



S1: today we're gonna be talking about, stage-structured and size-structured models. again, uh matrix models, following up on the, the, topics we've uh, discussed earlier. mostly we've been talking about, age-structured models or it's Leslie matrix models. now we're going to um, talk about some variations on that theme. <P :06> first we're gonna talk about, a relatively simple, modification of an age-based or Leslie matrix model. for long-lived species, an aged-based Leslie matrix model can be very large. for example uh, some of the sea turtles, can live to over fifty years. so if you have uh, an um, Leslie matrix that has one entry for each, year class that can be a fifty by fifty matrix. which is not only, w- uh, a big matrix, and uh, m- moderately cumbersome to to work with. it can be hard to uh, supply the data for that. um, if you wanna estimate the transitions for, specific age classes, you'll need some replication. so you need to be able to age the animals well and get sufficient, uh data on each age class to accurately estimate, the transition. often that's quite hard and so simplifications are made to, to average use average survival rates for the older age classes. and sometimes for younger age classes as well. <P :05> um, also for long-lived species, the oldest age can be hard to measure accurately... if the population of is of relatively modest size, and especially if it's if it's difficult to measure ages, it can be h- it can be hard to tell, what the actual oldest age is, uh is likely to be just because there's, can be a lotta chance variations in the survival especially of the older groups and there's gonna be small numbers. so it can take a while before you g- you um, very accurately identify, what the oldest age is and therefore what the, the size of your life table and your, your matrix model should be... well if annual adult survival is similar, across, for example adult ages, you can create a single, adult, stage. so we're l- we lump them all, into uh into one group. i'll show you an example in a second here, wh- this will simplify the analysis, and you need less, less data. uh, you can use averages for adult survivals... for example here's some, here here's a a, matrix model for painted turtles, based on some data collected uh, here in Michigan, in southeast Michigan and in this, this uh, based on a Leslie matrix model all the adults have been lumped into a single age class, in the last column. <P :04> there are, seven columns in the matrix. the first six you notice there are zeros in the top row. so these this population of painted turtles did not begin to reproduce until, um age class seven. in this typical Leslie matrix model, we have our d- survival probabilities on the sub-diagonal. so this is this entry point-eight-two here represents the probability, that uh the first individuals in the first year class will survive to the start of the second year class. and likewise here from the second to the start of the third and so on. notice this last entry at the bottom here, is on the d- main diagonal... so what this represents is the probability, that that adults, who are now in the in the age class seven the probability that, that adults will survive one year, and remain in that adult age class. so here we have the the annual survivals leading up to, adulthood. here's age class six, and the estimate is about eighty-two percent surv- of the six-year-olds survive to become adults. and then the, survival of adults is approximately eighty-two percent as well. <P :04> it'd be pretty unusual if it was, exactly eighty-two all the way along there so you can tell that uh there was some averaging done to estimate these per- percentages. notice also that the, the number of, uh female, hatchlings produced per female, is uh pr- is pretty low. so for for the females to replace themselves it's gonna take several years of reproduction, to on average be able to reproduce themselves. <P :05> okay so the the thing to notice then is that this is this is no longer a straight, age class model, because now we're lumping individuals of several ages together in this final category, uh cup- an adult stage. <P :05> we can still use our, uh our m- matrix analysis techniques. and calculate the stable age distribution, of painted turtles, and in this case we get, the stable stage distribution for that that final stage. because their survival was eighty-two percent all the way across, these bars now, steadily, decline. from the, whatever numbers there were in the first age class the second one is eighty-two percent of that and so on and so on and so on. <P :04> and there's continuing eighty-two percent survival, in the adult class. but that goes down every year, because of survival, it's but it's added to because these, eighty-two percent of these, um, age six, uh turtles move into the adult class. is that, does that make sense? <P :05> nobody's spoken up yet here. okay. <P :06> i don't know how many of you have, uh canoed down the Huron River, or walked by some, some local ponds, you often see turtles, sunning themselves on logs or on the bank. y- casual impression is that uh large turtles seem to be more common than smaller ones. why is that? what uh, why why might you not see more small turtles? first let me introduce another term here cohort. a a cohort of individuals are those individuals born in the same year... as you could, see from the previous graph each cohort of turtles has a steady decline in numbers with age... but in casual observation we tend to group turtles by size not age when we look at 'em sunning themselves on a on a log, uh, out in the river... smallel smaller turtles are growing with age. so they're getting they're getting older but they're increasing in size fairly rapidly, so the uh, older juveniles also appear to be relatively larger. so that the numbers pile up in the