Let p(t) be the ratio of dextral snails in the population of snails. p = 1 means
ID: 3004231 • Letter: L
Question
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 Which has no left -right bias. Locate the equilibria of p and determine their stability. Suppose that at t = 0 (which is a very long time ago, perhaps a few hundred million years ago), p(0) = 1/2; 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) = 1/2 as t oright arrow infinity (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 mosttly sinistral snails now).Explanation / Answer
(a) for equilibrium points :
dp/dt = 0
?p. (1-p). (p-1/2) = 0
p = 0,1,2.
Hence equilibrium points are p = 0; 1; 2.
The issue is their stability. Since this is just a one-dimensional dynamics, we can see from the sign of dp/dt what the stability property of each equilibrium point is. The result is that both 0 and 1 are stable, while the intermediate point 1/2 is unstable.
(b) Consequently, for any initial condition near to 1/2 - that is, with a nearly equal distribution of left and right handed snails initially- the evolution under this governing equation will be toward either all left handed or all right handed snails. Which limit is approached is determined by the sign of p0 -1/2, in other words where there is a very slight majority one way of the other. Finally, the model itself does not distinguish between right and left handed preference, since the governing DE is completely symmetric. The initial condition breaks this symmetry.
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