Dr. Richard Rebarber
Richard Rebarber is a Professor of Mathematics at the University of Nebraska, Lincoln. He received his Ph.D. from the University of Wisconsin, Madison in 1984, with thesis work in control theory. He spent his first 20+ years at Nebraska working on Control Theory, and has recently been working with biologists on population dynamics and other topics in Mathematical Ecology. He has been the long-time director of an applied mathematics REU Site and has extensive experience mentoring undergraduate research.
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[00:00:00.719]Well thanks very much, thanks to Kevin
[00:00:03.422]and your department for inviting me.
[00:00:06.253]That's the first time I've heard mathematics
[00:00:08.229]described as the real world.
[00:00:14.480]We're a very precise world.
[00:00:17.237]But I don't think, it's very much the real world, and
[00:00:22.099]I am a mathematician and my background
[00:00:24.260]is very much mathematics, not in modeling,
[00:00:27.536]however I will not be proving anything,
[00:00:29.381]that would be kind of foolhardy to do.
[00:00:31.261]It's actually foolhardy to give proofs
[00:00:35.288]in a mathematics talk too
[00:00:36.718]because they're usually so specialized.
[00:00:38.713]So there's not gonna be any proofs in here.
[00:00:42.881]Okay, so this project started as a capstone course
[00:00:47.201]in mathematics at UNL, and it's called Math in the City.
[00:00:50.915]For each class we get data from a non-academic source,
[00:00:53.937]and design projects around that data.
[00:00:56.991]So in fall of 2017 we obtained data
[00:00:59.943]from Nebraska Game and Parks
[00:01:02.504]about a white perch population in Branched Oak Lake.
[00:01:05.324]White perch are not native to Nebraska
[00:01:07.104]and are invasive in Nebraska even though they're not
[00:01:09.885]necessarily invasive in other locations.
[00:01:13.083]The population exhibits stunting, which means that
[00:01:16.201]fish are smaller than they should be for their age.
[00:01:18.705]And populations are dominated by smaller,
[00:01:20.878]less desirable fish.
[00:01:22.173]So that's a problem, not only do the white perch
[00:01:24.378]crowd out other more desirable fish,
[00:01:26.688]but they're also crowding out,
[00:01:28.400]the small ones are crowding out the large ones.
[00:01:32.258]'Kay, and so
[00:01:35.006]the people at Game and Parks sent me to a paper,
[00:01:40.595]from out of this department,
[00:01:42.997]Chris Chiziski, Kevin Pope, and Wilde.
[00:01:47.733]A modeling approach to evaluate potential management actions
[00:01:50.603]designed to increase growth of white perch
[00:01:53.644]in a high-density population.
[00:01:56.753]And I knew Kevin before from, he served on,
[00:02:01.026]as an outside person on some PhD committees
[00:02:05.867]in the math department, or at least one.
[00:02:08.441]And then there's another paper
[00:02:10.432]that was a little more theoretical,
[00:02:13.535]but these are the two basic papers about...
[00:02:17.737]So I'm getting used to this thing.
[00:02:19.755]About stunted populations.
[00:02:24.816]The papers are in discrete time, age structured.
[00:02:30.947]So they're like, if these means anything to people,
[00:02:35.762]But it's coupled with a fecundity that depends upon length.
[00:02:40.376]And then there's a density dependence
[00:02:44.738]that the probability of fish growing
[00:02:47.202]is a decreasing function of the total population biomass.
[00:02:51.628]So the more biomass there is,
[00:02:53.633]the less likely the fish is going to grow,
[00:02:56.164]which seems sensible.
[00:02:59.586]So my colleagues and I,
[00:03:04.274]they're not at University of Nebraska, they're part of,
[00:03:07.435]so I guess I'm gonna digress here,
[00:03:08.836]they're part of the a program
[00:03:12.434]that is for people like me
[00:03:15.271]to mentor early career faculty at
[00:03:19.997]colleges that don't have graduate students.
[00:03:22.712]And this was part of that kind of program,
[00:03:24.828]and while their area's not typically analysis
[00:03:29.869]or applied math or anything like that,
[00:03:31.091]they took to this quite nicely.
[00:03:33.168]And so we started working on this before
[00:03:37.376]the students in Math in the City
[00:03:40.066]got their hands on the data.
[00:03:41.983]And so we proposed a length-based model.
[00:03:44.798]Reasons for this would be,
[00:03:46.488]it's easier to collect length data than age data.
[00:03:50.298]Managers might be more interested in size distribution
[00:03:52.744]rather than age distribution.
[00:03:55.445]Life history parameters might be lengths-based,
[00:03:58.329]for instance larger fish can hold more eggs.
[00:04:01.364]And this model is
[00:04:04.499]more tractable to mathematical analysis,
[00:04:06.948]it's a simpler model than the age-based one
[00:04:12.708]that we were given before.
[00:04:14.825]And we want to do mathematical analysis on it
[00:04:17.190]not just simulations, so this is an easier model.
[00:04:21.597]The other one we really couldn't do
[00:04:22.982]the mathematical analysis on.
[00:04:25.588]Okay, so we're starting with a population vector.
[00:04:30.985]So you've got different,
[00:04:34.815]P0 is the population in class zero.
[00:04:38.040]P1 is the population in class one.
[00:04:40.616]PN is the population in class N.
[00:04:43.403]So there are N plus one stages.
[00:04:48.629]Or not stages, length classes.
[00:04:52.718]They can be called stages,
[00:04:53.990]but we're defining by length classes.
[00:04:56.179]So our assumptions, and there's quite a few
[00:04:57.654]of assumptions to make this work.
[00:04:59.012]Newborn fish cannot reproduce
[00:05:00.466]in their first time step of life
[00:05:02.216]and we place them in the zeroth class.
[00:05:06.439]The time step is constant, determined by behavior
[00:05:10.252]of the species, or by data collection.
[00:05:12.997]That's not very restrictive.
[00:05:15.192]After one time step, surviving newborn fish enter stage one.
[00:05:20.042]And in one time step a fish can either stay
[00:05:23.046]in its length class, or grow into the next length class.
