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Wolves in Sweden

Wolves in Sweden. A Mathematical Analysis of the Threat of Extinction for Sweden’s Wolf Population From 1980 to 2001 Mark Hughes and Laura Gray. Endangered Wolf Population. Wolf population has been scarce since the 1940’s In 1980 there were only officially 4 known wolves left in Sweden

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Wolves in Sweden

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  1. Wolves in Sweden A Mathematical Analysis of the Threat of Extinction for Sweden’s Wolf Population From 1980 to 2001 Mark Hughes and Laura Gray

  2. Endangered Wolf Population • Wolf population has been scarce since the 1940’s • In 1980 there were only officially 4 known wolves left in Sweden • Why is the Population so Low? • Inbreeding depression • Low density • Human Activity • Swedish “Preditor Report” in 1999 reported that 6790 wolves were killed by humans in Sweden between 1827-1839 • Historically wolves were killed for bounties • In 1991 the population started to rapidly increase due to the introduction of a Russian wolf into the breeding pool • In 2001 there was officially 92 known wolves left in Sweden

  3. Mean Population of Wolves 1980-2001 1991

  4. Pure Birth Process For n Individuals, we have

  5. Ran a binomial function to determine whether an individual gave birth (1) or not (0) • If (1) then ran a Poisson function to determine litter size of the reproductive event • Poisson mean

  6. Pure Birth Process Simulations

  7. Pure Death Process Simulation

  8. Birth Death Process Simulations

  9. Imperfect Data • Unfortunately data collection was not perfect • Numbers from year to year did not add up • Two possibilities: • (1) Wolves are migrating in and out of the country • (2) Additional unrecorded births and death • We assumed that discrepancies in the data were due to unrecorded births and deaths • However, the treatment of migrating wolves would be similarly be accounted for in our model

  10. Revised Birth and Death Rates Revised Poisson Mean

  11. Birth and Death Process Equations The master equation for this process is The corresponding differential equation is With initial condition

  12. Using generating functions, the previous series of PDE’s yield functions for the E(population) and var(population) From our Data (Revised) we Obtained Pre 1991 Post 1991

  13. Extinction The Probability of Extinction is: If then Meaning that the population will eventually go extinct with probability 1 If then which is greater then zero, meaning even if the birth rate is larger then the death rate the probability of eventual extinction is positive

  14. Proportion of Wolf Population that Goes Extinct

  15. Complete Model Simulation (1980-2001)

  16. Further Analysis • statistical analysis could be performed to compare means and variances of simulation vs. continuous model • t-distribution test could be performed to determine whether constants rates could be assumed • additional (more detailed) data would aid in further in depth analysis

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