Chaotic attractors in Atkinson-Allen model of four competing species

We study the occurrence of chaos in the Atkinson-Allen model of four competing species, which plays the role as a discrete-time Lotka-Volterra-type model. We show that in this model chaos can be generated by a cascade of quasiperiod-doubling bifurcations starting from a supercritical Neimark-Sacker bifurcation of the unique positive fixed point. The chaotic attractor is contained in a globally attracting invariant manifold of codimension one, known as the carrying simplex. Biologically, our study implies that the invasion attempts by an invader into a trimorphic population under Atkinson-Allen dynamics can lead to chaos.Peer reviewe

Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system

Lotka-Volterra systems have been extensively studied by many authors, both in the autonomous and non-autonomous cases. In previous papers the time asymptotic behaviour as t → ∞ has been considered. In this paper we also consider the “pullback” asymptotic behaviour which roughly corresponds to observing a system “now” that has already been evolving for a long time. For a competitive system that is asymptotically autonomous both as t → −∞ and as t → +∞ we show that these two notions of asymptotic behaviour can be very different but are both important for a full understanding of the dynamics. In particular there are parameter ranges for which, although one species dies out as t → ∞, there is a distinguished time-dependent coexistent state that is attracting in the pullback sense.Ministerio de Ciencia y Tecnología (España). Dirección General de Investigación Científica y TécnicaRoyal Society University Research Fello

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

On the asymptotic shape of solutions to Neumann problems for non-cooperative parabolic systems

We consider a class of nonautonomous parabolic competition-diffusion systems on bounded radial domains under Neumann boundary conditions. We show that, if the initial profiles satisfy a reflection inequality with respect to a hyperplane, then global positive solutions are asymptotically (in time) foliated Schwarz symmetric with respect to antipodal points. Additionally, a related result for (positive and sign changing solutions) of a scalar equation with Neumann boundary conditions is given. The asymptotic shape of solutions to cooperative systems is also discussed.Comment: 30 pages. Revised versio

Solving fractional-order competitive lotka-volterra model by nsfd schemes

In this paper, we introduce fractional-order into a model competitive LotkaVolterra prey-predator system. We will discuss the stability analysis of this fractional system. The non-standard finite difference (NSFD) scheme is implemented to study the dynamic behaviors in the fractional-order Lotka-Volterra system. Proposed non-standard numerical scheme is compared with the forward Euler and fourth order Runge-Kutta methods. Numerical results show that the NSFD approach is easy and accurate for implementing when applied to fractional-order Lotka-Volterra model.Publisher's Versio