Introduction to Population Models and Logistic Equation (Differential Equations 31)

hi and welcome to another video in this video we do something exciting not that the rest of men unexcited super exciting but in this video we start to look specifically at some some very neat applications of differential equations now we've talked a ton about how to solve them now and we've done solutely some examples of some applications on what they're used for but we're going to talk about some very specific things in these next few videos we're going to talk about population growth we're gonna talk about how difference of equations do it a fairly good job at modeling population growth and decline we're also gonna look at some different techniques to look at this we're gonna look at at what are called equilibrium solutions and stability and we're gonna look at critical points and how all those play together but it starts with an understanding of how populations really work and how they can be modeled by differential equations so in this introductory section I'm going to show you that I'm gonna show you two very common ways that populations can grow and decline will talk about extinction will talk about population explosion will talk about limiting the carrying capacity and limiting populations so those are the things that I want to discuss in this video and then we're gonna map that over the next several videos we'll start with some very simple examples and deal with some ones later there that are quite advanced so let's run through exactly what it is that we're gonna be talking about and how a difference of equations can model these populations so we all know that populations change and barring an influx of immigration or people leaving like an exodus or something populations generally change depending on birth rates and death rates so we all know that the kids are born and and people die and and our population changes or or any sort of the population where things are born and dying it changes over time and so because their population changes due to births and deaths it makes sense that how our population is changing can be expressed by births minus deaths now know note something because population is a change in population and not population itself or with a rate of change with respect to time or a time rate of change it sounds kind of funny but it says how the population is changing in a certain amount of time and that's what we're going to deal with the increment for just a little bit so so think through this does a population change is that based on births and deaths well it seems so so how a population is changing is let's say births minus deaths great all right well well we typically have a birthrate and a death rate in any population but here's what we mean when we say a birth rate for a death rate what we mean is a birth rate would be the rate of births per unit of population per unit of time that sounds kind of funny and let's we understand what that is so let me explain this to you so when we say like it's a birth rate as this we say a birth rates let's suppose five births per thousand people per year does that make sense so it's kind of based it's not kind of it's based on the population and a time period and so this birth rate is a birth rate per unit of population per unit of time it depends on both things and so we're gonna say all right well if the birth rate depends on population population change also depends on time so it's related to both of them in other words we say hey this birth rate is related to both population and the population is related time it's going to include both of these ideas same thing with the date the death rate so our death rate is all right how many deaths per thousand people per year or per hundred people per month whatever it is it's per unit of population we're talking about per unit of time I hope this make this makes sense see let's do it a little bit of a recap here we're going to be talking about how population changes that's it's gonna be a derivative in just a minute with respect to time how the population changes is related to the births we have and the deaths we have and the difference between them so if births are outpacing deaths if the birth rate is outpacing death rate our population is going to climb if the other way is true if the birth rate is is below death ray probably it's going to fall you talk about that so um so let's let's look more specifically about this so if our birthrate is this this beta and our death rate is as Delta and these are all remember what this is this is the rate of births per union population per unit time well then let's let's think about that so our births themselves not the birth rate but the births themselves the number of births that we have since the births would be rated based on the rate of our births per unit of population per unit of time and the population is whatever it happens to be so amount and time is whatever increment of time we're considering when the number of births would be the rate at which our our things are our group of subjects is growing so the birth rate times whatever population we have at that time times however much change in time we're considering now let's go through this real quick so because a birth rate is okay number of the birth rate is per unit of population per unit time let's make sure these units match up so birth rate is per unit of population so if I say hey this is a this is five births per thousand and I have thousands here some amount of population units per year well I have a change in years here or days or seconds or whatever we're considering so these units would match up to the number of births and needs to get that so if this is per population and then per unit of time we're going to get the number of births ear burn three times population times time the same thing would happen with our deaths so the deaths would be the the rate at which things are dying her unit of population per unit of time some sort of thinker now these are incremental so this is considering every small that will increment well if we put this together if the way our population is changing is related to our births minus deaths and births can be represented by hey a birth rate per unit of population per unit time and deaths the same thing but let's take this and say hey the way that our the amount of change of our population the way our population is changing can be represented by births minus deaths only we're going to use this Earth's minus deaths let's do that right up here so then our incremental