ES100: Community EcologyWhat Controls Population Size and Growth Rate (dN/dt)?Types of InteractionsCompetitionSpecies InteractionsSlide 6Slide 7Competition: Lotka-Volterra ModelSlide 9PowerPoint PresentationSlide 11Outcomes of Competition ModelPredator-preyPredator-Prey RelationshipsSlide 15Slide 16Slide 17Predatory-PreyMutualismSlide 20CommensalismSlide 22Assumptions of Lotka-Volterra ModelsSummary of Interaction Equations:Test you knowledge!Problems with Simple Logistic Growth1. Separate Births and DeathsSlide 282. Refine Carrying CapacityRemaining ProblemsGeneral Notes on Using ModelsCommunity DynamicsTrophic CascadeHow would we Model the Fox Population?Slide 35Slide 36Slide 37Simplified Temperate Forest Food Web What happens to when it’s a WEB instead of a CHAIN?Food Web doesn’t account for Keystone SpeciesSummarySlide 41ES100:Community Ecology8/22/07What Controls Population Size and Growth Rate (dN/dt)?•Density-dependent factors:•Intra-specific competition•food•Space•contagious disease•waste production•Interspecific competition•Other species interactions!•Density-independent factors:•disturbance, environmental conditions•hurricane•flood•colder than normal winterTypes of InteractionsCompetitionPredator-PreyMutualismCommensalismCompetitionNatural Selection minimizes competition!Species Interactions•How do we model them?•Start with logistic growth = r * N (1 – )= r * N (1 – )N KdN dt = r * N ( - )= r * N ( - )N KdN dtK K = r * N ( )= r * N ( )dN dtK-N KUse this equation for 2 different speciesSpecies Interactions•Population 1  N1•Population 2  N2•But the growth of one population should have an effect the size of the other population = r= r11 * N * N11 ( ) ( )dN1 dtK1-N1 K1 = r= r22 * N * N22 ( ) ( )dN2 dtK2-N2 K2Species Interactions•New term for interactionsa12  effect of population 2 on population 1 a21  effect of population 1 on population 2•Multiply new term by population sizethe larger population 2 is, the larger its effect on population 1 (and vice versa) a12 * N2 a21 * N1Competition: Lotka-Volterra ModelIf two species are competing, the growth of one population should reduce the size of the otherPopulation 1  N1Population 2  N2 = r= r11 * N * N11 dN1 dtK1 - N1 - a12 N2 K1 = r= r22 * N * N22 dN2 dtK2 - N2 - a21 N1 K2CompetitionIf two species are competing, the growth of one population should reduce the size of the otherPopulation 1  N1Population 2  N2 = r= r11 * N * N11 dN1 dtK1 - N1 - a12 N2 K1 = r= r22 * N * N22 dN2 dtK2 - N2 - a21 N1 K2Because this is a negative term, K is reducedBlue Area = Bluejay’s Carrying CapacityIt takes 1squirrel to use the portion of the carrying capacity occupied by 4 bluejays.aBS = 4 Interspecific competition regulates bluejay population⎟⎟⎠⎞⎜⎜⎝⎛−−=BSBBBBBKNNKNrdtdN4COMPETITIONGreen Area = Squirrel’s Carrying CapacityIt takes 4 bluejays to use the portion of the carrying capacity occupied by 1 squirrel.aSB =.25 Intraspecific competition regulates squirrel population⎟⎟⎠⎞⎜⎜⎝⎛−−=sBssssSKNNKNrdtdN 25.COMPETITIONOutcomes of Competition ModelMany possible outcomes, depends on the balance of:r1 vs r2K1 vs K2a21 vs a12a12 > 1 Interspecific competition dominates population size of species 1a12 < 1 Intraspecific competition dominates population size of species 1a12 is the per capita effect of species 2 on the the pop’n growthrate of species 1, measured relative to the effect of species 1.Predator-preyPredator-Prey Relationships•Prey defenses: avoid conflict!•coevolution•as predator evolves, prey evolves to evade it•warning coloration and mimicry•CamouflageRed = Fox’s Carrying CapacityIt takes 10 rabbits to support 1 foxaFR =.10 ⎟⎟⎠⎞⎜⎜⎝⎛+−=FRFFFFFKNNKNrdtdN 10.Predator-PreyPredator-PreyYellow = Rabbits Carrying CapacityIt takes 10 rabbits to support 1 foxaRF = 10 ⎟⎟⎠⎞⎜⎜⎝⎛−−=RFRRRRRKNNKNrdtdN 10Predator-PreyPredator-Prey•Bottom-up vs. Top-Down control•Predators can promote diversity by keeping competition in checkPredator-Prey RelationshipsPredatory-PreyIf it is a predator-prey relationship, then the two populations have opposite effects on one anotherPrey (N1)Predator (N2) = r= r11 * N * N11 dN1 dtK1 - N1 - a12 N2 K1 = r= r22 * N * N22 dN2 dtK2 - N2 + a21 N1 K2Because this is a negative term, K is reducedBecause this is a positive term, K is increasedMutualismBoth species benefitMutualismIf it is a mutually beneficial relationship, then the two populations increase each other’s sizePopulation 1  N1tiPopulation 2  N2 = r= r11 * N * N11 dN1 dtK1 - N1 + a12 N2 K1 = r= r22 * N * N22 dN2 dtK2 - N2 + a21 N1 K2Because this is a positive term, K is increasedBecause this is a positive term, K is increasedCommensalismOne species benefits, the other is unaffectedCommensalismIf the relationship is commensalistic, one species benefits (the commensal) and the other is unaffected Population 1  N1Population 2  N2 = r= r11 * N * N11 dN1 dtK1 - N1 + a12 N2 K1 = r= r22 * N * N22 dN2 dtK2 - N2K2Because this is a positive term, K is increasedBecause there is no a21 term, K is unchangedAssumptions of Lotka-Volterra ModelsAll assumptions of logistic growth model… plus:Interaction coefficients, carrying capacities, and intrinsic growth rates are constant.Summary of Interaction Equations:Competition: (- , -) Predator/Prey: (+, -)Mutualism: (+, +)Commensalism: (+, 0)⎟⎟⎠⎞⎜⎜⎝⎛−=121211111?KNaNKNrdtdN⎟⎟⎠⎞⎜⎜⎝⎛−=212122222?KNaNKNrdtdNTest you knowledge!What type of relationship– what equation to use?A coati eats tree fruit.Your dog has a fleaYou use a fast bicyclist to “draft” off ofProblems with Simple Logistic Growth1. Births and deaths not separated-you might want to look at these processes separately-predation may have no effect on birth rate2. Carrying capacity is an arbitrary, set value3. No age structure1. Separate Births and Deaths= Births - DeathsBirths = b*NDeaths = d*NdN dtBirths and deaths may be density dependent1. Separate Births and Deaths= Births - DeathsBirths = b*NDeaths = d*NdN dtBirths rate may be density dependentDeath rate may be dominated by predator effectsExample:Births = b*N(1-