Logistic growth

it's mr. Andersen and in this video I'm going to talk about logistic growth you should probably watch my video on exponential growth if you haven't done so first but logistic growth is what happens after in other words after a population eventually maybe crashes or finds a natural kind of a carrying capacity and so let's start by talking about population growth in general imagine we have two rabbits our n value is 2 so n is always going to refer to the population size now it's going to change over time but to start let's say n is equal to 2r is going to tell us how fast that population is going to change and we call that growth rate it really is determined by two things births and deaths let's say that these rabbits have to baby rabbits and so we'd have two births minus zero deaths so that's going to be 2-0 or two and then we divide a by our original n which was too so it's going to be 2 over 2 or 1 and if you have a growth rate of one things are going to go really really quickly but we could have had a growth rate of point 5 how could we get that let's say two rabbits are born minus one dies so that would be 2 minus 1 or 1 over 2 and that's going to be point 5 the take-home message about exponential growth remember is if if R is ever greater than zero we're going to show fast exponential growth and so let's say it's one what's going to happen while those four rabbits are going to become eight and then we're going to take eight in this case times that growth rate of one and so we're going to now have 16 and now we're going to have 32 and now we're going to have 64 and we're going to have 128 and we're going to have 256 and pretty soon my whole screen is going to be filled with rabbits and I heard my fan come on on my computer because it's really taxing to put that many rabbits on the screen and so does that occur in nature no and Darwin noted that as well he looked at instead of rabbits elephants elephants take a long time to reproduce in other words they have to be 30 before they can reproduce and they live tell about 90 and what he found is that even though they might produce three pair of offspring during that period of time if you just let them keep breeding over and over and over again after five centuries or 500 years later you're going to have 50 million elephants and so exponential growth goes really really quickly and we know that it can't last forever eventually we're going to hit a limit and so let's look at those four rabbits again eventually they're going to reach what's called k or carrying capacity it's the maximum amount in a population that it can be supported in a general kind of an ecosystem or in an area and so why is that well rabbits eat things and that is grass and when they run out of grass they die now I don't have the ratio right you have to weigh more producers then you're going to have consumers and so let's put our rabbits on the grass what are they going to do they're going to eat the grass and as they do so that grass is going to go away and they're going to have to move on to another area and so luckily there's more grass for them to find but eventually they'll eat their way into a corner so this rabbit right here has a problem if this grass didn't grow back it would really be stuck and so if we had much more than four rabbits we would have exceeded the carrying capacity and in class I ask my students to build models that showed logistic growth and a lot of them used paper and pencil but one creative group use minecraft and what they did is they constructed a little chamber here and they put one chicken and every time the chicken would step on one of these platforms two doors would open up and two more chickens would come out so pretty soon we have three chickens and they'd step on platforms and pretty soon we have five chickens and it just went really really really quickly pretty soon there were chickens everywhere and I asked them well you've gotta show logistic growth and these bright students Gabe Ethan and David said well watch what happens eventually as they stand on the platforms it becomes so crowded they can't get off anymore and so they can't release any more chickens into the area and so eventually it reaches kind of a carrying capacity of chickens in this one container now let me show you another model this is a net logo model called rabbits grass and weeds so let me launch that for a second so in this model the rabbits are going to be these little white things with ears and then the grass are going to be these little green squares and so we start with five now the rabbits are going to find grass if they can and if they can they're going to get energy and if they get enough energy they can breed but if they don't find enough grass then they're going to die and so if we let it go for a second it goes really really quickly so let me stop it there for a second so let's watch what happened and so this red line here is going to represent the number of rabbits over time and so again we started with five and pretty soon we add 350 rabbits what happened to the grass while the grass started to reproduce as well but it crashed because the rabbits were eating all the grass and so let's watch what happens now if we let it roll for a second so we're going to see a big drop in the number of rabbits and then it's going to finally hit a limit so we're going to find like a perfect amount of rabbits that we have so we're reaching what's called carrying capacity and you can see that if we let it go over and over and over the grass and the rabbits are going to go up and down but we're going to reach a limit now let's say that we gave the grass more energy so if there's more energy in the grass let's increase that we're going to see exponential growth again and then we're going to have even more rabbits or let's say we said that the grass has less energy as we decrease that and we're going to have way fewer rabbits and so that