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Problem 17 Medium Difficulty

Consider a population $ P = P(t) $ with constant relative birth and death rates $ \alpha , \beta, $ respectively, and a constant emigration rate $ m, $ where $ \alpha, \beta, $ and $ m $ are positive constants. Assume that $ \alpha > \beta. $ Then the rate of change of the population at time $ t $ is modeled by the differential equation
$ \frac {dP}{dt} = kP - m $ where $ k = \alpha - \beta $
(a) Find the solution if this equation that satisfies the initial condition $ P(0) = P_o. $
(b) What condition on $ m $ will lead to an exponential expansion of the population?
(c) What condition on $ m $ will result in a constant population? A population decline?
(d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was $ 1.6% $ of the population. Because of potato famine in the 1840s and 1850s, about 210,000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?

Answer

a) $P(t)=\left(P_{0}-\frac{m}{k}\right) e^{k t}+\frac{m}{k}$
b) Population will grow If $m < k P_{0}$
c) Population will remain constant If $m=k P_{0}$
Population will decline If $m > k P_{0}$
d) The population is declining.

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Video Transcript

Hey, it's clear. So when you read here Server for partying, we have dp over. DT. This is equal to keep p minus m. When we calculate for a d. T. We got a DP over. Keep he minus. Um, we're gonna substitute X kee p minus, um, and then four. And then we're gonna integrate both sides so we get one over K. The integral of K C P or her keep p minus M is equal to the integral d t we substitute this be we're going to get won over K. Shelton of X. It's equal to t. Yeah, so we only substitute back. You get one over K l and KP minus um is equal to t plus c. So to find the constant we're gonna substitute T is equal to zero and P is equal to p sub zero. So we get one over K l n. Okay. He subzero minus m is equal to see and we get one over k felt and of K P minus, um, this equal to t plus one over K felt and of K piece of zero minus m. We end up getting l on of KP subzero Linus. Um over K piece up zero minus m is equal to Katie. So we raised the power on the base e for both sides to get k p minus m over K p sub zero minus m is equal to e the k t. And we end up getting p of tea to be equal to piece of zero minus, um, over k times e the k t plus m over for part B. We're gonna use the equation we got from part, eh? Were given that dp over d t is equal to k T minus m. So using the equation, we're gonna substitute the value of P. Then we end up getting deep. He over DP is equal to K times piece of zero minus and over King B to the K T plus, um, over K minus. We get this to be equal to K piece of zero minus um times e Katie. You know that one dp over DT is positive than P increases asked. He increases and wanted negative. Then p decreases as tea increases. So we know that the population will grow exponentially if dp over d t is bigger than zero. This implies that this equation would as well be bigger than zero. So we get Hey, Humpy. Subzero minus M is bigger Zero. So we get the population will grow if m is less than K comes piece of zero. So we're gonna erase party now. And when we look at part see, we're going to use the equation Read arrived from a Let me reiterate that. So we get we got p of tea is equal to piece of zero minus m o over k times e to the k t plus em over King So we know that it's given that d p over d t is equal to k p minus m so we just substitute the values to get dp over. DT We did the same thing as we did here, which we ended up getting king He sub zero minus m times e to the k t. We said that the conditions when dp over DT is positive and negative. But when it zero, then we know that pee is gonna be constant. S t increases. We know that the population is constant when dp over d t is equal to zero. So this implies that this has to be equal to zero as well. So we get K piece of zero minus M is equal to zero and we end up getting m is equal to K comes piece of zero. So that's when the population will remain constant. We know that it declines if d pee over DT is less than thorough. So he used this equation and we just make this less than zero. So yet, kay piece of Serra minus m iss less than zero to get em is bigger than Cape piece of Zero. So that's when the population will decline me. Erase this part. So we're gonna look at the population now, uh, 18 47. So we know that the population was eight million in Ireland, so we just get the equation. P subzero is equal to eight times tend to the sick power. So we know that K is equal to the difference between birth and death. Rates over the population were given that the difference is 1.6%. So Kay has to be equal to 0.16 We know that there's 210,000 people, um, per emigrated from Ireland so M is equal to point to one times 10 to the sixth power. So we just yet K times piece of zero is equal to 0.16 times eight times Tend to the sick when we get 0.1 to 8 times turned to the sixth and we see that M is bigger than K times key subzero. So looking at part, see, we know that the population is declining.