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0:04
Hi all Dr. Clark here again and today for forest management when I talk about population growth or the dynamics of it we've talked about age calculations and back calculating age we've talked about tree hides and other things these are measurements that are important but the overarching overall overall goal of a population manager or forest manager is to manage your population so you go to know population trends so that's kind of why I want to talk about population
0:44
growth we're also going to talk about some other components to population as we progress but this isn't very important that you know how to calculate whether your population is growing declining and at what point can you expect your population to kind of stabilize okay so that's what we're gonna get at now this all population growth all really came from a period of human population growth and I don't know if he was the first but he's one of the
1:21
very first people to write about population growth and that was Thomas Malthus if you remember from maybe general biology or some other courses that you might have had Thomas Malthus was the individual who wrote a book on human populations and basically stated that human populations has the capability of outgrowing their food resources Darwin read this book Darwin used thomas malthus's ideas about humans and population growth and he basically transcended that on to all organisms and
2:00
said that well it's not just humans that had the possibility of over reproducing but all organisms have the possibility of over reproducing and thus came out his theory of natural selection in other words the individuals that are best fit for the given environment are the individuals that live and survive and reproduce okay so really that came from human population biology but regardless of that okay we can look at human population biology you know now okay we're sitting at 2020 and we're
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really close to eight billion people in the world hey the way that we look at population biology or population growth of human populations is by trends okay you know that really our population started exploding right after World War two with the Industrial Revolution and this is where you could start you know better farming better industrialization them you know better use of resources or I shouldn't say maybe not better use of resources more abundant use of resources
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and you just see a huge spike in population growth of humans I mean if you look we were sitting at about 1 billion ok this graph isn't really showing it but we're sitting about 1 billion from 1850 all the way to like 1550 so you know close to 300 years the world's population was 1 billion and you know we haven't even added 200 years since then but when we get add 200 years since 1850 and we get to 2050 were expecting 9 billion so that means it we
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sat at 1 billion for 300 years and before that we're less than 1 billion the last 200 years we've added about 8 billion people to the planet and more importantly the last hundred years we've added 5 billion people or we will have added 5 maybe 6 billion we've added fiver now but by the time we get to 10 20 50 ok from 1950 to 2050 the last hundred years we will have added roughly five billion six billion people just depend
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on where the population lands that's a big deal right now the reason why that's a big deal is we don't quite know what the carrying capacity is what's the top out for humans now in the 1970s I it was thought to be about 9 billion and with better technology better medicine more advances we think now that it could be close to 12 billion but there will be a point where the population on the planet reaches its carrying capacity you really you can't add any more to the positive
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growth of the population based on this curvature and based on this trend it looks like you know it's going to be close to that 12 billion because the the curve the graph is starting to slope off we don't really know if it is 9 billion you can expect to reach 9 billion by 2050 so for most of you in your lifetime in the next 30 years um it will be possibly close to that carrying capacity if it's you know 12 billion then most of you don't have to worry about it and
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that's gonna be way out there but regardless when you get close to that cutting capacity things start happening ok death rate has to increase because you're reaching a point where your resources or something is making it so your population can't grow as fast ok we'll come back to that when we start looking at other examples so again Malthus did this to begin with he described this mathematically that's what we're going to do we're going to
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use an equation couple equations to predict what a population is going to be or to track how much population growth a group of trees might have or some kind of vegetation might have first of all a capitalized in in most most ecology or population examples the capital n means you're talking about the total population and in some cases you know the total population in other cases that's what you're trying to figure out is the total population if you remember
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a definition of population it's the same number or the total number of species in a given region that have the capability of reproducing okay when we're talking about population we're talking about a single species population we're not combining species okay so population is species specific the rate or little R is a rate of growth the rate of growth is determined by lots of different things it's determined by how many offspring an organism can produce
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how many of those offspring are going to die and so birth rate death rate how many individuals are coming into your population how many individuals are leaving your population immigration emigration and a lot of birth rate and death rate is determined by lots of different situations or characteristics okay time time depends on what organism you're talking about okay if we're talking about an organism that reproduces every five years then your
