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Recall that the Logistic Growth Equation is given by P'=k P (M-P), where M is the carrying capacity of the environment. Suppose a species of fish in a lake is modcled by a logistic population model with relative growth rate of k = 0.05 per year and carrying capacity of M = 10000. For best results in this problem, let P be the number of fish in thousands. Compute M and P. accordingly. a. Write the differential equation describing the logistic population model for this problem b. Use the Geogebra slope field generator or other such tool to sketch the direction field for this model. Draw several solutions (not the slope field) corresponding to initial conditions P(0) = 1, P(0)=3, and P(O)=5. c. If 2500 fish are initially introduced into the lake, solve and find the analytic solution PO) that models the number of fish in the lake after t years.

Recall that the Logistic Growth Equation is given by P'=k P (M-P), where M is the carrying capacity of the environment. Suppose a species of fish in a lake is modcled by a logistic population model with relative growth rate of k = 0.05 per year and carrying capacity of M = 10000. For best results in this problem, let P be the number of fish in thousands. Compute M and P. accordingly. a. Write the differential equation describing the logistic population model for this problem b. Use the Geogebra slope field generator or other such tool to sketch the direction field for this model. Draw several solutions (not the slope field) corresponding to initial conditions P(0) = 1, P(0)=3, and P(O)=5. c. If 2500 fish are initially introduced into the lake, solve and find the analytic solution PO) that models the number of fish in the lake after t years.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Recall that the Logistic Growth Equation is given by P'=k P(M-P), where M is the carrying capacity of the environment. Suppose a species of fish in a lake is modeled by a logistic population model with relative growth rate of k = 0,05 per year and carrying capacity of M = 10000. For best results in this problem, let P be the number of fish in thousands. Compute M and Po accordingly. a. Write the differential equation describing the logistic population model for this problem b. Use the Geogebra slope field generator or other such tool to sketch the direction field for this model. Draw several solutions (not the slope field) corresponding to initial conditions P(0) = 1, P(0)=3, and P(0)=5. c. If 2500 fish are initially introduced into the lake, solve and find the analytic solution PO) that models the number of fish in the lake after t years. d. Use part c to estimate the number of fish in the lake after 5 vears. Add vour solution to vour plot in part b. e. How long will it take for the population to reach 8000 fish?
Transcribed Image Text:Recall that the Logistic Growth Equation is given by P'=k P(M-P), where M is the carrying capacity of the environment. Suppose a species of fish in a lake is modeled by a logistic population model with relative growth rate of k = 0,05 per year and carrying capacity of M = 10000. For best results in this problem, let P be the number of fish in thousands. Compute M and Po accordingly. a. Write the differential equation describing the logistic population model for this problem b. Use the Geogebra slope field generator or other such tool to sketch the direction field for this model. Draw several solutions (not the slope field) corresponding to initial conditions P(0) = 1, P(0)=3, and P(0)=5. c. If 2500 fish are initially introduced into the lake, solve and find the analytic solution PO) that models the number of fish in the lake after t years. d. Use part c to estimate the number of fish in the lake after 5 vears. Add vour solution to vour plot in part b. e. How long will it take for the population to reach 8000 fish?
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