Assume that the growth of peacocks obey the rule dP : p³ – 17P² – 38P dt - Here P represents the population of peacocks and t represents the time variable. (a) Find all biologically reasonable equilibrium points. (b) Determine the stability of each biologically reasonable equilibrium. (c) Draw a phase-line diagram. (d) If there are initially 19 peacocks, how many will there be eventually?

A second-order chemical reaction involves the interaction of onc molccule of a substance P with one molecule of another substance Q to produce one molecule of another substance X. Suppose that p and q are the initial concentrations of P and Q and let r(t) be the concentration of X at time t. Then p- I(t) and q -r(t) are the concentrations of P and Q at time t, and the rate at which the reaction occurs is given by the equation dr *(I - b)(I – d)o= dt where a is a positive constant. Solve the initial value problem with r(0) = 0 assuming p # q. Answ: r(t) = Pq(e®(q-p)t _1 qea(q-p)t-p , you may nced to manipulate your answer to get this.

1) Find all equilibrium solutions of the equation (1 − x) (x² − 4) - x = and classify each one in terms of stability. Draw a phase space diagram and sketch by hand several typical solution curves. Describe the long term (t → ±∞) behavior of the solutions.

1. Sketch the nullclines in the xy phase plane, identify steady states, find the linearized system at each steady state and determine stability of each steady state. (a) dx = y² - x°, dt dy = x-1. dt