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# Consider the differential equation$\frac {dP}{dt} = 0.08P (1 - \frac {P}{1000}) - c$as a model for a fish population, where $t$ is measured in weeks and $c$ is a constant.(a) Use a CAS to draw direction fields for various values of $c.$(b) From your direction fields in part (a), determine the values of $c$ for which there is at least one equilibrium solution. For what values of $c$ does the fish population always die out?(c) Use the differential equation to prove what you discovered graphically in part (b).(d) What would you recommend for a limit to the weekly catch of this fish population?

## a) see graphb) When c is greater than 20, the asymptote is always at 0 while that isn't the case when c is less than 20.c) When $c \in(-\infty, 20),$ there are two equilibriums.d) The weekly limit for catching fishes must be $20 .$

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Differential Equations

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