Consider the following system of differential equations.

=()+() (1) =()+() (2)

Assume that >0,<0,0<<1, 0<<1

Answer the following questions in brief. Please show all necessary calculations. Ignore any steady states where both , are equal to zero.

a. Derive for each isocline and show their signs.

b. Draw each isocline on a separate diagram. In each graph, plot () on the vertical axis and () on the horizontal axis. Be careful to label the isoclines.

c. On the graph of each isocline, draw the arrows of motion. Show calculations.

d. Now bring the two isoclines together on the same graph. If there are any stable arms, draw them.

e. Suppose now that we assume >0 and <0. Other parameters have same definitions as before. Quickly show what type of equilibrium you will have. Your reasoning/method must be different from the steps followed in parts (a) - (d) above.

[(5+5) + (5+5) + (5+5) + 5 + 5= 40 points]