lecture08

# lecture08 - Sections 3.2 and 3.3 Population Modeling and...

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Sections 3.2 and 3.3 Population Modeling and Newton’s Law of Cooling Feb 10, 2009 Population Modeling and Newton’s Law of Cooling

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Announcements A Summary of This Session: (1) Modeling population growth, exponential (Malthusian) and logistic models; (2) Modeling temperature using Newton’s law of cooling. Population Modeling and Newton’s Law of Cooling
Population growth/decay When we model population growth, the simplest model is the exponential (or Malthusian) model. Basic ideas: P = P ( t ) = population size as a function of time. (1) Rate of population growth is proportional to the difference between the birth rates (inflow) and death rates (outflow) (2) Birth rate and death rate are each proportional to the population size P ( t ) at time t . Equation: dP dt = bP - dP . Here we assume that b and d are constants.In fact, we let k = b - d . So the equation becomes dP dt = kP . Population Modeling and Newton’s Law of Cooling

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Population growth/decay, cont’d k > 0: exponential growth k < 0: exponential decay How to solve dP dt = kP subject to P (0) = P 0 (initial condition) Answer: Separation of variables. dP P = kdt So integraldisplay dP P = integraldisplay kdt Therefore ln P = kt + C Population Modeling and Newton’s Law of Cooling
Population growth/decay, cont’d P = Ce kt Since P (0) = P 0 , one gets C = P 0 , or P ( t ) = P 0 e kt So all we need for this model are the values of k and P 0 .

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