S1: mark them i want you to put them back again.

SU-f: (xx) do we have (to number them,) then the total number (xx)

S1: okay, we are gonna count the number that are marked.

SU-f: well what (about) (xx) (count the total number of) (xx)

SU-f: how are we gonna mark them?

S1: okay <SS: LAUGH>

SU-f: oh, more than a hundred

SU-f: am i being difficult?

S1: no you're not being difficult.

SU-f: (good)

S1: as always, you just look ahead. it's_ i'm not used to it.

SU-f: i'm not gonna eat this.

SU-f: (xx) (you're just) patient.

SU-f: patient? <SS: LAUGH>

SU-f: i would like, (xx)

S1: first of all, leave her alone. <SS: LAUGH>

SU-f: (xx) patient.

S1: leave her alone... <SS: LAUGH> oh i like that. oh definitely the happy face is the best mark so far. it has to be doesn't it? okay, what i need to know is how many people marked, Beth?

SU-f: six.

S1: six.

SU-f: three.

S1: three.

SU-f: eight.

S1: eight.

SU-m: five.

S1: five.

SU-f: five.

S1: five.

SU-m: five.

SU-f: four.

S1: five four.

SU-f: five.

SU-f: zero.

SU-m: three.

S1: five

SU-f: i wonder if (xx)

S1: five. three?

SU-m: se- seven.

S1: seven.

SU-f: three.

S1: three.

SU-f: seven.

S1: seven.

SU-m: three.

S1: three.

SU-f: eight.

S1: eight. you took a big handful didn't you? <SS: LAUGH>

SU-f: yeah. i ate a few of the yellow ones.

S1: i'm still waiting on Sarah. two okay, then [SU-f: (oh) three. ] thr- oh i'm sorry three. okay. please put them all back in the bowl and stir them all together.

SU-f: (xx) why does it matter how many we took?

SU-f: remember the (grashopper) (xx)

S1: ah you've done you've done one already have you?

SU-f: we did (xx) we did like an actual experiment.

S1: it's easy with M-and-Ms it works with M-and-Ms. <SS: LAUGH> okay. um, someone with a calculator Steven yours, you have a calculator don't you?

SU-m: no, actually, <SS: LAUGH> someone took it, actually.

S1: oh does anyone have a calculator?

SU-f: i do.

S1: just if you could total the number for me it would make my life easier.

SU-f: seventy-seven.

SU-f: seventy-seven.

S1: seventy-seven.

S1: it works. okay. now, if they're all mixed up, what we do is, we now, wh- what we'd do normally, um is you go out and you would catch a bunch of fish and you would mark them. you mark them in a number of ways. one way is to clip a fin. typically if it's a trout you would clip the adipose fin, or maybe a little piece of the dorsal fin or the anal fin. usually not, whi- or or the paired fins for that matter. uh you you take a clip out. and um, then you put the fish back. and, then, you come back on another occasion, say a day later a week later a year later or whatever, and you recapture a bunch of fishes. so now we go back and we'll start from the, other end of the class this time, you take a small handful, and what you do is you count the number that have been, uh you tot- count the total number that are in the um the total number that have been um, captu- recaptured the total number of yellow ones, and the numbers that were marked amongst those yellow ones. so now, we resample, the population, and we keep a total of the, the total, that we catch, and the number which are marked. plus, i- and the number that are marked. okay? so now what you need to do quite simply, um is to, count the totals and the numbers that are um that are marked. um and obviously when you recatch the fish if you're recapturing them you look for the number that are marked and uh and and so on. often, if you, go to a um_ if you're of the, boat dock stand round the lakes here, you'll find fishermen will bring in fish that are marked. and Petoskey there is frequently a D-N-R officer who is actually interviewing people about their fish, and checking to make sure they haven't exceeded the fishing limits, and uh they will give him the marks and so on, cuz a lot of fish are marked, and, you can mark fish in different ways. you know if you got fish from say if you're f- uh planting trout into the Great Lakes, and they've come from different hatcheries, different hatcheries may mark fish before they go into the water. and then they might recapture them, and they can get estimates of the survivorship of fish from different hatcheries. the survivorship of fish from hatcheries is pretty low. um, fish that are raised in hatcheries, haven't been raised in the hurly-burly, of the real world. so they typically tend to be not as good competitors, they tend to be much better as prey items from the perspected of predators and so on, so they, um it's generally considered that uh by many, that the bulk of the production from hish fish hatcheries goes to increase the growth of the resident populations, hopefully the wild populations. so, we have some numbers here. um, maybe if i could write these up you could plunk them in for me. [SU-f: sure ] um, i hope this will work. given my life these days who knows? okay, so what we're going to do is to find a pen, and start off. Sarah, how many did you recapture?

