 optimization tool and now here are the problems that can be solved with that tool but nevertheless I mean there's this is not going to exhaust all the possibilities so I think in the well I'm not supposed to comment on this until the competition is over at 6 p.m. but I'll say that that you know the more kind of the more tools you know and especially if you know partial differential equations then you can model kind of waves vibrations and okay so the company is called comap and if you look at the contests called MCM math contests and modeling and these are the contest problems this year so and again it's still ongoing so there's one problem of this sweet spot of a baseball second problem has to do with some sort of profiling geographically profiling the location of a criminal based on the crimes that he or she committed and the last one is has to do with this garbage patch of in the specific so it's kind of interesting usually there's a description of this kind of interdisciplinary problems take you know 20 pages so lots of data lots of things to to look at and try to come up with a model that explains and I think they also refer to a paper here so but what I want to say is that you can look at previous contests here if you're just kind of curious and you will see lots of I mean a wide variety wide variety I think last year there was a one that had to design a you know traffic circle which in the United States is more of a curiosity right but you know the parts of the world is really crucial that you have efficient traffic circles energy cell phones I think that's a problem that UCS team chose last year had to design a cell like from scratch like if you have a whole new country that you have to build the infrastructure do you build landlines or do you just rely on cell phones you know very open any questions there's no wrong answer and in fact nobody has been graded on those reports saying this is wrong you know but a lot of the papers have been are kind of disregarded because it says presentation doesn't is not appealing you know there's no there's no sequence of thoughts right or the model is not built well or something like that right so it's anyway so we could people that have good experience with you know computational tools usually have an edge on this competition so it's as usual you know in real modeling it's that's an essential essential part so you can you can take a look at this if you want so let me come back to any questions before we start any questions about homework that I turned in yeah no thank you for so I pushed it back to let's say to next Monday so there are three problems it shouldn't take you till Monday but I figured if you need a little bit of extra time so I'm sorry let me so okay so I the homework is due Monday I'll tell you one thing about chapter 4 and chapter 5 are pretty much our can be read lumped together so you know these three problems you know I would say if you see if you finish them after Wednesday maybe we talk a little more on Wednesday then you can start talking working on now in chapter 5 and once that we should start talking about chapter 5 as well right now I left it the same yeah so today I'd like to talk about two examples of dynamical systems we started talking about last time a little bit so let me let me remind you what dynamical systems model is are I just want to copy another so I have two codes here two new codes one talking about a continuous system and the other talking about a discrete system so we'll see what difference are the differences are okay so I'm gonna start with talking about even an example of a 2d well 1d we thought we saw last time so I said 1d model it would be a logistic model that usually comes from pop-population growth and it says that if I have x as a function of t it's monitoring the population at time t then let's see so I'm gonna put here in parenthesis the rate of change the growth rate of the population is has this expression so this is rx plus or minus excuse me minus r over kx squared right so this is just a constant think about it as a positive constant a right so this every time you build a model you have to kind of take into account certain effects in a population growth there is an intrinsic growth rate so that's that's assuming that the bigger the population the faster it grows and that applies to I don't know bacteria population or things that so individuals that don't kind of compete with themselves it's just kind of unrestricted growth so this is an intrinsic growth rate so that's the reason for that is if you only look at that effect so you ignore every other effect then what you see here is an exponential growth and r plays a role of that growth rate so the r is the if you want to sort of a well is the ratio between the growth rate and the current population okay so that's the that's called interesting growth rate but if you have other effects for instance if you have a limited environment in which your population kind of lives and and thrives and you take into account competition between species between the same individuals of the same species then there is simplest model is to introduce a term quadratic term that comes from product of x with x right and array you know constant in front of it meaning that the competition kind of inhibits the growth right so the growth rate is decreased by at a rate proportional to the square of the population size right so again the higher the population the faster the I mean the bigger the effect the negative effect on the on the growth okay again this is not the only one and you will see we'll see other models but this is considered to be sort of the next simplest from exponential growth okay so this would be just exponential model all right and why do we prefer to write it in this form rather than in like in this form which is probably again more intuitive the reason we write in this form is that if you look at this steady states of this population we say you know what would have what would the population have to be initially so that it stays at that value forever so that would be a constant solution to this equation remember