The rate of population change is proportional to both the population and a factor involving 1 minus P/m.
• ✅ To solve the logistic equation, variables are separated and integrated.
• 🔢 The equation is manipulated to isolate terms involving 'P', the population variable.
• 📉 The process involves multiplying by 'm' to combine like terms and simplify the equation.
• 📚 Partial fractions are used to break down complex fractions for easier integration.
• 🤔 The 'cover-up' method is introduced to determine constants in partial fractions.
• 📈 Integration of both sides of the equation leads to the use of natural logarithms.
• 🔄 The derivative of M - P with respect to P is considered to simplify the equation further.
• 🌱 The final form of the logistic growth equation is P = m / (1 + C * e^(-K*T)), where 'C' is a constant determined by initial conditions.
• 📋 The constant 'C' can be calculated using the initial population and the carrying capacity.

Q & A

What is the logistic differential equation?

The logistic differential equation is a model for population growth that includes a carrying capacity, denoted as 'm'. It states that the rate of change of the population with respect to time is proportional to the population itself and to the factor (1 - P/m), where P is the population size.

What does the term 'carrying capacity' represent in the logistic differential equation?

The 'carrying capacity' (m) represents the maximum population size that the environment can sustain indefinitely.

How does the logistic differential equation account for population growth?

The logistic differential equation accounts for population growth by incorporating a term that reduces the growth rate as the population size approaches the carrying capacity, thus preventing unlimited growth.

What is the significance of the term '1 - P/m' in the logistic differential equation?

The term '1 - P/m' represents the fraction of the carrying capacity that is not yet occupied by the population. It ensures that the growth rate decreases as the population approaches the carrying capacity.

How is the logistic differential equation solved?

The logistic differential equation is solved by separating variables and integrating both sides. This process involves manipulating the equation to isolate terms involving 'P' on one side and terms involving 'dt' on the other.

What is the role of the constant 'K' in the logistic differential equation?

The constant 'K' is an integration constant that arises during the separation of variables and integration process. It represents the initial condition of the differential equation.

What does the term 'partial fractions' refer to in the context of solving the logistic differential equation?

In the context of solving the logistic differential equation, 'partial fractions' refers to a technique used to decompose a complex fraction into simpler fractions that can be integrated more easily.

How does the video script describe the process of integrating the logistic differential equation?

The video script describes the integration process by first multiplying both sides by 'm' to combine like terms, then using partial fractions to decompose the complex fraction, and finally integrating both sides to find the solution.

What is the final form of the logistic growth equation as presented in the video script?

The final form of the logistic growth equation presented in the video script is P(t) = m / (1 + C*e^(-K*t)), where P(t) represents the population at time 't', m is the carrying capacity, C is a constant related to the initial population, and K is the growth rate constant.

How is the initial population size related to the constant 'C' in the logistic growth equation?

The initial population size (P_0) is related to the constant 'C' by the formula C = (m - P_0) / P_0, where m is the carrying capacity and P_0 is the initial population size.

What does the 'C' term represent in the logistic growth equation?

The 'C' term in the logistic growth equation represents the ratio of the difference between the carrying capacity and the initial population to the initial population itself.

Outlines

00:00

📐 Introduction to the Logistic Differential Equation

The video introduces the logistic differential equation, a mathematical model used to describe population growth with a carrying capacity. The equation is presented as the rate of change of population with respect to time, which is proportional to the population itself and a factor (1 - P/m), where m is the carrying capacity. The video demonstrates how to solve this equation by separating variables and integrating both sides. The process involves multiplying through by m to combine terms and then using partial fractions to integrate. The solution process includes dealing with complex fractions and finding constants through the cover-up method. The result is an expression involving natural logarithms, which is then simplified to reveal the logistic growth equation.

05:01

🔍 Deriving the Logistic Growth Equation

This paragraph delves deeper into solving the logistic differential equation. The process involves integrating with respect to time and dealing with absolute values. The video explains how to handle the absolute value by introducing a plus-minus sign and taking reciprocals to isolate terms involving P. The solution leads to a form of the logistic growth equation, which is expressed as P(t) = M / (1 + C4 * e^(-KT)), where P(t) represents the population at time t, M is the carrying capacity, and C4 is a constant. The video also discusses how to find the constant C4 using the initial population, P0, and shows that C4 = (M - P0) / P0. This part of the video is crucial for understanding how to apply the logistic growth model to real-world scenarios.

10:03

🌟 Applying the Logistic Growth Equation

The final paragraph of the script focuses on applying the logistic growth equation to find the carrying capacity and initial population. It emphasizes the importance of being able to rearrange the logistic growth equation to solve for these values. The video provides a formula to calculate the constant C, which is crucial for understanding the initial conditions of the population growth model. The presenter also encourages viewers to subscribe to the channel and thanks them for watching, indicating the end of the tutorial. This part of the script serves as a practical guide for viewers to apply the concepts learned in the video to their own problems.