[00:05:26.976]So that tells us if we're gonna do something like this,
[00:05:30.238]if we want a model that's gonna work like this,
[00:05:32.408]we have to be very careful about the length classes,
[00:05:36.678]how large the length classes are.
[00:05:38.785]Because we need the length classes to allow something like
[00:05:43.795]in one time step a fish can either stay in its length class
[00:05:46.285]or grow into the next length class.
[00:05:48.135]The length classes can't be so small that in one time step
[00:05:51.940]you can jump over one.
[00:05:55.461]So it's not so much restrictive as it does say
[00:06:00.605]something about the modeling you're gonna do.
[00:06:03.958]Okay, so the parameters that we'll need here.
[00:06:08.045]The average length of a fish in the Ith stage,
[00:06:11.867]we're gonna call LI, L sub I.
[00:06:14.666]The survival rate of stage I fish in each time step
[00:06:18.518]is S sub I, so we're assuming that the survival
[00:06:21.862]doesn't change over time.
[00:06:26.367]But there are things changing over time
[00:06:27.875]but not the survival.
[00:06:29.033]Same with the fecundity, the fecundity of stage I fish
[00:06:33.706]each time step is FI.
[00:06:35.848]And now we're making an assumption here.
[00:06:37.845]Assume that the SIs and the FIs are increasing in I.
[00:06:44.324]Now this is often true.
[00:06:48.937]Because larger fish can often procreate more
[00:06:55.979]often have better survival probabilities,
[00:06:59.125]but it's not of course always true.
[00:07:01.823]For instance, one glaring example where it would not be true
[00:07:04.566]would be in the presence of angling,
[00:07:06.820]where the anglers actually want the larger fish
[00:07:10.004]rather than a smaller fish, so that's an assumption
[00:07:12.334]that restricts us somewhat.
[00:07:21.441]So I'm trying to use this, here we go.
[00:07:25.080]Okay, now what's gonna change with time,
[00:07:28.762]let P sub T be the probability that at time,
[00:07:32.774]that at time step T
[00:07:35.109]a fish grows into the next stage if it survives.
[00:07:40.772]So here's our model.
[00:07:42.464]Now, this might look a little intimidating,
[00:07:44.067]but it's actually a pretty simple model.
[00:07:46.282]Let's spend a little time on this.
[00:07:48.251]Okay, so in the first row are just the fecundities.
[00:07:53.425]So if you're used to Leslie matrices
[00:07:57.537]this will look awfully familiar to you.
[00:07:59.457]So you've got...
[00:08:03.756]This says that the first stage class
[00:08:09.279]generates F1 times the population in stage one fish.
[00:08:15.785]Plus F2 in the population of stage two fish, and so on.
[00:08:21.207]Okay, so now
[00:08:23.630]here we've got the survival probabilities.
[00:08:25.825]Stage zero, it just survives to stage one.
[00:08:34.461]it will either, a fish in stage one
[00:08:37.900]will either survive
[00:08:40.997]and go to the next stage
[00:08:43.681]and the probability that that happens is P sub T
[00:08:46.980]or it will survive and not go to the next stage.
[00:08:51.273]And so the probability of it not going
[00:08:54.380]is gonna be one minus P sub T since P sub T
[00:08:56.646]is a probability, it's a number between zero and one.
[00:08:59.463]So this says
[00:09:01.745]that stage one fish either move on
[00:09:06.832]with a certain probability
[00:09:09.719]or stay with one minus that probability.
[00:09:12.962]And that's true at all the other stages
[00:09:15.461]until we get to the end.
[00:09:17.690]And in the end when we've gotten to
[00:09:20.138]the largest class of fish where it's unlikely for fish to,
[00:09:25.821]to get larger, then we just say there's the probability SN
[00:09:30.170]that it continues.
[00:09:34.387]Okay, so now that probability, that's kind of,
[00:09:39.064]so what happens with the model
[00:09:43.152]is that you've just got a basic, you have a,
[00:09:47.281]if the probability of moving on is one, it's just a basic
[00:09:54.220]and is easy to analyze.
[00:09:56.557]However the probability is not always one.
[00:09:59.973]The more crowded it gets the smaller the probability is,
[00:10:02.490]and crowded is going to be measured by biomass.
[00:10:05.750]So we'll assume that the mass of a fish of length LI
[00:10:09.954]is WI is equal to
[00:10:12.741]a mass length coefficient times LI, I cubed.
[00:10:16.429]So the idea that all animals are spherical.
[00:10:19.338]Which you guys have probably seen.
[00:10:23.086]The population biomass at time step T is just
[00:10:28.309]you add up
[00:10:32.792]the masses of the fish
[00:10:35.926]times how many fish there are in each class.
[00:10:38.461]So this thing is the total biomass.
[00:10:43.553]How much in stage class I
[00:10:46.643]times the number in stage class I,
[00:10:48.414]how much they weigh in stage class one,
[00:10:50.379]times the number in stage class one and so on.
[00:10:54.993]Now, there's a nonlinearity that's strictly decreasing
[00:10:58.136]and continuous, so the probability,
[00:11:02.687]well let's first talk about this, this is fairly easy.
[00:11:05.416]I'm using mathematical notation here,
[00:11:08.517]and they're just saying G is a function
[00:11:11.972]where you can plug in any biomass
[00:11:14.267]and get a number between zero and one.
[00:11:16.815]The reason it's between zero and one is because
[00:11:19.752]it's gonna turn out to be a probability.
[00:11:23.309]And so it starts at one when there's no biomass.
[00:11:28.496]Sorry, let me.
[00:11:30.142]When there's no biomass the probability
[00:11:34.033]of moving on is one, it's not crowded.
[00:11:37.096]The fish will always move on.
[00:11:40.942]As the biomass gets really large
[00:11:43.540]it gets more and more crowded
[00:11:45.040]and the probability that you're gonna move on
[00:11:47.345]is going to be, is gonna approach zero.
[00:11:51.609]And so we now have this dynamical system.
