Make sure that you show clearly the reasoning you use to solve the
problems. It is also advisable to keep a photocopy of the assignment you hand
in. You are free to use any reference you wish as long as you cite your source.
However, you must define and explain all notation or concepts used that were not
covered in the lecture or a prerequisite course. Each question is worth 25 points.
Q1. Consider the Wright-Fisher diploid model,
p(t + 1) = wAAp(t)2 + wAap(t)[1 −p(t)]
wAAp(t)2 + 2wAap(t)[1 −p(t)] + waa[1 −p(t)]2 ,
where p(t) is the frequency of allele A at the generation t and wij is fitness of
genotype ij (i,j = A,a).
(a) Reduce the number of parameters.
(b) Derive an allele frequency change in one generation, ∆p = p(t + 1) −p(t).
(c) What are equilibria of p?
(d) Show the parameter condition in which an internal equilibrium (i.e., 0 < ̄p < 1) exists and is stable. (e) Describe in words how selection can maintain genetic variation (i.e., how the internal equilibrium can exist and be stable). 1 Q2. Consider the prey evolution model presented in class, dN dt = N ( r0 + r1x − N K − xP 1 + ̄xhN )∣∣∣∣x= ̄x , dP dt = P ( c ̄xN 1 + ̄xhN −d ) , d ̄x dt = Gx ∂ ∂x ( 1 N dN dt )∣∣∣∣x= ̄x , where N is the prey density, P is the predator density, ̄x is the prey mean trait, r0 is the basic growth rate, r1 is the coefficient for the defense cost, K is the density- dependence parameter, h is the handling time, c is the conversion efficiency, d is the mortality rate, and Gx is the additive genetic variance. (a) Derive dynamics of the mean prey trait, ̄x. (b) Write an equilibrium value of the mean prey trait, ̄x, by using ̄N and ̄P. (c) Conduct nondimensionalization for reducing the number of parameters. (d) What is an internal equilibrium (i.e., ̄N, ̄P > 0)? How does increasing the
predator mortality (d) affect the predator equilibrium density?
(e) Show the Jacobian matrix of the model.
(f) Calculate the Jacobian matrix of the model at the internal equilibrium.
(g) How does increasing the genetic variance Gx affect stability of the internal
equilibrium?
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