The system (dx)/(dt)=x-x^(2)-xy,(dy)/(dt)=4y-2xy-7y^(2) represents two species x and y that are competing for resources.
a. Find the x^(')=0 and y^(')=0 nullclines.
b. Plot a phase plane diagram showing the nullclines. Use two different colors - one for the x^(')=0 nullclines and one for the y^(')=0 nullclines. Indicate the direction of growth across the nullclines with arrows.
c. Give the equilibria of the system.
d. Compute the Jacobian matrix for the system of differential equations.
e. Evaluate the Jacobian at each equilibrium. Then, find the eigenvalues and use them to determine whether the equilibrium is stable or unstable.
f. Use Euler's method with Delta t=0.1,x(0)=0.2,y(0)=0.1 to approximate the solution on the interval 0<=t<=20. Attach or sketch a plot of both x and y on a common time axis.
g. If x(0)=0, what happens to x and y in the long term?
h. If y(0)-=0, what happens to x and y in the long term?
d. Compute the Jacobian matrix for the system of differential equations
e. Evaluate the Jacobian at each equilibrium. Then,find the eigenvalues and use them to determine whether the equilibrium is stable or unstable.
Use Euler's method with t = 0.1,x(0 = 0.2,y(0) = 0.1 to approximate the solution on the interval 0 t 20.Attach or sketch a plot of both x and y on a common time axis
g. If x(0) = 0, what happens to x and y in the long term?
h.If y(0)= 0, what happens to x and y in the long term?