Keywords

💡differential equation

A differential equation is a mathematical equation that relates a function to its derivatives. In the context of the video, it is used to model the rate of change of a population over time. The logistic differential equation specifically is a type of differential equation that includes a carrying capacity, which limits the growth of the population.

💡logistic differential equation

The logistic differential equation is a specific type of differential equation that models population growth with a carrying capacity. It is defined by the rate of change of the population being proportional to the population size and a factor that accounts for the carrying capacity. The video script explains how to solve this equation step by step.

💡carrying capacity (m)

Carrying capacity refers to the maximum population size of a particular species that the environment can sustain indefinitely. In the logistic differential equation, 'm' represents this carrying capacity, which is a crucial factor in limiting population growth as explained in the video.

💡separable variables

Separable variables is a method used to solve differential equations by rearranging the equation so that all terms involving one variable are on one side and all terms involving the other variable are on the other side. The video script describes this process as a way to solve the logistic differential equation.

💡partial fractions

Partial fractions are used in calculus to decompose a complex fraction into simpler fractions. In the context of the video, partial fractions are mentioned as a technique to simplify the integral when solving the logistic differential equation.

💡integration

Integration is a fundamental operation in calculus that finds the accumulated value of a function over an interval. The video script discusses integrating both sides of the equation to solve for the population function over time.

💡natural log

Natural log, or the logarithm to the base 'e', is a mathematical function used in various scientific fields. In the video, natural log is used in the integral form to solve for the population function.

💡derivative

A derivative in calculus represents the rate of change of a function with respect to its variable. The video script mentions taking the derivative of 'M - P' with respect to 'P', which is a step in the process of solving the logistic differential equation.

💡initial population (P0)

The initial population is the starting value of the population at time zero. The video script explains how to relate the initial population to the logistic growth equation by using it to find a constant in the equation.

💡logistic growth equation

The logistic growth equation is a model that describes the growth of a population that is limited by the carrying capacity of the environment. The video script provides the general form of this equation and explains how it is derived from the logistic differential equation.

💡function notation

Function notation is a way to represent functions in mathematics, often using a variable followed by arguments in parentheses. In the video, function notation is used to express the population as a function of time, denoted as P(t).

Highlights

The video introduces the logistic differential equation, a model for population growth.

The logistic differential equation includes a carry capacity factor, representing the maximum sustainable population size.

The rate of population change is proportional to both the population and a factor involving the carry capacity.

The video demonstrates solving the logistic differential equation by separating variables.

A constant K is introduced during the separation of variables.

The equation is manipulated to group terms and prepare for integration.

Partial fraction decomposition is required for integration, involving complex fractions.

The video explains multiplying by 'm' to simplify the equation and combine terms.

The process of integrating both sides of the equation is outlined.

The use of the cover-up method for partial fraction decomposition is detailed.

The video clarifies how to find constants for partial fraction decomposition.

Integration leads to the natural logarithm of the population terms.

The derivative of the carry capacity minus population is discussed.

The video shows how to isolate P to graph the logistic growth function.

The logistic growth equation is derived in a form that includes an exponential term.

The video explains how to handle the absolute value in the logistic equation.

The final form of the logistic growth equation is presented, relating population to time.

The video provides a formula for finding the constant C in the logistic equation using initial population.

The importance of the initial population in determining the logistic growth curve is emphasized.

The video concludes with a summary of the logistic growth model and its practical applications.

