294 research outputs found
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
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
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
Dynamics of a competitive Lotka-Volterra systems in R3
We describe the dynamics of the 3-dimensional competitive Lotka-Volterra systems x˙=x(a−x−y−z), y˙=y(b−x−y−z), z˙=z(c−x−y−z), providing the phase portraits for all the values of the parameters a, b and c with 0 < a< b< c in the positive octant of the Poincaré ball
Proper rational and analytic first integrals for asymmetric 3-dimensional Lotka-Volterra systems
Agraïments: The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013.We go beyond in the study of the integrability of the classical model of competition between three species studied by May and Leonard [19], by considering a more realistic asymmetric model. Our results show that there are no global analytic first integrals and we provide all proper rational first integrals of this extended model by classifying its invariant algebraic surfaces
Traveling plane wave solutions of delayed lattice differential systems in competitive Lotka-Volterra type
99學年度楊定揮教師升等代表著作
100學年度研究獎補助論文[[abstract]]In this work we consider the existence of traveling plane wave solutions of systems of delayed lattice differential equations in competitive Lotka-Volterra type. Employing iterative method coupled with the explicit construction of upper and lower solutions in the theory of weak quasi-monotone dynamical systems, we obtain a speed, c *, and show the existence of traveling plane wave solutions connecting two different equilibria when the wave speeds are large than c *.[[incitationindex]]SCI[[booktype]]紙
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