Transcript Stochastic lattice models for predator

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 → B+B) with rate σ; both species interact via
predation: A eats B and reproduces (A+B → A+A) with rate λ.
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, σ = 0.1, λ = 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, σ = 0.1, λ = 0.1.
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.
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.
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.