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.

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Math 2241, Spring 2016