Population Growth

Lecture: Population Growth

Prof. Stephen C. Stearns

Yale Univeristy

Overview

The growth of populations is held in check by several factors. These can include predators, food and other resources, and density. Population density affects growth rate by determining how likely is it that an organism will interact with a member of its own species compared to an organism of a different species. Population growth studies rely on the mathematics of logs and exponents.

Part I:

Part II:

(Video source: http://open.163.com)

Population Growth

Prof. Stephen C. Stearns

（(Video source: http://open.163.com)

(Chinese and English subtitles: 俞欣玥）

So let’s do an example.Here’s a life table that actually is roughly that of a small bird.So the guys that fly into my bird feeder in Hamden,the house sparrows,the chickadees and so forth,they could have a life table that looks something like this.They don’t live very long.So here are the probabilities of surviving to age 1,2 and 3.Here are the birthrates.I’ve set them constant for the three age classes.Here’s LxMx.This is our population growth rate.It’s 1.2 per generation.So just based on this,we can make several interesting statements.There population is growing.Okay?And that’s because 1.2 is greater than 1.And it is multiplying actually 1.2 times per generation;not per year,but per generation.We can calculate the generation time with this formula.So we just divide sum of LxMx times x,by R0.That turns out to be 1.67 years.And the meaning in words of generation time,in demography-it’s a technical meaning.So that is the average age of mother of a newborn.The little-r can be estimated,and this is an estimate,it’s not precise-by taking the log of big-R and dividing it by the generation time; and that’s about 0.11.So this population is growing at a compound interest rate of about 11% per year,and we can use that to go back,and use our doubling time calculations to figure out.Oh, it’s doubling about every 6.3 years.If I’m putting the bird seed out there in Hamden,I had better be ready for large expenditures.And I’d like you to note that R0 is calculated on a different time period,generation time than little-r.So when big-R0 is equal to1,little-r is equal to 0,and the population is then-the word is used,stationary;it’s just replacing itself.A stationary population is replacing itself,but a population with age structure,that’s growing,can be in stable age distribution.Stable age distribution refers to the ratios between the age classes;stationary refers to whether or not it’s just replacing itself,or whether it’s growing or declining.So that’s a rough sketch of simple demography,and an introduction to the different ways that ecologists and demographers conceptualize growth rates.No I want to criticize-not really criticize,but comment on one of the basic assumption of this.The first way that you can complicate simple dynamics is by putting in age structure,but the second way you can complicate it is by putting in density.Here is growth of a bird was introduced to Great Britain,the turtledove,and took off from a very small population,and it grew exponentially.Remember, if you have a straight line on a semi-log plot-so y-axis here;numbers are on a log scale;x-axis is on an arithmetic scale.So this is a straight line, and it’s going like gangbusters,it’s growing exponentially.And about every,let’s see,one,two,three-it’s increasing in size about 10 times every 3 years.So these doves are really pumping out the babies,and they’re surviving pretty well.But something happens up here.This is a real point here,and it starts to level off.So the question is,what does increasing population density do to the demography,the birth probabilities,and the death probabilities of individuals?Can we understand that in terms of the kinds of concepts we are already covered this morning?Well you’ve seen this plot before.So as density-we can think of this being a high density population,and this being a low density population,with rapid growth,rapid individual growth-now not population growth,but rapid individual growth increase of size with age at low density,and slow individual growth at high density.And I think by now you’re familiar with the idea that there is a reaction norm for age and size at maturity.And if you move from low density to high density,basically what happens,in many cases,is you get organisms that mature later,at a smaller size.So one of the very basic characteristics that determines population growth is age at maturity.That’s basically the interval over which the compound interest is being calculated,and that’s responding plastically to density.So as the density of the population goes up,the organisms delay maturity,they mature later-here,rather than here-and they matures smaller-so they are less capable of making babies,and they’re doing so later in life at a slower rate.The other dramatic thing that happens with density is that mortality increases.Here is an experiment that was done with trout.This is a log scale here and a log scale here,and the flat line here basically indicates that growth was stopped,or that density has stopped increasing.And if you take an experimental stream bed,and you seed it with baby trout with a lot of times,at different initial densities per square meter.So this would be 10 raised to the 2.5 per square meter.So this is about 300 trout per square meter,and this is 10 trout per square meter over here.So in arithmetic terms,we’re going to 10 per square meter,up to about 300,and we get out,at the end of the experiment,is roughly 10 per square meter.So there’s an enormous amount of mortality which is going on,over on this end.And you can think of that as sort of runt of the litter