8,901 research outputs found
Lotka-Volterra predator-prey model with periodically varying carrying capacity
We study the stochastic spatial Lotka-Volterra (LV) model for predator-prey
interaction subject to a periodically varying carrying capacity. The LV model
with on-site lattice occupation restrictions that represent finite food
resources for the prey exhibits a continuous active-to-absorbing phase
transition. The active phase is sustained by spatio-temporal patterns in the
form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional
lattice are utilized to investigate the effect of seasonal variations of the
environment on species coexistence. The results of our simulations are also
compared to a mean-field analysis. We find that the parameter region of
predator and prey coexistence is enlarged relative to the stationary situation
when the carrying capacity varies periodically. The stationary regime of our
periodically varying LV system shows qualitative agreement between the
stochastic model and the mean-field approximation. However, under periodic
carrying capacity switching environments, the mean-field rate equations predict
period-doubling scenarios that are washed out by internal reaction noise in the
stochastic lattice model. Utilizing visual representations of the lattice
simulations and dynamical correlation functions, we study how the pursuit and
evasion waves are affected by ensuing resonance effects. Correlation function
measurements indicate a time delay in the response of the system to sudden
changes in the environment. Resonance features are observed in our simulations
that cause prolonged persistent spatial correlations. Different effective
static environments are explored in the extreme limits of fast- and slow
periodic switching. The analysis of the mean-field equations in the
fast-switching regime enables a semi-quantitative description of the stationary
state.Comment: 17 pages, 20 figure
Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes
Spatial variation in population densities across a landscape is a feature of many ecological systems, from
self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of
environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of
populations. However the ways in which abiotic and biotic factors interact to determine the existence
and nature of spatial patterns in population density remain poorly understood. Here we present a new
approach to studying this question by analysing a predator–prey patch-model in a heterogenous
landscape. We use analytical and numerical methods originally developed for studying nearest-
neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns
emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a
rich and highly complex array of coexisting stable patterns, located within an enormous number of
unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable
basins of attraction, making them significant in applications. We are able to identify mechanisms for
these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby
landscape heterogeneity can modulate the spatial scales at which these processes operate to structure
the populations
Invasion controlled pattern formation in a generalized multi-species predator-prey system
Rock-scissors-paper game, as the simplest model of intransitive relation
between competing agents, is a frequently quoted model to explain the stable
diversity of competitors in the race of surviving. When increasing the number
of competitors we may face a novel situation because beside the mentioned
unidirectional predator-prey-like dominance a balanced or peer relation can
emerge between some competitors. By utilizing this possibility in the present
work we generalize a four-state predator-prey type model where we establish two
groups of species labeled by even and odd numbers. In particular, we introduce
different invasion probabilities between and within these groups, which results
in a tunable intensity of bidirectional invasion among peer species. Our study
reveals an exceptional richness of pattern formations where five quantitatively
different phases are observed by varying solely the strength of the mentioned
inner invasion. The related transition points can be identified with the help
of appropriate order parameters based on the spatial autocorrelation decay, on
the fraction of empty sites, and on the variance of the species density.
Furthermore, the application of diverse, alliance-specific inner invasion rates
for different groups may result in the extinction of the pair of species where
this inner invasion is moderate. These observations highlight that beyond the
well-known and intensively studied cyclic dominance there is an additional
source of complexity of pattern formation that has not been explored earlier.Comment: 8 pages, 8 figures. To appear in PR
Cross-diffusion driven instability in a predator-prey system with cross-diffusion
In this work we investigate the process of pattern formation induced by
nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra
predator-prey kinetics. We show that the cross-diffusion term is responsible of
the destabilizing mechanism that leads to the emergence of spatial patterns.
Near marginal stability we perform a weakly nonlinear analysis to predict the
amplitude and the form of the pattern, deriving the Stuart-Landau amplitude
equations. Moreover, in a large portion of the subcritical zone, numerical
simulations show the emergence of oscillating patterns, which cannot be
predicted by the weakly nonlinear analysis. Finally when the pattern invades
the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude
equation which is able to describe the shape and the speed of the wave.Comment: 15 pages, 5 figure
Spatiotemporal dynamics in a spatial plankton system
In this paper, we investigate the complex dynamics of a spatial plankton-fish
system with Holling type III functional responses. We have carried out the
analytical study for both one and two dimensional system in details and found
out a condition for diffusive instability of a locally stable equilibrium.
Furthermore, we present a theoretical analysis of processes of pattern
formation that involves organism distribution and their interaction of
spatially distributed population with local diffusion. The results of numerical
simulations reveal that, on increasing the value of the fish predation rates,
the sequences spots spot-stripe mixtures
stripes hole-stripe mixtures holes wave pattern is
observed. Our study shows that the spatially extended model system has not only
more complex dynamic patterns in the space, but also has spiral waves.Comment: Published Pape
- …