Ecology modeling
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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

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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 growth

Exponential Growth


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 growth

Logistic 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

Lotka-Volterra


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


Species 1

Species 1


Species 2

Species 2


4 competition scenarios

K1/α12

K2

4 competition scenarios


Case 1 species 1 outcompetes species 2

Case 1: Species 1outcompetes species 2


Case 2 species 2 outcompetes species 1

K1/alpha12

k2

Case 2: species 2outcompetes species 1


Stable equilibrium

Stable Equilibrium


Unstable equilibrium

Unstable Equilibrium


Predator prey

Predator-Prey


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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