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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 ff erential equations which arise through physics. Indeed, Newtonian Mechanics (a.k.a. Classical Mechanics) is where di ff erential equations began. We will spend the remainder of the lecture considering di ff 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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