 Hi everyone one more one more lecture before our long weekend cruising right along in our semester I invite you to spend the weekend catching up on your work making sure you're getting the semester in the year off to a good start work hard play hard right but this is a great time to get ahead in your reading by a 1b is going to catch up with you really quickly when the labs get rolling the workload is going to increase dramatically for you and so now's a great time to get a step ahead for some of you ecology is going to be the easy part this stuff will come naturally thinking in cycles and levels and interactions but for others this is going to be the most difficult people are really people do vary in in the success that they have in the different modules in this course so if it's easy now it may not be in the second or third module and so be careful with that okay it's a long semester it is a marathon not a sprint in here so try to keep keep your energy levels high all semester long if you are aiming to do well I'll eventually maybe next week put up some practice questions on b-space do that next week to give you a sense of how questions are asked on the exams those questions that I put up are not meant to be as hard or as easy as the questions will be on the exam inevitably when you put up questions like that you give the exam and people say oh it's so much harder than your practice questions okay then the questions I'm going to put up on b-space are going to be easier than the questions on the exam how about that I will avoid that problem by saying that now really I put them up to give you a sense of how a multiple-choice question in here is structured the other professors will write their questions differently but all of the exams in the lecture portion of the course are multiple choice there's a guide to successful taking of multiple choice exams on the course website written by John Lotto a former professor of this course and you're welcome to look at that if you want to to try to develop some techniques for good test taking but it's not unlike so many of the exams you've taken to this point in your lives in the multiple choice and the strategies that are important for doing well I emphasize the conceptual material on the exams you do not need to know the date of birth of an individual who founded a particular topic in ecology in species names are absolutely not relevant for your studying for exams they may be used a species may be used in a test question but you will be given the name you are not to not it's not necessary to memorize those types of things right we're really memorization itself is somewhat de-emphasized in this module we're going to be testing on conceptual understanding if the test is written well it is going to determine if you understand these concepts and for some of you that'll be hard for some of you coming through a science track and if this is your first course that's going to be that's going to be challenging it's because you can't just rely on your memorization you need to understand the material and the litmus test of whether you understand it or not is whether you can discuss it whether you can communicate about it with someone else thus the value of your discussion sections and talking about these issues great to form study groups in preparing for the exams smaller large so that you're talking about the things when these words cross your lips having been formulated the concepts by your brains then you'll get a sense of whether you understand it or not okay sitting down with the book and memorizing it and then coming cold without having talked about it at all into the exam you're really not likely to do as well you might do just fine you know everyone's different but that's my advice to form study groups and to talk about these things as much as possible okay any questions on those things feel free to email me and I'll address them we are going to cover a lot today so let's get right into it we are continuing our discussion of population ecology and we will be looking at population growth models and the phenomenon of exponential growth initially then we will talk about constraints on population growth and density dependent effects on populations and the logistic growth model we'll be talking about exceptions to the standard models in a brief but important discussion of Ali effects and if we have time at the end we'll start talking about predator prey dynamics if we don't get to that today it'll be a great benefit just fine in our next lecture on inter specific relationships remember our figure of a population of sparrows from last time let's start to become a bit more mathematical in our presentation of populations in our modeling of populations begin represents the number of individuals in a population that was introduced previously and we discussed how population numbers can increase through birth in immigration and populated population numbers can decrease through deaths and emigration so using capital letters to symbolize those phenomena you can generate a simple equation where the change in population numbers delta n equals births plus immigration minus deaths plus plus emigration let's simplify that and assume that immigration and emigration don't exist or they cancel each other out and we'll focus on the phenomena of births and deaths let's talk about population change with respect to time with respect to concrete intervals of time in a discrete sense in the sense of discrete population growth either across the lives of individual organisms during the lifetime of an individual or across generations of reproduction from one generation to the next some discrete interval that we represent by delta t the change in time so big delta n over big delta t equals births minus deaths a lot of this I'm going to really simplify the presentation even relative to what's given in your book which is already pretty simple in your labs with your GSIs I hope they'll add complexity to the presentation your labs that you'll perform for the this section actually are more complex in some of the mathematics that's great the GSIs are going to present it however they want and some of them will give more complexity than I will I'm giving you a very simple presentation I you know it's not to say that's easy but it's it's almost as simple as you can get for a population ecology introduction going back to our life tables because it's from these tables that we we generate much of the data necessary for this modeling work remember our cute little squirrels from the Sierra Nevada and and the phenomenon of a life table mapping survivorship and mortality across cohorts of squirrels in these different age classes