kinds of things,baby pigs competing in a litter,or you can simply think of it as interspecific competition.These trout are competing for food,and the ones that are best at finding it are going to be the ones that survive.There’ve been lots of expereiments done with plants.Here again we get a log scale on this axis,and a log scale on this axis,and this is the density of surviving plants here.So this is the number per square meter.On this axis we have the amount that were planted,and that’s going to range from-down here we’re probably at about 12 or 15,up to about 1000 per square meter,were planted.And these different lines here are showing you what the density of plants is 22 days after planting,39 days after planting,61days after planting,and 93days after planting.And this is the demographic process that basically leads to constant final biomass.So if you go on,all the way through to harvest,you are going to end up with about 100 plants per square meter,and by the way,the way that plants regulate their individual growth,u essentially get the same amount of leaf area or root area.So if u plant a lot,you are going to end up with fewer leaves and less roots,and you are going to get smaller plants,but the total amount of plant material sitting on that square meter will be roughly constant,irrespective of initial planting density,if you just let this process go on long enough.The other thing that happens is that as density goes up,fecundity goes down,and that’s because as density goes up,competition for food increases,individuals are not getting fed as much,and they are therefore less able to make babies.In this panel over here,we see basically natural variation in song sparrows on Mandarte Island,which is a small island off the coast of Vancouver Island;it’s not too far from Victoria.If you take the ferry from Vancouver,over to Victoria,you go within about a kilometer of Mandarte Island.And this is the number of young that females were able to fledge.These are the years.So that’s 1980,1981,1982.This is the number of breading females on this island.And what you see is that as the number of breading females increases,the number they can fledge starts to decrease.And in the year 1985 they did an experiment,where they added food-so they wanted to see whether this natural variation could be manipulated,just by putting food out on the island-and those who were fed were able to rear nearly 4 offspring;and at high density,up here,in 1985.Those who were not fed were only able to rear one.So there was a difference per female of nearly 3 offspring,and that was what the population density was doing to them,and the manipulation experiment showed that it was the density and that food was the mechanism.If you look at a dune plant in the Netherlands-this is Vulpia,which is a grass,and this is growing near Leiden,on the dunes on the North Sea in the Netherlands.The number of seeds per plant,number of flowing plants per quarter square meter;the number goes down.So the more crowded they get,the fewer seeds they can make.So this is variation that’s really going on in Nature,and it shows you that the effects of density are quite real,and they are pretty dramatic.These effects can be combined.So if we want to combine number of births and numbers dying,this would be density,over here,going from 0 up to a number K.The numbers dying increase as we go across this,and the numbers being born decrease,and at some intermediate point,you have the greatest difference between the births and the deaths,and that difference between the births and the deaths is the recruitment.So the recruitment inferred from this graph looks like this,and at 0-you get 0 recruitment when you hit this number K;and that number K is called carrying capacity.So the population,under density-dependent regulation,will increase until it hits carrying capacity,and then it stop growing.Now as density is taking its effect,the size distribution of individuals in the population often shifts.So what I’m basically doing here is I’m showing you all the things that density does to be individual lives of the organisms,that are experiencing this density increase.And this is a very interesting one,because basically what’s going on here-and this is measured in flax.If you start at low,medium or high density,2 weeks after emergence the distribution of body sizes in the population is pretty similar;it’s a nice bell-shaped curve.This is now 6 weeks from emergence.So a bunch of other replicates have now been gathered.And you can look at plant weight here,you can see that they’re bigger, the scales are changing.But you’ll notice that at the medium and at the high-density treatments,you’re starting to get quite a bit of shew.You are starting to get a few big ones and lots of little ones;that’s what this kind of histogram means here,lots of little ones and a few big ones.And at the final harvest,which is probably about 12 weeks after planting,you’re starting to get a little bit of skew at the low density treatment,but at the medium and high density treatments,you’ve got a lot of skew.Now this is something that was going on,by the way,in these trout experiments;you were getting a few big ones and a lot of little ones.In the plants living in the Dutch dunes,you were getting a few big ones and a lot of little ones.It’s pretty widespread pattern that happens as density increases.There’s going to be inter-specific competition.Some individuals are going to be better than others.And it’s going to result in these kinds of size and growth distributions.So the more competition there is,the greater the skew will be between lots of little ones and a few big ones.So that means that competitions is producing an asymmetry,and this asymmetry,where