Transcripts

okay this video show us another

differential equation to model

populations and this right here it's

called the logistic differential

equation which is says the rate of

change of the population with respect to

time is proportional not only to the

population itself but also to this

factor namely 1 minus P over m where m

is what we call the carry capacity it's

pretty much like the maximum number that

the population can reach and this video

showed us how to solve this we can do

this by separating the variable I want

just multiplied it here on both sides

and I will just keep the constant K

right here I will have to divide this

and that to the other side so all the

pieces will be together as you see on

the left hand side we have the P world

so these are now will be in the

denominator and I'll just write down P

times 1 minus P over m and that's 1 over

that and that's what we have and we

should try to integrate both sides but

in a P world you see in the denominator

here we have P times 1 minus P over m

well we have to do some partial

fractions but this right here it's a

complex fraction so that's changes

little bit so let me just just multiply

this by mm like this so we can combine

like terms

both are stuff combine the fractions so

here you were pretty much having 1 over

P times M minus P and then this is over

m right but let me just put the N on the

top like that and then we have the DP

like this how's that so once again you

just multiply this by mm and you have

this denominator and you just bring this

M up up so you have this right here and

you have this is equal to K DT and

before intergrade we have to do partial

fraction but let me just put integral

sign first on both sides okay so in

water for me to integrate that I will

have to do partial fraction I will have

to what this is factor they already

power water to break this apart as some

number over P and then adding with

some other constant over m minus P and I

will just do this the quick and dirty

way name D the cover-up method I want to

figure out what constant goes up right

here alright notice this is just a

linear term M to the first power so on

the top it has to be a constant I will

go back to the original and then cover

this factor up and you have to ask

yourself how can you make you a hand

equal to zero well you have to plug in P

cos you're here right

well cover this up by you plugging 0

into this P the remaining P when you do

that you have M on the top over a minus

0 which is just 1 so here you have a 1

right here and we do the same thing I

want to figure this out I will come back

to the original and I'm going to cover

up the same factor on the denominator

and you have to ask yourself how can you

make this equal to 0 and minus p 2 p 0 P

has to be M so you're plugging M into

that P and now you have M over m again

so you have 1 again it's not my fault

it's just how it is

so one on one okay it's not that bad I

told you and we still have to integrate

K DT right here and now we can't agree

here we get natural log absolute value

of P and if you know beforehand P is not

positive for population growth right the

amount of ends were ranked ooh so ever

but if you want to talk about well no no

just keep that fog right just just just

do it that's for you anyway right here

we also have natural log absolute value

of M minus P like this well we have to

ask ourselves what's the derivative of M

minus p with respect to P the derivative

this is negative one so be sure divided

by negative 1 so Y actually is a minus

here it's actually a minus here okay on

the right hand side we have K T where

you integrate K in the T world so you

have that and now that's at C 1

and from here I want to get a P by

itself so I can graph that function well

we have two L ends and this is

subtraction so I can just write natural

log of absolute value of P over m minus

P like that and this is equal to K T

plus c1 and to carry of the Ln B to e to

the power e to this power so that is

send out we cancelled it and we see this

is pretty much P over m minus P in the

absolute value and now e to the KT is

just a function part we maintain to be

this and remember where you have e to

the sum of a power right here you can

look at this as e to the KT times e to

the C 1 power

but since he wants a constant is also a

constant a constant to a constant is

another constant let's put down C 2

alright so far so good but I want to

carry off the absolute value it's okay

seriously just carry off the absolute

value but be sure you attach a plus

minus on the other side and now we have

P over m minus P and that's equal to

this right here but I want to get a P by

itself and an easy way to do is that I

will just take the reciprocals on both

sides because that way I will just end

up with one term in the denominator so

let me do that so I will have this

upside down which is M minus P over P

and I'm going to put this in the

denominator as well the reciprocal of

that so already done as 1 over well plus

minus C 2 it's just another constant so

I'll just write RC 3 putting this done C

3 in the denominator

likewise I'll write this down as e to

the KT down in the denominator but I

want to do some fix because here you see

I will write down there a minus P over P

Part C is a constant 1 minus EC

it's another constant so I'll just write

our CFO for that and e to the KT

in the deep nominator I will just give

that to be a negative exponent so

already says e to the negative KT like

that so far so good and the beauty of

doing this is that when we have one term

let me show us multiply P on both sides

it's easier this way so I have M minus P

and that's equal to C for p e-

KT like this and to continue I will just

add a p on both sides so we have m and

that's equal to once I add a P on both

sides I will have P Plus this term and

this term of course we see the P right

here so we can factor out a P and let me

put that at the end right here and we

have that 1p so I will have 1 plus C 4 e

to the negative KT and I like to put a P

at the end it's because this is

multiplication sometimes when you put a

pea in the front it seems like P as a

function of Y ever that's no good well

right here at the emphasize that this

was P times the parenthesis of M minus P

right so this was a multiplication so

sometimes you just have to emphasize any

way I am about to tip I process by this

term so we get P by self and this is the

population right and that's equal to M

on the top over 1 plus C for e to the

negative KT and this is usually the form

for what we call the logistic growth

equation alright so it's really cool and

here is the function notation I want to

include we usually write P of T to be

this and this is the general form

alright and once you have this done you

don't need too spicy for you can just

write down the C the legitimacy if you

like but they meet here guess what is

this right here this does not represent

the initial population

let me show you what this represents

suppose you were given the initial

population and let me just read yes P of

0 is equal to P naught like this

this right here it's a function notation

P of 0 that means when T is equal to 0

this right here it's peanut so let's see

what we get when we have that I will

just pull down peanut right here so we

have peanut like this and that's equal

to N on the top over 1 plus just to see

you don't have four anymore so you don't

see and we have e to the negative K

times 0 and this is a multiplication

right that's a bit sort of parentheses

sometimes but it's ok she's not that bad

P naught is equal to M on the top over 1

plus this is pretty much just going to

give you 1 so we have 1 plus C like this

and on the stuff of C so perhaps I can

do the reciprocal of both sides again

and I will multiply the M to the other

side so I get and now let me just minus

1 both sides so I guess C equals 2 and

of course for the 1 I can write this

Pina

over Pina like this so in the end we see

C is equal to M minus P na over P na so

if you were given the initial population

right here what you have to do is

looking for the carrying capacity is and

then minus the initial population and

then divided by the initial population

and this right here will be for this C

right here

so I will actually pass this right here

for you guys and I also include this for

you guys sometimes maybe your

differential equation is set a

differently but if you can get that

differential equation to being this form

then you can just quote this formula

right here and you can use this to help

you to find out what the series right so

anyway hopefully you guys saw like this

video and if you are new to my channel

please subscribe thank you guys so much

and as always that's it