Competing SpeciesCoexistence and Chaos in Complex EcologiesUniversity of Arizona, March 25, 2010J.C. Sprott, J.A. Vano, J.C. Wildenberg, M.C. Anderson, J.K. NoelGroup MembersDavid DeCesariJennifer KanemaruDaniel WeissDaniel WeissCarolyn WiseMentor: Sarah MannUniversity of Arizona, March 25, 2010Modeling SpeciesCompetition in the Real WorldWhy Use Models?Predict instabilityParameters are chosen in a variety of waysUniversity of Arizona, March 25, 2010••• Can model relations with equations:••For Example: IN THE WILDUniversity of Arizona, March 25, 2010•For Example: • Owl, Snake, Frog, CaterpillarPopulation GraphFrogUniversity of Arizona, March 25, 2010OwlSnakeCatepillarTimePopulationWhat You Can’t See8AdaptationOccurs every 20 time stepsClampingOccurs at 10-6 to prevent extinxtionTimePopulationUniversity of Arizona, March 25, 2010Lotka-Volterra modelLotka-Volterra equations: University of Arizona, March 25, 2010x = prey, y = predator, t = timeVariation of Lotka-Volterra equations xi= Population size of species idxi/dt = Rate of change in size of population iri = Growth rateaij= Competition matrixUniversity of Arizona, March 25, 2010The Numerical MethodDiscretizeDevelop difference equation (Forward University of Arizona, March 25, 2010Develop difference equation (Forward Euler Method)Implement in MatlabDifference Equationsy would represent an University of Arizona, March 25, 2010represent an animal populationy0 would represent the initial conditionsApproximation of time derivative of x(t):Exact time derivative of x(t) from DE:Forward Eulerdx/dt ≈ (xn - xn-1) / ∆tExact time derivative of x(t) from DE:The iterative method:University of Arizona, March 25, 2010xn≈ xn-1+ f∆tMatlab ImplementationInitialization of population vector and competition matrixClamping at 10-6University of Arizona, March 25, 2010Clamping at 10-6AdaptationStep sizeWhy Forward Euler?•Biomass – The total mass of living organisms in a certain ecosystemBiomass and Biodiversity•Biodiversity - The diversity of plant and animal life in a specific habitatUniversity of Arizona, March 25, 2010Biomass (with adaption)TimeOur GraphUniversity of Arizona, March 25, 2010TimeTimeTheir GraphBiodiversity (with adaption)Our GraphUniversity of Arizona, March 25, 20101Their GraphBiodiversity vs BiomassTheirs (without adaptation) Ours (with adaptation)University of Arizona, March 25, 2010University of Arizona, March 25, 2010BiomassTime2*10^6With Adaptation:Without Adaptation:University of Arizona, March 25, 2010TimeTime2*10^4Without Adaptation:Biodiversity2*10^6With adaptation:University of Arizona, March 25, 20102*10^4Without adaptation:TimeResultsWhat effects do the following have on Biomass/Biodiversity?ClampingAdaptationWhat does this all mean?Why are our results relevant?AdaptationUniversity of Arizona, March 25, 2010Modifications Different changes in adaptation Changes in mutation (different number of time steps to implement mutation) University of Arizona, March 25, 2010of time steps to implement mutation) Changes in clamp sizeConclusion•Applications of models:•Competition for resources•Current ResearchUniversity of Arizona, March 25, 2010•Objects prone to crashesAcknowledgmentsSprott, J.C., J.C. Vano, J.C. Wildenberg, M.B. Anderson, and J.K. Noel. "Coexistence and chaos in complex ecologies." Physics Letters A335. (2005): 207-12. Web. 23 Feb 2010.University of Arizona, March 25, 2010335. (2005): 207-12. Web. 23 Feb 2010.Fox, J. W., and D. A. Vasseur. 2008. Character convergence under competition for nutritionally-essential resources. American Naturalist 172:667-680http://people.revoledu.com/kardi/tutorial/DifferenceEquation/WhatIsDifferenceEquation.htmQUESTIONS?University of Arizona, March 25,