note that there's the column here called death rate which is the number of individuals that die in a particular interval relative to the total number of individuals present that that column of information will be carried over in some of our modeling work and recall also that we distinguish females from males in this process because they're the statistics for these populations differ the statistics for these sexes differ within the population I very briefly mentioned reproductive tables where the life table focuses on survivorship and mortality the reproductive table emphasizes natality and birth again structured in terms of cohorts we have columns representing the proportion of females that produce a litter of offspring in a particular interval age interval so squirrels under one year old are not producing any offspring the average size of a litter of offspring and in that case the female bears those offspring of course but those offspring are both include both males and females and both are counted here specifically the number of females in a litter represented here and finally the average number of female offspring per individual for that interval that age interval this is often the case that only females are focused on in this work after all it's only the females that are bearing the young in these sexually reproducing species and only the females need be relevant to the calculations that will use so that's that's often the case and things are simplified in that way so this this data these data are also used in the modeling work that will that will focus on now and of course this is for squirrels other organisms are going to differ in the timing of these events some mice some that live right here outside the building will start breeding in a few weeks of age so your time intervals would change and you know at three weeks they're producing a litter so this is going to be organism-specific in part based on the biology of the specific organism it may also be environmentally influenced and those two themes will carry through the discussion here the intrinsic biology of the organism as it influences life tables and reproductive tables the life histories of these organisms as well as environmental and external influences on these life histories and their patterns both are important intrinsic phenomena and extrinsic phenomena so let's introduce two more terms and these are these are terms that are derivable from our life tables and reproductive tables the columns that I higher highlighted the per capita birth rate little b and the per capita death rate little d per capita right you've encountered that at some point before by head per capita so according to the number of individual offspring sorry according to the number of individuals present that's what we mean by per capita right so a per capita birth rate is the rate of birth relative to the number of individuals present and the phenomenon of the total number of births in an interval or the total number of deaths is equal to that per capita birth and death rate times the number of existing individuals so big b equals little b times big n and big d equals little d times big n substituting back into the equation from the previous slide our change in numbers for a particular time interval is equal to bn minus dn and moving forward from there well we need to we need to introduce another term and this is little r our per capita rate of increase for interested in modeling populations over time we're really interested in the phenomenon of the number of deaths relative to the number of births and subtracting the deaths from the births we arrive at a an estimate of how the population is changing in time little r our per capita rate of increase or it's variously known the intrinsic rate of increase I think your book uses per capita rate of increase we'll stick with that note that if r equals zero if births and deaths are equal if birth rate and death rate if birth rates and death rates are equal r will equal zero and total number of individuals in the population won't change over time there may be a lot going on there may be lots of individuals being born and lots of individuals dying but if they cancel each other out total numbers of individuals will not change on the other hand if ours greater than zero the population will be growing and if it's less than zero declining so if it's negative it will be declining positive growing this is all very handy there's our equation from the previous slide let's just take n out of that equation and by substituting in r the per capita rate of births minus the per capita rate of deaths substituting r for this term we arrive at this equation where the change in population number relative to time an interval of time is equal to r times n so let's look at what what this implies for the shape of a growth curve but first let's make it a little more realistic because growth doesn't happen in simple intervals for most organisms growth happens population growth happens instantaneously it's happening births and deaths are happening all the time sure there are very typically breeding seasons reproductive seasons and seasons of higher and lesser mortality but they're not occurring in strict strictly fixed intervals typically so we really need an instantaneous rate of change and to do that we need to use calculus but you don't we're not going to I'm not going to ask you to rely on calculus for this given that you're all over the math in terms of your your math backgrounds maybe your GSIs want you to that's great if they do but for for I'm just going to gloss that step over and and just give you the new term terminology for handling instantaneous rates of change using a differential equation and so let's this here's our step in doing so just to replace our big delta with with little d right so we have dndt in the differential equation equals our instantaneous the instantaneous rate of increase times total population number and there your book gives you a couple of definitions of r ecologists have struggled with the concept of r for a long time and there's a lot of very active debate on it I'm going to gloss over that whole debate and I'll just ask you to for for a single understanding of r and a single simple understanding of r not even divided into an instantaneous r or a maximum r you can just work with a single r single our concept the intrinsic rate of increase and so this will be the equation we can rely on for the basic modeling of population growth in the absence of constraint and I'll explain what I mean by that absence of constraint now okay so dndt equals r times n in our instantaneous rate of population growth model of population growth and this is