You are a consultant who has been hired to help the province of Nova Scotia establish regu- lations for commercial cod fishing in their territorial waters. The problem is that you never actually know how many fish are out there or which species will dominate in a given year. People who depend on fishing for a living depend on you to set the limits high enough so they can make a decent living, but, if you set the limts too high, there is the danger that the fishery could collapse and the cod become extinct. In this exercise you will examine the logistic equation, an equation that models the fish population with no human intervention, and then compare it with two models for harvesting the fish, the constant-harvest model and the proportional harvesting model. You will then make a recommendation to the Novia Scotia Fisheries Commission about the benefits and problems with each of the models. You will then explore the role of bifurcation in this scenario in question 6. Population Models If P(t) denotes the size of a population at some at time t, a model for exponential growth dP is = kP where k is a constant and k > 0. This is not a very realistic model as ex- dt tended exponential growth is not sustainable over the long run due to limited environmental resources. To account for the expectation that model such as dp/dt P dp/dt P will decrease as P increases in size, we can try a d dt = f(P) or (1) The model given by (1) assumes that the rate at which the populations grows (or declines) is dependent only on the population present and not on any time-dependent mechanisms such as seasonal variations. This is called a density dependent hypothesis. dP dt = Pf(P). The logistic equation supposes that the environment is capable of supporting no more than a fixed number of individuals K. The quantity K is called the carrying capacity of the environment. If f(0) = r and f(K) = 0, then assuming a linear relationship, equation (1) can be written as P (-2). K = Pr 1 (2) The solution of this equation can be found by separating the variables, and using partial fractions to integrate the left hand side. The complete solution can be found in our textbook. Questions to be uploaded to Gradescope (a) Logistic Equation i. Without human intervention, the population of cod can be modeled by the logistic equation, equation (2). Use the values r = 1 and K = 4900 find to the critical points of this autonomous DE. You will need to solve a quadratic equation, labeling the roots P+ and P_ will prove helpful later on. ii. Use a phase line analysis (a one dimensional phase portrait) to give the stability of each critical point. iii. For each region on the phase line that you found in part (a) above, sketch a typical solution curve. (b) Constant Harvest Model: A Specific Case In the constant-harvest model fisher- men are allowed a constant take, regardless of the time required for that. The parameter h indicates the constant harvesting rate 2- +P (1-P)- K dP (3) i. Our goal is to understand what happens to the equilibrium values as h increases. For r = 1 and K = 4900 find the equilibrium values, P and P_ of equation (3) for each of the values h = 300, 600, 900, and 1225. - h h> 0. dt ii. For each of the values of h above draw a phase diagram and classify the stability of the equlibrium values P and P_. iii. What happens to P and P. as h increases? iv. A. For which value of h is there only one critical point? Draw a phase line for this case. B. What does this mean in terms of the sustainability of the fishery? C. What fraction of the carrying capacity is the h-value that yields a single critical point? (c) Proportional Harvesting Model In the proportional harvesting model, the amount of fish a fisherman can take is pro- portional to P, the population present. Thus only a fraction of the fish are allowed to be harvested every year. If the constant of proportionality is a, where 0 a 1, then we can write, dP =rP (1-2)-ap K h > 0. i. Compute the critical points of the proportional harvesting model in equation (4) and use a phase line analysis to classify their stability. You can let r= 1 and keep K = 4900. ii. What happens to the cod population as a increases? iii. Does the Proportional Harvesting Model predict extinction? If so, for what values of P and a? (d) Conclusions Now it is time to prepare your report and suggest a policy for cod fishing to the Nova Scotia Department of Fisheries. Which would you recommend and why? (e) Bifurcations The model dealing with constant harvesting provides an example of bifurcation. A bifurcation is essentially a dramatic change in the qualitative structure of the phase line, such as the appearance or disappearance of a critical point. A critical point of an autonomous DE ' = f(x) is said to be hyperbolic if small pertubations of f(x) do not change the qualitative structure of the DE and nonhyperbolic if arbitrary small changes do change the nature of the system. A bifurcation diagram for a family of DEs is a graph that shows the location and stability of the critical points for each parameter, in our case h the harvesting rate. Consider our example r = = 1 and K 4900. The critical points satisfy = dP dt P ( - / / ) - K = TP 1 - h = 0, and these can be solved using the quadratic formula. i. Plot the graph of P vs h from (5) for 0