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

Exponential Growth

dN/dt=rN ◦r=intrinsic rate of increase ◦N=population size Why is this unrealistic? ◦In 24 hrs one bacteria would turn to 10 24

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)

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

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

Species 1

Species 2

4 competition scenarios K 1 / α 12 K2K2

Case 1: Species 1 outcompetes species 2

Case 2: species 2 outcompetes species 1 K1/ alph a 12 k2k2

Stable Equilibrium

Unstable Equilibrium

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 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

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 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