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

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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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