[00:11:55.668]We've got this model, the population at time T plus one
[00:12:00.943]is equal to this matrix
[00:12:04.116]times the population at time T.
[00:12:05.772]I want to go back to that matrix.
[00:12:07.723]Oops, wrong direction, want to go back to this matrix.
[00:12:10.746]So this is the matrix A sub P sub T.
[00:12:14.765]So it's this model, but then we're saying,
[00:12:21.257]This thing is anti-intuitive to me.
[00:12:25.894]It's going exactly the wrong direction,
[00:12:27.880]I don't think it's gonna go, there we go.
[00:12:30.427]So we're saying, and then the probability of moving on
[00:12:34.708]is going to be this decreasing function of biomass.
[00:12:39.781]So this thing takes into account the more biomass there is,
[00:12:43.206]the less likely it is for a fish
[00:12:45.640]to move on to the next class.
[00:12:49.967]Now, I'm gonna be talking about,
[00:12:52.646]mathematicians like to talk about norms in vector spaces.
[00:12:59.663]And in this case it's really easy.
[00:13:01.539]When you talk to biologists you can just simply say,
[00:13:04.023]the norm is gonna be the total population.
[00:13:07.538]Now I'm gonna make sure that we're on the same page
[00:13:10.176]about terminology but this is,
[00:13:12.710]at least among mathematicians fairly standard terminology.
[00:13:15.465]A system is persistent if for every nonzero,
[00:13:18.893]nonnegative P of zero, so if you start with some fish,
[00:13:23.182]there exists some number so that the population
[00:13:26.549]stays above that number.
[00:13:31.548]So the population survives.
[00:13:38.637]The next one's pretty intuitive.
[00:13:40.929]A system is bounded if for every nonzero,
[00:13:43.590]nonnegative P of zero there is a bound.
[00:13:48.524]There's an M so that the population stays less than M
[00:13:51.400]for all T greater than or equal to zero.
[00:13:53.756]Now that seems pretty realistic.
[00:13:55.685]If a model predicts a population that's unbounded
[00:14:00.656]there might be something wrong with the model.
[00:14:03.843]Because populations tend not to get unbounded.
[00:14:07.425]They can get large.
[00:14:09.896]And then a population vector, so there's a vector,
[00:14:13.176]is globally asymptotically attracting if
[00:14:17.698]for every nonzero, nonnegative starting point
[00:14:20.639]the population approaches that
[00:14:28.641]So this is an equilibrium vector.
[00:14:33.632]And typically when you have density dependence
[00:14:36.323]you expect, for many models,
[00:14:39.902]for the population to
[00:14:43.040]increase to some vector.
[00:14:47.006]And what we're saying is that it's gonna be the same vector
[00:14:49.630]no matter where you start.
[00:14:51.183]That might seem fairly restrictive
[00:14:54.257]but for most population models that's true.
[00:14:58.568]Okay, now I got to get into
[00:14:59.686]a little bit of mathematics here.
[00:15:03.433]We're gonna talk about something called
[00:15:04.449]the spectral radius of a matrix A.
[00:15:07.216]Now, I'm gonna give you the definition of it.
[00:15:10.880]If you know about eigenvalues it makes sense.
[00:15:13.156]It is the largest eigenvalue of A.
[00:15:16.096]I'm cheating here a little bit,
[00:15:17.819]it's the length of the largest eigenvalue.
[00:15:20.837]But it determines the asymptotic behavior
[00:15:23.764]of your dynamical system.
[00:15:32.161]If you don't know about eigenvalues,
[00:15:33.332]you don't need to understand it that much
[00:15:36.391]except we want to show, want to talk about
[00:15:38.765]what is the significance of this for biological systems.
[00:15:42.515]If the largest eigenvalue is less than one
[00:15:45.604]then the population crashes.
[00:15:51.267]If the largest eigenvalue is greater than one
[00:15:55.809]then the population goes to infinity.
[00:15:58.606]Remember I just said that populations don't go to infinity.
[00:16:03.653]But here we have a model,
[00:16:06.268]this does not have any density dependence in it.
[00:16:09.356]This is just a straight ahead linear model
[00:16:12.837]with nothing stopping the growth.
[00:16:15.977]So if the eigenvalue is greater than one
[00:16:18.287]the population blows up.
[00:16:21.345]And now the most interesting case,
[00:16:26.118]If the spectral radius is equal to one
[00:16:29.765]the population approaches the eigenvector
[00:16:34.636]for the largest eigenvalue.
[00:16:37.145]So if you don't know eigenvalues and eigenvectors
[00:16:39.880]just think, if this measure here is equal to one
[00:16:43.556]the population stabilizes.
[00:16:46.423]It doesn't go to zero, it doesn't go to infinity,
[00:16:48.665]it just, it stabilizes somewhere.
[00:16:51.050]And as I said, this is not a realistic model.
[00:16:55.892]It might be for,
[00:16:58.529]for small populations,
[00:17:02.315]but it might be in the first case.
[00:17:05.054]And the third case is weird because
[00:17:08.523]row of A is equal to one, that's very specific.
[00:17:12.674]The probability of that actually happening in a field
[00:17:15.371]is of course is actually zero,
[00:17:18.375]but mathematically it's a useful thing to think about.
[00:17:24.496]So here's the main theorem.
[00:17:26.618]And I think we can safely ignore this first one,
[00:17:30.657]and it has to do with the leading eigenvalues.
[00:17:32.759]But here's where it gets interesting.
[00:17:35.593]If row of A1 is less than one,
[00:17:38.967]then the zero population
[00:17:41.884]is globally asymptotically attracting.
[00:17:44.237]Okay, let's see what that says.
[00:17:49.478]If when the probability, this is the best,
[00:17:52.662]two is the best case for the fish.
[00:17:56.662]It's when the probability of moving on for the fish,
[00:18:00.067]of getting bigger, is...
[00:18:04.871]So A1 is the best case for the, I should to be clear,
[00:18:07.755]A1 is the best case for the fish.
[00:18:09.852]That is, the probability of moving on is one.
