fish hatchery employed a mathematician to design a model to predict the -opulation size of fish that the hatchery can expect to find in their pond at any iven time. The mathematical model that the mathematician created is: (1-25) a) Draw a one dimensional phase portrait of the autonomous differential equa- tion. What does this differential equation predict for future fish populations for various initial conditions? Describe the various cases in a few sentences interpreting your one-dimensional phase diagram. dP dt = 2P

Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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1. A fish hatchery employed a mathematician to design a model to predict the population size of fish that the hatchery can expect to find in their pond at any given time. The mathematical model that the mathematician created is:

\[ \frac{dP}{dt} = 2P \left(1 - \frac{P}{25}\right) \]

(a) Draw a one-dimensional phase portrait of the autonomous differential equation. What does this differential equation predict for future fish populations for various initial conditions? Describe the various cases in a few sentences interpreting your one-dimensional phase diagram.

(b) Solve the differential equation. Does your solution support your answer to part (a)? Why or why not?
Transcribed Image Text:1. A fish hatchery employed a mathematician to design a model to predict the population size of fish that the hatchery can expect to find in their pond at any given time. The mathematical model that the mathematician created is: \[ \frac{dP}{dt} = 2P \left(1 - \frac{P}{25}\right) \] (a) Draw a one-dimensional phase portrait of the autonomous differential equation. What does this differential equation predict for future fish populations for various initial conditions? Describe the various cases in a few sentences interpreting your one-dimensional phase diagram. (b) Solve the differential equation. Does your solution support your answer to part (a)? Why or why not?
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