(3) Application: In many ecological systems, when the population drops below a certain level, it cannot...

 

Transcribed Image Text:

(3) Application: In many ecological systems, when the population drops below a certain level, it cannot support itself anymore. For example this happens when mates cannot be found, lack of cooperation, being too exposed to predators, etc. This effect is commonly referred to as the Allee Effect. A simple model for the Allee effect can be cast in the following form: dP = g(P) dt where g(P)=-a P (P-Pm)(P-M), (3) where a > 0 is a constant [related to r in Eq. (2)], M is the carrying capacity as before, and Pm is another special population size such that Pm <M (as depicted in the plot). g(P) (a) (i) What are the roots of g(P)? (ii) lim g(P) = P400 3 lim_ g(P) = P-0 (iii) Sketch g(P). (Use the limits!) (iv) Draw, as before, the direction in which the trajectories move on the P-axis. (v) What can you say, from the sketch, about the stability of the fixed points: P=0 is P= Pm is P = M is Po M (b) Now, use the punchline in (1.e) together with the graph of g(P) to corroborate the stability of the fixed points: g'(Pi)_0⇒P is g'(P₂)_0⇒ P is g (P)_0⇒ P is (c) Write an interpretation of the model for this Allee effect in terms of Pm and M. What is so special about Pm? (3) Application: In many ecological systems, when the population drops below a certain level, it cannot support itself anymore. For example this happens when mates cannot be found, lack of cooperation, being too exposed to predators, etc. This effect is commonly referred to as the Allee Effect. A simple model for the Allee effect can be cast in the following form: dP = g(P) dt where g(P)=-a P (P-Pm)(P-M), (3) where a > 0 is a constant [related to r in Eq. (2)], M is the carrying capacity as before, and Pm is another special population size such that Pm <M (as depicted in the plot). g(P) (a) (i) What are the roots of g(P)? (ii) lim g(P) = P400 3 lim_ g(P) = P-0 (iii) Sketch g(P). (Use the limits!) (iv) Draw, as before, the direction in which the trajectories move on the P-axis. (v) What can you say, from the sketch, about the stability of the fixed points: P=0 is P= Pm is P = M is Po M (b) Now, use the punchline in (1.e) together with the graph of g(P) to corroborate the stability of the fixed points: g'(Pi)_0⇒P is g'(P₂)_0⇒ P is g (P)_0⇒ P is (c) Write an interpretation of the model for this Allee effect in terms of Pm and M. What is so special about Pm?