2. Consider the nonlinear ODE dN N = N 1 H N > 2.1 as a model of a shery. If H = 0, we assume the sh population N (t) [individuals] of a certain species, at time t [time], grows according to the logistic equation, which was discussed in the lectures. Harvesting sh is modelled by adding the term H to the logistic equation, which has the interpretation that the sh are harvested (and therefore the rate of change of population is decreased) at a constant rate H [individuals - time1]. (a) Plot the graph of N (on the horizontal axis) vs. dN/dt (on the vertical axis), for three different values of H that give qualitatively different solution behaviours. For each of these three different values of H, nd all equilibria (in terms of the parameters H, 'r, K) that are relevant for the model, determine the linearized stability of each such equilibrium, and sketch the phase portrait. Determine the critical value Hc where a local bifurcation occurs, and classify this bifurcation. Interpret the results of your analysis from parts (b) and (c), in terms of how the rate of harvesting affects the possibility of extinction of the population of sh. Explain what could happen (according to this model) if sheries managers thought the optimal rate of harvesting is He (since this theoretically gives the maximum harvest over the long term) and shers exceeded this rate of harvesting, even slightly