Stochastic lattice models for predator-prey coexistence and host-pathogen competition Uwe C. T

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Stochastic lattice models for predator-prey
coexistence and host-pathogen competition
Uwe C. Täuber, Virginia Tech, DMR-0308548
Research The classical Lotka-Volterra model
(1920, 1926) describes chemical oscillators,
predator-prey coexistence, and host-pathogen
competition. It 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. This can be
encoded into the following reaction scheme
() Predators die (A ? 0) spontaneously with
rate µ prey produce offspring (B ? BB) with
rate s both species interact via predation
A eats B and reproduces (AB ? AA) with 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 near a stable equilibrium state (top
figure). Monte Carlo computer simulation
results Top figure Time evolution of the
predator (a) and prey (b) densities for the
stochastic model () on a 1024 x 1024 square
lattice, starting with uniformly distributed
populations of equal densities (0.1), µ 0.2, s
0.1, ? 1.0. Predators and prey coexist, and
display stochastic oscillation about the
center. Bottom figure Space (horizontal)-time
(downwards) diagram for the predator and prey
densities (purple sites contain both species) a
simulation on a one-dimensional lattice with 512
sites with initial densities 1, µ 0.1, s 0.1,
? 0.1.
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Stochastic lattice models for predator-prey
coexistence and host-pathogen competition Uwe
C. Täuber, Virginia Tech, DMR-0308548

• Education and Outreach
• Postdoctoral associates Mauro Mobilia (partially
  funded through a Swiss fellowship), Ivan T.
  Georgiev, undergraduate research student Mark J.
  Washenberger, and IAESTE exchange student Ulrich
  Dobramysl (Johannes Kepler University Linz,
  Austria) crucially contributed to this
  interdisciplinary project.
• The PI has given invited talks and lecture
  courses at
• International 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
• Rudolf Peierls Centre for Theoretical Physics,
  University of
• Oxford (U.K.), October and November 2005
• Arnold Sommerfeld Center for Theoretical
  Physics, Ludwig
• Maximilians University Munich (Germany),
  December 2005
• Workshop Non-equilibrium dynamics of interacting
  particle
• systems, Isaac Newton Institute, Cambridge
  (U.K.), April 2006
• ASC Workshop Nonequilibrium phenomena in
  classical and
• quantum systems, Sommerfeld Center Munich,
  October 2006.
• 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 a stochastic
Lotka-Volterra system (, no local population
restrictions). Starting from a uniform spatial
distribution, islands of prey and pursuing
predators emerge, which grow into merging and
pulsating activity rings. The steady state is a
dynamic equilibrium, displaying erratic
population oscillations.