Math Insight

Quiz 7

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Total points: 1
  1. Bird flu (Avian influenza) is affecting chicken farms in Minnesota and you've been called in to help model the outbreak.

    Suppose you decide to use a metapopulation to model the fraction of chicken farms that currently have a bird flu infection as $\frac{dp}{dt} = C - E$ where $p$ is the proportion of farms that are infected, $C$ is the rate at which new infections occur and $E$ is the rate as which farms are able to eradicate influenza. Time $t$ is measured in months.

    From some preliminary assessments it appears that new infections occur when chickens are being imported from outside Minnesota (rather than new infections arising from chickens being transported from an infected Minnesota farm to an uninfected farm). Infected chickens are imported from outside farms at a rate of $0.1$ per month.

    What form should the infection rate $C$ take?
    $C=$

    Your plan to control bird flu is to eradicate the disease from infected farms at a monthly rate $x$, meaning $x$ is the fraction of infected farms from which the flu is eradicated per month. What form should the extinction rate $E$ take?
    $E =$

    Find all the equilibria:
    $p = $
    (If multiple answers, separate by commas.)

    Is it possible to completely eradicate the disease from Minnesota chicken farms?

    If so, how fast must you be eradicating the disease from individual infected farms to be able to completely eradicate the disease from Minnesota? (Your expression should be an inequality involving $x$.) If it is not possible, enter 'impossible'.

  2. To model the influence of habitat destruction of an endangered species, we let $p(t)$ be the fraction of habitat that is occupied by this species at time $t$, measured in days. Assume that species migrate from an occupied region of the habitat to an available region at rate of $0.14$ per day. Since each habitat region can continuously support the species for only fairly short time, any occupied region becomes unoccupied (by either the individuals dying or migrating to another region) with probability $0.12$ per day. In addition, due increased human activity, regions of this species habitat are being destroyed. Let $L$ represent the fraction of the original habitat that has been destroyed.

    1. In terms of $p$ and $L$, what fraction of the origin habitat is available for the species to migrate to?

      Therefore, what is the rate at which new habitat regions are being colonized by this species?

    2. Write a metapopulation model for the dynamics of the fraction $p$ of habitat regions occupied.
    3. What are the equilibria of the system?
      $p_{eq} = $

      (If multiple equilibria, separate by commas.)

      What are the biologically realistic equilibria?
      If $L \lt$
      , the biological realistic equilibria are
      $p_{eq} = $
      .
      If $L \gt $
      , the biological realistic equilibria are
      $p_{eq} = $
      .

    4. Which equilibria are stable?
      If $L \lt$
      , the stable equilibria are
      $p_{eq} = $
      .
      If $L \gt $
      , the stable equilibria are
      $p_{eq} = $
      .
    5. What happens to the fraction $p(t)$ of habitat occupied by this endangered species?
      If $L \lt$
      , then $p(t)$ approaches

      .
      If $L \gt $
      , then $p(t)$ approaches

      .
    6. The population of this endangered species will go extinct once
      percent of the habitat has been destroyed.

      Must all of the habitat of the endangered species be destroyed for it to go extinct?

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Math 2241, Spring 2023