Each of 1 through 5 can be interpreted as describing the interaction of two species with population densities x and y. In each of these problems carry out the following steps. (a) Draw a direction field and describe how solutions seem to 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 the trajectories in the neighborhood of each critical point. (e) Draw a phase portrait for the system. (f) Determine the limiting behavior of x and y as t and interpret the results in terms of the populations of the two species.dx/dt = x(1 + 2.5x 0.3y x2)dy/dt = y(1.5 + x)

Applications to Geometry Friday, September 16, 2016 12:03 PM 5.4: Special Products Friday, September 16, 2016 12:09 PM Observations: