Question

Problem 3. Consider the following system of two equations, in terms of the parameter 7, which describes the respective populations of two species: 1 = *(- zv- ) yz ( +x). y' 2 (a.) List all critical points and discuss the stability at the points in relation to the parameter 7. (b.) What values of the parameter y will allow a stable solution in which both species have positive populations.

Answer 1

- Ans X'- x13}y-x) 4 =82 - 1 3 +x) a Critical points x'=0,4 = 0 x 13-4y - x) = 0 and rx ( - 1+x)=0 E R=0, 3 - 1 4 - x = 0 and x=0, x=ļ N=0, 4-3 x=1 3.4 = ? 10,3) ($, 2) are critical points Now we find Jacobian matrix associated system J = 8 With ixe əxl xe ie The XC skil he - 12 x Ir+Qxr 2 for (03) J/ 10,31

for x=0, 4=3 2. then system of eq uation have constant solution Stability does not dependon r. for x = 0, y = 2 Q - 4. J1 (212) O 815) Now eigen varee of J - brd 4 J-10- ord > tr 8. 2 O=x² + + 2 / 8. x = -1/2 = √ (+)² - 4x1xt Stability of system depend on r b) for stable solution either both eigen value should negative or on eigen values are imaginany with negative real parts. for negative eigen value X=- 1 - / Cuse I r 2 4 2 + Ź- 2

3 4 for negative iz Vtr>0 and * - 0 and t - r o ſtr 2 - 4 < ro et ļ ir NIK 2 o<r < Case I for complex eigenvalue 4 & Lo t - r o <r 2.
thumbs up