Logistic growth in India The population of India was 435 million in 1960 (t=0) and 487 million in 1965 (t=5). The projected population for 2050 is 1.57 billion.
b. Use the solution of the logistic equation and the 2050 projected population to determine the carrying capacity.

Verified step by step guidance

1. Recall the general form of the logistic growth model: \[ P(t) = \frac{K}{1 + Ae^{-rt}} \] where \(P(t)\) is the population at time \(t\), \(K\) is the carrying capacity, \(r\) is the growth rate, and \(A\) is a constant related to initial conditions.
2. Use the initial condition at \(t=0\) (year 1960) where \(P(0) = 435\) million to express \(A\) in terms of \(K\): \[ 435 = \frac{K}{1 + A} \implies A = \frac{K}{435} - 1 \]
3. Use the population at \(t=5\) (year 1965), \(P(5) = 487\) million, and substitute \(A\) into the logistic equation: \[ 487 = \frac{K}{1 + \left(\frac{K}{435} - 1\right) e^{-5r}} \] This equation relates \(K\) and \(r\).
4. Use the projected population for 2050, which corresponds to \(t=90\) (since 2050 - 1960 = 90), where \(P(90) = 1570\) million (1.57 billion). Substitute into the logistic model: \[ 1570 = \frac{K}{1 + \left(\frac{K}{435} - 1\right) e^{-90r}} \]
5. Now you have two equations involving \(K\) and \(r\) from steps 3 and 4. Solve this system of equations simultaneously to find the carrying capacity \(K\).

Key Concepts

Logistic Growth Model

The logistic growth model describes population growth that starts exponentially but slows as it approaches a maximum limit called the carrying capacity. It is represented by a differential equation where growth rate decreases as population nears this limit, modeling realistic constraints like resources.

Carrying Capacity

Carrying capacity is the maximum population size that an environment can sustain indefinitely given available resources. In the logistic model, it is the horizontal asymptote that the population approaches over time, representing the stable equilibrium point.

Using Logistic Equation Solutions to Estimate Parameters

Solving the logistic differential equation yields a formula for population over time involving parameters like growth rate and carrying capacity. By substituting known population values at specific times, one can solve for unknown parameters such as the carrying capacity.