Consider the system dx dt = x(1 ? x ? y), (iii) dy dt =
Question and Solution
Consider the system dx dt = x(1 ? x ? y), (iii) dy dt = y(0.8 ? 0.6y ? x), which appeared in Example 2 of Section 7.3. I
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Consider the system dx dt = x(1 ? x ? y), (iii) dy dt = y(0.8 ? 0.6y ? x), which appeared in Example 2 of Section 7.3. If constanteffort harvesting is applied to species x, then the modified equations are dx dt = x(1 ? x ? y) ? Ex = x(1 ? E ? x ? y), dy dt = y(0.8 ? 0.6y ? x). (iv) (a) For E = 0, the critical points of Eqs. (iv) are as in Example 2 of Section 7.3. As E increases, some critical points move, while others remain fixed. Which ones move and how? (b) For a certain value of E, denoted by E0, the saddle point originally at (0.5, 0.5) coincides with the asymptotically stable node originally at (1, 0). Find the value of E0. (c) Draw a direction field and/or a phase portrait for E = E0 and for values of E slightly less than, and slightly greater than, E0. Estimate the basin of attraction for each asymptotically stable critical point. (d) Consider the asymptotically stable node originally at (1, 0). How does the nature of this critical point change as E passes through E0? (e) What happens to the species x for E > E0?
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