we said that constant solution means the right hand side has to be equal to zero so having it as a product is just convenient okay because you can see the right hand side is zero if x is zero or if x equals k right so we saw that in the direction field we saw that zero and k were the two steady states so this is steady state if the right hand side at x star is zero right so for logistic model x one minus x over k zero gives you those two solutions x star equals zero and x star equals k okay not only that but we saw from the direction field we saw that and also from the solutions remember we kind of computed the solutions we could compute the solutions explicitly and we saw that over time so for x zero excuse me x of zero so that's the initial condition as positive then x of t approaches this value k as t goes to infinity one x if the initial condition was not zero but was negative then you have so this value k has a meaning of well in the population dynamics k is called maximum sustainable population so it's just saying that that environment you know in that environment so with those two effects a population of size less than this value will thrive but we'll never will kind of approach that value and a population that that starts it's overcrowded you know with a value that's greater than that that particular number it's gonna decrease it's gonna the the competition is is stronger than the actual intrinsic growth so there's gonna be a decrease and again to that value right so sustainable and if you start at that value you're gonna stay at that value okay so yeah yeah population dynamics right yeah so in the in the example that we're gonna be talking about the next few I guess lectures we're gonna see more of the population dynamics than economics like we like a manufacturing right so so you'll see this word this term quite a bit sustainable population and that kind of means there's a logistic model underneath right so here's a here's an example in 2d competing species model so I'll write down the system and then we'll we'll see you know what where it might come from so I have two variables that I'm tracking over time and those are population of two different species so this is a population of species one I'm gonna use blue and thin whales blue whales versus fin whales so let's say population of blue whales is x1 and population of fin whales is x2 at time t at a given time t now obviously this this numbers are are integer right so there's not not like a half of half of a whale so obviously as time evolves you know you know more whales are born and some you know some die so this this is not gonna be a continuous variable right but the assumption here is that we're gonna be talking about a large period like hundreds of years right thousands of years and then a large population right so there's gonna be sort of some sort of an approximation to say that over that large period of time this this variables can be considered to be continuous so when we talk about a rate of change we can talk about a derivative and not only that but we can think about each population has sort of a logistic growth and again remember this is this are kind of the simplest possible well I guess the simplest would be to consider exponential growth okay but for large animals there's always gonna be some sort of over overcrowding effects so so exponential growth works for really tiny right tiny individuals so population of bacteria or something when we're talking about a large-sized animals and we have to have some some sort of maximum stable population so these are sort of the simplest two models right the simplest model for population growth and now in addition to that we're gonna introduce a competition effect so so right so this term would be sort of competition between individuals of the same species but now if they both coexist then there's gonna be some term that that inhibits the growth of each population so that is a negative term to the to the rate of growth right and it's proportional to the number of interactions between the two species so the product between x1 and x2 yeah no this is competition yet still yeah we're gonna talk about predator you know next I guess but predator prey so so this would be a different model I mean a different setup you would have a system where there's some sort of intrinsic growth right in absence of the other species then the predator would be some sort of a positive term because prey helps predator grow right and the prey would be a negative term because predator kill prey so so it's just kind of a different sign here so it's not I mean setting it up is not different but the dynamics could be quite different okay so here here we're just gonna use this one this simplistic model and even the even the fact that we use same constants you know can be debated right so the proportional constant of proportionality here for this competition speed term doesn't have to be the same it could have different ones right but when you think about about putting this I'm matching this with a real with a real system like let's say in the Pacific right I think that's a yes it's quite a big big issue in marine biology and figuring out how much you know how much fish is in the ocean right which species are being in danger you know endangered should we stop you know what we do I mean should we repopulate and so forth so so just fitting the such models with the real stuff is extremely difficult because the you don't know not I mean you know nothing about this even this intrinsic growth rate you know very little about the maximum sustaining population of each of each population right you know even little less about what what the presence of one species does on the other right so we really would need data that spans you know hundreds of years to get good fits for these things right and we don't have that data that data I think just I mean even now it's hard to collect data but whatever data is is is available for the last I don't know 50 hundred years at most unless the Eskimos were collecting data in their own way and we we just don't know how to interpret that so so again