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Question: 1 2 pts warmup consider the onedimensional autonomous differential equation...

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1. (2 pts) warm-up: Consider the one-dimensional autonomous differential equation y-y y 1) y 2 Sketch e function f(y)-y(y 1)(y 2) in the plane, and provide a drawing of the phase line associated with the given ODE We will now discuss how adding in a parameter will affect our phase line. For the purposes of this activity, we will consider the one-dimensional autonomous ODE y- f(y), and we will refer to the function f(u) as the feedback function. However, we will now consider how adding a parameter will affect our ODE. Often, as we vary a parameter, there will be particular values of this parameter that will meaningfully alter our solution behavior. In the context of one-dimensional equations, this means that equilibria will be created or destroyed. These points where such changes occur are called bifurcation points. For a basic example of this consider the ODE y-y2-α, where a is a parameter. Note that our particular choice of α will affect how our feedback function looks, and thus how many equilibria we have. For example, if α > 0, well have two equilibria-one sink and one source. If α 0, we will have one node. If α < 0, we will have no equilibria. Due to this drastic change, we would call the value a - a bifurcation value. Finding bifurcation values will often rely on being able to sketch the parameterized feedback function and the phase line associated with it. We will show the power of bifurcation analysis via a specific example. Suppose that the population of a certain trout in a certain lake is given (in some units, dont worry about it) by the logistic model Here, p(t) >0 is the population of fish at time t, and 4 represents the carrying capacity of the lake. 2. (2 pts) Provide a sketch of the feedback function (p)-p(4-p), and the associated phase line. Make sure to classify all equilibria as sinks, sources, or nodes.
Suppose we want to introduce some amount of legal fishing in the lake. With fishing, p(t) will satisfy the parameterized differential equation where c is a positive constant that quantifies how much fishing is allowed. Note that if we set c too high, the population of trout will be decimated and will quickly shrink to zero. 3. (4 pts) What is the largest sustainable fishing level - that is, how large can we make c without causing the fish population to decrease to 0? What is the long-term trout population with this level of fishing? (Hint: I you dont know where to start, try drawing the feedback function for some specific values of c. What does the phase line look like for these values of e? Look for significant changes in solution behavior.) 4. (2 pts) Suppose that we have made a mistake and set c too high. The trout population in the lake is decimated. and is extremely close to zero. We want to scale back fishing levels to allow the fish population to recover Reducing c to the level you found in the previous question will not fix the problem- we have drastically reduce or possibly even ban fishing, at least for a little while. Explain this fact, using phase lines for a few different values of c as evidence.
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