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1) Sketch the family of solutions p(t) as po varies, including examples of all the different qualitative types of behaviour. ) What is the expected long term behaviour of the population if (i) po = 100, (ii) po = = 200?

1) Sketch the family of solutions p(t) as po varies, including examples of all the different qualitative types of behaviour. ) What is the expected long term behaviour of the population if (i) po = 100, (ii) po = = 200?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do the questions d e f g Please do the questions d e f g Please do the questions d e f g
1. The Australian bilby is a desert dwelling marsupial that is becoming endangered because of habitat loss and the effect of predators, such as foxes and feral cats. Figure 1: A Greater Bilby. Image credit: https://commons.wikimedia.org/wiki/File: Macrotis lagotis_-_bandicut_conejo.jpg A sanctuary set up to help save the bilbies is enclosed by a predator-proof fence and most predators have been removed from inside. An initial population of po> 0 bilbies is placed in the enclosure and the population p(t) of bilbies after t years is modelled by the differential equation (*) where k = is the net birth rate, m = 800 is the carrying capacity of the enclosure, and h = 32 is the death rate per year from predators. dp = dt kp(1-P) - - h (a) Determine the equilibrium solution(s). (b) Sketch the phase plot for the ODE. (c) Describe the stability of the equilibrium solutions(s). (d) Sketch the family of solutions p(t) as po varies, including examples of all the different qualitative types of behaviour. (e) What is the expected long term behaviour of the population if (i) po = 100, (ii) po 200? (f) Assume instead that all remaining predators have been trapped and removed from the enclosure before the original po bilbies are introduced. Write down a differential equa- tion modelling the population of bilbies in this case, and describe the expected long term behaviour of the population if (i) po = 100, (ii) po = 200. (Give brief explanations.) (g) How realistic are the assumptions in the model (*)? Can you think of other factors that should be taken into account?
Transcribed Image Text:1. The Australian bilby is a desert dwelling marsupial that is becoming endangered because of habitat loss and the effect of predators, such as foxes and feral cats. Figure 1: A Greater Bilby. Image credit: https://commons.wikimedia.org/wiki/File: Macrotis lagotis_-_bandicut_conejo.jpg A sanctuary set up to help save the bilbies is enclosed by a predator-proof fence and most predators have been removed from inside. An initial population of po> 0 bilbies is placed in the enclosure and the population p(t) of bilbies after t years is modelled by the differential equation (*) where k = is the net birth rate, m = 800 is the carrying capacity of the enclosure, and h = 32 is the death rate per year from predators. dp = dt kp(1-P) - - h (a) Determine the equilibrium solution(s). (b) Sketch the phase plot for the ODE. (c) Describe the stability of the equilibrium solutions(s). (d) Sketch the family of solutions p(t) as po varies, including examples of all the different qualitative types of behaviour. (e) What is the expected long term behaviour of the population if (i) po = 100, (ii) po 200? (f) Assume instead that all remaining predators have been trapped and removed from the enclosure before the original po bilbies are introduced. Write down a differential equa- tion modelling the population of bilbies in this case, and describe the expected long term behaviour of the population if (i) po = 100, (ii) po = 200. (Give brief explanations.) (g) How realistic are the assumptions in the model (*)? Can you think of other factors that should be taken into account?
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