Math Insight

Review problems for exam 3

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  1. The curcurbit beetle (Diabrotica speciosa) show reduced mating behavior in the presence of falling air pressure, so their behavior could be used to predict weather like rain. Imagine that it rains 10% of the days. Imagine that on days when it does eventually rain, you observe reduced mating activity 90% of the time. On days when it never rains, you observe reduced mating activity only 15% of the time. You now observe reduced mating activity in the beetles. What is the probability it will rain today?

  2. An invasive plant has begun to invade a patch of wildflowers. Assume that the each plant divides into two with probability per hour of 0.003 and that the probability per hour that it dies is 0.001.
    1. Write down a continuous time stochastic model for the dynamics of $N(t)$.
    2. Write down a deterministic model that approximates the dynamics of $N(t)$.
    3. Solve the deterministic model assuming that when $t=0$, there are 7 plants.
    4. How close to do expect the deterministic model to be to the stochastic model? What important features of the behavior of $N(t)$ are missing in the deterministic model?
    5. Solve the deterministic model assuming that when $t=0$, there are 950 plants. In this case, how close do you expect the deterministic model to approximate the stochastic model. Justify your answer.
    6. Assume that the patch of land could hold at most $K=1170$ plants. Assume that this carrying capacity $K$ modifies both the birth rate and the death rate, multiplying them by the factor $\left(1-\frac{N(t)}{K}\right)$.

      Modify both the stochastic model and the deterministic model to include the effect of a carrying capacity of $K=1170$ plants.

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Math 2241, Spring 2016