[00:18:14.435]the spectral radius of this best case is less than one,
[00:18:17.940]if the best the fish can do is still crashing,
[00:18:21.077]then this nonlinear system also crashes.
[00:18:24.544]I think when put that way it doesn't seem that,
[00:18:28.235]that surprising at all.
[00:18:30.589]By the same token, A0.
[00:18:33.366]A0 says, the probability that the fish moves on is zero,
[00:18:37.021]you're never moving on, it's the bad case for the fish.
[00:18:40.243]If the worst case for the fish
[00:18:43.504]has a largest eigenvalue greater than one,
[00:18:47.085]well then the population's gonna blow up.
[00:18:51.678]Because if the worst case has the population blow up
[00:18:53.877]the nonlinear system is going to blow up also.
[00:19:00.742]And so as I said, I don't think this is that practical.
[00:19:04.486]This of course is very practical.
[00:19:06.720]I mean, you could have populations that are crashing.
[00:19:10.580]This next one is, if the largest eigenvalue
[00:19:17.472]worst case is less than one,
[00:19:19.324]but the best case is greater than one,
[00:19:23.143]and that's like when we were using data from
[00:19:27.209]Branched Oak, that's what we were getting.
[00:19:30.021]So if the worst case is less than one,
[00:19:31.791]the best case is greater than one,
[00:19:33.331]then it turns out that the system
[00:19:35.073]has a unique nonzero equilibrium.
[00:19:38.454]So there's an equilibrium population.
[00:19:42.414]And in the same situation, the situation which we,
[00:19:47.351]which I think it's most likely to happen,
[00:19:49.919]at least for an invasive species,
[00:19:51.748]then the system is bounded and persistent.
[00:19:53.932]So if you have this stuff happening
[00:19:57.615]the worst case the eigenvalue is less than one,
[00:20:00.027]best case eigenvalue is greater than one,
[00:20:01.945]then the population is bounded but also stays away from zero
[00:20:06.260]so it neither blows up nor crashes.
[00:20:10.044]And here's one if,
[00:20:13.680]and in this same case,
[00:20:15.617]worst case less than one, best case greater than one,
[00:20:19.259]then for every nonnegative state the biomass converges
[00:20:23.295]to the equilibrium biomass, okay.
[00:20:26.447]So what this is saying that mathematically we can prove
[00:20:30.330]that in this case
[00:20:33.290]that no matter where you start the biomass is going to
[00:20:37.964]approach the equilibrium biomass.
[00:20:41.595]So we know exactly where the biomass is headed.
[00:20:46.314]Now, I'm not so sure if it's apparent to this group,
[00:20:49.547]a mathematician would look at this and think
[00:20:51.463]there's something missing.
[00:20:54.369]The following appears to be true
[00:20:55.403]based on numerical simulations,
[00:20:57.000]but we've been unable to prove it.
[00:20:59.076]Okay, in this basic case,
[00:21:04.701]we had proved in this basic case that no matter
[00:21:08.914]what your population starts at,
[00:21:11.258]it's going to approach an equilibrium biomass.
[00:21:16.622]What we'd like to be able to say, that the population
[00:21:18.981]actually approaches the population vector.
[00:21:22.263]And we've done a lot of simulations
[00:21:24.399]and that appears to be true.
[00:21:26.290]However we've been unable to prove it.
[00:21:33.120]Okay, now I want to talk about stunting a bit.
[00:21:36.156]So, in those papers I mentioned earlier
[00:21:38.350]stunting in the population is observed
[00:21:40.348]in the presence of density dependence.
[00:21:43.556]Now our length-based model doesn't keep track of length
[00:21:47.429]as a function of age, so we can't actually talk about
[00:21:50.924]stunting, at least my understanding of stunting.
[00:21:53.663]Although I've read differently, my understanding of stunting
[00:21:56.417]is that fish are smaller at a given age than they would be
[00:22:00.633]if there wasn't the density dependence.
[00:22:02.992]However this length-based model can show whether or not
[00:22:05.503]the population is dominated by small fish.
[00:22:08.767]And the form of the matrix shows that
[00:22:10.620]as the biomass increases
[00:22:12.546]the probability that fish transition
[00:22:14.268]to larger stages decreases.
[00:22:16.461]So the higher the biomass,
[00:22:18.080]you'd expect that's pushing down the fish, making the fish
[00:22:24.360]So the higher the biomass the more we would expect
[00:22:28.914]the population is dominated by small fish.
[00:22:33.814]Now we can make this precise
[00:22:35.292]in the case where the population is bounded and persistent,
[00:22:38.128]that good case.
[00:22:40.598]So there's an asymptotic biomass
[00:22:43.229]and there's an equilibrium vector.
[00:22:46.324]Don't mind this T here,
[00:22:47.631]I'm just being sort of pedantic mathematically.
[00:22:51.212]This is supposed to be a column vector,
[00:22:52.253]I wrote it as a row vector.
[00:22:54.828]This is where we believe the population
[00:22:57.357]is going to be ending up.
[00:22:59.288]The limiting population.
[00:23:03.418]So let me just put it out here.
[00:23:05.505]Part A of the following theorem shows
[00:23:07.637]that the higher the biomass,
[00:23:09.410]the more the equilibrium population is dominated by fish
[00:23:12.952]in the first reproductive class, okay.
[00:23:16.480]And the theorem is here, it's a little bit weird.
[00:23:18.882]If you look at this part here,
[00:23:23.438]it says that X
[00:23:26.776]divide, so let's say X1 divided by X2.
[00:23:31.537]The population in the first class divided by
[00:23:33.896]the population in the second class
[00:23:35.657]is an increasing function, it gets more dominated
[00:23:38.199]by the smaller one as the equilibrium biomass goes up.
[00:23:44.423]And this is true for all of them
[00:23:46.896]except it goes the opposite direction in the X1 X0 case.
[00:23:54.039]So part A shows that the higher the biomass
[00:23:56.502]the more the equilibrium population is dominated by fish
[00:24:00.125]at the beginning, and more it's dominated in early stages.
