A constant-yield model, applied to species x, assumes th
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A constant-yield model, applied to species x, assumes that dx?dt is reduced by a positive constant H, the yield rate. Fo
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A constant-yield model, applied to species x, assumes that dx?dt is reduced by a positive constant H, the yield rate. For the situation described by Eqs. (i), the modified equations are dx dt = x(1 ? x ? y) ? H, (vi) dy dt = y(0.75 ? y ? 0.5x). (a) For H = 0, the x-nullclines are the lines x = 0 and x + y = 1. For H > 0, show that the x-nullcline is a hyperbola whose asymptotes are x = 0 and x + y = 1. (b) How do the critical points move as H increases from zero? (c) For a certain value of H, denoted by Hc, the asymptotically stable node originally at (0.5, 0.5) coincides with the saddle point originally at (0, 0.75). Determine the value of Hc. Also determine the values of x and y where the two critical points coincide. (d) Where are the critical points for H > Hc? Classify them as to type. (e) What happens to species x for H > Hc? What happens to species y? (f) Draw a direction field and/or phase portrait for H = Hc and for values of H slightly less than, and slightly greater than, Hc.
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