As populations increase in size, the growth rate of the population tends to slow as the en...

As populations increase in size, the growth rate of the population tends to slow as the environment inhibits the growth rate. The logistic growth curve is one means of trying to model this. This curve is given by a differential equation of the form

where k is a constant, P is the population, and C is the maximum population the environment can sustain, called the carrying capacity of the environment. Notice that if P is small relative to C, the differential equation for logistic growth is approximately the same as the one dP/dt = kP modeling exponential growth since (C – P)/C is close to one. As P approaches C, (C – P)/C is close to zero so that the growth rate in logistic growth is close to zero. Find the logistic growth curve of a population P if the carrying capacity of the environment is 1 trillion and the initial population is 10 billion.