Question

3. 1101 Consider the following population model: dN dt (a) [3] Rewrite your model using nondimensional population size r and and time unit T to time τ, and choose a proper population unit simplify your model to the following form dx: dt 22 ad identify α as a cornbímnation of the original parameters b, d and a. (b) [41 Show that, if 0

Answer 1

Answer:

(a). When K is quite large and x is very small then d.r dt In this case population has exponential growth d.r dt If r K and h-0 then0 In this case there won't be much change in the population

When h is quite large in comparison to r, and K then 0 dt In this case there is decreasement in the growth rate .

(b) If r-5, K 10 and h -1 then dr dt 10 d.r dt d.r dt For equilibria-=0 r2

$\\\\\Rightarrow 8x-x^{2}=0 \\\\\Rightarrow x(x-8)=0 \\\\\Rightarrow x=0,8$

Thus, equilibria are x=0 and x=8. Now, we have f(x)-4T f' (x) = 4-x -- Thus, x=0 is unstable. f' (8) -4 <0 Thus, 8 is asymptotically stable

(c) In the given model, the maximum sustainable yield is where the population size r is equal to the half of the carrying capacity K Thus, maximum sustainable yield occur at

Putting 2, _ in the equation rZ' (1-_ H we get H-which is the required maximum sustainable fishing vield * 2 K7

(a). When K is quite large and x is very small then d.r dt In this case population has exponential growth d.r dt If r K and h-0 then0 In this case there won't be much change in the population

When h is quite large in comparison to r, and K then 0 dt In this case there is decreasement in the growth rate .

(b) If r-5, K 10 and h -1 then dr dt 10 d.r dt d.r dt For equilibria-=0 r2

$\\\\\Rightarrow 8x-x^{2}=0 \\\\\Rightarrow x(x-8)=0 \\\\\Rightarrow x=0,8$

Thus, equilibria are x=0 and x=8. Now, we have f(x)-4T f' (x) = 4-x -- Thus, x=0 is unstable. f' (8) -4 <0 Thus, 8 is asymptotically stable

(c) In the given model, the maximum sustainable yield is where the population size r is equal to the half of the carrying capacity K Thus, maximum sustainable yield occur at

Putting 2, _ in the equation rZ' (1-_ H we get H-which is the required maximum sustainable fishing vield * 2 K7