change in population would be about equal to our burdens minus our deaths now let's let's clean this up a little bit we we know that our birth rate can change with respect to time we said this is birth weight per population per time and it can change over time and so can our death rate the so current population is are all changeable with respect to time that all rates of change with respect to time so I'm gonna drop the T here here here and here we're gonna keep that delta T that the incremental change in time for just a little while just to clean this up a little bit so our change of population is remember birth rate times population times time minus our deaths death rate times population times time and we're gonna factor out that delta T and we're gonna divide so if we factor this out so factoring out that Delta team gap bird three times population - death rate Amish population and then we get this delta T let's let's divide that so we're going to buy the delta T and we have this birth - death times population if I continue to factor that piece so all we've done here is said all right you know what change in population should be the way that our births and deaths are related so let's subtract them and say the change in population would be births minus deaths we can get positive population growth or negative population growth that's a decline we say all right the birth weight what that means is the rate at which our were being born per you know population per you'd at a time no problem so our total births will be given them here birth rate times population depends on death rate times population times time would give us our debts at a moment of time that's what that means so if our population is changing comparing our birth to deaths then we can subtract our births and our deaths no problem that that's all we've done we have factored out our delta T we've divided by our delta T and we haven't we have an estimate with for how our population is changing with respect to time and incremental change if we take the limit then that error should go to zero and we have this this relationship that the rate at which our population is changing with respect to time is birth rate minus death rate times the population now this is very interesting because this is how a lot of populations actually change is in some way they are related to April they're proportional to the population this is this is really stranger babies in or two really important cases here in just a bit so what I want to talk about right now is if our births and our deaths are commonly proportional to our population we're gonna look at two case on how that is so one last little recap all the way through then we're gonna talk about these two cases and see some really really cool things about it I'm gonna go super in-depth on to why it is that sometimes we get is what's called a carrying capacity or a limiting capacity we have a threshold oh sorry we have a bounded situation with our population and sometimes we don't sometimes you have extinct extinction or sometimes we have explosion so in one of those two cases so we're gonna talk about that for a really long time this will be a long video as far as my explanation is concerned now our birth rate and our death rate can change that they're typically always changing and there's two really important cases that we're going to consider one and there's more than this there's more than just two but these are two very important cases one is oftentimes as our population increases the rate of births decreases map now that that can be due to intelligence of the species saying hey there there's not a lot of resources here so if we make more things they're not gonna be able to eat or just sometimes naturally this just happens and where the the rate of births is decreasing as the population is increasing on a linear type of scale we're gonna look at that the other case is what if we don't have something that that's that's thinking about that we're considering that is not following that model sometimes birthrate is just straight-up proportional to population so as the population increases the birth rate also increases as the population decreases the birth rate also decreases that could be due to just availability so let's say that there are lots of males and lots of females and they're getting together and they're doing the bunnies do that population could explode oh but but wait a minute what the population is declining well then there's less males and there's less females and they're not meeting is often that population become extinct we're gonna look at that right now so two important cases are what if the birth rate is decreasing as the population is increasing on a linear scale I want to also mention in both these cases we're going to consider the death rate to be constant so now that doesn't mean that for a billion bunnies we only get five deaths so what that means is that the rate of deaths per unit of population per unit of time is staying the same so old age people things are died of old age or something or some other factor that's in there but we're not going to consider that to be changing for these two examples can can of change of course and I will show you some examples of that there's lots to this which is going to take several videos so two important cases we're going to consider whether the death rate is constant is what if the birth rate is decreasing as our population is increasing on a linear scale so so how would that look well then our our birth rate would be some initial birth rate minus some birth rate that's related to our population so can you see how this would became first we can see it's linear based on the variable P yeah can you see that we would start somewhere can you see that we would have a negative slope so our birth rate will be declining as our population increases the population is a positive number so if V sub 0 is positive so we start with some sort of birth rate that's positive and we have some sort of population that's growing remember this number is this positive number is well our birth rate is going to be declining notice will still have births but the rate at which they're born per unit of population for every time period is going to be declining as our population it's bigger and bigger as our population gets smaller and smaller it would increase so they're they're inversely related that's the idea okay well and our death rate would just be C what would our death rate be well ever if our rate of death is not not changing if it's constant well let's