725 research outputs found
Reduction to a single closed equation for 2 by 2 reaction-diffusion systems of Lotka-Volterra type
We consider general models of coupled reaction-diffusion systems for
interacting variants of the same species. When the total population becomes
large with intensive competition, we prove that the frequencies (i.e.
proportions) of the variants can be approached by the solution of a simpler
reaction-diffusion system, through a singular limit method and a relative
compactness argument. As an example of application, we retrieve the classical
bistable equation for Wolbachia's spread into an arthropod population from a
system modeling interaction between infected and uninfected individuals
Global attractors and extinction dynamics of cyclically competing species
Transitions to absorbing states are of fundamental importance in nonequilibrium physics as well as ecology. In ecology, absorbing states correspond to the extinction of species. We here study the spatial population dynamics of three cyclically interacting species. The interaction scheme comprises both direct competition between species as in the cyclic Lotka-Volterra model, and separated selection and reproduction processes as in the May-Leonard model. We show that the dynamic processes leading to the transient maintenance of biodiversity are closely linked to attractors of the nonlinear dynamics for the overall species' concentrations. The characteristics of these global attractors change qualitatively at certain threshold values of the mobility and depend on the relative strength of the different types of competition between species. They give information about the scaling of extinction times with the system size and thereby the stability of biodiversity. We define an effective free energy as the negative logarithm of the probability to find the system in a specific global state before reaching one of the absorbing states. The global attractors then correspond to minima of this effective energy landscape and determine the most probable values for the species' global concentrations. As in equilibrium thermodynamics, qualitative changes in the effective free energy landscape indicate and characterize the underlying nonequilibrium phase transitions. We provide the complete phase diagrams for the population dynamics and give a comprehensive analysis of the spatio-temporal dynamics and routes to extinction in the respective phases
Extinction in Lotka-Volterra model
Competitive birth-death processes often exhibit an oscillatory behavior. We
investigate a particular case where the oscillation cycles are marginally
stable on the mean-field level. An iconic example of such a system is the
Lotka-Volterra model of predator-prey competition. Fluctuation effects due to
discreteness of the populations destroy the mean-field stability and eventually
drive the system toward extinction of one or both species. We show that the
corresponding extinction time scales as a certain power-law of the population
sizes. This behavior should be contrasted with the extinction of models stable
in the mean-field approximation. In the latter case the extinction time scales
exponentially with size.Comment: 11 pages, 17 figure
Coevolutionary dynamics of a variant of the cyclic Lotka-Volterra model with three-agent interactions
We study a variant of the cyclic Lotka-Volterra model with three-agent
interactions. Inspired by a multiplayer variation of the Rock-Paper-Scissors
game, the model describes an ideal ecosystem in which cyclic competition among
three species develops through cooperative predation. Its rate equations in a
well-mixed environment display a degenerate Hopf bifurcation, occurring as
reactions involving two predators plus one prey have the same rate as reactions
involving two preys plus one predator. We estimate the magnitude of the
stochastic noise at the bifurcation point, where finite size effects turn
neutrally stable orbits into erratically diverging trajectories. In particular,
we compare analytic predictions for the extinction probability, derived in the
Fokker-Planck approximation, with numerical simulations based on the Gillespie
stochastic algorithm. We then extend the analysis of the phase portrait to
heterogeneous rates. In a well-mixed environment, we observe a continuum of
degenerate Hopf bifurcations, generalizing the above one. Neutral stability
ensues from a complex equilibrium between different reactions. Remarkably, on a
two-dimensional lattice, all bifurcations disappear as a consequence of the
spatial locality of the interactions. In the second part of the paper, we
investigate the effects of mobility in a lattice metapopulation model with
patches hosting several agents. We find that strategies propagate along the
arms of rotating spirals, as they usually do in models of cyclic dominance. We
observe propagation instabilities in the regime of large wavelengths. We also
examine three-agent interactions inducing nonlinear diffusion.Comment: 22 pages, 13 figures. v2: version accepted for publication in EPJ
Construction and application of exact solutions of the diffusive Lotka-Volterra system: a review and new results
This review summarizes all known results (up to this date) about methods of
integration of the classical Lotka-Volterra systems with diffusion and presents
a wide range of exact solutions, which are the most important from
applicability point of view. It is the first attempt in this direction. Because
the diffusive Lotka-Volterra systems are used for mathematical modeling
enormous variety of processes in ecology, biology, medicine, physics and
chemistry, the review should be interesting not only for specialists from
Applied Mathematics but also those from other branches of Science. The obtained
exact solutions can also be used as test problems for estimating the accuracy
of approximate analytical and numerical methods for solving relevant boundary
value problems
Order out of Randomness : Self-Organization Processes in Astrophysics
Self-organization is a property of dissipative nonlinear processes that are
governed by an internal driver and a positive feedback mechanism, which creates
regular geometric and/or temporal patterns and decreases the entropy, in
contrast to random processes. Here we investigate for the first time a
comprehensive number of 16 self-organization processes that operate in
planetary physics, solar physics, stellar physics, galactic physics, and
cosmology. Self-organizing systems create spontaneous {\sl order out of chaos},
during the evolution from an initially disordered system to an ordered
stationary system, via quasi-periodic limit-cycle dynamics, harmonic mechanical
resonances, or gyromagnetic resonances. The internal driver can be gravity,
rotation, thermal pressure, or acceleration of nonthermal particles, while the
positive feedback mechanism is often an instability, such as the
magneto-rotational instability, the Rayleigh-B\'enard convection instability,
turbulence, vortex attraction, magnetic reconnection, plasma condensation, or
loss-cone instability. Physical models of astrophysical self-organization
processes involve hydrodynamic, MHD, and N-body formulations of Lotka-Volterra
equation systems.Comment: 61 pages, 38 Figure
Geometric fluid approximation for general continuous-time Markov chains
Fluid approximations have seen great success in approximating the macro-scale
behaviour of Markov systems with a large number of discrete states. However,
these methods rely on the continuous-time Markov chain (CTMC) having a
particular population structure which suggests a natural continuous state-space
endowed with a dynamics for the approximating process. We construct here a
general method based on spectral analysis of the transition matrix of the CTMC,
without the need for a population structure. Specifically, we use the popular
manifold learning method of diffusion maps to analyse the transition matrix as
the operator of a hidden continuous process. An embedding of states in a
continuous space is recovered, and the space is endowed with a drift vector
field inferred via Gaussian process regression. In this manner, we construct an
ODE whose solution approximates the evolution of the CTMC mean, mapped onto the
continuous space (known as the fluid limit)
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