Question

Exercise 3.3 The x- and y-isoclines are given by a-bx-cy = d-ex - fy- 0 in int R2. These are straight lines with negative slopes. (a) If the isoclines coincide, then show that xy-k (with k = a/d) is a constant of motion. (b) If the isoclines do not intersect in int RA, one of the species (which one?) F tends to extinction (see fig. 3.1). One species, in this sense, is dominant. Fig. 3.1.

Answer 1

The x and y - isoclines are given by a - b x - c y = 0 in int IR^2/1 d - e x - f y = 0 These are straight lines with negative slopes As isoclines coincide So, a = b x + c y d = e x + f y b x + e y = a e x + f y = d Solving this Rightarrow y = a e - b d/c e - b f b x = a - c a e - b d/e c - b f = a c e - a b f - a c e + b c d/e c - b f b x = b (e d - a f)/e c - b f Rightarrow x = c d - a f/c e - b f C y = a - b e c - a b f/c e - b f Rightarrow C y = a c e - b c d/c e - b f Rightarrow y = a e - b d/c