Evolutionary dynamics - solutions

  1. Verify that $x(t)=x_0e^{at}\text{ where }x_0=x(0)$ is the solution to the following differential equation: $$\frac{dx}{dt}=ax$$

    This follows directly through differentiation:

    $$\frac{dx}{dt}=x_0ae^{at}=a\frac{dx}{dt}$$

  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: $$\frac{dx}{dt}=ax\qquad\frac{dy}{dt}=by$$

    Similarly, follows from differentiation and substitution:

    $$\frac{d\rho}{dt}=\frac{\frac{dx}{dt}y-\frac{dy}{dt}x}{y^2}=\frac{axy-bxy}{y^2}=(a - b)\frac{x}{y}=(a-b)\rho$$

    Using question 1 gives the required result.

  3. What is the mean fitness in a constant sized population with:

    This question is just a substitution exercise asking for $\phi=ax+by$

    1. $x=.6$, $y=.4$ and $a=5$, $b=12$: $\phi=5\times.6 + 12\times.4=7.8$.
    2. $x=.2$, $y=.8$ and $a=6$, $b=3$: $\phi=6\times.2 + 3\times.8=3.6$.
    3. $x=.5$, $y=.5$ and $a=53$, $b=1$: $\phi=53\times.5 + 1\times.5=27$.
  4. Show that the following system of differential equations has 3 potential stable solutions (what are they?) for $x+y=1$: $$\frac{dx}{dt}=x(a - \phi)\qquad\frac{dy}{dt}=y(b - \phi)$$

    $$\frac{dx}{dt}=x(a-ax-b(1-x))=x(1-x)(a-b)$$

    The three stable solutions are solutions to the equation $\frac{dx}{dt}=0$:

    • $x=0$
    • $x=1$
    • $a=b$
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