Biomathematics — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Each part of questions is of 7.5 marks. (a) State the law of mass action. Use the law to derive a system of equations modelling the chemical reaction (b) Describe the Verhulst-model for the discrete population growth. Show that the equilibrium states for this model are P = 0 and P = K. (c) Determine the rest states in Volterra-Lotka model of predator-prey interaction dx If predator and prey are present in the steady state, show that the coefficients a, b, c, e and f must satisfy the constraints (a) Discuss the direction of travel near the null-clines of the following modified SIS model: Also sketch the phase plane diagram. (b) Explain the SIS (Susceptible-Infectives-Susceptible) epidemic model. State its assumptions, write the differential equations, and explain the parameters used. (c) Describe the SIR (Susceptible-Infectives-Recovered) epidemic model with schematic representation and show that the epidemic outbreak will not take place if S(0) <*, where a is the infection rate and B is the per-capita recovery rate. (a) Consider the predator-prey model: (b) Determine the nature and stability of the equilibrium point (0, 0) for the system: * = 5x + 2y, j= 2x +2y, by determining the eigenvalues of the Jacobian matrix associated with the equilibrium point. (c) Use Poincaré-Bendixson theorem to show that the system dV VO has a periodic solution for 0 < 8 < iz. 5. (a) What is bifurcation? Show that dx - = | - y(x + 1), has an equilibrium point, which is a stable node, for u < 1, that becomes a saddle-point as u passes through the bifurcation point u = 1. Also show that there is an additional equilibrium point when u > 1, which is a stable node. (b) What is Hopf bifurcation? Show that Hopf bifurcation holds for the following system (c) Discuss the stability of limit cycle when Po, P1, Pz, ... are the points on Poincare line and Pn is at a distance of sn form Po and such that Sn+1=f(Sn), where Po is the fixed point. (a) Show that ỳ=M - Y(2x + 1), has an equilibrium point, which is a saddle node for 5 < 4 < 1 and a saddle focus for u > 1. Prove that j = - is a bifurcation point. (b) Write the transition matrix for Jukes-Cantor model of base substitution stating the significance of each entry. In what proportion of the sites will each base appear after one time step? What proportion of sites will have a base A in the ancestral sequence and a base T in the descendent sequence one step later? (c) Explain Kimura 2-parameter and 3-parameter models along with their corresponding distance formulas. Write the expression of the log-det distance between So and S1.