larger size categories. so in reality there is a steady decline with age. um, lemme go back to the, the previous graph here. if we had a, a co- complete, age-structured model, this graph would continue, and we'd have seven-year-olds eight-year-olds nine-year-olds ten-year-olds and so on up to, twenty or so years is probably getting close to the maximum age for painted turtles. and we'd see a steady decline in these in these histograms out to about age twenty. but now we're lumping those all together, and it it stacks up to be a pretty tall, pretty tall bar here. so there's maybe, seventeen percent of the population, represented in the, the smallest turtles. um, over thirty percent of the population, are classified as adults. so when you when you walk around or paddle down the river, looking at turtles, large ones, seem to be more common. and that's the explanation. it really is the case that there are more young ones than each age class of old ones, but they pile up in the size categories. <P :10> let's talk about elasticity, of the population growth rate. we've mentioned this before lemme, uh say a couple more things about it and uh, show an example. for the same proportional change in each matrix element, elasticity, measures the proportional change in the growth rate lambda... because we're talking about proportional changes in the growth rate, of each factor, the elasticities sum to one... the interpretation is that, that uh the elasticity reflects the proportional contribution of each matrix element, to the population growth rate lambda. so high elasticity, means a big effect, of that, parameter on the population growth rate. <P :06> for painted turtles, adult survival has the greatest effect on population growth rate and fecundity has the least. the paper by Heppell, sh- she, uh, added together the elasticities for, what she called juvenile turtles, juvenile survival, which was age classes one to three, and the subadults, which were age classes four five and six, and, this column represents the, elasticity of adult survival... the first bar here represents elasticity of the fecundity. so if the proportional changes in each of the elements of that matrix were the same, fecundity has a, relatively small, and the smallest of these, these uh four here, the smallest effect on the population growth rate. adult survival has the largest. in many other turtle populations, it takes a lo- even longer time, fo- be- before the females become mature. Blanding's turtles it's something like, uh thirteen to fifteen years, in uh, snapping turtle populations it can range from thirteen to nineteen years, for the age of first reproduction. <P :05> in that case, in those cases, adult survival, is even more dramatic. uh the the, a comparable histogram for those situations would be much taller and the elasticity would be approximately seventy percent. so, adult survival is overwhelmingly the biggest contributor to population growth rate. which means that, mortality of uh, adult turtles, is really critical to keeping those populations viable. <P :07> elasticity analysis can help evaluate management options... suppose you had two different, management plans you were, considering. under one plan you would estimate that, adult turtle survival would increase by five percent. and another plan, juvenile survival would be, increased by twenty-five percent... which would be the, most effective in helping that population go, grow. well, suppose we're talking about a population that's declining at five percent per year, so our lambda value is point-nine-five. and suppose, we're talking about one of those longer-lived species of turtles where the adult survival elasticity is point-six. juvenile survival elasticity is point-two, <P :04> we can use this approach to compl- compare the different plans' effects on lambda. <P :05> we can rank the plans by their potential effect on population growth rate. so for plan A... w- we multiply the adult elasticity by, five percent, which is the projected, increase in, adult survival. we get point-six times, point-oh-five or a three percent, increase in alpha. Ivan?
S2: um, at some point i take it painted turtles become non-reproductive right? when they reach the upper age of the, adult class. is that right or is that? i mean cuz that cuz the initial graph the number looked really low for, you know point-five-two-eight seems like a fairly low value. so it i- i mean is that_ am i reading that wrong or? 
S1: um the- uh there's kind of two separate things going on here. i think there is, s- at very old ages, there is some senescence and, uh, perhaps a reduction in fecundity. but that's not why that number is low. mainly, it's because there's approximately i think for that population something like uh six-point-six female eggs per female. but the survival of those eggs, uh, in that first year was very_ the eggs and the hatchlings the survival was very low so that, only about half an egg, half of a female egg survived. so actually there's, twelve or thirteen, eggs laid. about half of those are females. but only, one egg every other year, uh survives that first_ one egg ev- one female egg every other year survives that first year of life. things like raccoon predation are really, really severe. so the numbers of eggs laid is higher than that term, reflects. it's largely due to first year survival. another question?
S2: oh no, i was just_ that meaning that makes sense. because i actually thought that they would uh have more eggs than six. it seems like a small group.
S1: right. that's six female eggs.
S2: okay okay female eggs okay the total yeah 
S1: the total clutch is uh, bigger than that. yeah and in in um, cases where it's been looked at, uh, looked at closely, there does tend to be an increase in uh, clutch size with female age. but sometimes it's hard to detect in in all the noise. 
