Question

(1-0) -- ty Question 1 (20 Marks) Consider the following model for the interaction between two species in a predator-prey relationship: dc = 40 (1 dt 40 dy ry -4y + dt 4 (a) Which variable represents the predator species? (b) Make a nulleline diagram and identify the equilibrium points. (b) Use linearisation to classify each of the equilibria. (c) Describe in words the behaviour of the solution with initial condition (20,yo) = (5,5).

Answer 1

a ) =4+ (1 - 2 - at Log S. lity to oly dt ted H- - prey population y- predator popelatin equilibrum point (b) At de dy dt at for ( 1 2 - 4 & -434 OR flagg 'تا8 = 40 S Izo OR eo 4 - 4 a) da 16 8 i 16x4_y au y = 12 40 a* , jk] = Coo) (400) €16, 12) Mullclines TI ME 2017 y 10 16 207 -80+ to 20 30 40

© to lineamised; f = te (1-7 kg for my -44 4 A of of Әу -af -an an ay Defa ata .afa 2y an 29 au of ax 4 al y 5 Ofi. ay OG Dita ду ol - 4 ar 4- Caty 5 # -4 قرار و تام At Coo) 14 we find eigen: Values O -44 d = 2 and do since di=tve and ds=ave Cool Unstable. the equilibrium post A (40,0) A - 4 we find eigen realle J

an b-dl efl (-6)(4+4 948=0 & À +41-61-24 1820 t -dd-16=0 di alt da=1-17 it is lastable. Since d = the and basale it At (16, 12 J ho J A=58 ک S S 3 -d ک S since ar S A x + 8 + 4 3. d40-4 P1148 5. dyz - 4 - 44041 both real part negative equilibrium point is stable. cos Initial condition of (5,5) and this means there is selemination of prey due to equal number of initial population and this effect the prey. popular Hon. bara means X.COJES y cod =s 9 possible may
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