just call this the initial death rate Delta sub 0 why why would that be a constant well it's not changing we can say whatever is in the beginning it's going to be there forever per unit of population per unit time now we're gonna do say all right provided that change of population is births minus deaths we figured out a birth rate of death rate so we figured out births and deaths in relation to level in terms of that birth rate and death rate and we factored some things we didn't limit we have this relationship the change in population with respect to time is birth rate minus 10 three times population guys that's got to make sense that's to make sense of the way of population is changing with respect to time would be the difference between the birth rates and the death rates types of population it's it's almost not even a long time through this but it's almost an obvious obvious statement so yeah the birth rate minus death rate gives you kind of a rate of change per unit of population so multiplied by the population you'd have the rate at which this is changing with respect to time all right so our if our birth search our birth rate is dropping as our population is increasing linearly and our death rates the same we're going to take this we're going to substitute this for Earth's this for our deaths so our change in population our time rate of change population in respect to time would be this initial beta initial birth rate minus some birth rate that's changing with our population - our death rate that's always constant times this population this part is this we've just said we have a initial birth rate and we're dropping as our population increases linearly and then our death rate is held constant man here's what we're gonna do we're gonna distribute all this stuff we're in a group some things together so let's look at the change in population inspect the time let's say all right so let's uh let's just read the P and then what I want to do is I want to group this together so that I have my alright so to have my my keys out front and my p squared the back so I'm gonna do this so we can see what factors in with is it I almost factor out the P from here I'm gonna put my initial birth rate my initial death rate together and then what we're gonna do is we're going to factor out a P so wouldn't let us sorry we're gonna factor out a this birth rate that's changing with respect that is affecting our our birth rate change with respect to the population we're gonna factor out this beta sub 1 and P for both of these and what's gonna happen we're gonna force this guy to factor so we're gonna be dividing that so I'm looking the same I can certainly factor out a P we'll put that out front I'm also going to factor out this right there that that birth rate that's that's affecting our total change of birth rate it's a cut it's going to be young constant so let's let's factor that out so we do that let me have this maybe someone population inside we're gonna force that to factor so factory means divide or divided the base of one that would be beta sub 0 I'm still this is your over base of 1 we factored out the P minus P let's make sure that this works man it's a lot of stuff so if we were to distribute this base of 1 we can so we get beta sub 0 double sub-zero and of teen that's this and then if we distribute here we did this is someone P squared that's this these are the same exact expression out now a lot of times in your textbook or whatever you're working out of you're not gonna see that you're gonna see this you can see suppose that this and this and this and this say they are all held constants if they're held constant then let's call this one k let's call this one m and you get something that you might have seen before you might you might see it now where did they get that they've got this expression and there's a lot of things hidden in there isn't there they've got this expression by saying this constant writing and it is a constant if I hold our initial birth rate initial death rate and the way that that's changed well the the thing that affects our guard told birth rate change if I hold this all be constant then this right here is the constant we call em this right here is a constant we call it K this right here is called the logistic equation so if we called these things to be constant and all positive then what this is is called the logistic equation you might want to you might want to note that for a second because this happens on a lot of population change is that we have something that our population the rate of births is decreasing as our population is increasing but death rates held constant it's going to yield something called a logistic equation now I'm gonna make a statement right now before we go on to the next thing so I'm gonna have to go through this one last time I'm gonna have to erase some of this and put the logistic equation right here and then I'm going to talk about what if birth rate is proportional population we'll talk about what happens here now I'm gonna go and if you don't want to watch it that's fine I'm gonna go behind the scenes in the list so I'm gonna talk about everything you really need and I'll talk about the the reasoning behind that in kind of a quasi proof sort of thing to verify what I'm saying so if you don't want to watch the rest of that you just want the bare bones basics here it is for the first part least when we did all this stuff we said the rate of population change is related to birds minus deaths we got all the stuff now then we said okay if our birth rate is decreasing as population increases but death rate is held constant it's going to yield here's all the math behind it something called the logistic equation this is the missus logistic equation if this this and this are all positive one suppose it means velocity that means that we start with a positive birth rate and we're subtracting something that is based on population this is always positive and this is positive then this means that our birth rate is decreasing as their population is increasing we have to have this positive otherwise if that was negative obviously this would be growing as population increases that's our next case and so this would say that if these are all positive our debt rates staying the same and the end is positive death rate that should make sense we have a positive initial birth rate that is actually positive and initial and in a way that our population is making that decreasing linearly that's going to yield that logistic equation we just end up calling this