7,447 research outputs found
On a predator prey model with nonlinear harvesting and distributed delay
A predator prey model with nonlinear harvesting (Holling type-II) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some
numerical simulations.Ministerio de EconomÃa y CompetitividadFondo Europeo de Desarrollo RegionalConsejerÃa de Innovación, Ciencia y Empresa (Junta de AndalucÃa
Biological control via "ecological" damping: An approach that attenuates non-target effects
In this work we develop and analyze a mathematical model of biological
control to prevent or attenuate the explosive increase of an invasive species
population in a three-species food chain. We allow for finite time blow-up in
the model as a mathematical construct to mimic the explosive increase in
population, enabling the species to reach "disastrous" levels, in a finite
time. We next propose various controls to drive down the invasive population
growth and, in certain cases, eliminate blow-up. The controls avoid chemical
treatments and/or natural enemy introduction, thus eliminating various
non-target effects associated with such classical methods. We refer to these
new controls as "ecological damping", as their inclusion dampens the invasive
species population growth. Further, we improve prior results on the regularity
and Turing instability of the three-species model that were derived in earlier
work. Lastly, we confirm the existence of spatio-temporal chaos
Modelling chemotaxis of microswimmers: from individual to collective behavior
We discuss recent progress in the theoretical description of chemotaxis by
coupling the diffusion equation of a chemical species to equations describing
the motion of sensing microorganisms. In particular, we discuss models for
autochemotaxis of a single microorganism which senses its own secretion leading
to phenomena such as self-localization and self-avoidance. For two
heterogeneous particles, chemotactic coupling can lead to predator-prey
behavior including chase and escape phenomena, and to the formation of active
molecules, where motility spontaneously emerges when the particles approach
each other. We close this review with some remarks on the collective behavior
of many particles where chemotactic coupling induces patterns involving
clusters, spirals or traveling waves.Comment: to appear as a contribution to the book "Chemical kinetics beyond the
textbook
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
Moving forward in circles: challenges and opportunities in modelling population cycles
Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer–resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research
Long delay times in reaction rates increase intrinsic fluctuations
In spatially distributed cellular systems, it is often convenient to
represent complicated auxiliary pathways and spatial transport by time-delayed
reaction rates. Furthermore, many of the reactants appear in low numbers
necessitating a probabilistic description. The coupling of delayed rates with
stochastic dynamics leads to a probability conservation equation characterizing
a non-Markovian process. A systematic approximation is derived that
incorporates the effect of delayed rates on the characterization of molecular
noise, valid in the limit of long delay time. By way of a simple example, we
show that delayed reaction dynamics can only increase intrinsic fluctuations
about the steady-state. The method is general enough to accommodate nonlinear
transition rates, allowing characterization of fluctuations around a
delay-induced limit cycle.Comment: 8 pages, 3 figures, to be published in Physical Review
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