Discrete Dynamical Systems — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

(c) Convert into a one-parameter family. : Discrete Dynamical Systems : Common Program Group 1. (a) Suppose a population, initially 836,900, has a birth rate of 9% and a death rate of 7.5% per generation. Construct the model that satisfies: (i) Immigration occurs at 6800 per generation. (ii) Migration occurs at 4340 per generation. (b) (i) The intrinsic growth rate of the first species is 1.6 and that of the second is 1.3; each species’ growth is reduced by 0.35 times the population of the other species. (ii) In addition, the first species receives a constant immigration of 3000 individuals per time step, while the second species experiences emigration of 1500 individuals per time step. (c) Construct a model that satisfies : (i) In a population of size 250,000, recovery is not possible, and if 1200 are ill, that number will double in the following week. + 2x n, find all fixed points and use the derivative to determine their stability and any oscillation around them. (ii) In characterizations of chaos define dense orbit. (c) Find a 2-cycle of xn+1 = 1 – x2 and determine its 6. (a) Define the intervals of existence and stability. For the price model , find the positive equilibrium point and its intervals of existence and stability. (b) In some kind of bifurcation two fixed points, one stable and another unstable come into existence simultaneously at the same location and for the same parameter value. Show that this occurs for each of functions given below : of parents in that region includes 35% who are college educated and 65% who are not, what will that distribution be three generations from now? (b) For the prey-predator model with constant immigration Pn+1 = 0.8P n – 0.3Qn + 8000; Qn+1 = 0.2P n + 0.9Qn + 2000. Find the fixed point and determine whether it is a sink, source or saddle why iterating and graphing in solution space the first few iterates for several choices of initial conditions. (c) Find the equilibrium population levels, if any, for Nonlinear Population model . Also discuss their stability. 5. (a) What is a Cobweb graph? Draw a cobweb graph for xn+1 = x2 n starting with x0 (ii) Each week 45% of those ill in a population of size 359,700 are still sick, and the maximum number of new cases possible in any week is 70,000. 2. (a) Construct and Explain Linear Price-supply Model. Construct a linear price-supply model by assuming instead that demand is either constant or reacts immediately to price : (b) The following are overlapping-generations models that have been converted into systems. Find the original equation for : (c) Construct the linear competition model with no immigration, migration or harvesting that satisfies : 3. (a) When a population has a birth rate b that its death rate d, then that population will decrease over time and may eventually become extinct. For a population with b=0.15, d=0.25 and initial size of (i) Find the equation for the population size P n after n generations;