Please answer ALL PARTS (a, b, and c). Write legibly for parts a and b and SUBMIT EVERYTHING REQUIRED FOR PART C -- you need MATLAB for part c. Thanks in advance!

If you only know a and b, then you can just solve those and I will repost c as computer science -- thanks!

The Lotka-Volterra equations we investigated in class were based on an assumption that the growth rate of the preys per capita is proportional, which is valid with adequate resource, but not very accurate within an environment of limited resources. In the latter scenario, a logistic growth for the preys is more proper to assume. It then leads to the following modified Predator-Prey model: a'(t) = rx (1 - 0) azy; y (t) = bxy - ky, where, as usual, z(t) is population of the preys; y(t) is population of the predators; a, b,r, k are positive parameters, C is the carrying capacity for the preys (C in place of the conventional choice K to avoid confusion with parameter k). CA (a) Show that if (Assumption I) (the environment cannot carry enough preys to feed predators), the predator will become extinct in the long run. Hint: take good use of the stability analysis result presented in class. (b) Show that, on the contrary, if (Assumption II) c76 then it could happen that the preys and predators co-exist (so no extinction happens). (c) Choose a set of parameters of the model satisfying Assumption II. Draw the Phase portraits, together with at least three solution curves (starting with different initial conditions), of system (1)-(2).1 For the solution curves, choose your initial conditions so that one can see the different curves clearly. Argue how the graph illustrates the theoretic result from part (b)