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Ecology Modeling. February 25-March 4, 2010. Models are not the whole picture They use assumptions Exponential growth Exponential growth Logistic growth Competition models Lotka-Volterra Predator-prey models Predator-prey Theta Logistic. Ecology Models. Exponential Growth. dN/dt=rN

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ecology modeling

Ecology Modeling

February 25-March 4, 2010

ecology models

Models are not the whole picture

    • They use assumptions
  • Exponential growth
    • Exponential growth
    • Logistic growth
  • Competition models
    • Lotka-Volterra
  • Predator-prey models
    • Predator-prey
    • Theta Logistic
Ecology Models
exponential growth1

dN/dt=rN

    • r=intrinsic rate of increase
    • N=population size
  • Why is this unrealistic?
    • In 24 hrs one bacteria would turn to 1024
Exponential Growth
logistic population growth

dN/dt=rN(K-N/K)

    • Carrying capacity
  • Continuous growth
    • Overlapping generations (humans)
  • Discrete growth
    • No overlapping generations (some insects)
Logistic Population Growth
lotka volterra competition

Two species using each other’s resources

    • Not directly hurting each other
  • Pg 50 has equations
    • What do dN/dt, r, N and K mean? (from last week)
  • Negative-negative interaction
Lotka-Volterra Competition
competition coefficients

Competition coefficient

    • How many of species 1 is equal to species 2
    • α and β
    • When α and β are 0, we have logistic growth
  • To visualize we use isoclines
Competition coefficients
predator prey assumptions

Growth of prey only limited by predation

  • Assumptions
    • Predator specialist that only exists with prey
    • Individual predators consume infinite prey
    • Prey and predators encounter each other randomly
Predator-prey assumptions
predator prey relationships

Prey

    • dN/dt=rN-CNP
  • Predator
    • dN/dt=gCNP-dP
  • r=rate of increase for prey
  • N=population size of prey
  • C=constant rate of prey being captured
  • P=population size of predator
  • d=exponential death rate for predator (predator starvation rate for the stupid or diseased ones)
  • g=a constant depicting conversion of captured prey to predator population growth
  • Does not include a carrying capacity
Predator Prey Relationships
theta logistic

Incorporates carrying capacity

  • Assumptions for theta-logistic
    • Predator population density does not affect an individual predator’s chances of birth and death directly
    • Number of surviving offspring produced by a predator is directly proportional to the amount of prey it consumes
Theta-logistic
theta logistic predator prey

Prey

    • dN/dt=rN{1-(N/K)θ}-fP
  • Predator
    • dP/dt=gP[f-D]
  • K=carrying capacity
  • θ=how birth and death change with changing population size prey
  • f=number of prey eaten based on prey density
  • g=minimum prey needed by predator to survive
  • d=minimum per capita prey intake for stable predator population

f uses C and h

Theta-Logistic Predator-Prey
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