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time is a five year period every five years if you're talking about an organism that can reproduce every month then you know it's monthly reproduction okay so you have to you got to know your organism because this population is species-specific the time unit is species specific the rate of growth is species specific and because different organisms reproduce different number of offspring does offering of a different capability of
9:13
survival okay and they would produce those offspring at a different period of time or unit of time okay so this using these three variables we can get a geometric rate of increase or what we would often call an exponential growth rate and so this is the ideal rate at which a population can grow or change under certain circumstances okay and when we say certain circumstances we say ideal circumstances so that means that your rate of growth is not based on you
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know food resources so nutrient levels so you're not having a lack of phosphorus or lack of nitrogen you know having a lack of water or when we're talking about forests okay if we're talking about other populations like animal populations you know have a lack of food resources those kind of things and so under ideal circumstances ideal situation the geometric rate of increase is this equation okay population at time whatever time this is
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okay so maybe we're interested what the population will be 20 years from now okay you have the current population at time zero the rate at which it grows and then by you know raised to the time at which you're talking about okay and so we'll look at some examples of this to make it a little more kick clear well because this is ideal circumstances this will result an exponential growth curve okay the growth is going to increase per unit of time
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okay so geometric increase per unit of time and it has no limit and so it's a constant rate so if you're adding 50 new individuals every single year okay then you're adding 50 new individuals every single year or better yet if each individual in your population is reproducing to offspring every single year then that's going to result in an exponential growth curve that expedience very quickly okay so goes from maybe a very small
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population to a huge population very fast that's exponential growth it's often represented as a J curve okay so let's look at a typical example of X a bit exponential growth we often use insects because they grow exponentially especially things like cockroaches and flies they can grow exponentially and often do their populations often grow exponentially other organisms like trees and other vegetation maybe some weeds weeds will
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grow exponentially okay but most of the vegetation that a forest manager might be interested in it's not going to grow exponentially and I'll talk about why that is okay but that's looking at exponential growth rate with cockroaches first so let's say that a cockroach reproduces 10 offspring and every three months so your rate of increase or your little art is 10 offspring okay your period your time period is 3 months that's what we need to then calculate
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the cell so let's say your original population you have two cockroaches your rate is 10 and so your time one in first three months is going to be your population is going to be 20 individuals so if we jump over to time to your new population is 20 that's what you had from the first three months now you're looking at the second three months your rate is still 10 nothing's changed but now you have 200 individuals after the you know after the
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second three-month period so in six months you went from two individuals to 200 individuals and nine months you made it to 2,000 individuals and twelve months you're at 20,000 individuals so this is how exponential growth rate works it takes a very little bit little input because of some resource is available at maximum level okay so that population is allowed to skyrocket and you know jump in number very quickly so if you can see here this
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is you know cockroaches you know starting off maybe two individuals and gradually increases to 220 to 200 okay and then 2,000 and then boom 20,000 and then if we did it again you know maybe we're at well whatever 20,000 times 10 is so 200,000 right yeah 200,000 so um that's how a populations can grow exponentially and you can see that J curve look right there alright so we know that populations don't typically grow continuously forever when they start off they might have that J curve that
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exponential growth curve but in reality something will something will make it so that population has to slow their growth rate has to slow okay so when we're talking about that original equation that we're talking about exponential growth rate this is another way to write it you can just write you know changing in change in time is equal to the rate times in okay so D is Delta which you know means change and again and is your
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population time is whatever time you're talking about R is your rate of growth times the population okay so the change in population / change in time is equal to the rate times population size okay that's how you can rewrite exponential growth curve and that's probably the equation that you would like to use or should use is r times n so again that's exponential growth right there's the equation right now I highly suggest that you memorize this equation or have it
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handy because you're probably going to have to use it on a quiz or an exam plus it's just an easy equation to talk about how a population can increase now remember that R itself is going to change its it's species specific so R so you know and cockroaches maybe this is 10 per individual maybe in maybe in a in a tree maybe it's a hundred thousand seeds per year but only to get germinated so just because you reproduce a hundred thousand seeds doesn't mean
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your R is a hundred thousand it's the number of offspring that survive that's our okay so you have to R is actually birth - the death plus immigration - immigration that's what our really is so you take your birth rate minus your death rate and so let's say you are reproducing a hundred thousand seeds but nine hundred ninety eight thousand or nine ninety eight thousand ninety nine sorry ninety nine thousand nine hundred ninety eight die so you're left