S2: because i th- i was thinking in harvesting that it it'd_ well i guess what i was getting at was i was thinking i know like you know you could place maybe size you could actually allow harvesting beyond a certain size if that's somehow correlated with age class if you could say those were no longer reproductive, [S1: ahh ] and that, you know thereby sort of work in an ability to sustainably harvest some turtles cuz i know, some cultures do want turtles, so. 
S1: yeah absolutely absolutely. in fact, i've, been getting some uh, e-mail messages from some of the, uh turtle experts around the state here. and uh, Southeast Asia is apparently a hot spot one of the hot spots for turtle evolution, and there's a large variety of species there. they're coming under increasing pressure from human consumption, and especially as the economies improve in Southeast Asia, uh and all over Asia, uh, turtles are harvested for meat and it's severely depleting the turtle populations. that additional source of mortality on those populations because they're taking the larger adults, puts a severe cramp on the, uh the population growth, over there. well and over here too. um, the poaching of turtles in, uh in in Michigan and actually just even incidental harvest is really hard on the, populations they just can't sustain much additional mortality on the adult stage... okay, good good question good question. let's uh lemme get back to, follow up this comparison here. the way to do this is to, uh have the adult elasticity value, times the percent change that you think is gonna be from this plan five percent and that's, works out to point-oh-three. so that's a three percent increase in survival, uh excuse me three percent increase in population growth rate from that plan. so, three percent of the, uh, nominal rate of point-nine-five brings it up to point-nine-seven-eight-five would be the value projected under plan A. under plan B you take the juvenile elasticity, times the twenty-five percent increase in, juvenile survival, gives you a point-oh-five, percent increase in lambda. so lambda would increase by five percent to this point-nine-nine-seven-five bringing it almost up to a stable population. <P :04> so under this comparison plan B would be better, for the population. but notice that it's, in order to, to have the effect on the juveniles, be better than the effect on adults it had to be m- a very substantial increase in the juvenile survival... mkay so this is an example how this kind of analysis can help you <COUGH> excuse me help you evaluate different different management options. <P :05> so this kind of lumping of adults, into an adult stage is usually helpful especially for long-lived species cuz it can simplify analyses, uh requires much less data, you can do a lot of averaging over the, the observations that you do have, uh and avoids the problem of the oldest age. notice that we didn't have to specify how big that matrix is, we don't really have to know what the oldest age of the individual is. we're just assuming that, that uh eighty-two percent of them keep surviving and surviving and surviving um by the time you get out to, say twenty or twenty-five years, it's a very tiny f- number that are actually, uh surviving. but you're able to include that um in the analysis. <P :05> uh, one thing you need to, be cautious of is that this can overestimate the speed of response to some perturbation... suppose you, you had a change in the population size and you wanted it to, to do a matrix projection and see how quickly it would come back to the same to that same stable age distribution, um, if you have all the adults lumped in a single stage, you aren't gonna see that that actual uh wave of, population changes as it ripples through those older ages. it's all gonna be lumped together in that single, adult stage. so it's gonna give you the impression that that the population has, has uh come back to its steady value, faster than it really does. the population actually takes longer to achieve a new stable age distribution stable stage, or excuse me stable age distribution than the lumped model would indicate. <P :07> okay. size matters. for many population an individual's size is more important, than its age in effecting survival and reproduction. and that's certainly true in, in uh many fish and in in trees. like in forests there can be a lot of, fairly old, uh s- saplings that just haven't gotten enough light. they can be old, but they're not at the right size to mature... for some species it's hard to determine age. a lot of invertebrates it's very difficult and in some plants it's hard as well... but many species have clear, stages into which you can lump the the different types of the population for_ in insects it's pretty clear, defined eggs larvae pupae and adults. for many other arthropods, eggs and juvenile instars and adults, are pretty- pretty easy to define and categorize them... uh much easier to do than telling how how old they are, what their age is. so you can use a model based on size or stage, rather than age class. i think we've mentioned the n- names before. but the certain names are often associated with different types of matrix models. Leslie matrix model is the one based on age and Lefkovitch matrix model is the one based on stages. here's a citation for uh, Lefkovitch's discussion of plant uh population growth, grouping organisms by stages.<P :06> here's an example of a stage-based model for an insect population... this is a, species of insect where you just have three stages egg to larvae and adult. um, we have a transition probability, a probability of survival from egg to larvae, and from larvae to adult. there's a couple extra loops here. uh, here's the probability of a larva surviving and remaining as a larva, during the specified time interval. and here's, the probability of, surviving the next time step and staying in the adult, category. and then there's the, off- offspring produced by those adults feeding back t- into the egg category. so this is the Lefkovitch or st- stage-based matrix corresponding to this life cycle graph. notice that there's, the term for reproduction is only in the adult category. we do have terms on the diagonal, and that's these loops of probability of surviving but remaining as a larvae or st- staying as an adult that's the two, terms on the diagonal. <P :04> does that make sense? that seem straightforward? good... so for both both age-based and s- size or stage-based models, the time unit is fixed. that's really one of the keys. that it's the time unit that's fixed for each time step, uh, the le- the width of the size categories is up to the biological investigator what he thinks is most appropriate, but the time step, uh is is uh, selected, depending on the data that you have. so exam- for example you can pick a time unit like one year for turtles or, many other animals. for ones with f- faster population growth you might choose a one week for insects or maybe one day for certain, uh daphnia in lakes. then you estimate the transition probabilities the probability that an individual is found in the next stage one time unit later. Dana?
