1,514 research outputs found
Fisher-KPP dynamics in diffusive Rosenzweig-MacArthur and Holling-Tanner models
We prove the existence of traveling fronts in diffusive Rosenzweig-MacArthur
and Holling-Tanner population models and investigate their relation with fronts
in a scalar Fisher-KPP equation. More precisely, we prove the existence of
fronts in a Rosenzweig-MacArthur predator-prey model in two situations: when
the prey diffuses at the rate much smaller than that of the predator and when
both the predator and the prey diffuse very slowly. Both situations are
captured as singular perturbations of the associated limiting systems. In the
first situation we demonstrate clear relations of the fronts with the fronts in
a scalar Fisher-KPP equation. Indeed, we show that the underlying dynamical
system in a singular limit is reduced to a scalar Fisher-KPP equation and the
fronts supported by the full system are small perturbations of the Fisher-KPP
fronts. We obtain a similar result for a diffusive Holling-Tanner population
model. In the second situation for the Rosenzweig-MacArthur model we prove the
existence of the fronts but without observing a direct relation with Fisher-KPP
equation. The analysis suggests that, in a variety of reaction-diffusion
systems that rise in population modeling, parameter regimes may be found when
the dynamics of the system is inherited from the scalar Fisher-KPP equation
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
Sensing and decision-making in random search
While microscopic organisms can use gradient-based search to locate
resources, this strategy can be poorly suited to the sensory signals available
to macroscopic organisms. We propose a framework that models search-decision
making in cases where sensory signals are infrequent, subject to large
fluctuations, and contain little directional information. Our approach
simultaneously models an organism's intrinsic movement behavior (e.g. Levy
walk) while allowing this behavior to be adjusted based on sensory data. We
find that including even a simple model for signal response can dominate other
features of random search and greatly improve search performance. In
particular, we show that a lack of signal is not a lack of information.
Searchers that receive no signal can quickly abandon target-poor regions. Such
phenomena naturally give rise to the area-restricted search behavior exhibited
by many searching organisms
The influence of dispersal on a predator-prey system with two habitats
Dispersal between different habitats influences the dynamics and stability of
populations considerably. Furthermore, these effects depend on the local
interactions of a population with other species. Here, we perform a general and
comprehensive study of the simplest possible system that includes dispersal and
local interactions, namely a 2-patch 2-species system. We evaluate the impact
of dispersal on stability and on the occurrence of bifurcations, including
pattern forming bifurcations that lead to spatial heterogeneity, in 19
different classes of models with the help of the generalized modelling
approach. We find that dispersal often destabilizes equilibria, but it can
stabilize them if it increases population losses. If dispersal is nonrandom,
i.e. if emigration or immigration rates depend on population densities, the
correlation of stability with migration rates is positive in part of the
models. We also find that many systems show all four types of bifurcations and
that antisynchronous oscillations occur mostly with nonrandom dispersal
Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka-Volterra Models
We study the general properties of stochastic two-species models for
predator-prey competition and coexistence with Lotka-Volterra type interactions
defined on a -dimensional lattice. Introducing spatial degrees of freedom
and allowing for stochastic fluctuations generically invalidates the classical,
deterministic mean-field picture. Already within mean-field theory, however,
spatial constraints, modeling locally limited resources, lead to the emergence
of a continuous active-to-absorbing state phase transition. Field-theoretic
arguments, supported by Monte Carlo simulation results, indicate that this
transition, which represents an extinction threshold for the predator
population, is governed by the directed percolation universality class. In the
active state, where predators and prey coexist, the classical center
singularities with associated population cycles are replaced by either nodes or
foci. In the vicinity of the stable nodes, the system is characterized by
essentially stationary localized clusters of predators in a sea of prey. Near
the stable foci, however, the stochastic lattice Lotka-Volterra system displays
complex, correlated spatio-temporal patterns of competing activity fronts.
Correspondingly, the population densities in our numerical simulations turn out
to oscillate irregularly in time, with amplitudes that tend to zero in the
thermodynamic limit. Yet in finite systems these oscillatory fluctuations are
quite persistent, and their features are determined by the intrinsic
interaction rates rather than the initial conditions. We emphasize the
robustness of this scenario with respect to various model perturbations.Comment: 19 pages, 11 figures, 2-column revtex4 format. Minor modifications.
Accepted in the Journal of Statistical Physics. Movies corresponding to
Figures 2 and 3 are available at
http://www.phys.vt.edu/~tauber/PredatorPrey/movies
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