Consider again the system dx dt = x(1 ? x ? y), (i) dy d
Question and Solution
Consider again the system dx dt = x(1 ? x ? y), (i) dy dt = y(0.75 ? y ? 0.5x), which appeared in Example 1 of Section 7
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Consider again the system dx dt = x(1 ? x ? y), (i) dy dt = y(0.75 ? y ? 0.5x), which appeared in Example 1 of Section 7.3. A constanteffort model, applied to the species x alone, assumes that the rate of growth of x is altered by including the term ?Ex, where E is a positive constant measuring the effort invested in harvesting members of species x. This assumption means that for a given effort E, the rate of catch is proportional to the population x, and that for a given population x, the rate of catch is proportional to the effort E. Based on this assumption, Eqs. (i) are replaced by dx dt = x(1 ? x ? y) ? Ex = x(1 ? E ? x ? y), dy dt = y(0.75 ? y ? 0.5x). (ii) (a) For E = 0, the critical points of Eqs. (ii) are as in Example 1 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 asymptotically stable node originally at (0.5, 0.5) coincides with the saddle point (0, 0.75). 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. (d) How does the nature of the critical point (0, 0.75) change as E passes through E0? (e) What happens to the species x for E > E0?
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