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with two seeds that Germany that's your are okay and it's often written as a percent so a certain percent will germinate a certain percent will survive maybe it's point zero zero one percent will survive so even though you might produce one hundred thousand two hundred thousand millions of seeds its point zero zero zero one percent survived that's your arm okay times that by your population size right that's how you get exponential growth but exponential growth always has its limits so even
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though that cockroach population looks like it's just skyrocketing gonna keep on going eventually it will reach its carrying capacity or its K that is determined by something okay it's not always the same thing so exponential growth will always be limited by something what those limits are is depending on the species it depends on the area so it could be that we're talking about tree growth we've clear-cut a forest we have you know left
19:29
a couple seed trees they deposit all their seeds some of those germinate you get lots of little saplings but you have you know poor nutrients you have very little phosphorus very little nitrogen in the region okay so you have poor nutrients so your population is not going J it's reaching a carrying capacity the nice thing about carrying capacity is that can change it changes over time okay or maybe you have a huge population
20:04
growth tons of saplings come up and now you have trees shading out other trees so now your carrying capacity is based on sunlight it's based on how much light can reach the bottom of the canopy to those trees below it could be how many predators are there in a population so how many herbivores are in the population to consume material could be based on disease it could be days based on water availability okay so caring capacity is always going
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to limit your population size okay so shortages of food or any other kind of resources can lead to the reduction of the population size so that's carrying capacity carrying capacity is the top end how much of some resource food water sunlight space how much of that resource can be dedicated to the population now your carrying capacity even though your carrying capacity is very species-specific you also have to take in the other species around because they
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determine your carrying capacity also so let's say your carrying capacity for a forest that is you know one acre by one acre and small little little patch of force may be the carrying capacity for that given acre by acre might be you know ten thousand trees small poplar trees ten thousand of them can fit acre by acre okay well what happens if you have other trees coming maybe you have some pines come in maybe you have some ground vegetation that comes in that
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starts stealing a water resources or nutrient resources and maybe you have some other species in there that's now competing now you're carrying capacity isn't 10,000 you're careful capacity might be 8,000 with these other trees available or in that region or this other vegetation okay so carrying capacity again is that top end now a lot of organisms forests included they overshoot their carrying capacity so when a seed when a tree reproduces seeds it doesn't logically think all I'm only
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going to reproduce a hundred thousand with the germination of five individuals right this year because we're getting close to carrying capacity no nothing that tree is going to reproduce as many seeds as it can as many seeds as it has nutrients to reproduce so maybe it would produced 2 million seeds and 50 individuals are going to germinate but because those 50 individuals are going to germinate now you've overshot your carrying capacity because all the trees
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are reproducing as much as they can maybe the amount of water was the limiting resource and you had very good rainfall that year so all these trees are all reproducing at their maximum and you overshoot cane capacity because you had you know this one fluke year where you had really good water and you overshoot and what happens when you overshoot is when you get back to the normal carrying capacity now your population is over it and the next year
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when water resources are at their norm that population crashes and you have the huge die off and the scarcity of some type of resource it could be water it could be nutrient it could be sunlight it could be you know there's lots of different situations could be that there's less predators in the population could be there is less squirrels or scavengers that are going to eat seeds or nuts or fruits or something like that and you're you have
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a higher germination rate because you have less competition for the seeds there's lots of things that can cause that okay overshooting can cause your population to crash now how bad the crash is depends on a lot of things okay first of all why did you overshoot how much did you overshoot and then secondly how bad or how much has the carrying capacity changed so then at the same time that you overshot that the next year is your K much lower then your
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crash can be really heavy and you can crash well below the can capacity yeah well look at this Oh a lot of populations do this often they're called boom and bust populations so they'll explode send out tons of seeds population grows very rapidly overshoots gained capacity crashes okay cockroaches fleas flies lots of weeds College eaten Russian thistle different types of grasses will do this crabgrass other kinds of plants will shoot out millions of seeds and then due to the carrying capacity
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they've way overshot the K so there's not enough nutrients not enough water not enough so something in the in that region for them to sustain and the part relation crashes okay so for long-lived organisms it's often not as drastic of crash because what happens is the populations reproducing and it keeps reproducing and reproducing at a given rate but then due to a lack of some resource that population growth starts to slow and that's what you're seeing
