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Unformatted text preview: Lecture 18 Derivatives and rates of change Rates of change in the natural sciences Population models We now consider the role of rate of change in modelling population growth. In what follows, we let n = f ( t ) (1) denote the number of individuals in an animal or plant population at time t . The change in population between times t 1 and t 2 > t 1 is given by n = f ( t 2 ) f ( t 1 ) . (2) The average rate of change of the population over the time interval [ t 1 ,t 2 ] is defined as n t = f ( t 2 ) f ( t 1 ) t 2 t 1 . (3) If we assume that the population is growing, i.e., f ( t 2 ) > f ( t 1 ) for all t 1 < t 2 , then the phrase average rate of change may be replaced by average rate of growth. That being said, we could ignore whether the population is growing or decaying and simply call the ratio in (3) the average rate of growth, with the understanding that negative growth implies decay. We now proceed as we have done before, letting t 2 t 1 , to yield the instantaneous rate of growth at t = t 1 , lim t n t = dn dt = f ( t 1 ) . (4) There is one point that should be addressed here. In the above discussion, we have simply proceeded as in the past, where the dependent variable n is treated as a continuous real number. Obviously, when we talk about populations, we are using integers to count the number of individuals there are no half-organisms. When the changes n to the population for small time intervals t are very small in comparison to the population itself, i.e., when the ratio n n (5) is very small, often written as follows, n n << 1 , (6) 125 then these changes may be viewed as continuous. For example, when one individual is added to a family of four, the percentage change in the number of people in the family is large, namely, a 25% increase. When one individual is added to a city of 100,000, the percentage change is very tiny and may be considered as infinitesimal. In the discussion that follows, we assume that sufficiently large populations are being modelled so that n may be considered as a continuous variable. Example: A culture of bacteria in a Petri dish Suppose that we have a culture of bacterial in a Petri dish, with sufficient nutrient in the dish for the bacteria to multiply over a number of generations. Experimentally, in such cases of infinite resources, it is observed that the population of the bacteria doubles after a roughly fixed time interval T . Let us assume that at time t = 0, there is a population of n (0) = n (7) bacteria in the dish. For simplicity, lets measure time in units of T this is essentially a rescaling of time. This means that the populations at future times t = k , k = 0 , 1 , 2 , , are given by n (0) = n n (1) = 2 n n (2) = 2 f (1) = 2(2 n ) = 2 2 n ....
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