what the shape of it shape of that equation looks like over time the size of the population increases according to a j-shaped curve like this according to this model of growth this is exponential growth so if you let bacteria go wild through fission you can see how quickly they will increase to extraordinary numbers if left unchecked that's what this phenomenon is this exponential growth phenomenon this is what this represents the steepening of this curve in time growing all completely out of proportion to extrinsic reality really because of course if if if organisms grow like this they can't do so forever or we would be swimming in them right we're swimming in humans on this planet right now practically and humans have followed an extraordinary growth sequence it's a little funny minus 8,000 years ago so 8,000 in before the common era I just have it labeled a little oddly here population growth during these early early times was increasing slowly steadily but then faster and faster and with the industrial revolution pretty coherently following an exponential logistic sorry an exponential growth model and here we are sitting you know somewhere somewhere up in here with blips related to mass mortality in the process but the recovery from those those insults and the continuation of growth I'll spend a much longer talking about human population growth in a later lecture our understanding of this phenomenon has changed so much in the last 15 years it's a really interesting history examples from other organisms a local example elephant seals you can go to the beaches here point rays out by the lighthouse or even better down at the beaches north of Santa Cruz like annual Nuevo and see beaches chock a block with elephant seals big big animals huge seals with these proboscis these noses these long noses that you know give them their name their elephant seals the males are much more massive than the females truly gargantuan creatures and down at annual Nuevo you need to make a reservation but you can walk on the beaches with these guys in close proximity with a guide and one of the things that's so remarkable about doing this is that within the last hundred years the numbers of elephant seals were reduced on the scale of the whole planet to a couple dozen by the best estimates there were maybe 20 individual Ellis elephant seals on earth as a result of over hunting in the early 1900s from even in the 1960s according to this data these data from those that original set we've had an exponential increase and the occupation of all these beaches along California and elsewhere of elephant seals yeah there's not much genetic variation if you're if you're back here with 20 individuals and you may have indeed experienced the type of bottleneck where genetic variation was so limited that you might expect to see problems in health of the population over time yeah so this creates a very fragile situation according to our understanding in the health of the population something I'd love to be building on here but you'll certainly get exposure to those concepts in the evolution section yes you're right nevertheless you walk among them and you look at these numbers and they look darn healthy thumbing their big proboscis at such concepts from evolutionary biology right there are other examples real world examples elephants in Kruger National Park in northern South Africa again they were they were shot out of the place hunted for ivory hunted for big game trophies over exploited as many mammals across the globe have been as many organisms have across the globe and maybe particularly the big mammals biggest mammals so elephants are of great interest to many people not least the tourist industry of a place like South Africa so conservationists and the general public fought hard to reintroduce and care for the elephant population in Kruger and look what has happened over time they've done all too well they've done so well that they're creating problems they are creating problems for the environment because of their capacity to alter the environment elephants knock down trees elephant an elephant can eat a whole tree can knock down the tree and eat the leaves and you strip the bark and kill the tree and you get enough elephants and you're going to radically alter the ecosystem for other organisms you can have too many elephants and so you had this agonizing situation in a place like this where you have conservationists very much interested in preserving elephants and animal rights activists who absolutely don't want elephants harmed in any way and politicians who dearly want to keep the tourist industry going strong and ecologists who are talking to all of these individuals and also trying to say look we have a real problem on our hands because of the effects that this great increase in elephant numbers is causing on other organisms and we need to do something about it we need to manage this what do you do do you go around and shoot elephants for many people no absolutely you do not do that that's absolutely inhumane do you go around and and try to sterilize them so they cease reproduction well you might try but try you just try it's an extraordinary challenge I mean these are these are vast spaces and these are mobile animals with complex life histories and you might try but more likely than not it's not going to work so you have to make political decisions about whether you're going to go in and shoot try to relocate if you're going to relocate thousands of elephants where are you going to put them it's a really a dramatic problem and they've struggled with this in recent years and you can you can read about it online but do something you must really there is a management necessity here in a situation like that so exponential growth is real in these systems when these organisms are proceeding largely unchecked by other factors their intrinsic capacity for growth being realized or being close to realized without check will give you these types of j-shaped curves and these explosions of numbers note here the effect that this per capita rate of growth or this intrinsic rate of growth has on the shape of the curve when it's higher it leads to a steeper curve and what dictates are is both partly a function of basic the basic biology of the organism but it's also a result of environmental factors as I'll try as I'll try to draw out here organisms do not increase exponentially forever or we would be truly swimming in them why not they yes there are limits to growth like food about availability so we will talk about the limits to growth in terms of density independent factors and density dependent factors don't worry about the slide for the moment let me first