whole thing which is constant a different constant and this our constant K so I'm going to talk about what happens in just a minute I'm going to erase this little bit I'm gonna rewrite that logistic equation alright so let's take a closer look at the logistic equation so let's take a closer look at what would happen if our birth rate is decreasing as our population is increasing on a linear scale me get this logistic equation what's going on let's suppose that our population our initial population starts with M if P equals M you are not going to get any change in your population you're going to get this equilibrium solution this thing that says you're not changing at all you have managed to keep your population at exactly this level forever your birth rates and your death rates would even out and since your populations not changing in this case then your birth weight can't change your death rates are constant it's gonna be the same exact population all of the time we call that when we get there a carrying capacity or limited capacity so we're destroyed in just a second but if your P equals your M there there's no change now what would happen if P is greater than M what would happen if P is greater than M well if P is more than M you get this negative do you see that if that's negative this is positive that's population this is positive we help them to be a positive constant then that would be a negative if P is greater than M that's a negative that means the rate at which you nation is changing with respect to time is negative that would be a declining population go okay so wait a minute if are populated so M is a really important number if our our population is greater than that M then this is going to be a negative and our population will decline and it's going to be a population decline what if our P is greater than s are less than M if our our P is less than M our population is less than M our initial population then what's going to happen well if our initial population is less than M then hey that's positive remember that's positive this is population it's always positive if P is less than M this is positive and we will get a population increase it says the rate at which our population is changing with respect to time is positive remember about about slopes about these these first derivatives and if it's positive our slope is positive and our function is increasing this would give you a population increase now I'm gonna verify this the next statement for you in the next part of this video but yes this would be a population decline so so imagine this we have this number and that's based on a comparison of birth rates and death rates if they'd is greater than P you get it positive here so if P is less than M and that would be a population increase and the opposite is true if M is less than P or P is greater than M we get a population decline but only until we reach em why I'm going to show you that in a little while but this would be only until we reach em so here's the idea we're gonna have this this M that's a constant constants are horizontal straight lines so we're gonna have this M like this if our initial population is below that M we are going to get a population increase until we hit n if our P is greater than our initial population is created than M we're going to get a population decrease but only until we get to em and this this number M it's called our limiting capacity if we reach it from the bottom so if our population starts below and we get up to it that's called our limiting population and if it's something that we fall down to it's called our carrying capacity so M is this value that's either a carrying capacity or limiting population explain that one more time it's just from looking at the the rate of change of population state the time if that is positive then it's growing but how far it's going to grow until you reach M because at the moment that you reach in you stop so right there you say okay and that's called our limiting population that case so if n is above P we're going we're going we're going but the closer we get like that the closer we get to M so piece below it piece of piece positive for sure but we're growing so if this if the population is initial population is less than this value and then our population is growing that's a positive value but the closer we get so the population is growing but the M is staying the same the closer we get the less and less our population is increasing to the point that right here when this P equals M we no longer changing it anymore we just we reach it and we stay there or we get really really really close well what if that is true what if we start up here if we start with our initial population that is greater than this value and well then what would happen well that's negative if this piece of zero is greater than M and that would be negative and we would say there are our populations declining that that makes sense but the closer we get to it the closer the population gets to this constant M but less and less and less we're declining to the point where we're asking hi to that we reach what's called a carrying capacity so depending on where we start we would call this M this thing that we're going to end up at either a limiting population or in carrying capacity I hope that makes sense to you I hope that I've explained it well enough that you understand this thing called the logistic equation that at some point no matter whether we're above or below that's this at this value and this constant value and based on birth rates and death rates that we're going to reach it this right here is called a funnel and we're going to talk about funnels and spouts in the next few sections coming up what were we also talked about stability and things like that but what this is saying is that if you start here or here you're going to eventually reach this this solution this is business owned this carrying capacity limiting population now what if this is not the case what if our our birth rates are not declining as our population is increasing well that's not that's not true our birth rates are not declining population increases that are proportional to it then our beta would be directly proportional to our population we're gonna go right back to here yes all right if that's the case then the one even our populations changing with respect to time would be well beta would be this proportional to a population we'd still get this Delta and we have still a population I want y'all I want to make sure that you're with me on here this