I'm trying to the solve the following system of differential equations:
eq (1) dS/dt = (1/10)S-(1/20)SN
eq (2) dN/dt = (1/100)N-(1/100)N^2-(1/100)SN
I want to plot the phase space (phase plane) sol's , I have determined that the phase plane and N and S have equilibrim points at N=2, and N+S=1, these divide the graph into three regions, the area above N=2 is labeled region III with N(dot) <0, and S(dot)<0 the region below N=2 and to the right of N+S=1 and above the S axis is called region II with N(dot)<0, and S(dot)>0, the region to the left of the line N+S=1 and (0,0) is region I with N(dot)>0, and S(dot)>0.
I want to use Mathematica to show that 1. the lines N=2 and N+S=1 divides the first quadrant into three regions in which dS/dt and dN/dt have fixed signs as I have stated. 2. show that every solution of S(t),N(t) of (*) which starts in either region I or region III must eventually enter region II 3. show what every solution S(t),N(t) of (*) which starts in region II must remain in there for all future time. 4. conclude that based on part 3 that S(t) as it approaches infinity for all solutions of S(t), N(t) of (*) wich S(t sub o) and N(t sub o) positive. conclude too that N(t) has a finite limit (< or = to 2) as t approaches infinity.