Stationary pattern of vortices or strings in biological systems: Lattice version of the Lotka-Volterra model

Tainaka, Kei–ichi

doi:10.1103/physrevlett.63.2688

Cited by 123 publications

(83 citation statements)

References 12 publications

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“…In the Rock-Scissors-Paper model introduced by Tainaka (1988Tainaka ( , 1989Tainaka ( , 1994, agents are located on the sites of a square lattice (with periodic boundary condition), and follow one of three possible strategies to be denoted shortly as R, S, and P. The evolution of the population is governed by random sequential invasion events between randomly chosen nearest neighbors. Nothing happens if the two agents follow the same strategy.…”

Section: Simulations On the Square Latticementioning

confidence: 99%

Evolutionary games on graphs

2007

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Game theory is one of the key paradigms behind many scientific disciplines from biology to behavioral sciences to economics. In its evolutionary form and especially when the interacting agents are linked in a specific social network the underlying solution concepts and methods are very similar to those applied in non-equilibrium statistical physics. This review gives a tutorial-type overview of the field for physicists. The first four sections introduce the necessary background in classical and evolutionary game theory from the basic definitions to the most important results. The fifth section surveys the topological complications implied by non-mean-field-type social network structures in general. The next three sections discuss in detail the dynamic behavior of three prominent classes of models: the Prisoner's Dilemma, the Rock-Scissors-Paper game, and Competing Associations. The major theme of the review is in what sense and how the graph structure of interactions can modify and enrich the picture of long term behavioral patterns emerging in evolutionary games.

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Section: Simulations On the Square Latticementioning

confidence: 99%

Evolutionary games on graphs

2007

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“…Such competition is metaphorically described by rock-paper-scissors (RPS) games, where "rock crushes scissors, scissors cut paper, and paper wraps rock" [8]. While non-spatial RPS-like models often evolve toward extinction of all but one species in finite time [9], their spatial counterparts are generally characterized by the long coexistence of species and by the formation of complex spatio-temporal patterns [10][11][12][13][14]. Recently, two-dimensional versions of a model introduced by May and Leonard [15] have received much attention [11][12][13][14].…”

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confidence: 99%

When does cyclic dominance lead to stable spiral waves?

Szczesny¹,

Mobilia²,

Rucklidge³

2013

EPL

134

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Species diversity in ecosystems is often accompanied by characteristic spatio-temporal patterns. Here, we consider a generic two-dimensional population model and study the spiraling patterns arising from the combined effects of cyclic dominance of three species, mutation, pair-exchange and individual hopping. The dynamics is characterized by nonlinear mobility and a Hopf bifurcation around which the system's four-phase state diagram is inferred from a complex Ginzburg-Landau equation derived using a perturbative multiscale expansion. While the dynamics is generally characterized by spiraling patterns, we show that spiral waves are stable in only one of the four phases. Furthermore, we characterize a phase where nonlinearity leads to the annihilation of spirals and to the spatially uniform dominance of each species in turn. Away from the Hopf bifurcation, when the coexistence fixed point is unstable, the spiraling patterns are also affected by the nonlinear diffusion.

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“…Under the scope of the generalized entropies S q , we will study in the present paper the temporal evolution of a cyclic Lotka-Volterra model in the lattice (LLV) [13,14]. This model is particularly rich as it exhibits interesting features, such as stationary states with spatial patterns [13] and fractality [15], which makes it a potential candidate for the applicability of the entropies S q .…”

Section: Introductionmentioning

confidence: 99%

“…This model is particularly rich as it exhibits interesting features, such as stationary states with spatial patterns [13] and fractality [15], which makes it a potential candidate for the applicability of the entropies S q . The LLV is one of the descendants of the original versions constructed by Lotka [16] and Volterra [17] to model autocatalytic chemical reactions and the prey-predator dynamics, respectively, and further adapted to many other situations from active transport by proteins [18] to social processes [19].…”

Section: Introductionmentioning

confidence: 99%

Entropy production in the cyclic lattice Lotka-Volterra model

Anteneodo¹

2004

Eur. Phys. J. B

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The cyclic Lotka-Volterra model in a D-dimensional regular lattice is considered. Its "nucleus growth" mode is analyzed under the scope of Tsallis' entropies Sq = (1 − i p q i )/(q − 1), q ∈ R. It is shown both numerically and by means of analytical considerations that a linear increase of entropy with time, meaning finite asymptotic entropy rate, is achieved for the entropic index qc = 1 − 1/D. Although the lattice exhibits fractal patterns along its evolution, the characteristic value of q can be interpreted in terms of very simple features of the dynamics.

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Stationary pattern of vortices or strings in biological systems: Lattice version of the Lotka-Volterra model

Cited by 123 publications

References 12 publications

Evolutionary games on graphs

Evolutionary games on graphs

When does cyclic dominance lead to stable spiral waves?

Entropy production in the cyclic lattice Lotka-Volterra model

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