Question

A slope field for the logistic differential equation dy/dt=(1/2500)y(1010−y) is shown in the figure below.

a) Sketch the solution curve that passes through the point (0,250), and sketch the solution curve that passes through the point (0,1500).
b) If b=250, what is the largest rate of increase in the number of frogs in the pond? Explain your reasoning, and indicate units of measure.
c) If b=750, what is the largest rate of increase in the number of frogs in the pond? Explain your reasoning, and indicate units of measure.
d) Let b=100. Use Euler’s method, starting at t=0 with two steps of equal size, to approximate F(4), the number of frogs in the pond at time t=4 weeks. Show the work that leads to your answer.
e) Determine whether the approximation found in part (d) is an underestimate or an overestimate for the number of frogs in the pond at time t=4 weeks, when b=100. Explain your reasoning.

Answer 1

The maximum frog population growth rate occurs at half of the carrying capacity. Using Euler's method, an approximation of the population can be made, but it will be an underestimate as it doesn't capture the initial accelerated growth of the logistic curve.

Step-by-step explanation:

The logistic differential equation dy/dt=(1/2500)y(1010−y) describes the rate of change of a population over time, subject to both natural growth and limitations due to resources or other factors. It is representative of the logistic population growth model, often visualized as an S-curve on a graph.

The largest rate of increase for the number of frogs in the pond would occur when the population is at half of its carrying capacity, which is when y = 505 frogs. For a population that starts with b = 250 frogs, this maximum growth rate occurs before the population reaches 505, thus the rate is less than the absolute maximum. When b = 750, the population surpasses half of the carrying capacity, and the maximum rate of increase would have already occurred at the point y = 505.

Using Euler's method for an initial population of b = 100 frogs, with a time step of 2 weeks, we can approximate F(4), the number of frogs after 4 weeks. However, since Euler's method provides a linear approximation and the population grows according to an S-curve, the estimate will be an underestimate of the actual population at t = 4 weeks as it does not account for the accelerating growth in the initial phase of logistic growth.