[00:24:03.650]Part B shows that the equilibrium population vector
[00:24:08.363]has decreasing components,
[00:24:10.893]so that means that there's more fish in the
[00:24:16.701]stage one than stage two,
[00:24:18.067]more in stage two than stage three.
[00:24:20.468]Again, except for the first one.
[00:24:24.572]That first one is a little bit strange.
[00:24:26.364]So this is a way of discussing stunting,
[00:24:30.508]make it very precise.
[00:24:33.118]Now so before I go on to this example,
[00:24:36.641]I want to know are there any questions?
[00:24:46.678]Well I had a couple.
[00:24:50.844]growth rates are often dependent on size
[00:24:53.100]but it seems like here you would account for that
[00:24:57.204]by choosing your length classes carefully, is that--
[00:25:02.526]So I'll show how it's done for the white perch.
[00:25:06.875]And then the other one had to do with,
[00:25:08.823]would you back up a slide or two?
[00:25:13.188]A bit farther back.
[00:25:16.170]What are we looking for?
[00:25:19.107]Yeah, so basically with this biomass thing.
[00:25:21.265]So you can get an equilibrium biomass.
[00:25:24.698]You can prove that there's an equilibrium biomass
[00:25:27.591]but not that there's an equilibrium population vector.
[00:25:31.673]Yeah, it's very frustrating.
[00:25:33.077]Yeah, I'll bet.
[00:25:34.937]I've spent many many hours.
[00:25:35.770]I recall being in a room with mathematicians
[00:25:37.686]unable to prove things, it wasn't comfortable.
[00:25:41.615]No, I guess what I was wondering is does that mean possibly
[00:25:45.140]that there's, is that partly because
[00:25:47.729]there's more than one population vector
[00:25:50.627]that can give you the same biomass.
[00:25:52.371]That's exactly where we run into trouble.
[00:25:58.408]Any other questions?
[00:26:00.089]We have the time, it just.
[00:26:01.852]Okay, so now we're gonna get to the white perch.
[00:26:05.360]And just so white perch are not native to Nebraska
[00:26:08.941]and can be considered invasive there in Nebraska lakes.
[00:26:12.790]They crowd out more, other desirable fish.
[00:26:15.007]The population is stunted, so small fish predominate.
[00:26:18.973]Okay, so we choose length classes so that after one year
[00:26:21.737]fish can either stay in their length class
[00:26:23.336]or move to the next length class,
[00:26:24.592]but can't skip to a larger class.
[00:26:27.932]In the absence of density dependence,
[00:26:30.291]fish always grow into the next length class each year,
[00:26:35.694]in which case our length classes reduce to age classes.
[00:26:40.274]Density dependence, in this case slowing of growth
[00:26:42.547]due to crowding, slows down the growth.
[00:26:44.650]Okay, so when we say no density dependence
[00:26:47.840]that just mean, that's the good thing for the fish.
[00:26:50.133]It's not crowded so the fish just keep on, just move on.
[00:26:55.265]Okay, now all this stuff I got from
[00:27:00.623]the two papers I was talking about,
[00:27:02.373]especially Chiziski et al.
[00:27:05.322]The average length of the newborn perch
[00:27:07.679]is L0 is equal to six millimeters.
[00:27:10.123]By the way, I mentioned getting data from Game and Parks
[00:27:13.509]and I was warned by both Kevin and by Aaron
[00:27:16.322]at Game and Park, the data's really messy.
[00:27:21.075]We wound up using the data in your paper
[00:27:22.776]rather than the data we got from Branched Oak.
[00:27:29.535]None of the S&R people here are gonna be surprised
[00:27:32.331]but the data we got was really difficult to work with.
[00:27:38.665]But I thought it was good for the undergraduates
[00:27:40.770]and for me to see that.
[00:27:42.985]Okay, the maximum a newborn white perch can grow in one year
[00:27:46.678]was 45 millimeters.
[00:27:48.737]So after one year of life,
[00:27:50.189]white perch have an average length of
[00:27:57.649]Half of this, plus this.
[00:28:00.858]Now, after one year of life,
[00:28:03.205]energy is devoted to reproduction,
[00:28:05.802]diminishing somatic growth.
[00:28:08.438]So the formula actually winds up being
[00:28:12.390]we assume that the average length (mumbles)
[00:28:13.973]LI plus one is LI plus delta max.
[00:28:18.768]And then it's gonna be divided by this number.
[00:28:25.909]Where this is a formula that is derived
[00:28:29.816]in I believe both of the papers.
[00:28:32.952]Okay, so L8 is 308.47 millimeters.
[00:28:38.264]This would take approximately eight years to accomplish
[00:28:40.255]which is approximately the lifespan of white perch.
[00:28:42.413]Thus we limit our model to eight length classes
[00:28:45.652]Okay, so one thing I should mention is we're trying to
[00:28:48.680]match the Chiziski et al. paper
[00:28:52.320]because we want to be able to compare what we get
[00:28:55.180]with a length-based model with density dependence,
[00:28:57.576]with what they get.
[00:28:59.249]And the comparisons were pretty comparable.
[00:29:04.504]The models gave pretty comparable predictions.
[00:29:08.420]So B of T is the biomass at year T.
[00:29:11.356]We model the probability at year T that a fish grows
[00:29:15.614]into the next length class in one year
[00:29:17.386]by a Beverton and Holt response function,
[00:29:19.255]again from Chiziski, Pope, Wilde.
[00:29:22.854]So we use this function.
[00:29:24.176]Our theorem doesn't require this specific function
[00:29:26.872]but we used it for comparison purposes.
[00:29:34.372]It is assumed that the survival rates across age classes
[00:29:37.754]is a constant .68.
[00:29:40.585]Again, we got it from that paper.
[00:29:42.253]Which we assume is the survival rate
[00:29:43.763]across length classes too.
[00:29:46.618]And it is assumed that the fecundity of white perch
[00:29:50.015]of length L sub I is this FI.
[00:29:53.749]Now I thought it was interesting
[00:29:55.029]in the two papers we were looking at.
[00:29:57.364]One had a cube here,
[00:30:00.906]because it was,
[00:30:04.224]one of the papers had a cube because
[00:30:05.502]it was mechanistically derived.
