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(Logistic growth with periodic harvesting) The equation dp

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QUESTION:

(Logistic growth with periodic harvesting) The equation

\(\frac{d p}{d t}=k p\left(1-\frac{p}{N}\right)-a(1+\sin b t)\)

is a non-autonomous equation that considers periodic harvesting. What do the parameters \(a\) and \(b\) represent? Let \(b=1\) . If \(a=a_1\), What will happen to the fish population for various initial conditions?

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QUESTION:

(Logistic growth with periodic harvesting) The equation

\(\frac{d p}{d t}=k p\left(1-\frac{p}{N}\right)-a(1+\sin b t)\)

is a non-autonomous equation that considers periodic harvesting. What do the parameters \(a\) and \(b\) represent? Let \(b=1\) . If \(a=a_1\), What will happen to the fish population for various initial conditions?

ANSWER:

Step 1 of 3

Consider the given equation,

\(\frac{d P}{d t}=k P\left(1-\frac{P}{N}\right)-a(1+\sin (b t))\)

Consider the formula corresponding to the “Logistic Growth with Periodic Harvesting model” as,

\(\frac{d P}{d t}=k P\left(1-\frac{P}{N}\right)-h\)

Here, h is the number of fish harvested each year due to fishing, \(P(t)\) is the fish population at any time \(t\), \(k\) is the growth rate, and \(N\) is the carrying capacity.

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