what's what's the most important thing is competition species competing species is the fact that this term is is negative right so we put negative and I put alpha to be positive okay so so what do we do with this well first thing that we want to do is we'd like to understand so the goal is to understand the dynamics of the two species depending on on the their initial populations levels if you want so x 1 of 0 is x 1 0 x 2 of 0 is x 2 0 okay and other things would like to find out steady states which in this case would be called or equilibrium and this are called coexistence states if x if x 1 star so I'm going to use this x 1 x star to be x 1 star x 2 star if if these things are positive so so very soon we're going to be looking at this phase plane in which we plot x 1 versus x 2 right and we're going to be tracking initial conditions I mean we're going to start with the initial condition here right the initial population is this much for x 1 is just much for x 2 and we're going to follow that trajectory and in that plane there's going to be special points the steady states for which if I start here I'm going to stay here right for all times and they're called coexistence because if if there is a valley here right if there's an equilibrium where both x 1 x 2 are positive this means that if you start with that with those two valleys they're going to stay forever right so we'll see an example in a minute but again there could be a coexistence equilibrium here so this would be x star and there could be other initial conditions that go to this right this would qualify this equilibrium steady state to be you know not only coexistence equilibrium but it would be a stable coexistence equilibrium right it says that I can be you know I don't know for sure the initial population what it is but if it's in the ballpark of this steady state then as time evolves it's going to just get even closer to those values right now there's actually possibility that you have you have something that's coexistence equilibrium but it's unstable that is well we'll see okay but keep in mind about this being positive it's important to be positive if you haven't if you haven't a stable or a steady state or an equilibrium that's for which x 2 is zero what does that mean and you have any and you have on you know could be stable or unstable right it says that the population x 2 is going to go to zero right so that's not coexistence but it's still a steady state yeah so yeah so coexistence if only if both are strictly positive okay so we're going to look at the stability so that so a stable equilibrium is such that initial conditions that are close to that equilibrium approach that equilibrium excuse me the solution starting at those initial conditions approach so if x of t approaches x star for x of zero which is x zero is close to x star okay and we're going to define basically unstable is is an equilibrium that's not stable okay and we'll see how that that can happen all right so let's kind of be a little bit more specific for this example for this coexistence equilibrium so let's start with the steady state first of steady state analysis I guess that would be sort of the first thing to look at so in general let's say I have a system of two equations but it could be in general you know n equations and n unknowns so I have these two equations then an equilibrium x star solves so let's call it x notice that I'm using capital X to indicate it's a vector of coordinates x 1 x 2 so this solves the system f 1 of x 1 x 2 equals 0 so where you set the right hand side equal to 0 simultaneously so I'm gonna emphasize this it's a system saying by you know we have to solve both f the first right hand side equal to 0 and the second right hand side equals 0 okay so in the whale problem in our whale problem coexistence modeling excuse me not cause competing species model what's the right hand so what's the left hand side the right hand side so so I have dx 1 dt equals r 1 x 1 1 minus x 1 over k 1 minus alpha x 1 x 2 and I want to set it I have to set this equal to 0 and dx 2 dt is r 2 x 2 1 minus x 2 over k 2 minus alpha x 1 x 2 so I have to set both equal to 0 so let me use two different colors here so I'm gonna set first one equal to 0 in red and the second one equals 0 and blue and I want to show you this in the plane x 1 x 2 I want to point on a pinpoint the locations where both both are equal to 0 so here's how we do it we're gonna take the first equation and we're gonna graph it or find the points x 1 x 2 for which the first equation is satisfied right I'm gonna use red to to indicate that and let's see how do you solve the first equation well you see x 1 is a common factor so you see that I can factor x 1 I get r 1 1 minus x 1 over k 1 minus alpha x 2 equals 0 right and this leads to either x 1 equals 0 or r 1 minus r 1 1 minus x 1 over k 1 minus alpha x 2 equals 0 right so what's the first one x 1 equals 0 is the x 2 axis okay so any point on the x 2 axis makes second equation is just a linear equation equation x 1 x 2 so the graph it's a line right how this line is we don't know until we figure out the exact you know the exact values for r 1 k 1 and alpha but it's something like let's say something like this right this is x 1 equals 0 and this is r 1 1 minus x 1 over k 1 plus alpha x 2 equals 0 okay let's do the same thing with the other one x 2 oh so so this point this point we call it to be the null client so let me write this word here so this is called the null client corresponding to the first variable so x 1 null client if you want right so this is going to be the red null client and and the blue null client blue is a blue blue whale so I don't want to let's use green instead the green null client is similar right so again unless we we know the exact values of this constants we cannot plot you know very correctly but at least that these are lines and we know the x axis and so forth so so x the green null client would be the x 2 equals 0 and the other one is something like this right okay so the question is what are the equilibria if you know this null clients the equilibria will be always