Question

In a certain area of the Louisiana swamp, a population of 2,500 alligators is observed. Given adequate amounts of food and space, their population will follow the Malthusian growth model. After 12 months, scientists observe that there are 3,000 alligators. Alarmed by their rapid growth, the Louisiana Wildlife and Fisheries institutes the following hunting policy for a specialized group of alligator hunters: Hunting is allowed in only an odd-numbered month, and the total number of alligators taken is limited to 80 . Assuming the limit is attained in each month allowed and uniformly over the month, determine a model that gives the population of alligators. Solve that model. How many alligators are there at the beginning of the fifth year? (Assume a population of 3,000 alligators at the beginning of the initial year.)

Answer 1

Using a modified Malthusian growth model to include hunting every odd month, the alligator population growth would start from 3,000 and increase exponentially, with a periodic reduction of 80 alligators. Without the exact growth rate and carrying capacity, we estimate linear growth between data points, which would likely require a logistic equation for precise predictions.

Step-by-step explanation:

To answer the student's question about the population growth of alligators in a Louisiana swamp, we need to establish a model that accounts for both the intrinsic Malthusian growth and the periodic hunting that occurs every odd-numbered month. The Malthusian model posits that populations grow geometrically when resources are abundant, leading to exponential growth. However, when a hunting policy is introduced that removes a fixed number of alligators every odd month, this shifts toward a logistic growth model, as the removal of alligators represents a limiting factor akin to natural resource constraints.

Given that 80 alligators are removed every odd-numbered month, we can adjust our model to subtract this amount from the population at those intervals. To solve for the population at the beginning of the fifth year, we would calculate the population exponential growth for each month and then subtract 80 alligators for every odd month over the four-year period.

Starting with a population of 3,000 alligators at the beginning of the initial year, we would expect the population to grow exponentially but with a periodic reduction of 80 alligators every other month. A simplification of such a model is to consider a net growth rate over each even/odd interval, applying the Malthusian growth for the even months and then subtracting the hunting factor for the odd months.

Without the specific rates of natural increase and carrying capacity of the environment, we can assume linear growth between data points and estimate the population growth rate based on the initial observations. However, this is a simplification and the actual model may require more sophisticated calculations using differential equations and the logistic equation (P(t) = K/[1 + (K - P0)/P0 * e-rt]), where P(t) is the population at time t, K is the carrying capacity, P0 is the initial population, r is the intrinsic growth rate, and e is the base of the natural logarithm. By incorporating the hunting policy into these calculations, the model would provide a more accurate prediction.