Consider the prey evolution model presented in class, 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 non dimensionalization for reducing the number of parameters.
(d) What is an internal equilibrium? 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?
