- #1
alane1994
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Hey MHB, I have this rather large and convoluted project that consists of two problems. I have finished the first problem that consisted of 4 parts. I am working through the second part right now. It has a preamble that is kinda important. I will post that as well as the problems that I am working on.
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)
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.
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!
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)
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.
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!