represents every suit every sort of situation right so we're gonna always come back to this and say the later population is changing with respect to time is some comparison of birth rate minus death rate proportional to the population okay so let's suppose that our birth rate is also directly proportional to our population it's not declining it's not declining as our population increases it's proportional to it all right let me as our population goes up the birthrate could also go up well that's crazy that makes sense that makes sense like if there's more things in an area they could breed much much more without a limiting factor this thing is going to explode or if there's less things in the area they might be breeding less this population might become extinct it's a different situation here so we're saying let's suppose the birth rate is proportional let's make this K times me let's hold that death rate to be constant okay we're going to do something just a little bit tricky so what we're going to do is I'm gonna move this P to the front what I'm going to do is I'm gonna force this thing to factor out a cave if I force this a factor okay then we're fine here we just get people what I have to divide factories right so I'm dividing out okay we're gonna get this this constant death rate divided by K which was actually remember that that was that V sub 1 beta sub water we had now if I define this to be an N so let's just call this n another constant then here's what's gonna happen in this case M is no longer called our carrying capacity or limiting capacity and is called your threshold and here's what happens it's wild but check this out if your initial population is greater than your M you have a positive changing population this remember this is slope man this is how our function is changing so if our population is greater than our threshold then that's a positive slope and it will never now its ability to increase and increase it increase imagine this so this is greater than M and if it's still increasing it's going to be even more greater than am been more greater than M than more greater than M your population your slope is just going crazy or sub is filled positive your population is exploding so it m is your threshold then if your population initial population is greater than M you're going to get a population explosion I think they call that like a doomsday so it just it explodes to so much then they're eventually won't be any food so I'm going to say explosion so a population explosion what if your population is below that below that threshold well if your population is below that threshold then look at this if your population is less than M populations less less than them so this would be negative the rate at which your populations change in this rate the time is negative its declining but it doesn't ever reach em it's always always below that if your population starts below an and you subtract from it well then it's even more below them your slope is even more negative you're going to get you're gonna get extinction what I'm going to do is I must spend some time to go through I'm going to show you this I'm going to show you why this is because a lot of times it's not explained a good a good teacher will probably just do what I just didn't say yeah you know what here's here's to here's a few things number one the way the population changes and related to births and deaths it's pretty obvious number two if if we have this rate at which your population is changing is a birth rate minus death rate times population so and somehow somehow it's proportional this and some men are speaking we have two very important cases there are a lot more cases but here's two really common ones number one our birth rates sometimes decreases in a population increases and we study all through that we've got the logistic equation and we said all right here's the thing about it is that if our population is less than our M we're going to eventually reach it we're gonna grow it's going to be a positive growth but we're gonna reach it eventually and as this gets closer to this this slows down slows down slows down slows down slows down until we finally hit it if though there's other ways true of our population is above it it well decreases decreases decreases but a decreasing rate and then we're going to get we're gonna hit that that carrying capacity anyhow so that's what happens with our logistic equation but what another case is true whatever with another case is that our birth rates just directly proportional to our population as it grows rhythmic roses and declines or declines well then what would happen here is that we have what's called a threshold and as your population is above your threshold the rate of change just keeps on increasing without bound this is bounded this one's not or if it's below the threshold it's decreasing and it just keeps on decreasing to an explosion or to an extinction a lot of times your class is gonna stop right here and that's okay that's okay if you want to know why this is and why this is I'm going to do some pretty hardcore math we're gonna talk about partial fractions I will take this and I will take this and we're gonna use it as a differential equation we're gonna solve for P and you're gonna you're gonna see why this is a carrying capacity gonna see why this is an extinction or an explosion idea so hang on for a little bit I hope that I've explained to you the way that population change is really the births and deaths I hope I've explained these two important cases there are more but the two important cases where if our birth rate has decreased as population increases we get a bounded situation I need that to be the case okay I also need it to need to understand that if we don't have that if we have the birth rate is always just proportional we have this unbounded situation should be kind of kind of makes sense right if if we have a birth rate decrease it's published in increases you're going to hit a balance at some point but if you have it growing it as before just directly proportional it's going to be unbounded we're square that just meant okay so now that we should have a fairly good idea about how population change is related to birth rates and death rates and we had two very important cases we have that logistically logistic equation would mean that the birth rate is increasing as populations decreasing or decreasing as populations increasing inversely related I hope that made it make sense that if that's the case then remember that your birth rates are slowing