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#### alane1994

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- Oct 16, 2012

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**Harvesting a Renewable Resource**

*Suppose that the population \(y\) of a certain species of fish (e.g., tuna or halibut) in a given area of the ocean is described by the logistic equation*

$$\displaystyle\frac{dy}{dt}=r(1-\frac{y}{K})y$$

If the population harvested is subjected to harvesting at a rate \(H(y,t)\) members per unit time, then the harvested population is modeled by the differential equation

$$\displaystyle\frac{dy}{dt}=r(1-\frac{y}{K})y-H(y,t)$$

Although it is desireable to utilize the fish as a food source, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level and possibly even driven to extinction. The following problems explore some questions involved in formulating a rational strategy for managing the fishery.

Problem 1)

$$\displaystyle\frac{dy}{dt}=r(1-\frac{y}{K})y$$

If the population harvested is subjected to harvesting at a rate \(H(y,t)\) members per unit time, then the harvested population is modeled by the differential equation

$$\displaystyle\frac{dy}{dt}=r(1-\frac{y}{K})y-H(y,t)$$

Although it is desireable to utilize the fish as a food source, it is intuitively clear that if too many fish are caught, then the fish population may be reduced below a useful level and possibly even driven to extinction. The following problems explore some questions involved in formulating a rational strategy for managing the fishery.

This problem was based upon effort involved in harvesting. I can post this question along with my work for it if you guys are interested.

Problem 2)

This is the problem that I am currently working on.

*Constant Yield Harvesting*

*In this problem, we assume that fish are caught at a constant rate \(h\) independent of the size of the fish population, that is, the harvesting rate \(H(y,t)=h\). Then\(y\) satisfies*

\(\displaystyle\frac{dy}{dt}=r(1-\frac{y}{K})y-h~~~~~~~(ii)\)

The assumption of a constant catch rate \(h\) may be reasonable when \(y\) is large but becomes less so when \(y\) is small.

(a) If \(h<rK/4\) shows that Eq.(ii) has two equilibrium points \(y_1\) and \(y_2\) with \(y_1<y_2\); determine these points.

\(\displaystyle\frac{dy}{dt}=r(1-\frac{y}{K})y-h~~~~~~~(ii)\)

The assumption of a constant catch rate \(h\) may be reasonable when \(y\) is large but becomes less so when \(y\) is small.

My work,

\(\displaystyle f(y)=\frac{dy}{dt}=r(1-\frac{y}{K})y-h\)

\(\displaystyle ry(1-\frac{y}{K})-h=0\)

\(\displaystyle -\frac{ry^2}{K}+ry-h=0\)

\(\displaystyle y^2-yK+\frac{hK}{r}=0\)

\(\displaystyle y^2-yK=-\frac{hK}{r}\)

\(\displaystyle y^2-yK+\frac{K^2}{4}=\frac{K^2}{4}-\frac{hK}{r}\)

\(\displaystyle (y-\frac{K}{2})^2=\frac{K^2}{4}-\frac{hK}{r}\)

\(\displaystyle y-\frac{K}{2}=\sqrt{\frac{K^2}{4}-\frac{hK}{r}}\)

\(\displaystyle y=\frac{K}{2}\pm\sqrt{\frac{K^2}{4}-\frac{hK}{r}}\)

(b) Show that \(y_1\) is unstable and \(y_2\) is asymptotically stable.

This part is giving me some trouble. I think I would just calculate the second derivative of the original diff.eq and then you would plug in the two points found just above right?

Any help would be appreciated!