Biomathematics — DU B.Sc. (Prog) Physical Sciences courses pertaining to Mathematical Sciences Previous Year Question Papers

(b) Let Ma,) be the Jukes-Cantor matrix with parameter ai and Ma,) be the Jukes-. Cantor matrix with parameter a,. Compute Ma,) Ma,) to show it has the form May). Express, the formula for 013 in terms of a, and ar. (c) Calculate dc(S,, S,) for the two 40-base sequences, showing all steps 1. (a) Develop a continuous model for population growth using the given data, postulating a linear relationship between the population growth rate and the population size itself. Utilize this model to forecast the U. S. population in the year 3000. Year Population (b) Find the equilibrium states of the population growth model Pats= emin, where K is carrying capacity. (c) Write the logistic growth model and find, out the equilibrium points. Also, classify them as stable and unstable. (b) Show that the iteration scheme has a stable fixed point *, = 1 for u < 1 and that # = 1 is a bifurcation point, where the fixed point is x, = 1/u appears. Show that the period doubling occurs as soon as u exceeds 3. : (c) Discuss the stability of limit cycle when Po, P,, 2.... are the points on Poincare line and P, is at a distance of s, from P, and such that where Po is the fixed point. 6. (a) Write the transition matrices for Jukes-Cantor model and Kimura-2 model of base substitution stating the significance of each entry. In the Jukes Cantor Model of base substitution (a) In, what proportion of the sites will each base appear after one time step? (b) What proportion of sites, will have a base A in the ancestral sequence and a T in the descendent sequence one step later? (b) Use Poincare'-Bendixson theorem to , show that : the system vz VO has a periodic solution for 0 < 8 < 12 (c) Discuss the nature of the fixed points of the following system and sketch the corresponding trajectories in the phase plane. 5. (a) Show that the non-linear conservative system ỳ = u sinx - x where 4 ≥ Ó has one equilibrium point for O ≤ M < 1 and three for M > 1. Also discuss the nature of these equilibrium points. 2. (a) A drug whose elimination constant is I hourst-l) is first administered at 1 A.M. at a dosage of C ug/ml and in same dosage every three hour afterwards. What is the drugs concentration at (b) A drug is available in doses of M mg, which raise the blood plasma concențration of an average adult by 10 mg I'. Once in the blood plasma, the drug concentration decays according to equation with t= 4 hour. Health regulations require that the concentration never exceed 10 mg I!! What is shortest safe time interval at which doses can be given regularly? What is the minimúm concentration if doses are given at this time interval? (c) State of Law of Mass Action. Use it to derive a mathematical model governed by the intermediaries. X and Y in trimolecular reaction: where A, B, D and E are initial and final products, and all rate constant are equal to 1. 3, (a) Discuss the direction of travel near the null clines of the following modified SiS model Also sketch the phase plane diagram.