Evolutionary dynamics - exercises

  1. Verify that $x(t)=x_0e^{at}\text{ where }x_0=x(0)$ is the solution to the following differential equation: $$\frac{dx}{dt}=ax$$
  2. Verify that $\rho(t) = \rho_0e^{(a-b)t}\text{ where }\rho_0=\rho(0)$ is the expression for $\rho(t)=\frac{x(t)}{y(t)}$ where: $$\frac{dx}{dt}=ax\qquad\frac{dy}{dt}=by$$
  3. What is the mean fitness in a population with:

    1. $x=.6$, $y=.4$ and $a=5$, $b=12$.
    2. $x=.2$, $y=.8$ and $a=6$, $b=3$.
    3. $x=.5$, $y=.5$ and $a=53$, $b=1$.
  4. Show that the following system of differential equation has 3 potential stable solutions (what are they): $$\frac{dx}{dt}=x(a - \phi)\qquad\frac{dy}{dt}=y(b - \phi)$$