Competing Species Coexistence and Chaos in Complex Ecologies J C Sprott J A Vano J C Wildenberg M C Anderson J K Noel University of Arizona March 25 2010 Group Members David DeCesari Jennifer Kanemaru Daniel Weiss Carolyn Wise Mentor Sarah Mann University of Arizona March 25 2010 Modeling Species Competition in the Real World Why Use Models Predict instability Parameters are chosen in a variety of ways University of Arizona March 25 2010 IN THE WILD Can model relations with equations For Example Owl Snake Frog Caterpillar University of Arizona March 25 2010 Population Graph Frog Owl Population Snake Catepillar Time University of Arizona March 25 2010 What You Can t See8 Adaptation Occurs every 20 time steps Population Clamping Occurs at 10 6 to prevent extinxtion Time University of Arizona March 25 2010 Lotka Volterra model Lotka Volterra equations x prey y predator t time University of Arizona March 25 2010 Variation of Lotka Volterra equations xi Population size of species i dxi dt Rate of change in size of population i ri Growth rate aij Competition matrix University of Arizona March 25 2010 The Numerical Method Discretize Develop difference equation Forward Euler Method Implement in Matlab University of Arizona March 25 2010 Difference Equations y would represent an animal population y0 would represent the initial conditions University of Arizona March 25 2010 Forward Euler Approximation of time derivative of x t dx dt xn xn 1 t Exact time derivative of x t from DE The iterative method xn xn 1 f t University of Arizona March 25 2010 Matlab Implementation Initialization of population vector and competition matrix Clamping at 10 6 Adaptation Step size Why Forward Euler University of Arizona March 25 2010 Biomass and Biodiversity Biomass The total mass of living organisms in a certain ecosystem Biodiversity The diversity of plant and animal life in a specific habitat University of Arizona March 25 2010 Biomass with adaption Our Graph Time Their Graph Time University of Arizona March 25 2010 Biodiversity with adaption Our Graph Their Graph 1 University of Arizona March 25 2010 Biodiversity vs Biomass Theirs without adaptation Ours with adaptation University of Arizona March 25 2010 Biomass With Adaptation Without Adaptation 2 10 6 Time Time University of Arizona March 25 2010 2 10 4 Biodiversity With adaptation 2 10 6 Without adaptation Time University of Arizona March 25 2010 2 10 4 Results What effects do the following have on Biomass Biodiversity Clamping Adaptation What does this all mean Why are our results relevant University of Arizona March 25 2010 Modifications Different changes in adaptation Changes in mutation different number of time steps to implement mutation Changes in clamp size University of Arizona March 25 2010 Conclusion Current Research Applications of models Competition for resources Objects prone to crashes University of Arizona March 25 2010 Acknowledgments Sprott J C J C Vano J C Wildenberg M B Anderson and J K Noel Coexistence and chaos in complex ecologies Physics Letters A 335 2005 207 12 Web 23 Feb 2010 Fox J W and D A Vasseur 2008 Character convergence under competition for nutritionallyessential resources American Naturalist 172 667680 http people revoledu com kardi tutorial DifferenceEquation WhatIsDifferenceEquation htm University of Arizona March 25 2010 QUESTIONS University of Arizona March 25 2010