
Concept explainers
Consider again the system
dxdt=x(1−x−y),dydt=y(0.75−y−0.5x), (i)
Which appeared in Example 1 of Section7.3. A constant effort 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 investedin 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 ofcatch is proportional to the effort E. Based on this assumption, Eqs. (i) are replaced by
dxdt=x(1−x−y)−Ex=x(1−E−x−y),dydt=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 moveand how?
(b) For a certain value of E, denoted by E0, the asymptotically stable node originally at (0.5,0.5) coincides with thesaddle 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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