6.5Logistic Growth ModelYearsBearsGreg Kelly, Hanford High School, Richland, Washington
We have used the exponential growth equationto represent population growth.0ktyy e=The exponential growth equation occurs when the rate of growth is proportional to the amount present.If we use Pto represent the population, the differential equation becomes:dPkPdt=The constant kis called the relative growth rate./dP dtkP=→
The population growth model becomes:0ktPP e=However, real-life populations do not increase forever. There is some limiting factor such as food, living space or waste disposal.There is a maximum population, or carrying capacity, M.A more realistic model is the logistic growth modelwhere growth rate is proportional to both the amount present (P) and the fraction of the carrying capacity that remains:MPM-→
The equation then becomes:dPMPkPdtM-=Our book writes it this way:Logistics Differential Equation(29dPkP MPdtM=-We can solve this differential equation to find the logistics growth model.