Mathematical biology

For the logistic map $x_{n+1} = 2.5\,x_n(1 - x_n)$, what is the non-zero fixed point, in percent of the maximum population?

For which growth rates $r$ is the non-zero fixed point $x^* = 1 - 1/r$ of the logistic map locally asymptotically stable? (Choose all that apply.)

In the simple SIR model, an infection has $R_0 = 4$. What percentage of the population must be immune for an introduced infection not to grow (the herd immunity threshold)?

Which statement about the basic reproduction number $R_0 = \beta/\gamma$ is correct?

At the coexistence equilibrium of the Lotka–Volterra predator–prey model the Jacobian has eigenvalues $\pm i\sqrt{\alpha\gamma}$. What can we conclude from the linearisation alone?

In the competition model $x' = x(1 - x - a_{12}y)$, $y' = y(1 - y - a_{21}x)$, which parameters give a stable coexistence equilibrium?

A fish stock grows logistically with $r = 0.4$ per year and carrying capacity $K = 10\,000$ tonnes, and is harvested at a constant rate of $H$ tonnes per year. What is the largest $H$ for which a positive equilibrium exists (the maximum sustainable yield), in tonnes per year?

In the SIR model with $R_0 > 1$ and almost everyone initially susceptible, when does the number of infected people reach its maximum?

Which Lean statement says that Volterra's quantity is conserved along a solution of the predator–prey equations at time t?

Write final_size(R0) returning the fraction $1 - s$ of a population eventually infected in the SIR model, where $s \in (0, 1)$ solves $s = e^{-R_0(1 - s)}$ (return 0.0 when $R_0 \le 1$). Use bisection on $g(s) = s - e^{-R_0(1 - s)}$. Since $g(1) = 0$ as well, bracket the non-trivial root in $[0, 1/R_0]$, where $g(0) < 0 < g(1/R_0)$.