Question

6. Right-handed snails D'Arcy Wentworth Thompson, a noted scientist of natural history, wrote in his book, On Growth and Form (1917): "But why, in the general run of shells, all the world over, in the past and in the present, one direction of twist is so overwhelmingly commoner than the other, no man knows." Most snail species are dextral (right-handed) in their shell pattern. Sinistral (left-handed) snails are exceedingly rare A plausible model for the appearance of such a bias in population handedness can be as follows: let p(t) be the ratio of dextral snails in the population of snails. p = 1 means that all snails are right-handed, and p 0 means that all snails are left-handed. A model equation for p(t) can be (-) p(1-p) 149 which has no left-right bias a. Locate the equilibria of p and determine their stability b. Suppose that at t = 0 (which is a very long time ago, perhaps a few hundred million years ago), p(0) = }; that is, the dextral and sinistral snails are evenly divided. Describe what will happen a few hundred million years later. Argue that we should not expect that p(t) = as t -» oo (i.e., an equal number of dextral and sinistral snails at the present time), and argue that our present state of affairs (mostly dextral snails) is an accident (i.e., we could just as well have mostly sinistral snails now)

Answer 1

a. d PCHDC) equilibria if 2:0, all points of pare equilibnice and are stable as a pict) = k for some da bitta in Lo, i] Let to We know, if fry) changes from thor as y increases across yo, yo a equilibrium point then y = yo is an asyurpatotically stable solution It fcy) ehanger forom - tot, it is unstable If fly doesn't change as y increase arrests • 4-е задолд - 1933. А. Now, equilibrium parts are ich hando o is stable, instable, it is unstable. it & To, o and I are unstable, while itshable here. Fix fto. b. W LOG, 270 is unstable stationary point. Hence, by definition of unstability, we can have solution pot. i (0) but we will have to do for every to to out. If (t) - 3 > G. So

clearly, PL Ab (A) = 1/2 may not be love Lt brit), if it exists can have any value in [ord]. And so our present situation, bel can change in future , could have been anything else as well, and can change in future.