Problem 21 Hard Difficulty

There is considerable evidence to support the theory that for some species there is a minimum population $ m $ such that the species will become extinct if the size of the population falls below $ m. $ This condition can be incorporated into the logistic equation by introducing the factor $ (1 - m/P). $ Thus the modified logistic model is given by the differential equation
$ \frac {dP}{dt} = kP (1 - \frac {P}{M})(1 - \frac {m}{P}) $
(a) Use the differential equation to show that any solution is increasing if $ m < P < M $ and decreasing if $ 0 < P < m. $
(b) For the case where $ k = 0.08, M = 1000, $ and $ m = 200, $ draw a direction field and use it to sketch several solution curves. Describe what happens to the population for various initial populations. What are the equilibrium solutions?
(c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population $ P_0. $
(d) Use the solution in part (c) to show that if $ P_0 < m, $ then the species will become extinct. [Hint: Show that the numerator in your expression for $ P(t) $ is 0 for some value of $ t. $ ]

Answer

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Video Transcript

Well, the first part. We can't inform the original question here into these equation, which is easier to manipulate. If we're the section, we can see that for this problem we're taking. Okay, Ben and Capital M greater than zero. And for the first part, we shall. This inequality Which means that capital m my nose he is greater than zero. You're positive on B. Mine is lower. Case him. It's positive. That means that here we have each term in the product is positive. Which means that the derivative is also positive. And when the derivative is positive, we have that the population have to be increasing. Four be less than Lord Kishan. Ah, we have here some positive constants. Things lower case M is listen Capital M he's also positive and this is negative. So the whole brother is negative under the relative has to be negative which means that the population is decreasing now for Barbie. If we take b, you wantto either capital M or lower case him. We get that the derivative Sarah zero, which means that the solution is constant. That gives us two and it could even solutions. And now we can start emulating the direction who so for the direction field we have await for B equal toe one hundred, two hundred No. On in steps off one hundred, we we have a light here and obtain, for example here my nose seven point two. So we draw a segment with that is lope on. We obtained these direction field. Now, given the direction field, we can draw the solutions. So here we get solutions like long this And here. If the population is greater than capital m this ocean opportunites a synthetically to Capitol Hill And if the population is less than lower, kiss him. Then it started the king. So either mentally reaches syrup. We will see that later. We have the constant soldiers here. No, in this case is a little hard to draw the solutions given that the slopes are very high. So I drew an example for K equals to zero point zero one. And here the solutions are like more easy to drop. And that's part B for party. We can see that we're working with inseparable difference. Your equation. So now we have to take the integral in both sides with respect to time. Here we go, we separate the differential question, and we're going to integrate here. So to avoid that intro, we have to solve the partial fractions problem which starts here. And we get the solution. So you compose a video and check each step, and then we get this. We get the original question, we do this a problem. Then we integrate each site. We use the change of arable touring, huh? Change of variable. And then we we used the separable, the partial fractions, the composition. We're with you. And we need to have a decent girls. So we we take this and this and we know that they're revelation. Is is this? And here it's easy, because it's just constant. And we get that glossy here does. He gets more fired when we multiplied equation by something. But we just keep up Flossie here. It's a common practice. Um, we do a few manipulations and were after these equation. Yeah, And in this equation, it's easier to Appalachia. T equals zero, and we get peace here, here. So we obtain a about you foresee, and we can't do a substitution here. No. So we got the constant and now we do a substitution here on. We obtained this equation from this equation. We just do a few manipulations, standard manipulations. And we get the general solution here. On with that, we have the answer to party. Now we want to solve part the so it does. If the zero this is the only working lower kiss him then the population becomes seemed so. That means these population because to Syria. But that means that the denominator here it has to be equal to zero. So we have a like that. We make it equal to Syria. We possum We moved by on we have here is a product bond. Each of these terms are positive here, here and here. They're positive, but this is negative. We have a minus sign, which means that the whole thing it's positive this whole thing so we can apply the natural hovering in both sides. And we obtained this equation the second one. So then we are right here. We got t equal to this with him. Whether this hero k greater than zero and my nose Same greater than zero. And we want to show that this is positive. Then after a lot of love. So we want to see that these here inside the library is greater than one. We can check it here. Yeah, each of these are equivalent. And we are right here. Toby's inequality, which is true. Which means this is true. Which means that this is the inner organs hero on, which means that the whole thing is rather concerned. And we have a positive time for which P ofthe team is equal to zero. Which means that there's a moment, a positive moment which makes the population go extinct. And we used that zero has to be listening. That was it. The problem ist