(b) The ODE in (a) is of the form $x' = f(x,h)$. Plot $x'$ vs $x$ for $h = 0$, clearly labeling all roots and the absolute maximum of the graph. Plot the phaseline, and state all equilibria and their stability. Assuming the initial population is nonzero, does the population ever become extinct?
(c) Using graph graph in (b), plot $x'$ vs $x$ for $h < 1/4$. Plot the phaseline, and state all equilibria and their stability. Assuming the initial population is nonzero, does the population ever become extinct?
(d) Using graph graph in (b), plot $x'$ vs $x$ for $h > 1/4$. Plot the phaseline, and state all equilibria and their stability. Assuming the initial population is nonzero, does the population ever become extinct?
(e) If the fish population in a lake has a growth rate of 0.2 per month, with a carrying capacity of 40 thousand, and is initially populated at its carrying capacity, how many fish can be harvested per day at most to assure that the population will not become extinct? What assumption did you make?