Introduction to Mathematical Modeling — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

Science) – DSE 2. (a) Suppose that the population of a country is modelled using the differential equation aN = 0.4N- 0.002X2, With initial population equals to 200. The unite of time is in month. Find the population of the country after 2 months. (7.5) (b) Many animal population have decreasing per-capita birth rates when the population density increases, as well as increasing per-capita death rates. Suppose the density-dependent per-capita birth rate B (X) and density-dependent death rate A(X) are given by Where K is the population carrying capacity, kz& ky is the intrinsic per-capita birth rate and death rate and k, where 0 ≤ k ≤ 1, is a parameter describing the extent that density dependence is expressed in births or deaths. Show that this still gives rise to the standard logistic differential equation ax=rx(1-2) (7.5) (c) Consider the potentially disastrous effects of overfishing causing a population to become extinct, some governments impose quotas that vary depending on estimates of the population at the current time. One harvesting model that takes this into account is Then show that the only non-zero equilibrium population is At what critical harvesting rate can extinction occur? (7.5) 3. (a) Describe an input-output diagram for an endemic model of a disease in a city or country over a long time-scale, where there is no reinfection. Write the differential equations governing the model. (b) State any three preliminary assumptions on which a Predator-Prey model is built. How are the differential equations of Predator-Prey model modified to include the effect of DDT? Write the modified equations and define the new parameters p1 and pz. (7.5) (c) In a particular epidemic model, where the infected individuals eventually recover, the population dynamics are governed by the following system of differential equations: Given the parameter values ß = 2 and y = 0.4 and assuming that the total population N is 1000, in which initially there is only one infected individual but there are 999 susceptible individuals within the population, discuss the following: (i). When will the disease become an epidemic. Justify your answer. (ii). What is the peak number of infected individuals at any given time during the epidemic? (7.5) 4. (a) For the predator-prey model with density dependence for the prey and DDT acting on both species, Show that (X, Y) is one equilibrium point, where = asts and y-Re P (b) Suppose two species, X and Y are introduced into a new habitat where they compete for the same food source. The system is governed by: (i). Identify the per-capita birth rates (B1 and Bz) for both species and determine which species would grow faster in the complete absence of the other. (ii). If the initial populations are Xo = 50 and Yo = 30, calculate the initial rate of change for species X i.e (at). Is the population currently increasing or decreasing? (7.5)