describing the interaction of two species with populations x and y. In each of these problems, carry out the following steps. a.Draw a direction field and describe how solutions behave. b.Find the critical points. c.For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable. d.Sketch trajectories in the neighborhood of each critical point. e.Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions. f.Determine the limiting behavior of x and y as t → ∞, and interpret the results in terms of the populations of the two species. 2.dx/dt=x(1.5−x−0.5y)dy/dt=y(2−0.5y−1.5x)
describing the interaction of two species with populations x and y. In each of these problems, carry out the following steps.
a.Draw a direction field and describe how solutions behave.
b.Find the critical points.
c.For each critical point find the corresponding linear system. Find the eigenvalues and eigenvectors of the linear system; classify each critical point as to type, and determine whether it is asymptotically stable, stable, or unstable.
d.Sketch trajectories in the neighborhood of each critical point.
e.Compute and plot enough trajectories of the given system to show clearly the behavior of the solutions.
f.Determine the limiting behavior of x and y as t → ∞, and interpret the results in terms of the populations of the two species. 2.dx/dt=x(1.5−x−0.5y)dy/dt=y(2−0.5y−1.5x)
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