where the null clients intersect right well it's where the null clients of different colors intersect for instance where where the green line meets the green line here that point would not be an equilibrium right because it has to be a point that leaves on both null clients null clients of different colors so that's why it's it's it's useful to use different colors because you can see where those equilibria will appear and of course on paper this is the the steady states x 1 equals 0 x 2 equals 0 that's that's one of them right what's the other well it could be x 1 equals 0 and this thing equals 0 right but effects if x 1 equals 0 and and this one has to be 0 well x 1 is 0 so this thing has to be 0 so x 2 has to be equal to k 2 that means there's no species 1 well the equilibrium has to be basically where the species 2 is the maximum sustainable population and the same if x 2 is 0 then x 1 is has to be k 1 so this is 1 2 3 and the 4th is hopefully where coexistence occurs right so this would be solving the R 1 1 minus x 1 over k 1 minus alpha x 2 equals 0 R 2 1 minus x 2 over k 2 minus alpha x 1 equals 0 okay and it's a system of of two equations with two unknowns so this is linear system actually so you could actually do it even by hand if the constants would be nice okay and it's not guaranteed that this is always going to be a coexistence equilibrium because it could actually very well be you have one positive and one negative value right I mean you solve in the systems before two by two and you've gotten positive and you know sometimes I mean they don't have to be positive numbers right the solution of the of a system of two equations and two unknowns okay so let's look at the whales code here and the numbers will not be very pretty for instance alpha is 10 to negative 7 that's pretty much a guess but the magnitude of it you know being so small it says that you know that two species really don't bother each other too much it's not like one is a predator well no it just says that the the the competition between the two species is not affecting the growth unless the size of the population would be you know of 10 to the you know product would be 10 to the 7 and so forth right let's see the intrinsic growth rate for the blue whales population is 5% and 8% is for the thin whales so I don't know which one's actually bigger animal size animal but possibly the blue whales is larger right the smaller the animal the higher the intrinsic growth rate would be also the maximum sustained population is 150,000 for blue whales and 400,000 for thin whales right has to say something about the size I guess and then right so this is the right hand size of that system so again if you use MATLAB then to find the steady states you just simply say close your eyes you know and say solve the system f1 equals 0f2 equals 0 and hope that you get everything right remember you work simple and symbolic well if you work in symbolic world and then you always have to be wary a little bit of the of the outputs so right now I'm just going to define the right hand sides and I'm going to solve it okay and because the system was even you know you could do it by hand if you had the time you can see that you really get the four points right nothing maximum sustained population 0,0 maximum sustained population and then the coexistence coexistence equilibrium okay so that would be the steady state analysis if you want okay it's pretty pretty basic but the very next thing is to say well if I start with an initial population that's not any of those four values where's the house you know how is the population the species going to evolve in time right so you have to solve that system of differential equations and I mentioned last time a little bit about this so so this would be the steady states now the phase portrait well first of all let's start with a direction field so direction field of a dynamical system is really can be well see the dynamical system can be written as a one single equation or metrics equation if you want or vector equation so we're going to put the so x is x1 x2 right and this is a function of time and f of x is excuse me it's a vector f1 if you want to vex and f2 of x right the right hand sides so the phase plane is the x1 x2 plane and what is the direction field in this plane x1 x2 have to be positive so we're only care about the first quadrant here so it's going to be basically just as an illustration here every point in the plane is going to be associated with a direction and the direction is so what what's you know if I if I pick this point and I want to I want to compute the solution of that system and display it here then I can do this I can compute I can I can measure f1 f2 and then I'm going to get that slope right because remember dx1 dt is f1 dx2 dt is f2 so unless it's one of them is zero let's say none of them is zero both are not zero then you could get that dx2 over dx1 is is what was the ratio of the derivative of dx of x2 with respect to t and derivative of x1 with respect to t right so that's be f2 over f1 so it'd be if I if I were to plot x2 versus x1 at this point then the slope of that curve or of the tangent line to that curve would be the ratio of f2 over f1 right so that's why every time you direction field is is displayed like this okay you measure f2 and f1 now we'll see what happens when you are when you happen to be on a null line so for instance if f2 happens to be zero at a point so this means you're in a I don't know on the green null line this means that the slope the direction is going to be horizontal right when you are on the null line where f1 equals zero the direction is always going to be vertical okay so let's see this okay and I have to say one word about how Matlab actually displays this vector field so of course we're going to sometimes use p-plane just for convenience but there's kind of a funny syntax here for instance if I want to well obviously you cannot display direction field for every single point of the plane so if you choose that you have to choose the window size so x1 say between zero and nine hundred thousand x2 between