[00:30:09.160]And the other had the 3.41 because it was matched to data.
[00:30:12.423]And they really predicted very similar things.
[00:30:16.096]This one seems,
[00:30:18.046]since we wanted to compare to Chiziski, et al.,
[00:30:22.502]we used this, with this egg viability rate.
[00:30:28.783]And then I'm just noting that the
[00:30:33.064]survivals and fecundities
[00:30:36.654]are all positive and non-decreasing, just what we wanted.
[00:30:40.801]All hypotheses wind up being met for the theorem.
[00:30:44.071]Well we've had a bunch of theorems, so it met.
[00:30:46.561]And we found that the best case scenario for the,
[00:30:50.996]the worst case scenario for the fish,
[00:30:53.115]the leading eigenvalue is .6872 so
[00:30:57.699]if the biomass was infinite the population crashes,
[00:31:01.531]if the biomass is zero the population blows up.
[00:31:03.936]So the theorem states that the system of white perch
[00:31:07.461]has a unique nonzero equilibrium.
[00:31:10.546]Here's the equilibrium.
[00:31:12.655]It's bound and persistent and for every nonzero, nonnegative
[00:31:16.125]initial state the biomass converges
[00:31:17.899]to the equilibrium biomass.
[00:31:21.573]And of course we did simulations,
[00:31:25.045]which seemed to show that this was globally attracting
[00:31:29.825]as a population vector.
[00:31:32.241]And the equilibrium occurs when the limiting probability
[00:31:37.165]is just .33 or so.
[00:31:40.555]So this P star
[00:31:44.200]is a limiting probability.
[00:31:48.465]And this is not a coincidence that the limiting probability
[00:31:52.506]yields a leading eigenvalue of A.
[00:31:57.059]A with that probability equal to one.
[00:31:59.886]That's not a coincidence at all, that's what happens
[00:32:02.615]as the population stabilizes, or the biomass stabilizes,
[00:32:08.798]the system tries to seek out a probability
[00:32:13.594]so that A at that probability
[00:32:16.340]has a leading eigenvalue of one.
[00:32:18.823]And here is the equilibrium biomass
[00:32:22.091]that's globally attracting.
[00:32:25.139]And so we see here the population is dominated by
[00:32:29.117]small fish, but we also, so
[00:32:32.793]it goes down here, but it did go up here.
[00:32:38.769]That's what our theorem talked about.
[00:32:43.702]And I'd like to
[00:32:46.519]conclude with numerical simulations
[00:32:48.707]to illustrate some more mathematical,
[00:32:51.165]some of the mathematical results,
[00:32:52.997]verify that our model yields similar predictions
[00:32:55.085]as the age-based model.
[00:32:57.012]Actually we didn't put in the paper the comparison.
[00:33:00.799]Or didn't put in the talk rather the comparison,
[00:33:03.010]and give additional information
[00:33:04.210]about the predicted size distribution of the population.
[00:33:07.277]Okay, so we did the model for the white perch
[00:33:11.878]and our model implied that there was
[00:33:18.193]dominated by small fish
[00:33:20.361]and that the population biomass stabilized
[00:33:24.603]and our simulations showed that the population
[00:33:27.544]tends towards the equilibrium.
[00:33:30.615]Okay, so now
[00:33:32.804]I decided, and maybe this was not good for the talk,
[00:33:36.787]I decided to talk about the pictures
[00:33:40.419]before showing the pictures.
[00:33:43.221]So figure one, which I'm gonna show later,
[00:33:46.497]uses the same data as we did in
[00:33:51.118]section four is not right, but we did for the white perch.
[00:33:57.356]Well that's basically the one,
[00:33:59.933]and so figure one just shows what we did before.
[00:34:02.257]Now figure two
[00:34:09.886]we're going to lower the survival rate
[00:34:11.740]for all length classes.
[00:34:14.205]And that's gonna push the,
[00:34:17.397]the best case probability less than one,
[00:34:20.086]and so the simulation is gonna show that
[00:34:22.599]zero is in equilibrium.
[00:34:24.333]So if we lower the survival rate for everybody
[00:34:28.526]it's gonna show that the population crashes.
[00:34:31.720]For figure three we assume that the survival rate
[00:34:34.936]for all length classes
[00:34:37.448]with tenfold the fecundity for every length class,
[00:34:40.019]and the rest of the data's the same.
[00:34:41.800]So we really crank things up and in this case
[00:34:44.652]the worst case scenario still has
[00:34:46.985]a leading eigenvalue greater than one.
[00:34:48.450]The simulation's gonna show that the population
[00:34:52.155]will lead to an unbounded fish population.
[00:34:57.879]Actually, what happens is interesting in that the,
[00:35:00.502]in this case when the population blows up,
[00:35:02.844]it doesn't necessarily blow up in every length class.
[00:35:08.224]In more simulations it appeared to just be blowing up
[00:35:12.847]in the small length classes.
[00:35:15.461]Of course, you can never, to a mathematicians satisfaction
[00:35:18.793]you can never actually prove
[00:35:20.148]simulations that something's blowing up because,
[00:35:23.754]because we can't take time going out to infinity.
[00:35:29.284]And then, actually let me just go to those,
[00:35:32.791]because they're pretty simple.
[00:35:35.370]So in this case
[00:35:38.791]we actually did some better graphs,
[00:35:41.729]unfortunately I'm not showing here.
[00:35:42.908]I couldn't figure out how to load them in.
[00:35:44.913]That with a shorter time frame,
[00:35:47.071]we see in this case we've got, where the,
[00:35:53.057]where we expect the biomass to stabilize
[00:35:56.130]and we've got all the stages appear to be stabilizing.
[00:36:02.902]And the smaller, it might not look like the smaller stages
[00:36:06.065]are larger until you look at the scale here.
[00:36:10.618]So you can kind of see where the scales,
[00:36:12.742]10,000, 10,000, 4000, 2000, 1000.
[00:36:16.934]But you can see clearly that
[00:36:19.139]the population appears to be stabilizing.
