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In Examples 1 we used Lotka-Volterra equations to model populations of rabbits and wolves. Let's modify those equations as follows:

$ \frac {dR}{dt} = 0.08R(1 - 0.0002R) - 0.001RW $

$ \frac {dW}{dt} = -0.02W + 0.00002RW $

(a) According to these equations, what happens to the rabbit population in the absence of wolves?

(b) Find all the equilibrium solutions and explain their significance.

(c) The figure shows the phase trajectory that starts at the point (1000, 40). Describe what eventually happens to the rabbit and wolf populations.

(d) Sketch graphs of the rabbit and wolf populations as function of time.

(A) $$5000$$

(B). The Equilibrium solutions are:

$R=0, W=0$

$R=5000, W=0$

$R=1000, W=64$

(C) The populations of wolves and rabbits fluctuate around 64 and $1000,$ respectively, and eventually stabilize at those values.

(D)

Differential Equations

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Hello there. Okay, so here we got a modification of one of the most important and models for predatory prey systems. That is a lot to Volterra equations. So let's get Volterra Equation describes how the the predator and prey populations interact between them. And here we got a modification to to these equations to see what happens. So here are stand for the rabbit population, and the view is stands for the wolves population. So we're describing rabbits and walls. Uh, population dynamics. Okay, so the first part we need to study what happened with the rabbit population if the world's population is equal to zero. So what happened with these equations in this case? Well, then, if this is true, what happened is that we are dynamical system just reduced to one differential equation. And that differential equation is equal to our prime t equal to 000 Sorry. Eight times are one minus 0.2 are no more. So now we got We described the population of the rabbits by using this differential equations. But we need to give some more information to what happened with the population of the rabbits and for that, we need to calculate the different points of this dynamical system. Very simple one. And in order to obtain the clearing point of these dynamical system or differential equation, we need to impose that need a relative is equals to zero. This means that we, uh, don't change anymore. We reach a maximum population or even a minimum population following these dynamics. So in this case, if we impose this condition here to find the equipment points of this differential equation, we obtain the following. So we obtain a very simple equation is a, uh 00 to our, which is actually a quadratic equation. And if we solve this this quadratic equation, we obtain that the population is either zero or 5000. What is the meaning of this? Well, uh, this kind of solution is trivial because if you don't have any population of rabbits, is, uh, clear that you're not going to have any dynamics of the population of the erupts. But this implies that the even if you don't have any predator on your environment, the population will reach a maximum. That is 5000. So this describe this, uh, is an area where you don't have any equals on your environment. Now let's see what happened. If you study the equilibrium point of the whole system, it means that we are going to include the world's population. So I'm going to regret here the equations really fast. Our prime is equal to 0.0 zero eight. Our time is one minus 0.0 00 to our minus 0.1 R w And here we got W Prime is equal to minus 0.2 absolve you plus 0.0 000 to our doctor just to give you a small explanation to what this mother is describing. One particular part of these equations is that these, um, term here where you've got these two, the two populations multiplying between each other. That means that there is. There exists some interaction between the population basically in the context of the change of the rabbit population. What this term here represents is that some wealth is even a rabbit, but with a probability equals to 0.1 So this is like a coupling constant. So basically, if that company coupling constant is equal to one. That means that for each wolf that you got, we're going to eliminate one rabbit. That is not quite real. Because on reality, the the wolves are not 100% um, assertive by hunting the rabbits. So, yes, you got here, like, some kind of probability to cut in a rabbit. And this second term over here represents how the if some wolf and have contact with the rabbit population that we can understand as some rabbit being hunted by by wolf. Uh then with this probability, this will influence on the reproduction of wolves. Okay, so that's why these two complex constants between the rabbit and the wolf populations are different. Okay, so that's just some explanation of what these, uh, dynamical system is describing, which is quite interesting. But now let's see what happened with the kill even points. So having an including point for the system means that our prime and w prom r equals to zero. There is no change, no more change on the system. So we reach an equilibrium. Yeah, So basically, or system of or dynamical system reduced to just solve too. Equations were two variables, so these equals to zero. There's also equals to zero. Here, we just need to copy the equations and here the same. So look what happened here is that you got well 22 equations with two variables so you can solve and you're going to pay three solutions. Three kinds of solutions. The solutions are the first, the wolf and the rabbit populations are equals to zero. Basically, this is also, as I mentioned, a trivial solution. This doesn't have any relevant meaning because it's clear that if you don't have any populations, then there is no dynamic. So yes, this is a solution, but it's not important. It's a trivial solution. The second solution is that the wolf population is equal to zero and the rabbit population is equal to 5000. We have observed this result previously is basically that if you don't you don't have any wolf population. Then your rabbits will reach a maximum. That is 5000. So on this kind of, uh, situation, what happened is that your walls are going to extend and then your rabbits are going to reach a maximum of population of 5000. The second possibility the third. The third possibility is that your system, which are in equilibrium point of 64 wolves and 1000 rabbits. So when these two populations reached these values, then the system stay on change. Basically, there is. There is. You don't add any rabbit to the population. You don't have any wealth on on this population dynamics. So you've got these three possible cases that are represented by the equilibrium points of this dynamical systems. And well, this part answered, Yes, the the the part v of the problem. Let's continue now. What we have to do is see what happened with the phase space. So the phase space give you some notion of how is these two populations interaction between them? Um, these allow you to study different kinds of, um, initial conditions. So in this case, the X axis represents the rabbit population on the Y axis represents the world's population. We are going to start on the point here. 1000 aren't 40 wolves. Okay, so we start This is our initial condition for, or dynamical system. And then we left that our system evolved following the equation that I described previously, and we obtain something like this, enter enrich this point over here, and basically, this point represents an egg living point. Okay, this is this is an a clearing point, because from this point, you can you cannot change anymore. Um, basically, this point is 1000 and 60 for one of the equipment points that we have predicted previously by equating the differential equations to zero. Okay, so this is the clearing point. And this curve described this trajectory on the face of space described what is happening with the populations here on this part of the population of the rabbits start to increase the same for the wolves until it reaches some maximum value here for the rabbits and then start to decrease and the population of the world start to increase under again. It reads like a maximum point and then start to decrease the population of the wolves and also the population of the rabbits, and then start to direct to go into this equipment point following these hell IX of like vortex. So they describe, like, kind of osceola torrey behavior on both populations. That is how these are interacting until they both reach and Cleveland point. And actually, this kind of plot we can observe if we plot both of the populations on time. Okay, So let's see what happens if we plot these these dysfunctions with respect to time and mentioned you before we obtained this kind of conciliatory behavior at the beginning. And then we start to decrease until we start to convert to some a clear point. The same happened for the wealth population. You cannot observe too much of a change for the world population because of the scale. But yes, this has some kind of conciliatory behavior Until it's, it starts to regional living point. The initial conditions for these simulations were was the one that I stayed before on the face of space 1000 for the population of the rabbits and 40 for the population of wolves. And yes. So this basically is the how the the populations that are going to both with respect to time

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Differential Equations