Although the model in Prob. 20.4 works adequately when population growth is unlimited, it...

Although the model in Prob. 20.4 works adequately when population growth is unlimited, it breaks down when factors such as food shortages, pollution, and lack of space inhibit growth. In such cases, the growth rate is not a constant, but can be formulated as

kg = kgm (1−P/P max )

where kgm = the maximum growth rate under unlimited conditions, p = population, and pmax = the maximum population. Note that pmax is sometimes called the carrying capacity. Thus, at low population density p « pmax, kg → kgm. As p approaches pmax, the growth rate approaches zero. Using this growth rate formulation, the rate of change of population can be modeled as

This is referred to as the logistic model. The analytical solution to this model is

Simulate the world’s population from 1950 to 2050 using (a) the analytical solution, and (b) the fourth-order RK method with a step size of 5 years. Employ the following initial conditions and parameter values for your simulation: p0 (in 1950) = 2,555 million people, kgm = 0.026/yr, and pmax = 12,000 million people. Display your results as a plot along with the data from Prob. 20.4.