[00:36:23.179]Here's the case where the population crashes.
[00:36:26.001]And as I said we have some better simulations
[00:36:29.172]I couldn't figure out how to load into this presentation,
[00:36:34.883]but you can see the population crashing.
[00:36:38.095]And we let it go for a long time.
[00:36:40.383]It crashes early and stays crashed.
[00:36:43.531]Next one, where the population is blowing up.
[00:36:46.557]Oops, sorry, there we go.
[00:36:49.574]Next one where the population is blowing up.
[00:36:51.924]Now, it takes a long time before the population blows up,
[00:36:54.653]although part of it, it's not really taking a long time,
[00:36:57.633]this is a very high scale.
[00:36:59.537]So this is blowing up, blowing up,
[00:37:02.411]and then the scales are going down
[00:37:04.443]and then you get the interesting thing
[00:37:06.191]that when the population is
[00:37:10.658]going off to infinity, it's just the total population.
[00:37:14.562]The larger stages are stabilizing to zero.
[00:37:18.746]Well you can't quite see zero here because
[00:37:20.215]these are large scales, they're stabilizing to zero
[00:37:23.057]where the, this stage, you can't tell it
[00:37:26.781]but this stage is stabilizing to nonzero
[00:37:29.340]and then this one is stabilizing to nonzero
[00:37:31.582]and then these don't appear to be stabilizing at all.
[00:37:35.477]So that's what we get when we just do those mod,
[00:37:38.982]we just do things that match the theorem.
[00:37:42.943]Then we decided, let's mess around with things a little bit
[00:37:46.062]in ways that don't necessarily match the theorem.
[00:37:53.313]So figures four through six show that reducing
[00:37:55.234]the survival rate for fish longer,
[00:37:57.997]so if you reduce the rate for fish that are longer,
[00:38:02.425]that eliminates the dominance of the small fish.
[00:38:06.084]Now, we have no theorems about that
[00:38:08.058]because our theorems are about what happens
[00:38:10.728]when the population is, when the population is...
[00:38:17.332]That rather, survivals are increasing or staying level.
[00:38:22.117]But still, it's interesting.
[00:38:24.037]So when we decrease the survival rates
[00:38:27.513]it eliminates the dominance of the small fish.
[00:38:30.135]So here we're assuming that,
[00:38:32.998]in figure four we're assuming that 2% of the fish
[00:38:36.457]longer than 133 milligrams would be captured.
[00:38:40.798]So we decrease the survival rate in figure four.
[00:38:44.074]In figure five we do it more extreme.
[00:38:49.956]In figure six, still more extreme.
[00:38:54.031]And in figure seven,
[00:39:03.935]and then decrease them
[00:39:06.956]so the population vector is actually,
[00:39:10.650]if you can look at this cursor, actually a decreasing,
[00:39:13.487]where the survival vector is actually a decreasing vector.
[00:39:17.427]In figures four, five, and six,
[00:39:20.517]you decreased from stages two on up
[00:39:26.438]and you left stage one the same.
[00:39:28.818]So now let's look at the pictures.
[00:39:32.442]Okay, so here we go.
[00:39:34.366]2% of fish longer than,
[00:39:37.992]2% of the longer fish captured,
[00:39:41.835]and you see
[00:39:45.581]that the population stabilizes
[00:39:49.000]but now the population vector
[00:39:53.775]or the equilibrium that it heads to
[00:39:58.600]is dominated by
[00:40:01.951]larger fish rather than smaller fish.
[00:40:09.202]Although we have to be careful
[00:40:10.187]because if you look at the scales it's 10 to the six,
[00:40:13.208]then it goes down to 200 and then it goes back up.
[00:40:19.565]And then more extreme, and we get more extreme results.
[00:40:26.392]And more extreme then, still more extreme results, and
[00:40:32.230]probability of capture increases with length
[00:40:34.208]so this would be like a fishing scenario.
[00:40:36.509]So this might suggest that,
[00:40:40.459]that making a point of fishing for the largest fish,
[00:40:44.539]decreasing their survival
[00:40:49.239]will get rid of the stunting.
[00:40:51.523]Will get rid of the dominance by the small fish.
[00:40:55.349]And I just wanted to mention this work was made possible.
[00:40:59.141]So this was the American Institute of Mathematics,
[00:41:03.052]which was funded by the NSF.
[00:41:07.219]And let's see, there's just a number of places
[00:41:11.549]that funded us.
[00:41:13.218]And I guess...
[00:41:16.874]Okay, I think that's all, I think that's all for that.
[00:41:19.117]And that's all for the talk.
[00:41:37.832]Well towards the end of your talk you told us about
[00:41:42.419]capturing the smaller fish and you would have
[00:41:45.005]like a different kind of dynamic afterwards.
[00:41:47.979]Like the pond will be dominated
[00:41:50.809]by bigger fish instead of smaller fish, right?
[00:41:55.015]I'm sorry I didn't understand, I didn't hear the question.
[00:41:58.479]So you said that if you captured the bigger fish,
[00:42:04.114]you would have a different,
[00:42:07.231]you would have a different dynamic in the pond,
[00:42:09.958]like more of the bigger fish will survive?
[00:42:15.469]I know it seems a little anti-intuitive.
[00:42:17.510]Well it's not so much more of the bigger fish will survive,
[00:42:20.016]it's that the population will be more dominated
[00:42:23.379]by the bigger fish, which is a little bit different.
[00:42:25.839]The population's gonna go down in all of them,
[00:42:29.381]it's just that the, it's just that the domination
[00:42:33.180]by the smaller fish will decrease.
[00:42:35.176]Okay, so my question is what if you introduced
[00:42:38.169]like a competition parameter in your model,
[00:42:41.629]and if so where would it be placed in.
[00:42:44.775]So you have your model--
[00:42:45.733]You're asking about competition?
[00:42:49.392]So like if you introduce a predator that only eats
[00:42:52.124]like bigger fish compared to smaller fish.
