Question

3. Consider the following population model of mountain lions in Montana: dy = 0.24(1 - 6 - 1); where y denotes the population size in hundreds and t gives the time in years. His- torically mountain lions were hunted for their furs, and then later because they were considered dangerous. Until 1963, the Montana state government used to pay a bounty for each mountain lion killed. Assume people kill a proportion k > 0 of the mountain lion population each year. The question we want to answer is what proportion can be killed without devastating the population. (a) Write a differential equation that gives a model for the mountain lion population with hunting. (b) Draw a bifurcation diagram for the differential equation from part (a). There are two bifurcation values for k, one positive and one negative (c) For this part, focus on the positive bifurcation value for k, since the negative one is not at relevant for the mountain lion problem. Do a qualitative analysis to describe the long term behavior of solutions for various initial conditions for the system when the parameter k is i. below this bifurcation value ii. at this bifurcation value iii. above this bifurcation value

Answer 1

ENERAL - (a) People kills a proportion K of mountain tion population (1) each year. So the rate of change of y ) Contains the term aky. They the model is given by : - Tý= 0.2 y (-9/10) (93-1) – kyla

(b) “co -> Yao 4 (1-7)61-1) – *= 0.-> (10-13 (9-3) — 30K=0 → ys 13+149–120k. They 0 has three equilibrium. for K < 4/20 - 's one m n k3 49/126. Thus K= 49/120 is bifurention pta PUNE

REFE RA f' Cy)= 0.2(o) -1.60.2y +. 6- +0.24. (1-7) / 3 - K. f'(O) = 0,2*1*(--)-'K = -0,2-k

TUS W US So y=a is stable for k -0.2 <k f onstable for k<-0.2 They at K= w ong these is also another bit us Cation : At K = 49/120, there is a saddle-hodet f K= -0.9 , in a transeriti Carl bifurcations SISI WARISA MENDES

tru bifurcation diagram,

IFE + V-13 ho_ok name 9 13-14 Pak t 9112,

— у= 3, St К= 44/.

on tantati a andriatern न 1-44/120