The [Mathematical Expression Omitted]
isocline shifts up because [Mathematical Expression Omitted], while the [Mathematical Expression Omitted]
isocline is unaffected since [Mathematical Expression Omitted].
Similarly, we divide the
isocline [L.sub.2] of message 2 into [L.sub.21] and [L.sub.22] as follows:
From P', vertical
isocline [l.sub.1] is monotone decreasing when N > ([epsilon]m[H.sub.N]/([alpha][beta] - em)), and P < 0 holds when N [member of] (0, ([epsilon]m[H.sub.N]/([alpha][beta] - [epsilon]m))); P > 0 holds when N [member of] (([epsilon]m[H.sub.N]/([alpha][beta] - [epsilon]m)), [N.sub.0]).
The
isocline and full system analyses show similar patterns when there is variation in performance (Figs.
If so, the grazer
isocline will shift upward and to the left (compare grazer
isoclines in Fig.
The
isoclines [[GAMMA].sub.i] and [[GAMMA].sub.2] are, respectively, a hyperbola and a straight line.
Divided by the vertical
isocline [r(1 - (x/K)) - y/(a + [x.sup.2])] = 0, the rotated direction of vector fields below the vertical
isocline is counterclockwise, but above the vertical
isocline, the rotated direction of vector fields of system (3) is clockwise.
Movements of each treatment vector across the
isocline surface reflect changes in patch size.
The
isocline [L.sub.1] : y = (1 - x)(x + a) intersects with the phase set [N.sub.2] at the point Q((1 - p)[h.sub.2], [y.sub.Q]), and Q is below [Q.sub.0].
The
isocline S'(t) = 0, that is, x = D([S.sub.i] - S)([K.sub.s] + S)[[delta].sub.1](([mu] + m[[delta].sub.1]) S + m[K.sub.s][[delta].sub.1]), which intersects the S-axis at points (-[K.sub.s], 0) and ([S.sub.i], 0), and x < D([S.sub.i] - S)([K.sub.s] + S)[[delta].sub.1]/(([mu] + m[[delta].sub.1]) S + m[K.sub.s][[delta].sub.1]), S'(t) > 0.
This means a shift downwards or to the left in the prey null
isocline (if the prey suppresses), or a shift downwards or to the right in the predator
isocline (if the predator suppresses).
Obviously, y = f(x) = r(1-x/K)(b+[x.sup.2])/a[x.sup.2] is a vertical line and x = [square root of (mb/([epsilon]a - m))] is a horizontal
isocline. By direct calculation, the equilibrium [E.sup.*] is locally asymptotically stable under the condition ([epsilon]a - 2m)K > -2m[x.sup.*] and the index of the equilibrium [E.sup.*] is +1.
