3. {H} The populations of crows and pigeons obey the competitive discrete time dynamical system 1 613+]. : cn H 5pm: 1 Pn+1 = \"Er-'41 \"l" Pn {a} (2 marks} 1ill-frite the dynamical system in the form vn+1 = .411" where on = [em p]T, and A is a matrix which you should specify. {hi (8 marks} Determine matrices P and D such that D is diagonal. and A = Pep-1. (c} (ti marks) Find explicit expressions for on and p" in terms of m. pa and n. {(1) (4 marks} Show that if the initial populations are equal [i.e. cu = pg}. then eventually both the crews and pigeons 1will go extinct. J. (n) The populations oft-we types of fish obey the discretetime dynamical system 4 1 1iii-n+1 : Ear: _ Earn 2 ii bn+1= _En + Ell-in {a} (2 marks] Write the dynamical system in the form on.\" = 2h)" where u\" = [an,hn]T. and A is a matrix which you should sptti'. {h} (8 marks} Determine matrices P and D such that D is diagonal. and A = PEP1. [c]: (7 marks) Find explicit expressions for an and a. subject to the initial oonditions e.\" = lm and bu = 300. {d} (3 marks] Find the longtime behaviour of the populations in the limit n. } oo