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Unformatted text preview: 9.4 Models for Population Growth 1. The Law of Natural Growth Definition 1.1. If P (t) is the value of a quantity at time t and if the rate of change of P with respect to time t is proportional to its size P (t) at any time, then dP = kP. dt This differential equation is called the law of natural growth. If k > 0, then the population increases; if k < 0, it decreases. Remark 1.1. Notice the above equation is separable. 2. The Logistic Model A population often increases exponentially in its early stages but levels off eventually and approaches its carrying capacity because of limited resources. Denoting K as its carrying capacity, we obtain the logistic differential equation dP P = kP 1 − dt K The solution to the logistic equation is K − P0 K , where A = P (t) = −kt 1 + Ae P0 with the initial condition P (0) = P0 . 1 Section 9.4 Models for Population Growth 2 Example 2.1 (problem 8). Biologist stocked a lake with 400 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 10,000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. (b) How long will it take for the population to increase to 5,000? ...
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This note was uploaded on 03/14/2011 for the course MAC 2312 taught by Professor Zhang during the Fall '07 term at FSU.

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sec9_4 - 9.4 Models for Population Growth 1. The Law of...

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