you have a really skewed size distribution,just basically means that a few individuals are going to be doing most of the surviving and reproducing,and a lot of individuals are going to be doing most of the dying,and not reproducing.In many cases the number of large individuals is relatively constant,while the number of small  individuals varies much more widely.That’s the law of constant final yield.Or the flat line that we saw with the trout that were planted at-were seeded into the streams,at many different densities,but you always ended up with roughly the same number.That was probably the number of the large surviving competitive individuals,and the small  individuals are being squeezed out.That has an evolutionary consequence.This is a direct tie now between population dynamics and natural selection.Not all those who are born will survive to reproduce;some are starving or are being crowded out.And not all who survive to reproduce are large and in good shape.Many of them,in fact,are in poor condition and having few offspring.So inter-specific competition produces great variation in capacity to reproduce and in reproductive success,and that means that wherever you have reproductive and mortality skew,that’s being produced by inter-specific competition,you are generating the conditions for natural selection.It’s as though there’s an internal process in ecology that just hands you the conditions for natural selection on a dinner plate.Now I previously mentioned the case of the collapse,of the sardine fishery in California.And the sardine fishery,which collapsed in California between about 1948 and 1955-and still hasn’t come back by the way-it collapsed because of conditions that were affecting juvenile sardines that were out in the plankton in the Pacific.It did not collapse because of a fishery that was operating on the adult sardines;and this is pretty well established now.When fewer and fewer juveniles were being recruited into the adult population,the individual adults were essentially experiencing a release from inter-specific competition.So their response was to grow faster;they could eat more,they could grow faster and they could make more babies.And just as the last boats in that fleet were putting in and deciding to stop fishing,they were catching a very few,very big sardines;the sardines were about 1 meter long.They had gone from that big to that big,and that was what the release from inter-specific competition had done to them.And if they were able to re-establish the fishery and have lots of babies recruit into the population properly,and grow up,the size should shrink right back down.That would be the impact of density dependent population growth on the California sardine population.Okay,populations are held in check by lots of things. They're not just held in check by food. They are held in check by breeding sites, by space, by lots of limiting resources. As populations increase in density, the individuals shift along reaction norms. They reduce growth, they are smaller; when they become adults they experience more variation in adult size; they have lower fecundity and more variation in fecundity; and they have higher mortality and more variation in mortality, as they encounter density dependence. Now if you have a successful species--okay? So it's a dominant species in its local habitat, in its ecosystem, then intra-specific competition, where these kinds of things are going on, is often a more important brake on its growth than inter-specific competition. And that's because, because it's dominant, it's numerically abundant, and the average interaction that an individual has is with another individual of the same species, rather than with some individual from another species. So that would be a circumstance under which we could expect that intra-specific competition, generating all of these effects, is likely to be important. If the species are rare, or if they're often at a low population density, then the opposite might be the case; then the interactions with other species may be generating these kinds of effects. But whether the effects are generated by interactions with other individuals of your own species, or individuals of another species, the impact of increased overall density is likely to be qualitatively similar. It's going to produce reduced growth, smaller adult size, lower fecundity, higher mortality, and more variation in all of these parameters. Okay, next time we're going to do competition between species. But we have a few minutes this morning, because I managed to get through this one a little more quickly than usual, and so I would be happy to take questions on anything, if you wanted to ask; and we have about five minutes. You know, a lack of questions means that everything is just so stunningly clear that everybody could explain it perfectly well themselves. Fantastic. I once had the experience, when I was in my first year of graduate school in British Columbia, of being asked to be a TA. I had arrived in January, and I was asked to be a TA in Spring semester, and I was a TA in a statistics course. And I stepped into this statistics course and discovered that even though the Fall semester had been taught by one of the greatest teachers at the University of British Columbia, a man who was revered for the clarity of his lectures and for the engagement of his speaking style, that the students, in fact, knew very little statistics. And the problem was that they had thought, because he was so clear, that they understood everything. That was not the case. They couldn't understand it until they explained it themselves, and they discovered they couldn't explain it themselves. Any questions? Have a good day.