The system is designed to inform us about the future populations for two species that are cooperative (that is both species benefit from interaction).

dx/dt=5x+2xy

dy/dt=4y+3xy

a-Explain in complete sentences how you can determine by reading the differential equa- tions that system is cooperative rather than competitive or a predator-prey model.

b-Find allx-nullclines, and find ally-nullclines.

c-Plot all nullclines, and be sure to indicate the direction of the vectors along each nullcline.

d- If initially whent= 0 we have (x, y) = (2,1), what will happen to the values ofxand y in the long run as t?Show how you used the plot of the nullclines in the previous part to determine your answers.

e-If initially whent= 0 we have (x, y) = (1,3), what will happen to the values ofxand y in the long run as t?Explain how you used the plot of the nullclines in the previous part to determine your answers.

f-Using your plot in (1c), identify the equilibrium solution(s), and indicate whether each is stable, unstable, or semi-stable. You should rely on the graph in your previous answers.

g- Write a paragraph, using complete sentences, summarizing your conclusions and any conjectures that you have about the long-term outcome for the two populations depending on the initial conditions.