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Problem 4: Let P(t) denote the size of a population at time t. As we saw in class, udner one mathematical model we may assume that the rate of change

of the population is proportional to the size of the population. That is, \frac{d P}{d t}=k P for some proportionality constant k > 0 (for growth). Note that this means that \frac{d p / d t}{P}=k or in other words, the relative growth rate (4/P) is constant. This isn't always the case. We can make this model more general by instead assuming that the relative growth rate is dependent on P(instead of just assuming it is constant). We can rewrite this as \frac{d P / d t}{P}=f(P) \Rightarrow \frac{d P}{d t}=P f(P) for some expression f(P). The model in (2) called the density dependent hypothesis and is used to model animal populations. (a) One of the simplest versions of this model is when we assume f(P) = c¡P+ c2, for some constants c1 and c2. We can also assume that there exists a population number K at which the environment can't carry any more population. We can include this assumption in our model by setting f(K) = 0. Finally, we assume f(0) = r for some number r. Use these two assumptions to solve for c and c2 in f(P). (b) Use f(P)as found in part (a)to find the general solution for the differential equation (2). (c) Assume the initial condition P(0) = 1. Use this to find a specific solution from part (b). (d) Given the solution on part (c), what will happen to the population in the long run?

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