rabbit populations are going to drop but it's going to find another carrying capacity so what happens as your population size increases you run up against these limiting factors and those limiting factors are going to slow your growth so let me quit that so let's get to the math now so if we're looking at exponential growth so exponential growth the equation is going to be this right here and so DN over DT means the change in n over the change in time and so it's going to be our times n where r is the growth rate and n is the population size and so if we started like we did at the beginning with exponential growth you're going to have two rabbits what's our change in n over change in T it's going to be too how did we get that we're multiplying our which is one remember times n which is two and i'm going to get to what happens to that change in n well that's going to give me my new population so I'm taking two here plus 2 here and now I have four what's my change in n it over time for the next one again since our R is one it's going to be four again and so we're going to have eight we're going to have 16 we're going to have 32 so if we were to graph that that's going to give us that j-shaped curve but what I want you to do is look at this equation right here and watch as I change it to an equation that shows carrying capacity and so the only thing that's going to change is what's in the parentheses right here and so that's going to be the factor thats related to carrying capacity and so i'm going to choose an arbitrary carrying capacity let's say it's 10 that the area can only support 10 rabbits let's watch what happens and so we start with two rabbits and so i'm going to show you what's in the parentheses e'er and so we're going to have k minus and / k well what's k what's our carrying capacity that's 10 minus n so that's minus 2 so we're going to have aid and then what's our carrying capacity it's 10 so it's going to be 8/10 or it's going to be point eight okay now we're going to figure out this and so what's this it's our which we said was 1 times n which we said was two times what's in this parentheses which is 0 point eight and so that's going to be 1.6 and so this whole equation tells us how many new rabbits were going to add and we're not going to round so I can show you what happens as we increase this but know this that we're going to increase it by 1.6 rabbits so let's look at time one now we have three point six rabbits where did I get that it's the two original rabbits we had plus the 1.6 okay what's going to be in the parentheses now it's going to be k which is 10 minus 3.6 so that's six point four divided by 10 now it's point six for you can see that that number became a smaller number we're going to multiply that times 3.6 again it's one times three point six times that value and we get 2.3 i'm going to add that to our original 3.6 and now we get 5.9 what's going to be our k minus n over K it's going to be 0 point for one and so what do we get two point four and so you can see that as the population is increasing this factor is becoming smaller and smaller and smaller and so now we're only adding 1.4 and now we're only adding point 29 and now we're only adding point zero zero nine and so what's happening to n you can see that it's increasing quickly right away but the closer we get to that carrying capacity it's kind of leveling off and it's reaching that limit at one point so this is just mathematical it's not a model but it shows that we're eventually going to approach that point and so again in review what's our R is going to be the growth rate what's k k is going to be the carrying capacity so if our r is ever greater than zero remember we're going to show exponential growth and if we ever have k if we ever have a carrying capacity we're going to show logistic growth but we also have species that specialize in these two we've got our selected species in k selected species so what does that mean in our selected species is going to be a species that loves to grow as quickly as it can look how many babies this frog is going to have thousands of those and so friar to take this frog and introduce it into a pond where there are no frogs we're going to have a bunch of frogs really really quickly we're going to have an R that's really really large and they're going to fill that area they're going to run out of resources and crash and they're going to boom and bus they're going to fill it really really quickly if we were to look at this which is a chameleon chameleon is going to only have two offspring so its growth rate is going to be really really slow what is it going to do it's going to give them a lot of parental care it's going to invest a lot of energy in just its few offspring and they're all going to survive if we look up here with our selected species most of those are going to die eventually but with this chameleon since it's taking so much care of them it's going to take very good care of it they're all going to survive this is going to gradually increase and find a nice carrying capacity so what are we what are humans well we invest a lot in our young a lot of my students are 17 years old and they still live at home their parents take care of them so we're really investing in them so what kind of a population curve are we going to see in humans we're going to see a gradual exponential increase and then a nice carrying capacity which we are eventually going to reach and we don't want to have this boom and bust cycle so it's not so cut and dry as that there are obviously species that are somewhere in the middle so what about a tree do you think tree grows really slow but they're going to produce a lot of offspring and so what are they it's probably no right answer for that and so again what is logistic growth it's it's growing quickly but eventually reaching a carrying capacity and that's based on limits in your environment and I hope that was helpful