Systems of Difference Equations

In this lab we will use simulation to study various systems of difference equations. Our main focus will be on interacting species models like predator-prey and competing species.ccollins@math.utk.edu1. Linear Predator-PreyLetx(n)be the number of rabbits andy(n)be the number of foxes afternweeks. The model for their interaction is:x(n+1) = x(n) + 0.1 x(n) - 0.25 y(n) y(n+1) = y(n) - 0.2 y(n) + 0.15 x(n)On their own rabbits would grow in number, foxes would decrease. Write a MATLAB script (or use the one below) to simulate this system for 100 weeks. Plot the number of foxes vs. the number of rabbits, i.e. y vs. x. Try starting with 20 rabbits and 4 foxes. Do both species survive? Try increasing the starting rabbit population. Is there any value which lets both species survive?2. Nonlinear Predator-PreyLetx(n)andy(n)be as in 1. Now we are going to model the interaction between the rabbits and the foxes by amixingterm:x(n)y(n). The model is thenx(n+1) = x(n) + 0.1 x(n) - 0.02 x(n)y(n) y(n+1) = y(n) - 0.2 y(n) + 0.01 x(n)y(n)On their own rabbits would grow in number, foxes would decrease. When the number of rabbits is low, it is harder for the foxes to find them. Write a MATLAB script (or use the one below) to simulate this system for 100 weeks. Plot the number of foxes vs. the number of rabbits, i.e. y vs. x. Try starting with 20 rabbits and 4 foxes. Do both species survive? What do you think would happen if you extended the simulation? Try different starting values. Is there anequilibriumvalue, i.e. are there starting values so that the populations never change?3. Competing Species ModelLetx(n)be the number of owls andy(n)be the number of hawks afternweeks. The model for their interaction is:x(n+1) = x(n) + 0.1 x(n) - 0.25 y(n) y(n+1) = y(n) + 0.05 y(n) - 0.15 x(n)On their own both would grow in number, and each are hurt by the existence of the other. Write a MATLAB script (or use the one below) to simulate this system for 100 weeks. Plot the number of hawks vs. the number of owls, i.e. y vs. x. Try starting with 10 owls and 10 hawks. Which species survives? You will need to zoom in to see what happens. Increase the starting number of owls, until they survive. Is is possible for both species to survive? There is a nonlinear version of this model:x(n+1) = x(n) + 0.1 x(n) - 0.02 x(n)y(n) y(n+1) = y(n) + 0.05 y(n) - 0.01 x(n)y(n)Write a script to simulate this system. Is there an equilibrium solution to this system?Scripts% script file to simulate the linear predator-prey model x(1) = input('Enter the initial number of rabbits:'); y(1) = input('Enter the initial number of foxes:'); for n = 1:100 x(n+1) = x(n) + 0.1*x(n) - 0.25*y(n); y(n+1) = y(n) - 0.2*y(n) + 0.15*x(n); end plot(x,y), xlabel('Rabbits'), ylabel('Foxes') % end of script % script file to simulate the nonlinear predator-prey model x(1) = input('Enter the initial number of rabbits:'); y(1) = input('Enter the initial number of foxes:'); for n = 1:100 x(n+1) = x(n) + 0.1*x(n) - 0.02*x(n)*y(n); y(n+1) = y(n) - 0.2*y(n) + 0.01*x(n)*y(n); end plot(x,y), xlabel('Rabbits'), ylabel('Foxes') % end of script % script file to simulate the linear competing species model x(1) = input('Enter the initial number of owls:'); y(1) = input('Enter the initial number of hawks:'); for n = 1:100 x(n+1) = x(n) + 0.1*x(n) - 0.25*y(n); y(n+1) = y(n) + 0.05*y(n) - 0.15*x(n); end plot(x,y), xlabel('Owls'), ylabel('Hawks') % end of script