Incollection,
Characterizing spatiotemporal patterns in host-parasitoid models
Abstract Book of the XXIII IUPAP International Conference on Statistical Physics, Genova, Italy, (9-13 July 2007)
Abstract
We study spatio-temporal patterns in a model of two interacting populations by
coarse-graining. Such patterns are ubiquitous in Nature, for example in surface catalysis
reactions and calcium signalling in living cells. Understanding their properties is of
importance in particular in ecology. Our results apply to many three-state models such as
the rock-paper-scissors model and the three-state voter model.
The patterns lead to a division of space into three kinds of regions. In spatially discrete
models with reactions limited to between nearest-neighbour sites, these can be directly
defined by the three states of the model. Without this restriction, coarse-graining is
needed to establish and extract meaningful patterns. Of this, examples are found in ecology,
where one uses particular dispersal distances. Vortices arise as the boundary points of three
domains, forming the endpoints of the three different separating domain walls. They allow to
study the statistics and dynamics of the patterns, and present interesting statistics, such
as the vortex number, velocities and the diffusion properties.
We consider a host-parasitoid model, equally applicable to prey-predator systems. It is
defined on a square lattice with periodic boundary conditions. Each lattice site is in one
of three possible states, empty ($e$), host ($h$), or parasitized host ($p$). The
populations evolve by parallel, stochastic updates via the cycle $e h p e$.
Both hosts and parasitoids spread with a coupling decaying exponentially with distance.
Death of parasitized hosts ($p e$) occurs independently at each site.
Two regimes exist in the parameter space: a homogeneous and a patterned one. In the former,
both populations are roughly uniformly distributed in space, no spatial patterns are visible,
and the population densities fluctuate around their averages. In the patterned case, the
spatial distributions can be characterized by typical lengthscales (for the $e$, $h$, and $p$
separately). The densities exhibit sustained oscillations with a slowly fluctuating
amplitude, also noticeable in the vortex number.
Poincaré maps of the densities reveal that the discrete temporal dynamics can be understood
by a locally linear manifold, in the vicinity of the fixed point. In the patterned state,
the associated eigenvalues imply a stable fixed point with a time-scale separation between
the oscillation and decay time scales. Together with noise this produces the observed
sustained oscillations. The fluctuating amplitudes have also a spatial interpretation which
relates to synchronization of locally coherent regions as coupled nonlinear oscillators.
It is an important question of theoretical ecology what kind of coarse-grained
Lotka-Volterra-description would apply to multi-species interactions. We answer to this by
considering the possible eigenvalue structures in the presence or absence of patterns.
