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Portfolio Type 2 Fish numbers

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        We were first given the equation image00.pngwhere image01.pngis the growth factor and image12.pngis the         population at yearimage22.png.  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 pairsimage39.pngand 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 coordinatesimage02.pngis:


        Therefore        image04.png

image05.pngimage06.pngis the linear growth factor in terms of image07.pngwith a slope,                               image08.png

  1. Since image09.pngand we figured image10.png, we can derive:


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                        Coordinates: image20.png




                When image19.png2.3:

                        Coordinates: image25.png




                When image19.png2.5:

                        Coordinates: image29.png




        Their respective graphs in order when image19.png2, 2.3, and 2.5 are shown below:



        All three graphs vary initially but over time level off and become stable.  Like the graph         with an initial growth rate image19.png2, all three graphs are logistic graphs.

5.  When image19.png2.9:

                        Coordinates: image37.png




        Also, its graph and table values are shown below:image43.pngimage42.png

        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.

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image53.pngand an annual         harvest of 9300.
  1. Similar to the difference equation in 6, since the harvest size in unknown we replace it with a variable for the harvest size,image54.png, and in order to achieve stability image07.pngand image55.pngwill be equal therefore can be replaced byimage56.pngin 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 image54.pngby 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         image59.png

  1. Knowing that image60.png, we are able to use different values of image07.pngother 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 image61.png


                When image64.png


        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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