address density independent factors for which I don't have a slide now let's do it after I'll ask you about it after since we're already here let's focus on density dependent factors first and by this we mean factors that influence growth in relation to the density of organisms present and someone just has mentioned food here and we can think of resources broadly because as numbers of individuals increase in a population competition for resources is typically heightened and for and resources may be food but they may be other factors such as space the availability of space in which to live or something an abiotic factor like water availability not exactly food but very often there's great evidence in natural populations for relationships like this with increasing density measured here by plants per meter squared on a logarithmic scale relative to some measure of fecundity or of fitness term you'll get introduced to you in the evolution section per individual you see a decrease in fecundity or fitness relative to numbers relative to density and that's seen here in the in plantain in that plant relative to some limitation on resources probably space water light things like that or here in birds and in the song sparrow relative to the number of females in a unit area the size of the number of eggs laid the average clutch size it's a linear decrease more or less in the size of the clutch relative to the number of females in an area why because because food resources are limited in this case in this study I forget this was an island environment where this study was conducted I forget where it was exactly but food resources were limited so that the more females the were there were the less food there was to go around and the fewer resources that could be dedicated to the production of offspring it's competition as a result of density density dependent competition and these things affect life history parameters including including birth rates and death rates an example from your book these heirloom sheep from that were taken off of the one island where they were continuing to exist this is the closest relative to our domesticated sheep they were put on to another island here to island in in the early 1900s and population ecologists have been following them ever since and closely studying them the percentage of juveniles the percentage of young sheep that are that are breeding and producing lambs goes down with increasing numbers so the more numbers of individuals there are in the population the fewer young individuals there are reproducing and that has a great effect on overall population numbers I'll stress that more later but that age at first reproduction the earlier it gets that can have dramatic effects on population number increases so if the young individuals are not reproducing as much that's going to have a big effect on population growth another example of density dependent effects so just to summarize for you some of the effects of density that might limit population growth I focused on resource competition before but think about the fouling of the environment as a result of numbers in a in a dish like this numbers could increase following more or less exponential curve but as a result of the production of waste the environment may become less and less suitable for growth and numbers will be curtailed as a result territoriality among individuals as density increases as numbers increase sometimes organisms become more and more aggressive in their defense of their territories and that can start to constrain population growth this you will probably have encountered the fact that in populations with greater numbers and higher density parasites and pathogens and disease may be more likely to propagate and and and those can have checks on population growth with increasing numbers of individuals if only because greater densities lead to the ease with which parasites and pathogens can be passed through sneezing or coughing phenomenon like that among humans predation if you as the numbers of individual organisms in a population go up they may become focused on by predators so they become less and you can think about this in terms of optimal foraging strategy of the predators themselves as the numbers of individuals in a prey population go up predators may turn their focus to those prey populations as a result of their greater ease of capture because of their abundance that can serve as a check on their part on the growth of those populations probably here I can mention density independent forces you can you can pretty much for my purposes relate the density independent forces that might provide checks on populations to environmental abiotic factors can someone give me an example of a force that might act to curtail growth that's not related to the density of individuals present yes please a volcano excellent a volcano erupts and it doesn't matter if there are 300 of you or 20 of you if you're there you will be scorched drought ditto yes droughts if if there is not enough water present it's probably not relevant how dense your population is if you don't have enough water you won't be able to survive someone else I mean we get the theme already of these environmental forces that that act without reference to density those are good ones it's often subtle though if you think of a cold snap a freezing event let's say relative to our our eucalyptus grove here eucalyptus here don't do that well with with very cold temperatures so if it if it freezes over the you know if it goes below freezing the temperature over the course of the night you might expect that that's going to influence all trees in a population equally because if their tissues reach a certain low temperature they will die they will not be able to survive it but think about that a little more if there are 100 eucalyptus packed into a small space might they not bumper the temperature effects in the interior relative to the edges say if there are 100 of them packed into a space rather than five if there are only five big eucalyptus they might all die because the temperature won't vary across that space but if there are 100 there may be a buffer so it's often a little bit subtle but you can tease that apart to the degree you like now I'm going to complicate the traditional story a little bit further your book introduces the phenomenon of alley an alley effect and I'll I think it's really important because that the classic model of increasing density and its effect on population numbers in a negative pattern does not always hold for organisms in two respects and this this effect is named after Warder Clyde Ali a very interesting ecologist from the University of Chicago he