S3: if you were using those really small time steps, like back in the diagram on the previous slide, why wouldn't you (want) [S1: whoop ] oh, sorry. [S1: whoa jumping ahead excuse me. uh, there we go. ] why wouldn't you account for the probability of the egg staying as an egg? 
S1: ah, good point good point that's right. if your if your uh time unit was sufficiently small then there might, there might be uh, a loop here as well. that's a good point right. in this one [S3: and also ] in this one it it must be the case that for this analysis the time step corresponds to the time it takes an egg to ha- to hatch. 
S3: where would that go on the matrix?
S1: that would go right here in this, this one, this first element. the probability that, the first stage remained in that first stage. good question good question. anybody else? <P :09> i hope you can see that the picture okay. one of the first stage-structured models for plants was developed for teasel. Dipsacus sylvestris and it's in the teasel family. you've probably seen these these heads they're often used in the dried flower displays. they're pretty neat. teasel's monocarpic. big bang reproducer. often found in disturbed areas. <P :04> where the soil is disturbed and the seeds that are in the ground get a chance to, to uh grow. it's often seen along, roadways, <P :05> looks a lot better on my screen than it does on the big screen here, but the, you can see the little, little dots if you look here closely. and as you're driving along, uh actually U-S twenty-three this is near U-S twenty-three and uh Washtenaw Avenue, um, there's uh, you see teasel quite a bit... the plants grow with their leaves in rosettes, this would be a rosette with the leaves coming off a central place here and here's my little Swiss Army knife uh to give you an idea of its size this is a relatively large rosette. <P :06> small rosettes grow into large ones. here's here's uh one with really big leaves. <P :06> when the rosette is large enough, the plant reproduces. it sends up a stalk, flowers and uh seeds are produced. so here's the, the uh, formerly the flower on the big stalk, um you can see some of the rosette leaves at the base and then there's a different kind of leaf that are produced on the, on the stalk itself. <P :04> the age at which teasel reproduces varies with the growing conditions. and size of the plant seems to be a better predictor of reproduction and survival, size is better than age. <P :05> Pat Werner and uh Hal Caswell did some studies where they s- seeds were sown in several plots and followed for several years to estimate the survival probabilities and the seed production. <P :06> they divided the teasel population into six, size categories. first one was dormant first-year seeds. second was dormant second-year seeds. and then small medium and large rosettes. the small ones were about an inch in diameter. medium rosettes, about an inch to eight or nine, uh inches in diameter and large rosettes bigger than that. and then the last category was flow- flowering plants. <P :09> each flowering plant, in year T can produce, one year later, over three hundred dormant seeds... it can also produce some seeds that will by one year later, produce about three-and-half small rosettes. each flowering plant also s- will have some seeds that will make about thirty medium rosettes. and some of those seeds, that land in really favorable conditions can actually produce a large rosette uh by the next year. and it's monocarpic, uh the flowering plant dies after reproduction there's no survival of the adults <P :06> this is what the life-cycle graph looks like for the teasel population. <P :04> we can start over here with flowering plants. they produce large numbers of dormant seeds that sit in the ground. some of those don't germinate and stay a second year in the ground as dormant seeds... the flowering plant also produces some seeds that become small or medium or large rosettes. uh, some of the dormant seeds can germinate and produce, small medium and large rosettes, and likewise for the second-year dormant seeds. and then there are loops, for groups, for the small medium and large rosettes indicating that they can stay as, small medium and large or grow to the next larger size, or, um, with an arrow pointing towards number six there, they can flower if they get big enough. <P :08> this is the corresponding size-based matrix model for teasel. it has a different look than the other, models we've looked at. there's a lot more of the elements that are filled in, with something besides zeros. <P :05> lemme point out a couple things in the far right column. this this last column is the one for flowering plants. notice that there's a zero at the bottom element there in the, lower right. that's because this is a monocarpic, uh species, it flowers, none of those flowering ones survive. so the survival to the next period is zero... this is in contrast to what we just looked at before for turtles where we, we actually we put a term in here to, account for the fact that those adults were staying in the population, here they're all dying. some of 'em, som- some of those flowering plants produce dormant seeds, they don't produce of course any