and if we start up here that your population would look like this your birth rate and the rate of change of your population your slope will be increasing as your population is decreasing do you see that your population is going down your birth rates going up or vice versa if our population is going up our birth rate is decreasing on the way to the rate at which our population is slow me down it's decreasing so increasing at a decreasing rate or decrease in an increasing rate compared to our population what that's going to do is always give you some sort of a founded situation it has to this one's not that case this one says if you have a threshold if you start up here you are increasing at an increasing rate or you are decreasing at a decreasing rate your slope is negative he just keeps on getting more negative that's explosion or extinction I'm gonna show you mathematically why that happens not just show you in a picture I'm sure he mathematically so we're gonna solve we're gonna take that we're in salt it's gonna take a lot of time alright so because that's a difference of equation you go man I know about that I know the been acceptable and you're right so let's go ahead and group our P's DP and our constants DT let's take an integral now look inside we're gonna get something that's partial fractions yeah partial fraction BAM done that a long time yeah we're gonna get partial fractions do you remember this how we can say this this fraction can be separated as a over P plus B over m minus P oh yeah okay over that I remember that one equals a times M minus people are just getting some common denominators and seven of equal to the numerator might have been a long time for you plus B P if I let P equals 0 then P is there let's see that's 0 this would be a 1 equal a times M and so a would equal 1 over m and if P equals M [Music] so people's M this would be 0 this would be M 1 equals B M and B would be 1 over m so in both cases we're going to get this one so ya 1 over m p & 1 over m times n P I'm going to show that right now so we've used partial fractions here if you don't remember that on this and put a link up there so you can follow partial fractions again and we're going to get that this equals let's see it is 1 over N so 1 over NP plus B is also 1 over N 1 over m times n minus P DP on the right hand side we get this KT plus some sort of a concept and yeah we do need it we don't need on the left hand side because we will absorb it but we do need at least one constant here we'll see why in just a bit so oh my gosh well hey you know what those both have a factor of 1 over m let's pull that out another thing that I'm going to do and this is gonna seem kind of weird at first until you realize why I'm why I'm doing it so what I'm gonna do is I'm gonna pull out a negative I'm going to factor out a negative as well so I'm gonna put this 1 over P plus 1 over n minus P DP what I'm gonna choose to do is I'm gonna factor out a negative let's say all right let's make this negative now what would happen if i factor a negative is I'm gonna give negative 1 over P plus this is going to give us 1 over P minus M not M minus P and then factor the negative and we change its sign now I'm actually ready to do this integral if I integrate this this would be a negative 1 over m and we would get let's see 1 over P that's LNS of AP but either negative so Ellen 1 over P gives me Ellen absolute value P they you front same thing happens here you can do use of but because we factored out that negative right there we factor negative changer signs because because we did that remember our variables P we don't care about the M we don't have to use a use of like it all for there's no negative here this would just be this plus Ln absolute value P minus m and yeah we have a constant but we can reabsorb so making sense apart now here's what I'm gonna do two things right now I'm gonna multiply both sides by negative and that's going to clear this that's going to give us something be multiplied here I'm also going to say do you remember that Ln of M minus Ln of n will give you Ln of n over n do you remember that we can combine two logarithms being subtracted by dividing we have a logarithm - a logarithm we're going to make that a quotient here so I'm multiplying my negative M I'm also combining this we're going to get Ln absolute value this over this and I've x ham I've also distributed we're getting their head then looks nasty Boober we're sort of hitting there now we're gonna make an exponent on both sides so P to raise to both of these as exponents gonna give us this P minus M over P equals I'm gonna do a couple of fancy things here if I do e here and E here do you remember that I can separate my exponent I can take e to both of those terms and this would give us this e to the negative n ket times e to the negative n see that right there is a constant if I wrap up my absolute value in a plus and minus this whole thing is going to be one giant constant so all of this gonna change and this is gonna give us a constant e to the negative n ket I know that that's that's kind of quick but we've done this so many times in this in this course so we know that this is gonna be e to the negative M KT we know that this is going to be e to the negative MC plus and minus from absolute value we wrap all of that idea into a concept that's being multiplied times e to the negative M K T now we have to solve for P so here's what we're gonna do what we're going to do is is to kind of weird face number one I want to get rid of that constant how the wording really continent I'm gonna do it here because every population has an initial value doesn't it so when we take a look at an example the population at time zero is going to be an initial population now I'm gonna plug that in right now I'm gonna get our C and our C is going to equal P sub 0 minus M over P sub 0 but I'm not gonna plug it in until later why is because it gets real nasty if we try to do that right now really really really crazy so what I've done I've got all the way through the part of fractions I've just integrated we got this trying to solve for P I want to also solve for C in terms of our initial population and so I'm going to plug in 0 for T I'm gonna plug in P sub 0 for P remember this is a function of P based on time population based on time that's a constant that's a constant that's a constant that's our independent variable and then our PS are deep in it very short of solving for P and right now I plug in 0 for T and get P Sub Zero for the P and relationship to that