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lecture_2 (dragged) 1 - Another way to say this is dP dt...

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2 MA 36600 LECTURE NOTES: WEDNESDAY, JANUARY 14 Figure 2. Slope Field of dv dt = g -4.8 -4 -3.2 -2.4 -1.6 -0.8 0 0.8 1.6 2.4 3.2 4 4.8 -3.2 -2.4 -1.6 -0.8 0.8 1.6 2.4 3.2 Figure 3. Slope Field of dv dt = g γ m v -7.5 -5 -2.5 0 2.5 5 7.5 10 -12.5 -10 -7.5 -5 -2.5 Population Models. So far we have considered di±erential equations which arise through physics. Indeed, Newtonian Mechanics (a.k.a. Classical Mechanics) is where di±erential equations began. We will spend the remainder of the lecture considering di±erential equations which arise in biology. Consider a population, and let P = P ( t ) denote its size at some time t . The rate at which this population changes is the derivative P ° ( t ). (Of course, the population is a discrete quantity, but we can approximate it using a continuous function. Hence the derivitate makes sense.) We make a simple assumption that “the rate of change of the size of the population is proportional to the size of the population.”
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Unformatted text preview: Another way to say this is dP dt ∝ P = ⇒ dP dt = r P for some universal constant r that does not depend on the size of the population or the time t . (This constant is positive because our population grows as a function of time.) We will see later that this implies P ( t ) = P e rt where P is the initial size of the population. Unfortunately, this is not really the correct way to model population growth. Indeed, we do not expect population to grow without bound inde²nitely because we must take into account that there are limited resources, so there must be a limiting sustainable size K of the population. If the size P is greater than K we expect there to be a decrease in population because the population has exceeded its resources. Hence, we may assume that...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue.

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