Complex patterns and fluctuations in stochastic lattice models for predator-prey competition and coexistence PI: Uwe C. T

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Complex patterns and fluctuations in stochastic
lattice models for predator-prey competition and
coexistence PI Uwe C. Täuber, Virginia Tech,
DMR-0308548
Research The classical Lotka-Volterra model
(1920, 1926) for predator-prey interactions
predicts regular population cycles, but is
unstable against perturbations. A realistic
description requires spatial structure,
permitting traveling pursuit and evasion waves,
and inclusion of stochastic noise. Reactions
predators A ? 0, death,
rate µ () prey B ? BB,
birth, rate s predation AB ?
AA, A eats B and reproduces, rate ?.
Computer simulations show that predator-prey
coexistence is characterized by complex patterns
of competing activity fronts (see movie) that in
finite systems induce erratic population
oscillations (bottom figure). Spatial constraints
(finite carrying capacities) may induce predator
extinction (top figure, left). Monte Carlo
computer simulation results Top figure Time
evolution of the predator (A) and prey (B)
densities for the stochastic model () on a
square lattice, starting with uniformly
distributed populations of equal densities, fixed
µ and s, but varying predation rate ?. Predators
and prey may coexist, with the final state being
approached either in a spiral (for large ?), or
linearly (intermediate ?) or the predators
become extinct A ? 0, B ? 1 (small ?
inefficient predation). Bottom figure Erratic
predator (A) population oscillations on various
square lattices, µ0.1, s 4.0, ?1.0. The
oscillation amplitudes decrease towards zero as
the system becomes larger.
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Complex patterns and fluctuations in stochastic
lattice models for predator-prey competition and
coexistence PI Uwe C. Täuber, Virginia Tech,
DMR-0308548

• Education and Outreach
• Two postdoctoral associates contributed crucially
  to this inter-disciplinary project Mauro Mobilia
  (partial funding through a Swiss National Science
  Foundation fellowship), and Ivan Georgiev
  (through NSF DMR-0414122 PIs B. Schmittmann and
  R.K.P. Zia). Our results have been or will be
  reported at
• NSF workshop The Role of Theory in Biological
  Physics and
• Materials, Tempe, AZ, May 2004
• NSF workshop Opportunities in Materials Theory
  WOMT04,
• Arlington, VA, October 2004
• APS March Meeting, Los Angeles, CA, March 2005
• Summer school Ageing and the Glass Transition,
• Luxembourg, September 2005
• Workshop Applications of Methods of Stochastic
  Systems and
• Statistical Physics in Biology, Notre Dame, IN,
  October 2005.
• The PI visited Computer Technology classes at
  Blacksburg Middle School, sixth and seventh
  grade, and explained how computers and the
  internet are incorporated into university
  teaching and research. Computer simulation movies
  for the stochastic Lotka-Volterra system were
  shown as illustration.

Movie Time evolution of the stochastic
Lotka-Volterra system () for efficient predation
(large ?). Starting from a uniform population
distribution, islands of prey and pursuing
predators form, which develop into growing and
merging activity rings. The steady state is a
dynamic equilibrium of moving prey fronts
followed by predators leaving behind empty
(black) sites for the next prey wave.