Question

#19 all parts

Problems 17 through 19 deal with competitive systems much like those in Examples 1 and 2 except that some coefficients depend on a parameter a. In each of these problems, assume that x, y, and a are always nonnegative. In each of Problems 17 through 19: (a) Sketch the nullclines in the first quadrant, as in Figure 9.4.5. For different ranges of a your sketch may resemble different parts of Figure 9.4.5 (b) Find the critical points (c) Determine the bifurcation points (d) Find the Jacobian matrix J, and evaluate it for each of the critical points (e) Determine the type and stability property of each critical point. Pay particular attention to what happens as α passes through a bifurcation point. (f) Draw phase portraits for the system for selected values of α to confirm your conclusions. ya -y-0.5x) 17, dx/dr-x--, dy/dt 18. dz/d-x(1-x-y), dy/dt y (0.75- y-0.5x)

Answer 1

19. x(1-x)-xy x, Let a 0.2 Then x' = x(1-x)-xy x-nullcline: is obtained by setting x 0, which gives x(1-x)-xy = 0 or, x(1-x-y) = 0 Two lines (i) x = 0, (ii) 1-x-y 0 Below we have plotted both the nullclines.

Explanation of x-nullclines Observe that (i) when + y > 1 and χ > 0 that is a point like (1,1), then x 0, that is the solution moves to the left. x(1-x-y)-0 15 () when x +y<1 and x > 0 that is a point like (0.5,0), then x >0, that is the solution moves to the right. 0.5 x+y-1 X-0 0.5 0.5 1.5 x-nullclines

In the region y<o.2+0.6x, dy/dt>o thus the solution increases in this region, that is at a point like (a,b), the solution increases. Solution moves Chart Title 4.5 c,d) 3.5 upward. 0.2-y + 0.6x = 0 In the region y>0.2+0.6x, dy/dt<o thus the solution decreases in this region, that is at a point like (c, d), the solution decreases. 2.5 Solution moves downward. 0.5 y=0 0 4 y-nullcline

y-0, y[o.2-y 0.6x]- 0 y-nullcline: is obtained by setting which gives Two lines (i) y-0, (ii) 0.2 -y 0.6x 0.2(1 +3x) -y-0 In the region y<0.2+0.6x, dy/dt>0 thus the solution increases in this region, that is at a point like (a,b), the solution increases. Chart Title 3.5 0.2 - y + 0.6x - (0 In the region y>0.2+0.6x, dy/dt<0 thus the solution decreases in this region, that is at a point like (c, d), the solution decreases. 2.5 1.5 (a,b) 0.5 V-0 y-nullcline

Chart Title -nulclines or different values of alpha 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.9 1.7 2.5 3.3 4.1 4.9 5.7 0.2 0.3 0.7 1.1 1.5 1.9 2.3 0.5 0.6 0.4 0.2 0.7 0.3 0.1 0.5 -0,9 1.3 1.7 2.1 2.5 2.9 3.3 0.9 0.1 -0.7 1.5 -2.3 3.1 0.2 0.4 1.4 0.5 2.6 3.2 3.8 4.4 1.6 0.5 0.5 0.6 -4.7 5.5 6.3 3.1 3.5 0.5 0.5 0.5 7.3 5.6 6.2 8.1 4.3

Critical points: x(1-x-y) = 0 y[0.2-y + 0.6x] = 0 Thus (0,0), lf.x = 0, y = 0.2, so, (0, 0.2) is a critical point, If y = 0, x 1, (1,0) is a critical point, If x # 0, y # 0, then x + y-1, and 0.6x-y =-0.2 Adding we get 1.6x-0.8, or x 0.5, and so y = 0.5 Thus (0.5,0.5) is a critical point Thus there are 4 critical points.