SU-f: five.

S1: five, and how many were samples?

SU-f: four.

S1: four. so [SU-f: (xx) ] yes, so here are the numbers recaptured... uh no that's that's the next one over. probably easier to use the arrows, cuz it's uh, [SU-f: okay ] yeah, okay, the nu- put in the number cap- the total captured which is five, and the number that were recaptured was four. and that estimates the population at ninety-six. how many did you say Michelle?

SU-f: i said a hundred.

S1: yeah, hundred. uh, what you're doing essentially is we know that, um, we know that, seventy-seven were marked. and we caught another, five, and we found that, um, five-fourths of these were marked. so if we multiply the original number by the proportion that are marked in the recapture. that is an estimate of how many fish there should have been in the entire population. cuz if we'd marked them all you see, this number would be five-over-five. um and if we mark very few, then um uh the- if we mark very few of the actual population then the number that are marked when we recapture is going to be very smaller and so on. so let's have some more numbers here.

SU-m: what if a bunch died in the interim?

S1: i'll_ let's get to the assumptions in a second okay? <PAUSE:04> Mary?

SU-f: three and one.

S1: three and one. oh, that gives two hundred and thirty-one as our estimate. next ones?

SU-m: nop- three and two.

S1: three and two. that gives an estimate of one hundred and sixteen.

SU-f: two and zero.

S1: two and whoa zero. what does that give us?

SU-m: none.

S1: no it tells us that the population size is infinite, that we have an infinite number of M-and-Ms. i think i'm gonna [SU-f: yes ] take that number and become an economist. <SS: LAUGH> or an M-and-M manufacturer. so this is gonna give us infinity, okay.

SU-f: uh six and three.

S1: six and three. hundred and fifty-four.

SU-f: three and two.

S1: three and two.

SU-m: uh four and one.

S1: four and one.

SU-m: ten and nine.

S1: ten and nine. <LAUGH>

SU-m: uh one and one.

S1: one and_ one and one that's exactly seventy-seven isn't it? your indi- your, you indicate that every single fish in the_ that every single yellow M-and-M in the population was marked. Jody?

SU-f: one and five.

S1: one and five? five and one. <LAUGH>

SU-f: fi- five and one. sorry.

SU-m: six and two.

S1: six and two.

SU-f: eight and four.

S1: eight and four.

SU-f: four and two.

S1: four and two.

SU-f: one and one.

S1: one and one.

SU-f: seven and four.

S1: seven and four.

SU-f: five and two.

S1: five and two. okay. so our numbers here, range from a population estimate of two hundred and thirty-one to seventy-seven. and this is a pretty simple situation. so it means the the average, the mean of our population size, the mean estimate is one hundred and sixty-seven yellow ones, but the standard deviation is eighty-nine. <SS: LAUGH> this is a pretty simple experiment.

SU-f: doesn't that just mean we have to do more samples?

S1: well, that's a good question. let's think of some of the assumptions now. first of all um, each one of these is an estimate of the population. it is considered that, if you have a minimum of seven, replica- sa- resamples, seven resamples, then you've got a good estimate of the population size. we have got substantially more than seven, and look at the variation. okay? now but what about the assumptions? the assumptions for a mark and recapture model were met in this experiment to the best of my knowledge. Michael said, what happens if there are some deaths? well if there are some deaths, then obviously if there is differential mortality, suppose clipping the fin is a stressor, that tends to give higher death rates of the clipped animals, um, you're going to get, wha- um, very sub- fairly substantial overestimates of the population size. and of course you are handling, those fish that you marked. now in your experience, what's the survival rate of a lot of the fish you handle?

SU-f: it depends.

S1: it depends on the species doesn't it? but if we get minnows on a hot day

SU-f: <LAUGH> not very good.