zero and six hundred thousand the window size has to capture the steady states right if you want to see the whole picture and then I choose ten points to describe I mean to display ten points by ten points so grid of a hundred points I'm going to see a hundred directions and then unfortunately well I think the easiest one it would be to kind of code once again the the right-hand sides to right the right-hand sides and preface that by this command mesh grid so mesh grid basically just prepares those computes those a hundred points x1 and x2 and then here it just computes the you know the two directions the f1 and f2 and the main command is quiver quiver is for you know for basically plotting the arrows and you have to display x1 x2 and f1 and f2 okay and then it just kind of put the axis here so let's see what what this does okay so that's these are a hundred points and the arrows now this is not perfect why is it not perfect but first of all the length of the arrows don't necessarily match with the exact values of the of the right I mean I think if you if you'd like to display exactly the size of each arrow I think it will look messy I think you have to put like I think zero let's see if zero works yeah see maybe I should clear clear the figure and redo it here so so this quiver you can you can force it to display the vectors with their exact lengths but then you don't see much right it could it could get really messy or you could you could ask it to just rescale so this is a kind of a rescale automatic rescale so you just see the directions you don't see the magnitudes okay but anyway this is the direction field okay and now the next thing to do is as I said to identify the steady state by drawing the null clients so for no clients it's fairly easy just say solve f1 equal to zero and plot it so and then hold on so and do the same for the second one and I don't think I've used different colors but you can certainly use different colors but these are and again it doesn't plot the like x1 equals 0 x2 equals 0 or maybe it did but it doesn't show here so from this you it's hard to see actually more than just the equilibria right you see the equilibrium I mean we knew this is 150,000 we knew that was a 400,000 right and we can figure out we can ask the computer to well actually it did plot right it did compute the steady state their co-existent equilibrium but I mean it's hard to say what happens with initial condition that's not one of those four points right so I make a comment here that if you wanted if you want to see what happens there are two options one is to code here for instance one of the OD solvers and just give the initial condition say what happens if I start at 400 I don't know 200,000 and 500,000 right then it's going to compute that by solving the system of equations but it's not too pleasant so that's why p-plane is actually much better here all right and when we do p-plane what we see is we see the actual solution starting close to that it co-existent equilibrium actually come approach that equilibrium as t goes to infinity right so we call that to be a stable equilibrium okay all right so I'm going to switch to p-plane in a second but what I'm here there are issues of sensitivity to the parameters remember those values of the parameters by no means are set in stone I mean there's certainly 10 to negative 7 is not by far anything that you should you know bet on so you can do sensitivity to this parameter alpha for instance and you know you won't be able to do anything like a direction field but one thing you can do is you can ask the computer to try to solve find the steady-state for for as a function of that parameter and you can see you know for instance with respect to alpha that's what it is what you do with this probably the best ways to plot because otherwise you don't it doesn't register what you know whoops I skipped I'm sorry I skipped a cell okay and yeah and then then what do you do with those a coexistent equilibrium with respect to the parameter alpha well so what is what does it mean sensitivity to that I mean of course a numerical value with no ratio of relative changes could could be something to do but I think it's more reasonable to ask you know for what value of alpha there's going to be no more coexistence so so here you see when I solve x1 of alpha x1 is a function of alpha I solve it equal to zero what am I doing or x2 alpha equals zero right so I get a value for alpha what is that right this value of alpha no so this value I obtained it by taking that expression that I found for x1 or x2 or let's say x1 and I set it equal to zero right so so this means that for this value the the fourth equilibrium is going to be kind of stuck on an axis right being on an axis means it's no longer a coexistence so this value it's there's no more coexistence this is well this is the value where x1 the blue fin the blue whales would be distinct right this would be the value for which the fin wells would be extinct it's probably easiest if you do this this graph but again this this is by by far just the only I mean the only way to do it so here I just plot x1 as a function of alpha and x2 as a function of alpha and you should really use different colors I'm sorry as if I'm versus alpha right so what this what it says is you know and you can see what we know for what values of alpha both are above zero that's that's what I mean it's coexistence so it's the it's for these values of alpha and for these values of alpha so it has to be either too few like a small competition smaller than this value or larger than this value right and I think it's fancy way of of doing this but by identifying the points where it's actually right so it's saying that for alpha greater than 5.3 10 to negative 7 the coexistence you know there's going to be a coexistence equilibrium right and for values for alpha less than this it's going to be a coexistence equilibrium I mean that's what it says is actually solving a system of two inequalities okay but not by hand by by using a computer okay any questions on this one advice is is don't necessarily take this as the