[00:42:57.672]Okay, so from the mathematician's point of view
[00:43:00.228]that makes the model much more complicated
[00:43:01.837]if you're trying to model the dynamics of the predator.
[00:43:08.363]But in answer to your question about introducing a predator
[00:43:12.664]I would have to assume without working out any details,
[00:43:16.259]and this is just a,
[00:43:19.449]I would have to guess that it would work pretty much like...
[00:43:25.639]Well except the competition's going to probably affect
[00:43:28.742]the smaller fish more.
[00:43:30.595]So I actually don't know.
[00:43:33.893]I'm not sure about that.
[00:43:35.919]Does anybody here have an idea?
[00:43:38.609]Kevin, you've sometimes worked with actually
[00:43:42.240]getting rid of these fish.
[00:43:45.574]So I was I guess pondering that
[00:43:47.551]'cause I think you've already got the,
[00:43:49.188]you've got the intraspecific competition
[00:43:51.447]already in the model
[00:43:53.168]with the probability of moving on or not.
[00:43:56.202]Right, but he was talking about predators introduced.
[00:43:59.092]And I think the predators,
[00:44:01.254]and I don't see angling as any different
[00:44:04.263]than any other predator, so you've taken predators,
[00:44:07.156]unless the predators are targeting
[00:44:08.668]small instead of large fish.
[00:44:10.387]Right, and that's where I can't answer the question.
[00:44:12.374]If the predators are targeting large fish
[00:44:14.816]like the anglers would, then it would be the same.
[00:44:18.013]But if they were targeting smaller fish
[00:44:19.487]I'm not so sure what would happen.
[00:44:24.916]Other questions for Richard?
[00:44:41.356]Just one, so I mean, why,
[00:44:46.058]why treat size discretely, I mean, you know,
[00:44:48.791]or is it just because you can get a solution when,
[00:44:51.143]I mean, you know, 'cause we're talking about
[00:44:52.438]integral projection models all the time, right?
[00:44:54.843]So why that choice rather than an integral projection model?
[00:44:58.336]In point of fact, one of Brigitte my graduate students,
[00:45:02.536]Matt Reichenbach is doing a model
[00:45:04.557]where we're not treating time discretely.
[00:45:07.741]And it's a lot harder to prove stuff there
[00:45:09.686]because functional analysis comes into play
[00:45:11.306]instead of linear algebra.
[00:45:13.071]But he is in simulations seeing some of the same things
[00:45:16.126]but not all of the same things.
[00:45:18.228]So you're right, it's actually, it actually makes more sense
[00:45:21.901]although it's quite a bit harder to treat size as a,
[00:45:27.212]as a continuous variable.
[00:45:29.026]And Matt gave a talk on that last week
[00:45:30.806]which some of the people here heard.
[00:45:34.518]The one reason I can think of would be data collection.
[00:45:39.325]At least like, when I got the data from Game and Parks
[00:45:42.540]they gave it to me in, they had already broken it into
[00:45:48.299]length classes, and they didn't have,
[00:45:50.758]or they didn't give me the data for the actual lengths.
[00:45:54.641]Right, but that's a failure of field ecology,
[00:45:58.784]I mean, they measured continuously in the field, right?
[00:46:02.816]One other thing I was wondering about now
[00:46:05.691]and then it just came in and went out again.
[00:46:12.239]No, it's gone, sorry (laughs).
[00:46:18.329]Somebody back there.
[00:46:29.522]So this might be a little bit similar
[00:46:31.268]to the earlier question, but in a lot of species,
[00:46:35.135]juveniles and smaller individuals have lower survival
[00:46:38.564]than the adult individuals other than
[00:46:42.118]from pressures such as predation.
[00:46:44.498]Do you have any predictions as to what would change
[00:46:47.377]in your models if you modeled survival as being lower
[00:46:51.424]for the smaller age classes than the larger age classes?
[00:46:56.812]Well no, it is large, it is small,
[00:46:59.289]except for those last simulations it is,
[00:47:03.663]the theorems are for the situation
[00:47:07.101]where the survival is smaller for the smaller fish.
[00:47:11.243]So I don't have to make a guess there,
[00:47:14.704]the theorem tells us what would happen.
[00:47:17.033]The simulations at the ends were a few cases where,
[00:47:23.146]where it wasn't the case where the survival went up
[00:47:29.053]for larger fish.
[00:47:31.201]I'm sorry, I guess I misunderstood.
[00:47:32.957]I thought that survival was constant at .68 for all--
[00:47:35.808]The survival was constant in the example.
[00:47:37.993]In the theorem we needed the survival to be non-decreasing.
[00:47:42.525]So either constant or going up.
[00:47:45.497]Which we thought of as kind of the,
[00:47:49.876]the most likely scenario.
[00:47:52.801]Which it sounds like you're saying.
[00:47:56.761]And then we gave an example,
[00:47:58.841]we gave an example there saying,
[00:48:02.089]let me just go to the beginning, that
[00:48:06.088]assume the SIs and the FIs are increasing in I.
[00:48:10.903]I actually wrote that wrong, it should be non-decreasing.
[00:48:13.790]They can be constant or increasing.
[00:48:16.730]This is often true, but not always,
[00:48:18.332]for instance in the presence of angling.
[00:48:22.637]So yeah, no it was only at the end that I kind of
[00:48:25.584]turned it around just because we got interesting, you know,
[00:48:29.704]we got more interesting management recommendations
[00:48:33.962]if you want to get rid of the smaller fish
[00:48:35.871]we had to cheat and not do it,
[00:48:38.731]do it in a way that doesn't match our theorem.
[00:48:44.102]Any more questions?
[00:48:48.216]I have a note I have to get to.
[00:48:50.193]I will add that I thought it was really intriguing
[00:48:54.353]and exciting to see that a mathematician
[00:48:57.129]came up with a counterintuitive response
[00:49:01.050]from a fisheries management perspective.
[00:49:02.616]Our typical approach in the past for stunting
[00:49:05.173]has always been to attack the small ones,
[00:49:07.339]not go attack the large ones, so, a cool outcome on things.
[00:49:12.269]Again, let's give Richard a round of applause.
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