Math Insight

Review problems for final exam

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  1. Give three examples of what models/theory can do.

  2. Give an example of something that models/theory cannot do.

  3. Describe one error in thinking about models that was discussed in the Caswell 1988 paper.

  4. Must all models have simplifying assumptions? Explain your reasoning.

  5. When would it be more appropriate to use a stochastic model instead of a deterministic one?

  6. When would it be more appropriate to use a discrete time model instead of a continuous time one?

  7. Consider two possible stochastic processes that describe how the quantity $X(t)$ changes with time.

    Process 1: \begin{align*} P(X(t + \Delta t) = y \,|\, X(t) = x) = \begin{cases} a \Delta t & \text{if $y=x+1$}\\ 1-(a+b)\Delta t & \text{if $y=x$}\\ b \Delta t & \text{if $y=x-1$}\\ 0 & \text{otherwise} \end{cases} \end{align*}

    Process 2: \begin{align*} P(X(t + \Delta t) = y \,|\, X(t) = x) = \begin{cases} a x \Delta t & \text{if $y=x+1$}\\ 1-(a+b)x\Delta t & \text{if $y=x$}\\ b x\Delta t & \text{if $y=x-1$}\\ 0 & \text{otherwise} \end{cases} \end{align*}

    1. Scenario A: Suppose $X(t)$ is the number of individuals in a population at time $t$. Which process would be the better model of the population growth? Why? Interpret what the model parameters ($a$ and $b$) mean biologically in this scenario.
    2. Scenario 2: Suppose $X(t)$ is the position of a molecule at time $t$. Which process would be the better model of the diffusion of the molecule? Why? Interpret what the model parameters ($a$ and $b$) mean biologically in this scenario.
    3. For each stochastic process, write down a deterministic approximation and solve the deterministic model given an initial condition $X(0)=c$.
    4. For scenario A, under what conditions is the deterministic model a good approximation to the stochastic model? Why?

      Under what conditions is the deterministic model a poor approximation to the stochastic model? What features of the stochastic model are completely missed by the deterministic model?

    5. For scenario B, if the molecule diffuses in any direction with equal likelihood, what must be true about the parameters $a$ and $b$? In that case, what does the deterministic model predict? Evaluate the merits of the deterministic model.

  8. Molecules are diffusing with diffusion coefficient $D=0.0052$ $\frac{mm^2}{s}$ in one dimension between $x=0$ mm and $x=0.062$ mm. Let $c(x,t)$ be the concentration at time $t$ (in seconds) and position $x$ (in millimeters). Imagine that at $x=0.062$ mm, the concentration is fixed at $c(0.062,t)=6456.68$ molecules per mm and that any molecule that reaches $x=0$ mm is removed. At time $t=0$, the concentration of molecules is approximately $c(x,0) = 970000x^2 + 44000x$.
    1. If a molecule starts at position $x=0.062$ mm, how long will it take, on average, to diffuse to $x=0$ (assuming it never moves above $x=0.062$ mm)?
    2. What is the flux at time $t=0$?

  9. Glanville fritillary butterfly
    The Glanville fritillary butterfly (Melitaea cinxia) which lives in meadow habitat patches on the Åland Islands in Finland is one of the best studied examples of a metapopulation.

    From some preliminary assessments it appears that new colonization events of empty meadows occur when butterflies travel between meadows (rather than traveling in from outside). Butterflies move at a rate of 0.4 per year.

    Let $p$ be the fraction of meadows that are currently occupied by butterflies.

    1. What form should the colonization rate C take?
    2. Assume that each meadow that is occupied by butterflies goes extinct at rate 0.3 per year, and that there is no ‘rescue effect’. What form should the extinction rate E take?
    3. Write down the full equation for $dp/dt$.
    4. Find all the equilibria.
    5. What fraction of meadows would you expect to see occupied by butterflies in the long term? Justify your conclusion.

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Math 2241, Spring 2023