way to represent your you know solution or your report I mean as far as the graphical okay you know that this is just one way of doing it in different problems you may actually look different right but sensitivity is to a parameter is really about you know at what at what point you know is that conclusion going to be different so in our conclusion was that if I have 10 to negative 7 then there is a coexistence right well if I if I if alpha is not that but just slightly more then there's no more coexistence equilibrium right so that would be sort of this the sensitivity okay so so see whales code so this would be stability done with p-plane I think right now it's p-plane 8 okay so let me oops I thought I had that on didn't I show you people I showed you people in last time a little bit right so somehow it's not on this machine so let me put it back on here oops you don't modify this code at all I just want to call it okay so if you look in the gallery here there is I think there is a competing species model already so okay it's using different values so I'm gonna just show you with these values but you can just change this numbers right and call it instead of XY call it X1 X2 and so forth okay but you can see ways of doing this for instance you can find you can show them all client it just shows them to you okay it's not perfect it's not going to be perfect sometimes it's kind of you can see this here it's like an artifact right so it's not a perfect graphical tool but you can actually let's see you can find equilibrium points I think I showed you last time so you can just pin points and it's gonna have to do it one at a time and you have to be close to whoops oh yeah that's a good example here is this an equilibrium point no this is same color lines right so this is not a intersection of two null lines right so I was hitting that point that and it wasn't giving me because that was but but this one is right okay and let me just say this so here the eigenvalues and eigenvectors we're going to basically in chapter 5 we're going to link this eigenvalues of the Jacobian metrics the sign of this eigenvalues with the stability properties so for instance the fact that both of these are negative says that the that equilibrium is stable right and it's called a sink because everything is kind of sucked into that point we didn't really watch what this was but you can see one is positive one is negative so this is not a stable equilibrium right it's not a sink yeah well it's a subtle point basically it says there's only one direction in which is is a stable and all on all the direction is actually going to be unstable it looks like a saddle so let's let's just see a few solutions here you can see it here okay now let me say this is not the whale right for the whale probably had different values there you had to put the different values and then what you would see you would actually see a sink you would see a stable equilibrium here the coexistent negative right but again the tool is the same okay and that's pretty wonderful right you solve a system by just clicking but this phase portrait as is called contains null clients several solution curves right the equilibrium so you can you can read a lot of things out of it you can read stability properties of the equilibrium you can also see what happens with certain initial conditions it by doing a keyboard input so you can say initial value is let me clear this so you can you can start with initial condition that I don't know maybe in this case would be 0.4 and 0.6 and it wasn't great because it went forward and backward in time so you would need to see the you need to see what oh you need to see the arrows here to see which direction it goes right or else you could do let me let me raise again you could do solution direction it could be just forward in time it's just a nice graphical interface so you don't have to code any of this but and then when you computer do you see you know it starts at that point and it goes towards that equilibrium right meaning this species it goes extinct right again so we'll talk about discrete systems on Wednesday but let me just say that for discrete systems the only difference is that typically it's not a grow a rate of change as a function of the current state but it's some sort of a discrete change so it's not instantaneous change it's like a discrete change so it's a change in the values over some fixed time for instance I don't know a day or a year right it could be a whale population model but you know the population changes once a year right based on the current state you know what's going to be a year from today so if it were the same direction field what you would see is actually you would just see starting with the initial condition whoops that's a bad example here but let me just display that so it has to do with in this case it's a docking problem it's kind of interesting but just to show you the so imagine that you would have a direction field that's like this if it's a continuous system you would fit curves that actually are tangent to this directions right whereas if it is discrete let's say over one year let's imagine is a whale problem right you would have this initial population and a year later it would be this based on this discrete model right now again why this big can when the the arrow was this small well turns out that the arrow was this big but this was a displayed scale down so you can see the arrows if you didn't scale down you would see just a forest of arrows here right but this is a this is one arrow that initiates at this point right then after a year it follows this right it still follows a direction field but in a discrete fashion and you can imagine that he's going to have different behavior possibly okay sometimes discrete models are required because of the nature of the problem you're talking about sometimes continuous models are good approximations okay so we're good we have homework that's still weak from today so but I would I would get you know I'd say get started on the two problems that are continuous and we'll talk about this one on Wednesday thanks