The Ten-Year Cycle in Numbers of the Lynx in Canada

Elton, Charles; Nicholson, Mary

doi:10.2307/1358

Cited by 634 publications

(405 citation statements)

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“…Upon ignoring any spatial fluctuations and correlations, which amounts to assuming the factorizations c AB (t) = a(t) b(t) and c BB (t) = b(t) 2 , after substitution into Eqs. (14) and (15) one recovers the deterministic mean-field rate equations (6) and (7) with ρ = ζ = 1. This implies that the site restrictions on a mean-field level correspond to a finite prey carrying capacity.…”

Section: Abmentioning

confidence: 81%

“…Notice that the diffusion rate D and the coordination number z do not appear explicitly in Eqs. (14) and (15). However, they would enter the equations of motion for the two-site probability distributions, i.e., the correlators c AB (t) and c BB (t).…”

Section: B Sllvm Equations Of Motion and Some Exact Propertiesmentioning

confidence: 99%

“…Even though it is not possible to solve Eqs. (14) and (15) in a closed form, owing to the emerging infinite hierarchy of higher-order correlations, we can still obtain some useful and nontrivial information on the phase portrait. Let us denote by a * and b * the stationary concentrations of the predators and the prey, respectively, and by c * BB and c *…”

Section: B Sllvm Equations Of Motion and Some Exact Propertiesmentioning

confidence: 99%

“…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%

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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: Abmentioning

confidence: 81%

Section: B Sllvm Equations Of Motion and Some Exact Propertiesmentioning

confidence: 99%

Section: B Sllvm Equations Of Motion and Some Exact Propertiesmentioning

confidence: 99%

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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“…The multiannual population cycles of certain mammals and their predators in northern latitudes have been studied extensively over the last century (Elton and Nicholson 1942, Keith 1990, Akçakaya 1992, King and Schaffer 2001, Korpimäki et al 2004, Tyson et al 2010, Krebs 2011. The importance of cyclic herbivores, such as snowshoe hares (Lepus americanus) and brown lemmings (Lemmus trimucronatus), to tundra and boreal ecosystems lies at the heart of this interest in oscillatory population dynamics (Krebs 2011).…”

Section: Table Of Contents List Of Figuresmentioning

confidence: 99%

Theoretical impacts of habitat loss and generalist predation on predator–prey cycles

Vitense

Wirsing

Tyson

et al. 2016

Ecological Modelling

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I used a reaction-diffusion-advection framework to investigate the relative and combined damping impacts of generalist predation and habitat loss with reaction terms taken from the May and Rosenzweig-MacArthur models. The models were parameterized using data from snowshoe hare and Canada lynx field studies to generate similar cyclic dynamics in the center of a single patch in the absence of generalist predation. I found that generalist predation has strong stabilizing effects for both models and may represent a threat to the persistence of specialized predators. The magnitude of cycle damping due to habitat loss depends on movement rates and model choice, but ultimately results in the loss of cycles. Differences in model carrying capacity may explain differences in model sensitivity to habitat loss, and cycle amplitude may or may not decrease monotonically with habitat loss, depending on model choice. Elevated generalist predation rates at patch edges and in matrix habitat hasten cycle attenuation in situations that lead to increased prey exposure to generalists, including small patch size, higher movement rates into the matrix, and increased prey density at patch edges. Habitat disturbances may therefore have myriad consequences for cyclic systems depending in part on the nature of specialist predator-prey interactions and the extent to which the disturbances increase generalist access to prey. Field data that clarifies the relationships between habitat loss and fragmentation, generalist density and behavior, and cyclic activity would be invaluable in informing future modeling efforts.

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Seasonal Time Series

Pawitan

2005

Encyclopedia of Biostatistics

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The Ten-Year Cycle in Numbers of the Lynx in Canada

Cited by 634 publications

References 11 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

Theoretical impacts of habitat loss and generalist predation on predator–prey cycles

Seasonal Time Series

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