MTH108: HOMEWORK 4 – LINEAR POPULATION MODELS
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MTH108: HOMEWORK 4 – LINEAR POPULATION MODELS
Section A – Warm-up questions
A1. Given the coupled system ( xn+1 = 3xn − 2yn + 5, yn+1 = xn + yn + 3 ) and initial values x0 = y0 = 1, find x1 , x2 , x3 and x4 .
A2. Given the coupled system ( xn+1 = 4xn − 2yn + 3, yn+1 = 2xn + 3yn − 1 ), find a second order linear difference equation satisfied by xn .
Section B – Main questions
B1. In a laboratory, there is a population of bacteria in a flask . At noon on Monday, the population consists of 200 cells . At 5pm, the population has grown to 600 cells . Construct a discrete time Malthusian model for the population . According to this model, how many cells will there be at noon on the next day?
B2. The density of a population of butterflies changes throughout the year . The rainy season lasts for 115 days and during this time the daily birth rate is Br and daily death rate is Dr . The dry season lasts for the other 250 days of the year and during this time the daily birth rate is Bd and daily death rate is Dd . Find an expression for the annual intrinsic growth rate (that is, over a 365 day period) in terms of the model parameters .
B3. In appropriate units, the biomass, xn , of coral in a reef satisfies the Malthusian model xn+1 = (9/10)xn . The population density of parrotfish, yn , which feed on the coral, satisfies yn+1 = (3/4)yn + (3/5)xn . Here n denotes time in years . Given initial conditions x0 = 20 and y0 = 30, find an explicit formula for yn .
B4. Consider a population of a species of beetle that can live for up to 3 weeks, passing through 3 life phases: larva, young adult and old adult, which last for one week each . Only 20% of the beetle larvae survive to become young adults but 50% of young adults manage to become old adults . Young and old adults each produce an average of B offspring per week . Formulate a set of linear difference equations to model the beetle population, and express it as a single difference equation . It is observed that over many years, the number of beetles remains approximately the same . Given this fact, find the value of B .
B5. A population of flies is described by the following system of coupled difference equations:
An+1 = An + Ln
Ln+1 = 4An
where An and Ln denote the numbers of adult flies and larvae (immature flies), respectively, on the nth day. According to the model, what is the average number of larvae produced per adult per day? What fraction of larvae survive to adulthood? Write a second order difference equation for An and find its general solution .
B6. Suppose two armies are in battle . Let xn and yn denote the number of soldiers in army X and army Y after n days . In each day, each soldier of army X kills 0 .16 soldiers in army Y and each soldier of army Y kills 0 .25 soldiers of army X . So the battle evolves according to the equations xn+1 = xn − 0 .25yn and yn+1 = yn − 0 .16xn . Given initial conditions x0 = 12000 and y0 = 10000, find an explicit formula for xn for the period of validity of the model, and show that the X army will be defeated in 10 days .
Section C – Modelling investigations
C1. Adapt the model in B5 to include the effects of daily reinforcements . How many reinforcements are needed to change the outcome of the battle?
2022-08-08