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# The logistic growth model

AP.CALC:
FUN‑7 (EU)
,
FUN‑7.H (LO)
,
FUN‑7.H.1 (EK)
,
FUN‑7.H.2 (EK)
,
FUN‑7.H.3 (EK)
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FUN‑7.H.4 (EK)

## Video transcript

in the last video we took a stab at modeling population is a function of time and we said okay well maybe the rate of change of population with respect to time is going to be proportional to the population itself that that rate will increase as the population increases and when you actually try to solve this differential equation you try to find an n of T that satisfies this we found that an exponential would work and exponential satisfies this differential equation and it would look like this visually it would look like this it would look like this visually where you're starting at a population of n naught this is the time axis this is the population axis and as time increases population increases exponentially now we said there's an issue there what if Thomas Malthus is right that the environment can't support let's say that the environment can't support me do this in a new color let's say that the environment really can't support more than more than k more than a population more than a population of K then clearly the population can't just go and go right through the ceiling they're not going to be able to have food or water or resources or whatever it might be they might generate too much pollution who knows what it might be and so this this this this first stab at modeling population doesn't quite do the trick especially if you are kind of in malthus's camp and that's where P F and once again I'm sure I'm mispronouncing the name Verhulst who is going to come into the picture because he read Malthus his work and said well yeah I think I think I can do a pretty good job of modeling the type of behavior that Malthus is talking about and he says you know what we really want what we really want is is something let me write it so the rate let's try to model let's set up another differential equation let's set up another differential equation and now let's say okay instead of you know if if the if if if n is substantially smaller than what the environment can support then yet that makes sense to have exponential growth that makes sense to have exponential growth but maybe we can dampen this or maybe we can bring this growth to 0 as n approaches as n approaches K and so how can we actually modify this maybe we can multiply it by something that for when n is small when n is much smaller than K this term right over here is going to be close to 1 and when n is close to K this term is equal as close to 0 so let me write that when n so this is our these are our goals for this term right over here when n n is much smaller so much smaller much smaller then then K so now the population is not constrained at all people can have babies and those babies can be fed and then they can have babies etc etc then this thing should be should be close to should be close to 1 and so then you have essentially our old model but then as n approaches K when n as n approaches K then then this thing should approach then this term this term or this expression should approach 0 and what that does is is n approaches the natural limit the ceiling to population then no matter what this is doing this if this thing is approaching 0 that's going to make the actual rate of growth approach 0 so they're going as food is going to be more scarce it's going to be harder to find things and so what what what can i construct here dealing with N and K that will have these properties and if you're if for fun you might actually want to pause the video and see if you can construct a fairly simple algebraic statement using N and K and maybe the number 1 if you find the need to 2 for an expression that has these properties well let's see what if we start with a 1 we start with a 1 and we subtract we subtract n over K we subtract n over over my K is in pink over K does this have those properties well yeah sure it does when n is really small or I should say when it's a small fraction of K then one mind this is going to be a small fraction this then this whole thing is going to be close to one it's going to be a little bit less than one and when n approaches k is n gets closer and closer and closer to K then this thing right over here is going to approach one which means this whole expression is going to approach zero which is exactly what we wanted and this thing right over here is actually then this this is used in the tons of applications not just not just in population modeling but that's kind of one of its first its first kind of reasons or motivations this differential equation right over here is actually quite famous it's called the logistic differential equation logistic differential equation logistic differential differential equation and in the next video we're actually going to solve this and it's actually a this is a separable differential equation you could you can actually solve it just using standard techniques of integration it's a little bit hairier than this one so we're going to work through it together and we're going to look at the solution the solution to the logistic differential equation is a logistic sometimes though is just well the logistic function which once again is really is essentially models population in this way but before we actually solve for it let's just try to interpret this differential equation and think about what the shape of this function might look like and to do that to do that actually let me let me get actually I can it's nice to see the faces so let me draw let me draw some axes here let me draw some axes here so that's my time axis that is my population axis let me scroll up a little bit because sometimes the subtitle show up around here and then people can't see what's going on so let's think about a couple of permutation a couple situations so if our initial if our initial if our knit if our n at time equals 0 remember n is a function of T if at time equals 0 n is equal to 0 so if n is equal to 0 then this term is going to be 0 and then your rate of change is going to be 0 and so you're not going to add any population and that's good because if your population is 0 how are you going to actually be able to add population there's no one there to have children so there's actually one solution to this differential equation which is just n of T that is n of T is equal to 0 and that's neat that this satisfies the logistic differential equation hey if your population starts at 0 if n sub-nought is zero then you're just going to be at zero forever well that that's actually what would happen in in real life there's no one there to have kids now let's think about another situation what if our population what if n naught is equal to K what happens if n naught is equal to what happens if n naught is equal so that's K right over there what happens if at time equals 0 this is our population well if n is equal to K then this is 1 minus 1 then this thing is 0 and so our rate of population change is going to be 0 so essentially if my population is 0 then after a little bit of time my population is still the same K if my rate of change of population is 0 that means my population is staying constant and so my population is going to stay there at K and that's actually believable Malthus would actually probably say that you're going to have you know maybe it goes a little bit beyond the capacity of the of the environment and then you have some you know some some flood or some hurricane or some famine and it goes around but for our purposes you can ever model anything perfectly for our purposes that's that's pretty good you know you're you're at the limit of what the populate of what the environment can handle so you just kind of stay there so that's actually another constant solution that n of T if it starts if it starts at now you can kind of appreciate why initial conditions are important if you start at zero going to stay at zero if you start at K you're going to stay at K so that is n of T just stays K but now let's think of a more interesting scenario let's assume a net let's assume an initial population that's someplace between 0 and K so this is going to be this is going to be I'm going to assume initial population that is some place it's greater than 0 so there are people to actually have children and it is less than K so we aren't fully maxing out the environment or the land or the food or the water or whatever it might be so what's going to happen so and once again I'm just going to kind of sketch it and then we're going to actually solve it in the next video so when n is a lot less than K it's a small fraction of K you're going to it's going to you know this term is going to be the main one that's influencing it because this is going to small fraction of K I mean even the way I drew it it looks like it's about of I don't know it looks like it's about a sixth or seventh or an eighth of K so it's one minus one eighth so it's 7/8 it's going to be 7/8 times times this so really this is what's going to dictate this is what's going to dictate what our rate of growth is and if this is kind of dictating it we're kind of looking more of a well let's just think of it this way as the population grows the rate of the rate of the rate of change is going to grow so it's going to look it's going to look something it's going to look something like this as our population gets larger our radar our slope is getting higher and it's getting steeper and steeper but then as n approaches K then this thing is going to become this going to be close to 1 - close to 1 and so this is going to become a very small number it's going to make this whole thing approach 0 so as n approaches K the whole thing the rate of change is going to flatten out and we're going to ask them towards K and so the solution the solution to the logistic differential equation should look something like this depending on what your initial conditions are if your initial condition is here maybe it does something like this if your initial conditions here maybe it does something like something like this and once again and this is what's fun about differential equations before even doing the fancy math you can kind of get an intuition just by thinking through the differential equation of what it is likely what it is likely to be down here or when when n is what's smaller than K its rate of increase is increasing as n increases and over here as n gets close to K its rate of increase is decreasing now let's actually in the next video actually solve for what n of a solution to this and see if that if that confirms what our intuition is