27:13
with the human population when you saw that curve okay we reached our maximum probably somewhere around 2000 maximum growth rate maybe maybe earlier than that 1990s maximum growth rate and then we started to kind of trickle still growing still going towards your carrying capacity but you're not at what we typically call msy or maximum sustainable yield okay we're going to talk about that in a second maximum sustainable yield is what a lot of
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managers try to keep their population at and I'll talk about it so this slowing of the rate of growth moving from an exponential growth rate to what we call logistic growth rate will change your growth curve from a J to an S okay and that's because your population is gradually reaching that carrying capacity or that population is gradually becoming more resource scarce and this is the typical for most populations on the planet even cockroach populations
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are going to reach a point where that population starts to flow now in some cases it's too late and they overshoot the carrying capacity and then they crash but for a lot of them as long as the growth rate slows or the their food source is slow you'll have your desk or your deaths in your births are going to be closer to each other so your rate of growth starts to slow and no reach a carrying capacity okay and so that's the logistic growth curve this is
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the mathematical formula for that the only real difference here that we're talking about is you have a K now okay so there are lots of ways to write this you're going to see other equations but K is included so your R it's the same thing as your rate of growth you're in as your population okay and your K is your carrying capacity so you have to know how many individuals can that given region support okay then at that point you can calculate what we call a
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logistic growth curve a logistic growth curve okay will often look sigmoid 'el okay this is showing you a typical logistic growth curve of a pop of most populations because what will happen is you have slow growth rate so let's say a new area happens it's guys let's say you have a fire you know sweep the landscape and it's it's an extreme fire it burns everything okay your first population is going to be those pioneer species etc okay but let's
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skip over those they're gonna have the same kind of growth as this but let's talk about some trees it will come in so let's say you're you know your first trees are going to be some Pines that are gonna come in so some ponderosa pines that are gonna come in and you might just have a few you know seeds that are carried in probably by some animals transferring them in etc from a from a forest nearby alright so you get few individuals population starts to
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grow expand because you have this open area there's really no fighting for resources all you're fighting against is some grasses and annuals maybe some little bit of perennials but what's the trees get up there they're stealing the sunlight they're stealing more nutrients their roots go deeper they can they can definitely get established fast okay and they start going and going on and on and at this point about half of K is the
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fastest rate of growth that's what we call the maximum sustainable yield this is where your AR is at its maximum is half of K okay we'll come back to that again but this population that keeps going keeps going keeps going and now it's starting to reach its resource decline so now you're starting to see that birth rates and death rates are getting closer to each other so your AR is starting to slow and you can start to see that slope is changing
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so it's going from a J curve to an S curve and even at that this population still can overshoot carrying capacity so it overshoots the carrying capacity it crashes okay and it overshoots again it crashes a little bit and this is what a lot of populations will do we expect this to be true of pretty much every population is once you reach carrying capacity if nothing else changes and it's pretty stable environment you should see a cyclic movement around
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carrying capacity up and down up and down up and down now some years is going to be a large up lift and then a very drastic down lift and then other years if K is very stable you might not even see the blip over okay it might just be pretty much a straight line okay but remember a very key component to this is carrying capacity changes your K will change your K is dictated by something a limiting resource water nutrients something sunlight space something okay
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so k can change now space if you only have certain money I'm out a spaceman but we're talking about an island okay unless water is rising and the island shrinking your K is pretty much constant but in other regions if it's space and you know you have a forest and then next to it you you might have some grass lines you could get some encroach Minh onto the grasslands or vice versa the grasslands could be encroaching on the forest so your cake could be shrinking
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or it could be growing okay again Kate can change nutrient levels can change often this occurs in new establish forests the nutrient levels are great because pioneer species came in they you know got established they died the nutrient load the soil trees come in suck up those nutrients okay and the more coniferous trees you have okay the worse your soils are going to be so your carrying capacity might be dictated by how poor your soils are because now you
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don't have new nutrients really coming in or not very much say so that might be your carrying capacity and unless you have some kind of new nutrients come in maybe that means that you have Aspen or poplar or some tree that drops it leave its leaves come in and get established Oaks come and get established start dropping leaves in that region increase in the level of nutrients in that region so you can get more Pines and your kin capacity can change okay so it really
