1. The California Department of Fish and Wildlife has decided to allow fishing of a certain species of fish in a small l

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1 The California Department Of Fish And Wildlife Has Decided To Allow Fishing Of A Certain Species Of Fish In A Small L 1 (232.86 KiB) Viewed 12 times

1. The California Department of Fish and Wildlife has decided to allow fishing of a certain species of fish in a small lake, but they're not sure how many fishing licenses to allow. They've contacted you, local ecologist and hobbyist fisher, for help. You've determined based off their records that the population P(t) of the fish species can be roughly modeled according to the logistic equation dP dt P = .75(1- :)P. 350 Using your own fishing experience, you expect that the average fisher might catch about 3 of these fish per year (they're a hard catch!). (a) Suppose that L fishing licenses are to be offered. Modify the logistic equation to produce a family of differential equations d = f(P) depending on the parameter L that models the described harvesting scenario. dt (b) Sketch a bifurcation diagram for your family of differential equations. Where, if anywhere, does bifurcation happen? Use your diagram to help you determine what the largest number of licenses LM that can be issued without risk of eventually killing the entire fish population is. Explain why your choice is correct. (c) Use a phase diagram to help you describe the long term behavior of the fish population, assuming that the LM licenses would only be issued once the lake has stabilized at it's maximum fish population. (d) As an ecologist, you realize that your model represents a "best case' environmental scenario. There are many real world factors that can also influence the population of the fish. For example, toxic waste from the local chemical plant might run into the lake if there is a particularly heavy rainstorm, killing off some fish. Given the possibility of unexpected changes in the fish population not accounted for in your model, what do you tell the department? Is it safe to recommend they issue LM licenses, or should they go with less? (Hint: think about how much "buffer room" your model has, and decide how bad of a disaster would be required to happen for the fish population to die off if fishing is allowed.)