second-year, dormant seeds because you have to be a dormant seed for one year before you can be a second-year dormant seed so that's what's this, uh transition over here. then uh, but it can produce some small or medium and large rosettes. uh, <P :05> and the rest of these terms indicate the, the uh transitions to the different sizes. the ones along the diagonal are the, probability of staying in that particular size. the ones on the sub-diagonal are the transitions to the next one... kay? is that_ you see how that corresponds to the uh life cycle, graph there? good. i'm seeing quite a few heads nodding. good... we can do an eigenvalue analysis with the, stage-based models just like we did with the age-based models... to remind you of the definition here a vector X with the property that matrix multiplication is equivalent to scalar multiplication, so that the matrix times that special vector is equal to, a scalar times that special vector, for some scalar lambda, it's called an eigenvector of the matrix and the scalar is the eigenvalue. <P :04> so the an- analysis is the same as for age-based matrices the right eigenvector is the stable, size distribution or stable, stage distribution of the population. the left eigenvector, gives the reproductive values for each size or stage, of the population and the dominant eigenvalue, is the discrete growth factor or lambda, for that population... so it's just the same, when we're talking about these stage-based matrices. for the teasel population th- the population growth rate, lambda is two-point-three-two-two per year. so it's more than doubling each, year. each female is producing, two-point-three, females per year. at the stable stage distribution ninety percent of the population, is in dormant seeds... the flowering plant has the highest reproductive value... so here's the numerical results of the teasel matrix analysis. there's the six stages and their names. here's the stable stage distribution. notice that, about sixty-four percent of the population is in the dormant, first-year dormant seed category. and uh, almost twenty-seven percent of the population is in the second-year dormant seed category... among the, the green part of the population, the largest fraction is in the medium rosette category. and the flowering plants are the smallest numerical category of that of that whole population. and if, once it reaches this stable stage distribution these would be the proportions that you'd expect. the last column shows the reproductive value for an individual by stage. and as you might expect, the flowering plants stage is has the highest reproductive value. <P :07> kay, any questions? <P :06> kay in summing up here lemme briefly mention some of the assumptions of these age and stage-based matrix models both assume that there's a l- a linear effect of the number, in each category, on the numbers of births and deaths. mkay we're we're having constant, constant values for survival, and reproduction, per individual. we're not including any kind of density dependence yet, in these. and the parameters are constant. but of course the real world is not really linear. at least not all the time. and conditions do change... so you have to be cautious in interpreting the results of these matrix manipulations we're, you only are calling this a true prediction about what's gonna happen if you believe that these things are true. that there's no density dependence and that the parameters are gonna remain constant you're g- are gonna have constant survival, and fe- fecundity, uh, if that's true you c- will be able to make a prediction. but if it's not gonna be true, it still represents, the description of the current population conditions and as th- as those numbers exist today, this is the direction in which that population is moving, and where it would be heading, if th- things did remain constant. so i think that's the, uh, a real take-home message here that, it's still valuable even if it's not a true prediction it still gives you insight into what's going on in this population, uh, what parts of the population are uh, contribute most to the population growth rate. give you some ideas about what's w- uh, important to consider in the management of those populations. <P :06> okay. i think that's, i think that's the last one. yeah, okay any any further questions? <P :04> have a good weekend. next week we'll talk about uh human populations and, allometry. 
S4: whenever you talk about those turtle populations i always think about the uh, study that grad student from Central Michigan was doing on wood turtles in Manistee National Forest, while we were up there. [S1: oh. ] she said she, didn't find a single nest that had survived raccoon predation, didn't find any turtles that were younger than ten years old. 
S1: oh boy. [S4: yeah... ] tha- that's just a real life style that depends on keeping trying, reproducing and eventually hope we get a, successful clutch.
S4: the raccoon, the raccoon population is just so high...
S1: yeah. i remembered this (person.) somebody else asked about that too and i can't remember who it was.
S4: i don't know. w- 
S1: have a- have a good weekend. 
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