Nonequilibrium phase transition in a lattice prey–predator system

Lipowski, Adam; Lipowska, Dorota

doi:10.1016/s0378-4371(99)00482-3

Cited by 63 publications

(103 citation statements)

References 16 publications

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“…Clearly, as noted by other authors [17,21,25,26,27], a more realistic description of the predator-prey interaction should include the possibility for the agents to move. In fact, in ecosystems a prey tend to avoid the interaction with an incoming predator, while the predators aim to pursue the prey.…”

Section: Introductionmentioning

confidence: 90%

“…[26], checked that the range of interaction did not affect the characteristics of the oscillatory behavior displayed in two dimensions by the model of Ref. [17]: Actually, the amplitude of the oscillations was always found to vanish in the thermodynamic limit.…”

Section: Introductionmentioning

confidence: 97%

“…Since Lotka and Volterra's seminal and pioneering works [1,2], many decades ago, modeling of interacting, competing species has received considerable attention in the fields of biology, ecology, mathematics [3,4,5,6,7,8,9,10,11,12,13,14,15], and, more recently, in the physics literature as well [16,17,18,19,20,21,22,23,24,25,26,27]. In their remarkably simple deterministic model, Lotka and Volterra considered two coupled nonlinear differential equations that mimic the temporal evolution of a two-species system of competing predator and prey populations.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, a variety of so-called 'individualbased', or stochastic, lattice predator-prey models have been investigated, e.g., in Refs. [16,17,18,19,20,21,22,23], employing largely mean-field-type approaches (including refined versions, such as the pair approximation of Refs. [16,22]) and Monte Carlo simulations, mostly in two dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…In addition, in Refs. [17,18] the oscillatory behavior displayed by the densities of the coexisting populations in some region of the active parameter space was studied as well. It was reported that in one and two dimensions the population densities showed characteristic erratic oscillations whose amplitude vanishes in the thermodynamic limit (even in the presence of longrange interactions), while it was argued that the oscillation amplitude may remain finite in three dimensions [17].…”

Section: Introductionmentioning

confidence: 99%

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Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

2006

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We study the general properties of stochastic two-species models for predator-prey competition and coexistence with Lotka-Volterra type interactions defined on a d-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-toabsorbing 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.

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Section: Introductionmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

2006

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Broad scaling region in a spatial ecological system

Roy

Pascual

Franc³

2003

Complexity

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The ubiquity of scale-free patterns in ecological systems has raised the possibility that these systems operate near criticality. Critical phenomena (CP) require the tuning of parameters and typically exhibit a narrow scaling region in which power laws hold. Here we show that an individual-based predator-prey model exhibits scaling properties similar to CP, generated by a percolation-like transition but with a broader scaling region. There are no drastic changes in ecological quantities across this critical point and species coexist broadly in parameter space. The implications of these findings for the stability of ecological systems "near" criticality is discussed.

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Adaptive control of nonlinear complexHolling II predator–prey system with unknown parameters

Aghababa

2015

Complexity

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In recent years, control of nonlinear complex predator–prey systems has attracted the attention of many researchers. The previous works have some weaknesses such as neglecting the consideration of the effects of both model uncertainties and unknown parameters and having an infinite time of convergence. To overcome the mentioned shortages, this article solves the problem of robust control of nonlinear complex Holling type II predator–prey system in a given finite time. It is assumed that the parameters of the system are fully unknown in advance and some uncertainties perturb the system's dynamics. To tackle the system unknown parameters, some adaptation laws are introduced. Thereafter, a robust switching controller is proposed to finite‐timely stabilize the predator–prey system. An illustrative example demonstrates the efficiency and usefulness of the proposed control strategy. © 2015 Wiley Periodicals, Inc. Complexity 21: 260–266, 2016

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Nonequilibrium phase transition in a lattice prey–predator system

Cited by 63 publications

References 16 publications

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

Phase Transitions and Spatio-Temporal Fluctuations in Stochastic Lattice Lotka–Volterra Models

Broad scaling region in a spatial ecological system

Adaptive control of nonlinear complexHolling II predator–prey system with unknown parameters

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