zero I'm gonna get my C but I'm not gonna plug it in yet so if I plug in 0 for T notices look at if I plug in 0 for T this whole thing would be 0 I started this whole thing we zero this would be 1 I would get P sub 0 minus M over P sub 0 equals C times 1 C is this right here so I'll write that to the side and say see is P sub 0 minus M or just M ever over P sub 0 we are going to use that in just a minute I promise but before we do it before plug it in there let's leave this where it is because it's much nicer to work with right now and let's solve for P we're going to multiply both sides by P we're going to improve our P terms on one side and our n term on the other so I'm going to subtract this from both sides and add our M so we've subtracted this we're going to add our M let's factor that P since we have two terms and let's do by the sulfur people now I promise you we're gonna come back to this but do you know there's a lot nicer dude that's see but we do have our see right here we're gonna now plug in that see because we want this in terms of P and a constant K with T they're variable remember the M is a constant K is also a constant we know what those things where we define them for the logistic equation and then that's C we now know what that is let's see is this piece of zero initial population - am a constant over P Sub Zero and plug that in now so our population with respect to time is some constant end based on birth rate and death rate 1 - C is that piece of zero - end of the piece of zero so that is C times e to the negative n ket now that might not look very good to you but we are almost done all we're going to do because this is a complex fraction and you know how to accomplish fractions right we know how to multiply my piece of 0 over T sub 0 just multiply by the LCD or denominator yes this is going to multiply give you something very nice n times P sub 0 this is going to distribute it will cancel this fraction this will have to go in parenthesis but it will also go to your 1 so we're going to get M times initial population over initial population - what I'm gonna do here is I'm gonna cancel this piece of zero we're gonna get piece of zero minus M e to the negative M K T now we're gonna do one other thing that's going to help us to see this a little teeny bit better what we're gonna do is we're going to factor out a negative here remember how it's better than that you're way back here and it made things easier we're going to sort of undo that right now so we're gonna factor that negative in other words we're just going to reverse these and change that sign so factoring that that would be a piece of zero plus and minus P sub 0 e to the negative and AD thank you world's Inc let me see anything i okay great this this is your population so we just solve a pretty heavy-duty differential equation with a lot of stuff you go ok we got down to there that's that's that's fantastic what's going on what what the rule would happen here well we're gonna look at is the relationship between m and your initial population what I told you was that in the logistic equation you have this M that is like this flat complex obviously constant so it's this horizontal flat line and I told you that if your initial population is below it and we logically went through it but I'm showing you right now if your initial populations below it you will have an increase in your population until you reach it if you have and there's no populations that above it you're going to have a decrease in your population until you reach that M it is called a limiting population or a carrying capacity depend on whether you're increasing to it or decreasing to it let's see why that is it's a very important year and we'll look at these functions understand what's going on this is your population it is what's going to happen now you know that an exponential doesn't matter what they said that exponential it is positive do you see that like that's a positive number now how the size of it I only carried on it's a positive number here's what I want you to look at what happens if P sub 0 if P sub 0 is less than M if P sub 0 is less than panel if P sub 0 is less than n then this is a positive think this through and minus something less than it will be positive that's already a positive then this right there will be a positive number what that means if P sub 0 is less and then this is positive right so you take this initial population that's positive you add something to it this would be then your population then this thing where this whole thing would be less than I can't write it there I'd have to write up your soo this would be less then the population times M over the population itself with it so if this thing is positive then this is a bigger denominator than this then this fraction would be less than this fraction and this would be less than that itself which means your population would be less than that so if your population so and this equals M if your initial population is less than M sure your increasing but your bounded we've just proven that this would be bounded right here you said man if this is positive we know that's positive then this is a number that's bigger than P Sub Zero what that means is that this fraction is less than M times P Sub Zero P Sub Zero that's an upper bound if we simplify that that upper bound is just M it says that yes if your population is below M it's going to increase your saw that in our differential equation but it's in again increased until you're bounded by ya that's what's gonna happen now now what would happen another case what if your piece of zero the initial population is greater than animal if your initial population is greater than M if this numbers bigger than this number this is positive if this number is bigger than this number this is negative negative times a positive is a negative you take something positive - another number this would be smaller than P sub 0 therefore this fraction would be bigger than M over and times P sub 0 P sub 0 this is bigger than that look at look again positive number times negative number P some sort of beaker and that would be a negative times a positive to negative something positive - negative balcony that's going to be this is gonna be smaller than P sub 0 therefore this fraction is bigger than this fraction but this is and that gives you a bounty gives you lower down so what's happening here this man I know there's a lot of stuff I hope I haven't confused you but what's happening here is we know that two things that if this was the case our