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    SilvaHoover PredatorPrey

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    ugwak
    The document discusses Lotka-Volterra predator-prey models. It covers: 1) The history of the models developed by Alfred Lotka and Vito Volterra in the 1920s. 2) The basic 2-species model equations involving prey and predator populations. 3) Assumptions of the models including unlimited prey food supply and predation being the only factor affecting predator population. 4) Examples of phase plots showing the oscillating relationship between prey and predator populations over time. 5) An extension of the model to include a third species as a super-predator.

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    SilvaHoover PredatorPrey

    Uploaded by

    ugwak
    62 views17 pages
    The document discusses Lotka-Volterra predator-prey models. It covers: 1) The history of the models developed by Alfred Lotka and Vito Volterra in the 1920s. 2) The basic 2-species model equations involving prey and predator populations. 3) Assumptions of the models including unlimited prey food supply and predation being the only factor affecting predator population. 4) Examples of phase plots showing the oscillating relationship between prey and predator populations over time. 5) An extension of the model to include a third species as a super-predator.

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    predator prey dependence

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    The document discusses Lotka-Volterra predator-prey models. It covers: 1) The history of the models developed by Alfred Lotka and Vito Volterra in the 1920s. 2) The basic 2-species model equations involving prey and predator populations. 3) Assumptions of the models including unlimited prey food supply and predation being the only factor affecting predator population. 4) Examples of phase plots showing the oscillating relationship between prey and predator populations over time. 5) An extension of the model to include a third species as a super-predator.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    62 views17 pages

    SilvaHoover PredatorPrey

    Uploaded by

    ugwak
    The document discusses Lotka-Volterra predator-prey models. It covers: 1) The history of the models developed by Alfred Lotka and Vito Volterra in the 1920s. 2) The basic 2-species model equations involving prey and predator populations. 3) Assumptions of the models including unlimited prey food supply and predation being the only factor affecting predator population. 4) Examples of phase plots showing the oscillating relationship between prey and predator populations over time. 5) An extension of the model to include a third species as a super-predator.

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    Lotka-Volterra

    Predator-Prey Problems
    By: Alexandra Silva and Dani Hoover
    Intro to Systems ESE 251
    11/24/09
    History
    Alfred Lotka

    -American -Italian mathematician


    biophysicist -Proposed the
    -Proposed the predator-prey model in
    predator-prey model Vito Volterra
    1926
    in 1925
    2-Species Models
    Equations and Variables
    X = ax bxy
    Y = -cy + dxy
    X: the population of prey
    Y: the population of predators
    a: natural growth rate of prey in the absence of
    predation
    b: death rate due to predation
    c: natural death rate of predators in the
    absence of prey
    d: growth rate due to predation
    Assumptions
    The prey always has an unlimited supply of
    food and reproduces exponentially
    The food supply of the predators depend only
    on the prey population (predators eat the prey
    only)
    The rate of change of the population is
    proportional to the size of the population
    The environment does not change in favor of
    one species
    Phase Plot of Predator vs. Prey
    Set parameters 3
    Phase plot

    a=b=c=d=1
    2.5
    Set initial conditions:
    x=2 (prey), y=2 2
    Equilibrium point (1,1)

    (predators) Y, predator
    1.5

    Equilibrium Point:
    1

    x=(c/d), y= (a/b)
    0.5
    Counter-clockwise
    motion 0
    0 0.5 1 1.5
    X, prey
    2 2.5 3
    Steady-State Orbit explanation

    A = Too many predators.


    B = Too few prey.
    C = Few predator and
    prey; prey can grow.
    D= Few predators,
    ample prey.

    http://www.vanderbilt.edu/AnS/psychology/cogsci/chao
    s/workshop/2DS.html
    Phase Plot: Case 2 Phase plot
    2

    1.8

    1.6

    1.4

    1.2
    When initial
    Y, predator

    0.8
    1
    conditions equal the
    0.6 equilibrium point:
    0.4

    0.2 Parameters:
    0
    0 0.2 0.4 0.6 0.8 1
    X, prey
    1.2

    Solution to L-V equations


    1.4 1.6 1.8 2
    a=b=c=d=1
    2

    1.8
    Prey
    Predator Initial conditions:
    1.6

    1.4 x=1 (prey), y=1


    1.2
    (predators)
    population

    0.8

    0.6

    0.4

    0.2

    0
    0 5 10 15 20 25 30 35 40 45 50
    time
    Phase plot
    2.5

    Phase Plot: Case 3 2

    1.5

    Y, predator
    When parameters are
    1

    changed:
    Parameters: 0.5

    a=c=d=1, b=2 0
    0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
    X, prey

    *Increase the death rate 4.5


    Solution to L-V equations

    Prey

    due to predation 4 Predator

    3.5

    Initial Conditions: 3

    x=2 (prey), y=2


    population
    2.5

    (predator) 1.5

    0.5

    0
    0 5 10 15 20 25 30 35 40 45 50
    time
    Phase Plot: Case 4

    When a species dies Phase plot


    600

    out:
    500
    Parameters:
    400
    a=1, b=c=d=1

    Y, predator
    Initial Conditions: 300

    x=50 (prey), y=500 200

    (predator) 100

    0
    -10 0 10 20 30 40 50
    X, prey

    Prey dies, therefore


    predator dies too.
    3-Species Model (Super-predator)
    Equations and Variables (for 3-
    species model)
    X= ax-bxy (prey-- mouse)
    Y= -cy+dxy-eyz (predator-- snake)
    Z= -fz+gxz (super-predator-- owl)
    a: natural growth rate of prey in the absence of
    predation
    b: death rate due to predation
    c: natural death rate of predator
    d: growth rate due to predation
    e: death rate due to predation (by super-predator)
    f: natural death rate of super-predator
    g: growth rate due to predation
    Phase Plot of Prey vs. Predator vs.
    Super-predator Phase plot Solution to L-V equations
    5
    Prey
    4.5 Predator
    Super Predator
    5
    4

    4 3.5
    Z, predator

    population
    3
    3
    2.5

    2
    2

    1 1.5
    2
    4 1
    1.5
    3
    1 0.5
    2 0 5 10 15 20 25 30 35 40 45 50
    0.5 time
    Y, super predator 1
    X, prey

    Parameters:
    a=b=c=d=1, e=0.5, f=0.01, g=0.02

    Initial Condition:
    X=1, Y=1, Z=1
    Problems with Lotka-Volterra Models
    The Lotka-Volterra model has infinite cycles
    that do not settle down quickly. These cycles
    are not very common in nature.
    Must have an ideal predator-prey system.
    In reality, predators may eat more than one
    type of prey
    Environmental factors
    Thank you
    Thank you to Anatoly for helping us with this
    presentation and helping us to make
    programs in MATLAB.
    Questions?
    Sources:
    http://www.cs.unm.edu/~forrest/classes/cs365/
    CS%20365/Lectures_files/lotka-volterra.pdf
    http://www.vanderbilt.edu/AnS/psychology/co
    gsci/chaos/workshop/2DS.html
    http://en.wikipedia.org/wiki/Lotka%E2%80%93
    Volterra_equation
    http://isolatium.uhh.hawaii.edu/m206L/lab8/p
    redator/predator.htm
    http://www4.ncsu.edu/eos/users/w/white/www
    /white/ma302/less10.PDF
    http://www.cs.unm.edu/~forrest/classes/cs365/
    CS%20365/Lectures_files/lotka-volterra.pdf

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