Portfolio Type 2 Fish numbers
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We were first given the equation where is the growth factor and is the population at year. A logistic model is one that grows and then stabilizes over time. This will be shown in the following set of problems. 1. The biologist estimates that if 10,000 fish were introduced into a lake, then the population of the fishes would increase by 50% the first year but then level off and never exceed 60,000 fishes. Therefore we are given a set of initial and eventual numbers that we can assign variables to: Thus we can fill the ordered pairsand these points are indicated on the graph below: The window and table values for the graph above are: The window of the graph shows the scale of the graph and the values give evidence for the points on the graph. The slope of the line with coordinatesis: Therefore is the linear growth factor in terms of with a slope, 2. Since and we figured , we can derive: . 3. In order to determine the fish population over the next 20 years, we can use the logistic function model found in 2 and graph this.
Like the graph with an initial growth rate 2, all three graphs are logistic graphs. 5. When 2.9: Coordinates: Also, its graph and table values are shown below: The peculiar outcome of the graph is noted and can be accredited to higher initial growth rate which makes it harder for the fish population to stabilize. From the table values one can observe that after the 14th year, the fish population fluctuates from 41910 and 70720 because of the higher growth rate. Looking past 20 years the population continues to fluctuate never seeming to stabilize. 6. Realizing the fish population has stabilized after 20 years, if the biologist and regional managers would like to see the commercial possibility of an annual 5000 harvest value the difference equation would then be The graph, and table values would be the graph shown below: Observing the graph and its values, one can see that the fish population will decrease and then slowly stabilize at a fish population of 50000 which is proved in the table value of the above graph below: After the 39th year, we can see that the fish population stabilizes at 50000.
8. Similar to the difference equation in 6, since the harvest size in unknown we replace it with a variable for the harvest size,, and in order to achieve stability and will be equal therefore can be replaced byin the below equation: Since the equation inside the square root must be greater than zero to satisfy the equation, we can solve for the highest value of by setting the value inside the square root having to be greater than zero. Therefore we can conclude that the maximum annual sustainable harvest is when 9. Knowing that , we are able to use different values of other than 60000. I chose initial populations of 20000 and 40000 because both show that with a harvest size of 8000 they become steady well before reaching 60000 fish. When When But, since these values both lead us to stable populations we must find the initial population which does not become stable. This is shown when the initial population is 19999 in the graph below: Thus, we can conclude that the politicians will have to wait until the population reaches 20000 before an annual harvest of 8000. They will only have to wait 2 years.
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