population was going to be increasing we saw that from regular put everything together for you I know if our initial population is less than M get this if our initial population is less than M this is smaller than this that's a positive number times a positive number positive then these are population is increasing okay so this case right here says population is increasing right it says that but it also says that our population has number bounded it says that you're increasing but you aren't only increasing until you get to M it can't give it any other way now what if the other case what if our initial population is greater than M this is bigger than that gives you a negative it says population is decreasing but we have another bound or decrease until we reach M so in either case we're a logistic equation we're going to reach M this carrying capacity or limiting population that's all they can they can happen now I'm gonna race this I'm going to talk about the next next situation but I'm not going to go through all of this work it's extremely similar you're just doing I'll show you in a minute but let me let me erase and stuff and we'll come back and talk about this alright so last little bit we're going to talk about the explosion and extinction we just learned about the logistic how we have this this limiting population or a carrying capacity because in relationship between an increasing population and a upper bound or a decreasing population and the lower bound it's going to happen that way up with analyzation equation now what if what if we have this other case where our birth rate is just directly proportional to the population so as population grows birth rate also grows without bound or dense population decreases birth rate also decreases without bound they're directly tied together you would go through exactly exactly the same idea all the way down and solve repeat just exactly showing like show you again same stuff but you end up with something like this I want to talk about why this population is going to explode or become extinct let's look at it what if your initial population is greater than n notice that's not negative so this is not this right here is still positive but it's it's growing pretty fast generally so let's look just here focus on this what if your initial population is greater than M as we go forward and go forward and go forward with our with this situation then if this is positive and this is getting bigger and bigger and bigger then the difference between your initial population and something that is positive look at this difference between an initial population something is positive this is going to get smaller under this would be positive positive and growing so we have seven ischl population that's getting then where the population shrinking smaller to only getting bigger the same thing smaller smaller smaller smaller smaller this is a constant so wait a minute you taking a constant as time goes on you're gonna get number over number that's shrinking this is going to go to infinity that's gonna go to infinity as T approaches infinity this that's it remember this is constant that's going to say this is positive this is going to get really really really big this is going to stay positive number - really big number as we get close to that population initial population this is going to get very very very very small dividing dividing concept by something very very very small you're going to get infinity it says that if you have a threshold of M and you start here with an initial population above it if the initial population is greater than M it is going to eventually explode there's no bound now the other other case if our initial population is less than n that's the same this would be yeah well this would be that would be something really large as T goes to infinity but this if P is less than M this becomes a negative a negative times a positive is a negative oh wait a minute you're taking a positive number and you're subtracting it you're adding you're adding to it so you're getting this this denominator that's growing bigger and bigger and bigger and bigger and bigger and bigger and bigger not getting closer to some initial initial about not getting closer to zero eating something it's getting bigger bigger closer to infinity take a constant and divide by something getting closer to infinity and you're going to go to zero that means that your initial population starts below your M you are going to get the population is decreasing and continues to decrease you're going to get extinction I really really hope this has been probably one of the most brutal lessons as far as theory that we've ever had so I understand that it's really heavy-duty I said a lot of things in this lesson and if you stuck with me good job you're going to be fine in the next few videos I'm gonna do a lot of examples to illustrate these concepts to show you from some very basic stuff some pretty complicated ideas what we've done in this video is I've shown you that the way that population growth relate I've shown you how population growth relates to birth rate and death rate I've shown you two very important cases where birth rate declines as population increases or birth rate increases as population declines I've also shown you how the relationship where birth rate is directly proportional to our population and we've seen that for the first example the the logistic equation that we are going to reach some sort of threshold what I want you to understand is that if birth rate is decreasing as populations increasing if it's inversely related that you are going to reach some sort of a bound the opposite if birth rate is directly related directly proportional to your population you're not you can have a threshold where if you're above it climb forever you're below you fall from voluntary zero in this case I hope that makes sense I was to you from the next video we're going to start just to to very basic examples on how to set problems up and then as we go forward we'll be talking about this and explore an extinction and an explosion and we'll also be exploring the logistic equation in several more videos so hope you enjoyed it that was crazy and we'll see you for another video you

Channel: Professor Leonard

Views: 34,692

Rating: 4.9768786 out of 5

Keywords: Math, Leonard, Professor Leonard, Differential Equations, Logistic, Logistic Equation, Population Models, Population Change, Limits, Extinction, Doomsday

Id: 7SPEvSmfK0w

Length: 64min 39sec (3879 seconds)

Published: Fri Apr 12 2019