TnRchnedrD tai
Over >
Jvdn< 0
Figure 25 Flue ndjusl €xprnlinin". toward the line @oun=u
Iblt *
(Jo: Jo). then Blue will change expenditures bringing them closer to its optimal line. In this section, we will investigate the effects of these motions and the possibility of stabilizing the non-military expenditures. This has meaning for the model, with these variables being inputted. We consider the behavior of solutions of the Richardson model when $ and v are nonnegative. Only the portion of the graph relating $ and v that lies in the first quadrant is considered.
D: Mutual Grievances
inteu lc
In this situation, each side has a permanent underlying grievance against the other side. Mathematically, this model assumes that the parameters both positive. Suppose that Blue and Red are completely disarmed, with the initial expenditure level at (0, 0). According to the Richardson model, the rates of change of expenditures at this instant would be:
Nu-M+97>
z6-n0+5=,>0
(21)
This means that each nation would start arming itself. Can this system be stable? In the case where both positive, the point of stability (r, y) will lie in either the first quadrant or the third quadrant (see Exercise 18). We will start with the case where (x, y) lies in the first quadrant. Then the lines and will be as pictured in Fig; 2.6. These lines split the first quadrant into four regions. Label them counterclockwise as I, II, III, and IV, with the origin in region I. If the initial armament expenditures are at (0, 0), then as we have seen, both and will increase. The net result will be to move the expenditures toward the point (x1, y1) deeper.