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just depends on you know the population know as a manager let's say you're managing your own forest if nutrients is your your carrying capacity if that's what's making it so you can't plant more trees person would then you fertilize your forest and that's what a lot of managers on plantation tree plantations do okay they don't want nutrients to be the limiting source so they just add nitrogen they add fight phosphorus to the soil those
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are the two typically the two limiting nutrients in a forest so you're just adding nitrogen and phosphorus to the forest so you can increase your carrying capacity now always you're going to reach when we're talking about trees you're always going to reach a carrying capacity with light penetration or available light okay you can only have trees so dense okay there's so much shading that occurs before the tree can't do enough photosynthesis to
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maintain itself okay so that's important to know but prior to reaching that point a lot of forests will be nutrient deficient or water deficient okay and you can add those two resources and often managers of plantations like that they'll do so okay so again I told you you can rewrite that equation a little bit differently okay and it's more meaningful in my opinion to write it this way as your growth is equal to your rate of growth
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or rate of change times your population okay times K minus n or your carrying capacity and minus your current population divided by K that K minus n is very important because as your population gets close to K eventually when K and n are the same this is going to be zero okay so your rate of growth is going to be zero if you're in overshoots k this is going to be negative and actually your rate of is going to be negative which means your
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population is actually declining okay so again these are the two models that we're talking about exponential growth okay that J model okay again exponential growth will always reach a carrying capacity and then sigmoid sigmoidal growth or logistic growth curve okay where you get this kind of trickle and you're slowly reaching carrying capacity again it's not going to be a straight line on carrying capacity it's probably gonna have some
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overshoot some a little bit of you know up and down cyclic around that K and then if Kate ranges you can see another blip you know where you got to reach that K or if K goes down you know you're gonna have a crash where your population it's gonna crash back down to that K all right so let's talk a little bit about maximum sustainable yield it's really important as a tool in wildlife management but forest managers use this also okay because if you know something
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about the trees that you're managing you know something about the carrying capacity that you have then you know what the maximum growth is going to be now it's not important because you're not harvesting these organisms at a really young age from what most most managers are not okay so you're harvesting these organisms at a certain size typically and that probably over this maximum sustainable yield but what this does for forest managers is it
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tells them at what point in time can you expect your force to be its most productive in germination and seed and growth and then at what point in time can you expect your rate of growth to basically be nil okay now if that rate of growth is nil then this might actually tell you how many trees you should leave on the landscape okay so even though this time period so let's say it takes 20 years for the population to get to a point
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where it's maximum growth and it takes another 20 years for that population to reach bearing capacity okay well you can cut out 20 years if you only knock your population back down to its maximum sustainable yield its population size here if you knock it down to here then your rate of growth can you know expand to that okay so you just take that 20 years off and you knock your population back to its maximum sustainable yield or maximum sustainable population size this
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is where they're going to be reproducing the most efficiently and they're gonna be adding back into into the population then when you get back to your K you remove all the large trees that were basically seed trees here and knock it back down it's in maximum sustainable yield and you keep doing that there are individuals there are forest managers that use these models to drop their population to half of its K and allow for it to bounce back up drop your
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population to half K allow for it to bounce back up you're cutting off years of your forests years of your harvest by doing so now it's a lot more intensive when it comes to cutting or removing of forest material because you're not able to go in and remove trees you know with machinery often you're having to you know go in and remove it with sauce and and actively remove trees that way so for some individuals this doesn't work at all
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because you know you're interested in cutting your entire plantation replanting it moving over to the next section of your plantation ok cutting that one replanting it etc etc and you just rotate through your acreage cutting your trees but for some individuals this is you know cutting years off of your forest because your growth rate is so high here ok all right so with that being said you can expect to see some homework or some assignment that you you
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get to do calculating population growth it's important to be able to calculate both exponential growth and logistic growth because in some situations you might need to know what's the exponential growth of this newly planted forest or you know this single seed tree that I left on this one acre what's the exponential growth rate going to look like okay or what's the logistic growth rate gonna look like 20 years down the road so with that okay we'll see you next time
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