What this question is checking? This question is checking ideas about modeling a system using difference equations. Once you decide on a model, this question asks you to compute the iterations using Matlab and then to calculate any steady state in the model and its stability.
a) It is now 35 years since dinosaurs first escaped from Jurassic Park. Scientists have now observed the interactions between two different species of dinosaurs: Stegosaurus (plant-eating) and Allosaurus (meat-eating). Allosaurus are predators for Stegosaurus but Stegosaurus can try to defend itself with its spiky tail.
If left to themselves, the population of Stegosauruses increases by 40% while the population of Allosauruses reduces by 50% over each year.
When two Stegosauruses meet, they have a child with a probability of 2%.
When two Allosauruses meet, they can fight. Four in five such fights end with one Allosaurus dying.
When a Stegosaurus meets an Allosaurus, a Stegosaurus dies with a chance of 50%.
If a female Allosaurus kills a Stegosaurus, there is a 2% chance that she will give birth to two baby Allosauruses. Male to female ratio among Allosauruses is 50:50.
Write a system of difference equations to describe how the population of Stegosauruses and Allosauruses changes from year to year, using the observations listed above.
b) Write a Matlab program to compute the number of Stegosaurus in five years time if we start with 10 Stegosauruses and 1 Allosaurus. Show the plot of the number of Stegosauruses and Allosauruses versus the number of years for the first 5 years.
c) Consider the difference equation
x_(n+1)=exp(-5x_(n)^(2))+eta
starting from the initial condition of x(1)=0.2. Knowing that eta takes only values between
-1,1, vary eta to explore the behavior of this equation. Identify values of eta when the dynamics
(i) oscillates between 2 values, (ii) oscillates between 4 values, and (iii) oscillates chaotically.
Show your results using time course and cobweb plots for each of the three cases, along with
the programs that you use to make the figures.
Hint: You can re-purpose the functions that we wrote in class (this is available in the Matlab
Programs folder in the Modules page of Canvas).
