Question

2. Consider the autonomous nonlinear system (such systems arise in competing species population models). (a) Find the equilibrium solutions. (b) Use the linearization theorem to determine if each equilibrium solution is locally asymptotically stable or unstable.

Answer 1

T So, the equilltorum solutions are 10,0),(012), (200) and Chil) d'= {(1-42-44) y'= $ 11- 42 -zyby equilibrum solutions 7 / (1-2x-2y)x=0 » X20 or 1-2x+2y=0|| and & (1-7-x-2y) Y=0 = 720 g 1-1/22-2y=0 1220 and y=0 $10,0) M 120 and 1-12x - 2y 20 | ③ uneorretion :: frain) = lert x2_t_xy) ge4.4) 414-494-533) Jacobian - J0903) - yang Jomiya 14(1-x-45) ** 7 lity & tx - 3) . - . / Ż 01-}] 2 y = 2 110,0) mm yno and 1-X 1-0 =) |-$430 TEK at coro) 3101072 segervatus hird, d2 = 1/2 since, dino ,d270 +) 10,0] is asymptotically unprable. 1 H) at (01²) £11-0-£) *(0) 7 J10,3) / 2 2 2 7210) - 3 x 3 ) -17210)] 31,$y=0 (4-1)X +C+4)320 -*+30 since edict , Az = n/a diso 1 Azco (1) is a saddle point hardy zo 1-fx-4420 1-720 1 = 1 41

At point(210) 123 12)-310) Julio) : | *(1-2-4(0) [ 72.ro) ¢ J1 2,0) = 1t ² Reformalus di- , d2 = 1/2 since dico ,d230. (2701 is a saddle point At point (dry) L (ro) -1-10) / - (T'TE ink crit of the ( ] - 2- p LÝ 202/208 96 takp +519 -Lo + f2ef Espenvalus 1 J-1)=0 - taking -ve sign 0) taking - Design dz -20 -1 288 ro 15-18) == (*)6*) --- 34+ $t tør + 12+ tap 20 de 48124208 +180 =) (111) is asymtorically stable soluna -20 + $ 400 - 192 96