Math Insight

Math 8540, initial test, version 1

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  1. A population of antelope of size $a(t)$ and a population of caribou of size $c(t)$ compete for grazing space on a island, where $t$ is time in years. If either population were alone on the island, the island could support a population of 600 antelope (i.e., the carrying capacity for antelope alone would be 600) or a population of 1200 caribou (i.e., the carrying capacity for caribou alone would be 1200). Since the animals don't have exactly the same diet, each additional antelope effectively decreases the carrying capacity for caribou by only $\frac{1}{5}$ and vice versa. The result is that the competition of the species can be modeled by \begin{align*} \diff{ a }{t} &= 0.2a\left(1 - \frac{ a + c/5 }{ 600 } \right)\\ \diff{ c }{t} &= 0.4c\left(1 - \frac{ c + a/5 }{ 1200 } \right). \end{align*} To analyze the result of this competition do the following.
    1. Calculate the nullclines and plot them on the phase plane. To help you, we show a phase plane plot below. Dotted lines are included as a hint, but they are not intended to be complete. Be sure to label the nullclines.
    2. Identify all equilibria. (Only equilibria where both population sizes are non-negative count.) Give their values and show them in the phase plane.
    3. Sketch a direction arrow in each of the regions of the phase plane divided by the nullclines. You only need to consider the part of the phase plane where $a(t) \ge 0 $ and $c(t) \ge 0$.
    4. Sketch a direction arrow on each segment of each nullcline.
    5. Sketch a plausible solution in the phase plane starting at the initial condition of 250 antelope and 1050 caribou, i.e., $(a(0),c(0))=(250,1050)$. The solution should be consistent with your direction arrows.

  2. Let $E(t)$ be the firing rate of an excitatory population and $I(t)$ be the firing rate of an inhibitory population. We model their evolution as \begin{align*} \tau_e\diff{E}{t} &= -E + \Phi(W_{ee}E - W_{ei}I + P_e)\\ \tau_i \diff{I}{t} &= -I + \Phi(W_{ie}E - W_{ii}I + P_i) \end{align*} where $\Phi(x)$ is a sigmoidal function with $\Phi'(x) \ge 0$, $\lim_{x \to -\infty} \Phi(x)=0$, and $\lim_{x \to \infty} \Phi(x) = M$. The $W_{jk}$ are positive coupling parameters, the $P_j$ are external inputs, the $\tau_j$ are positive time constants, and $M$ is a positive maximum firing rate.

    Assume that $(E,I) = (E_0, I_0)$ is an equilibrium.

    1. Calculate a condition for $(E_0,I_0)$ to be a saddle point.
    2. Calculate a condition for a Hopf bifurcation of the equilibrium $(E_0,I_0)$, i.e., where the equilibrium changes from a stable spiral to an unstable spiral.

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