1. (a) The population p(t) of Loompaland (home to the Oompa Loompas) in 1920 and 1970...

 

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1. (a) The population p(t) of Loompaland (home to the Oompa Loompas) in 1920 and 1970 was 530 and 2310, respectively. Predict its population in 2020 and 2070 using the exponential model for population growth dp dt (1) [10 marks] (b) The population of Loompaland in 2020 is actually 7600 Oompa Loompas. i. Correct your prediction for the the population of Ooompa Loompas in 2070 using the logistic model of population growth dp dt rameters: = =rp kp. rp(1 – R). (Help: The relative growth rate r for the logistic model (2) is r = -0.031476.) ii. What is the carrying capacity N of Loompaland? iii. In what year will the population reach 99% of N? [16 marks] (c) The infamous chocolatier Willy Wonka reopens his chocolate factory and starts hiring Ooompa Loompas, who emigrate from Loompaland at a rate proportional to the population p. X= (2) i. Modify the logistic ODE (2) to include emigration at a rate proportional to the population p. ii. Non-dimensionalise your model in (c)i. by introducing non-dimensional pa- P N T = rt. iii. Using the linear stability criteria and phase lines, analyse your non-dimensional emigration model for all possible scenarios. iv. What do your results tell you about the future population of Loompaland? 1. (a) The population p(t) of Loompaland (home to the Oompa Loompas) in 1920 and 1970 was 530 and 2310, respectively. Predict its population in 2020 and 2070 using the exponential model for population growth dp dt (1) [10 marks] (b) The population of Loompaland in 2020 is actually 7600 Oompa Loompas. i. Correct your prediction for the the population of Ooompa Loompas in 2070 using the logistic model of population growth dp dt rameters: = =rp kp. rp(1 – R). (Help: The relative growth rate r for the logistic model (2) is r = -0.031476.) ii. What is the carrying capacity N of Loompaland? iii. In what year will the population reach 99% of N? [16 marks] (c) The infamous chocolatier Willy Wonka reopens his chocolate factory and starts hiring Ooompa Loompas, who emigrate from Loompaland at a rate proportional to the population p. X= (2) i. Modify the logistic ODE (2) to include emigration at a rate proportional to the population p. ii. Non-dimensionalise your model in (c)i. by introducing non-dimensional pa- P N T = rt. iii. Using the linear stability criteria and phase lines, analyse your non-dimensional emigration model for all possible scenarios. iv. What do your results tell you about the future population of Loompaland?