Final answer:
The growth rate constant (k) is 0.02, and the carrying capacity, when rounded to the nearest whole number, is 13 thousand individuals. These values are determined by comparing the given logistic growth equation to the standard form and solving for k and the carrying capacity.
Step-by-step explanation:
The student is asking about the logistics model of population growth, specifically the values of the growth rate constant (k) and the carrying capacity in the logistic growth equation dP/dt = 0.02P − 0.0016P².
To find the value for k, we can compare the given logistic equation to the standard form dP/dt = rP(1 − P/K), where r is the intrinsic growth rate and K is the carrying capacity. By matching coefficients, we get k = 0.02, which is the coefficient of P in the equation, and this value is already rounded to four decimal places as requested.
To find the carrying capacity, we look at the coefficient of P², which will be r/K. Therefore, we can solve for K using the equation K = r/0.0016. Plugging r = 0.02 into the equation gives us K = 0.02/0.0016 = 12.5, which rounds to the nearest whole number as 13 thousand individuals. This represents the largest population size that the environment can sustain without further growth.