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Differential Equations Modeling

Differential Equations Modeling. A Brief Introduction. Hares and Lynxes. One of the only truly long-term sets of ecological data comes to us from the Hudson Bay Trading Company.

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Differential Equations Modeling

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  1. Differential Equations Modeling A Brief Introduction

  2. Hares and Lynxes One of the only truly long-term sets of ecological data comes to us from the Hudson Bay Trading Company. They kept very good records, for over a century, of the number of lynx and hare pelts that they received from trappers in the region surrounding Hudson Bay. This data clearly suggests periodic increases and decreases in the hare-lynx populations. Mathematical population models reproduce this natural cycling tendency in predator–prey systems.

  3. Single Population, P What controls the rate at which a population grows? The size of the population! Simplest Model: The growth rate of the population P is proportional to P. Why does this Differential Equation say? What does it predict? What are its limitations? “Exponential Growth” P0

  4. Single Population, Better Model A Better model If P is very small compared to C, then The population of grows exponentially. P0

  5. Single Population, Better Model A Better model If P is very small compared to C, then The population of grows exponentially. . . . for a while! P0

  6. Single Population, P A Better model On the other hand, as P nears C, the population growth slows down C and the population stops growing altogether. The graph levels out at P=C. P0

  7. Single Population, P Logistic Growth C C is called the carrying capacity of the environment. It is the number of individuals that the environment can sustain before overcrowding and hunger limit the size of the population. This model of population growth is called “logistic growth.” P0

  8. A Simple Predator-Prey Model. Two interacting populations: Prey population--- “hares” --- H = number of hares Predator population --- “lynxes” --- L = number of lynxes. Robert M. May described a system of differential equations that models the interaction between the two species. The rates at which H and L are changing.

  9. A Simple Predator-Prey Model. No Lynxes? Logistic growth term---in the absence of lynxes, the hare population grows Logistically.

  10. A Simple Predator-Prey Model. Interaction terms---when hares and lynxes meet up, hares die and lynxes thrive

  11. A Simple Predator-Prey Model. Why the product of H and L? The total possible number of meetings of hares and lynxes is the product of H and L. Coefficentα is the percentage of the total possible number of meetings in which a rabbit dies. Coefficient β? (Not equal to α !) Note: α«1; β«1.

  12. A Simple Predator-Prey Model. The “death term”

  13. A Simple Predator-Prey Model. No Hares? In the absence of Hares, the Lynxes die off at an exponential rate. Why is this coefficient negative?

  14. A Simple Predator-Prey Model. Taking some reasonable values for the parameters and some initial conditions, we have . . . Lynxes Hares 2500 2000 1500 1000 500

  15. A Simple Predator-Prey Model. • What does this model predict about the populations of Lynxes and Hares • in the short term? • in the long term? • Notice the way that the cycles “interact.” Lynxes Hares 2500 2000 1500 1000 500

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