S1: yeah, um, okay so so tha- so if in the time period that we've looked at there's been differential mortality, then we'll get uh unusual population, numbers or i- inaccurate population numbers. and one of the assumptions of the method is that there is no differential mortality, and you know, from your experience, that that is sufficiently close to a load of hogwash as to be um, hogwash. what happens if there are births in the interim? [SU-f: (xx) ] that's gonna have the opposite effect at increasing the numbers. and for this kind of reason you try to take a relatively short time period. you know um a few days and so on in doing these estimates. when you're trying to estimate population sizes if you possibly can. but what about immigration? if we mark the fish on the big shoal here, where are the fish gonna go? are they gonna stay there? what happens if they decide to go and see their relatives on the other side of the lake? what of the other hand if there is um, (what'd i say immigration?) suppose there's immigration and emigration? obviously that is going to affect the numbers. so, there're a lot of assumptions, that are contained in this particular model. yet it's probably, the best direct measu- method, that we have available. and you can see, that, with something as simple as this we've got numbers that are, very b- very very wide-ranging. now you can refine the method a- at least in theory you can refine the method by doing multiple markings. for example if you collect a sample of fish say seventy-seven if we were to go around and take all these and put a second mark on each one, you would then know which were marked originally which were marked the second time and you could begin to estimate things such as mortality. and if you put multi- as you add multiple marks on the fish, um you're able to get better and better estimates of mortality. mind you, fish have only got so many fins to clip. and you know from what we talked about in terms of functional morphology and locomotion, that if you clip too much off a fin, you're gonna impair its ability to swim you're gonna impair its ability to avoid predators, you're gonna impair its ability to catch prey. there are other ways of marking the most common method actually now, is to shoot a tiny little piece of metal into the nasal capsule. there's cartilage around the nasal capsule, (i) can shoot a tiny little piece of metal into the nose capsule. um, no fish has ever complained so presumably it doesn't hurt the fish. and, this is um this is coded electromagnetically, so that you can get the fish and you can sort of stick it into this electromagnetic thing you get a beep and you get that kind of information. it's a superior method. um, for some larger fish they put external tags on. because often the source of sampling are fisherpersons. and they don't usually have these devices for putting the noses of their fishes in this electromagnetic sensor and so on. so they put tags on the outside and the- they are gonna impair the fish. they're nice little flags that other predators can see and those kinds of things. so they send very definite um, signals. but um, this then, is an example of the mark and recapture method. and um, do feel free to sample the catch. <S1: LOUD EATING SOUNDS>

SU-m: how many were in there?

S1: i've no idea. oh there were a hundred and sixty-seven plus or minus eighty-nine... anyone, please don't eat the other ones. Phil wants to count them all later. <SS: LAUGH> okay. so, we can estimate numbers. but when we're dealing with population dynamics, we're not interested in, just how many there are, now, but how the numbers vary with time. so, lemme spend a little bit of time and i mean a little bit of time, talking about um a little bit about the continuous methods and that is, the models. and, lo and behold, you already know the model. <PAUSE:06> for, fishes, well for populations in general, the most common model that is used to look at population numbers is the logistic model. <PAUSE:06> well you'll recall, R little-R, is the symbol that's used for the growth rate of the population, and K is the carrying capacity. and you'll recall, that if you plot the numbers as a function of time, then this rises to some asymptote, where the asymptote here is K, the carrying capacity. and you'll remember that the explanation for the shape of this curve, is intraspecific competition. competition amongst the individuals at some uh high population numbers so that the per capita resource base becomes sufficiently small, the population numbe- uh growth rate declines. birth rates go down death rates go up um, and so on. bear in mind, that the population dynamics people, believe that the importance of competition predation and so on, comes from its effect on demographics. and here, we had an explanation, of this curve in other words the population dynamics, in terms of a mechanism. the mechanism was important in terms of the population dynamics. in talking further on this and linking it to reproduction and talking about the R-to-K continuum, we recognize that population, another process, its impact was on demographics, and the the ecological consequences in terms of say releasing populations from competition direct and indirect in- fe- effects and so on, were mediated, via these population um effects and so on. i want to add one factor to this model. there are many many things that you can put into play with this. you can put in different growth rates for predators and for prey and you wind up with a Lotka-Voltera models. you can put in competition amongst individuals and build models that way. you can put in delays so that say if fish breed once a year what's gonna be the effect and so on. and these models can get, these continuous models can get increasingly complex to try and describe, the sort of average kinds of effects that are going on and so on. i want to just think about one, and that is an issue which is important in terms of, uh conservation biology, threatened species and so on, and this is the idea of the minimum, population size. the basic logistic curve makes a very simple assumption. and that is that the per capita growth, the growth rate if you like, is constant at every single population density. it doesn't matter if there are ten fish or a hundred fish, if there are ten fish the growth growth rate may be one fish a year, if there are a hundred fish it will be ten fish per year, but the rate of growth

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