MAT136H1 Lecture Notes - Relative Growth Rate, Logistic Function, Differential Equation
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MAT136H1 Full Course Notes
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Differential equation can be used to model population growth based on the law of natural growth, logistic equation and a few other applications. Assumption: population grows at a rate proportional to the population size: , where ( ) is the value of at time , and is a constant. For , population increases; for , population decreases. Taking integral of both sides of the differential equation: representing an arbitrary constant. Thus, ( ) , meaning the initial population. Rewriting the equation means the relative growth rate is constant. Sometimes population grows exponentially initially then levels off. After the carrying capacity of , the population growth rate becomes negative, thus it represents maximum population environment is capable of sustaining in the long run. ) this is known as the logistic differential equation. Rearranging, then taking integral of both sides: (