wasn't actually that small that's the only picture I could find of him very very great thinking ecologist who saw that for some populations when they went below some critical threshold in their numbers they could suffer increasing mortality or reductions in natality that might drive them to extinction remember from the traditional models if population numbers are low we should expect increased growth that's what those curves all look like in part as a result of the availability of resources in populations with low numbers but what if your population numbers are so low that it's just hard to find a mate and that could be true for plants or or animals if you think of plants with pollinators if plants are so widely separated and so so rare the pollinators may not be able to take the pollen to the female organs of another plant for reproduction or if you have two tigers in a forest one's a male and one's a female great but if that forest is you know a thousand square kilometers they just might not be able to find each other so below a certain threshold populations may be so low in numbers that there's a negative that over time population numbers further decline I could give you other examples and if I have time I will that's the way your book presents it as this negative phenomenon but there's another aspect to the alley effect in another part of this curve that's also relevant barely showing up here is your your line of standard negative density dependence that's this dotted line okay note that above this threshold increasing density has an actually a positive effect on our on the per capita growth rate so there's a negative effect in this region but a positive effect in this region so the alley effect does not always only involve a negative effect let me give you another example first let me let me just give you a definition here on the board that might help with you with this because this will this will cause a bit of confusion but I want this compute confusion to be productive because it's going to get you to think about about these growth phenomena and and to recognize the simplicity of the traditional the traditional introduction so this region of the negative effect this is all under the alley effect okay the region of negative effect so below some critical threshold and you can call it the alley threshold if you want little r decreases with population size or density in this region of positive effect are increases with density until it's it's checked until it's overpowered by by negative feedback until checked okay I gave you the example of maybe the problem of finding mates can someone think of another example of where alley effects might be important in a population something not related to made finding at all yes sorry up here yes great example for predatory behavior if an organism hunts and packs wild dogs in Africa have been studied in this regard wild dogs are you know fairly small dogs that run very fast that hunt and packs to overpower much larger animals when their numbers get below a certain density when they when the populations become below a certain threshold they just don't have the numbers to be able to hunt and they can spiral toward extinction locally as a result great example anyone want to try another one yes in defense exactly yeah that's another good one it's it's sort of the opposite of the predation situation in defense he said schooling fish in defense if a school of fish the numbers become so low they don't have the proper defensive mechanisms and are more easily picked off as individuals and may spiral to local extinction that can be true for many other types of organism so you can see how that is related to the negative region of effect but also the positive region because once numbers increase to a certain point then they do have that capacity to hunt well or to defend themselves and will increase further and please try to see how that's really counter to the traditional the traditional rate relationship between density and are or some other measure that looks like that okay so food for your thought hopefully we will just get to cover the a the logistic growth model here at the end and then you are all off on a great weekend I hope but hang with me for five more minutes so we'll introduce a new term K and you know again really simple here we're really dumbing it down by saying it's the maximum population size that an environment can sustain but that's fine the case our carrying capacity it's it's not a fixed value as we'll see it's it varies over time and it varies over space it's very important in our modeling work here we have our previous equations to build on so let's modify our differential equation with this extra term the carrying capacity minus the total number of individuals in the population divided by the carrying capacity and let's try to see the effect that the addition of this term has on the shape of our curve here's our J shaped curve up to this point by adding the following our typical exponential as K is approached as the carrying capacity of the environment is approached note how the population growth becomes an acetote to asymptote to that line how the curve goes from being J shaped to S shaped this is the shape of the curve modeled by that equation let's try to bear out what that means a little bit as we go so that's just a more up close view of the same so think about it if K is at this level say it's a hundred so if K is a hundred as n increases the number of individuals in the population increases from 50 to 60 to 70 the numerator here is going to get smaller and smaller to the point where the numerator will equal will to the point where the numerator will go to zero and render this side of the equation zero such that the increase in numbers in time will be zero and you will see zero population growth so as the carrying capacity is approached by population numbers you will reach this asymptote of zero population growth and no increase in population size okay two more minutes gotcha for two minutes you guys so in this is observed this type of growth as exponential growth has real-world examples and can be observed in nature and then the laboratory logistic growth has been observed in both settings in lab ecosystems I mean a whole variety of organisms have been shown to follow a curve like this here's an example from paramecia but various other organisms often it's not as simple as an s often there's a time lag in the effect of whatever countering forces they are that draw the population towards some carrying capacity so there's often an overshoot as a result of a time lag before the population settles back down to toward K and we'll leave it at that have a great weekend