A non-linear analysis for spatial structure in a reaction-diffusion model

Ochoa, Francisco Lara; Murray, J. D.

doi:10.1016/s0092-8240(83)80069-x

Cited by 7 publications

(6 citation statements)

References 27 publications

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“…The above linear analysis regarding the Turing instability and the emergence of Turing patterns is qualitative in nature, and it fails to capture any quantitative information of the resulting patterns. On the other hand, weakly nonlinear analysis acts as an efficient method to quantify the emerging patterns sufficiently close to the Turing bifurcation threshold [10,20,25,28,39]. This method enables us to derive an evolution equation for the amplitude of the resulting pattern through the technique of multiple-scales, since the pattern evolves on a slow time-scale for parameter values close to the bifurcation threshold [10,20,25,28,39].…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

“…On the other hand, weakly nonlinear analysis acts as an efficient method to quantify the emerging patterns sufficiently close to the Turing bifurcation threshold [10,20,25,28,39]. This method enables us to derive an evolution equation for the amplitude of the resulting pattern through the technique of multiple-scales, since the pattern evolves on a slow time-scale for parameter values close to the bifurcation threshold [10,20,25,28,39]. In this section, we aim to perform a weakly nonlinear analysis to derive the amplitude equation of the pattern for the spatial system…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

“…Now, imposing the Fredholm solvability condition for the equation (3.17), we obtain S 3 , U 1 = 0 which results in the following cubic Stuart-Landau equation for the amplitude A [10,20,25,28,39]:…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

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Spatiotemporal pattern formation in a prey–predator model with generalist predator

Manna

Banerjee

2022

Math. Model. Nat. Phenom.

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Generalist predators exploit multiple food sources and it is economical for them to reduce predation pressure on a particular prey species when their density level becomes comparatively less. As a result, a prey-predator system tends to become more stable in the presence of a generalist predator. In this article, we investigate the roles of both the diffusion and nonlocal prey consumption in shaping the population distributions for interacting generalist predator and its focal prey species. In this regard, we first derive the conditions associated with Turing instability through linear analysis. Then, we perform a weakly nonlinear analysis and derive a cubic Stuart-Landau equation governing amplitude of the resulting patterns near Turing bifurcation boundary. Further, we present a wide variety of numerical simulations to corroborate our analytical findings as well as to illustrate some other complex spatiotemporal dynamics. Interestingly, our study reveals the existence of traveling wave solutions connecting two spatially homogeneous coexistence steady states in Turing domain under the influence of temporal bistability phenomenon. Also, our investigation shows that nonlocal prey consumption acts as a stabilizing force for the system dynamics.

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Section: Weakly Nonlinear Analysismentioning

confidence: 99%

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

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Spatiotemporal pattern formation in a prey–predator model with generalist predator

Manna

Banerjee

2022

Math. Model. Nat. Phenom.

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“…Hence, r and /J3 are real from (56) or (58). Equations (57) may now be recast into the following system of equations where X\ = x\r + ixu, and X2 = x2r + ix2i-dx\r dr dx u dr = x2r,…”

Section: Basic Equationsmentioning

confidence: 99%

Quasiperiodicity and chaos in the nonlinear evolution of the Kelvin-Helmholtz instability of supersonic anisotropic tangential velocity discontinuities

Choudhury¹,

Brown²

2003

Quart. Appl. Math.

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Abstract.A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, strongly anisotropic, magnetized, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations.For modes near the critical point of the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein-Gordon equation.The nonlinear coefficient turns out to be complex, which is, to the best of our knowledge, unlike previously considered cases and leads to completely different dynamics from that reported earlier. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability. In particular, the complex nonlinearity allows the existence of quasiperiodic and chaotic wave envelopes, unlike in earlier physical models governed by nonlinear Klein-Gordon equations. In addition, numerical diagnostics reveal the onset of chaos as a consequence of modulation of the external magnetic field.

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“…To consider the nonlinear evolution within the usual weakly nonlinear Ž theory Dodd et al, 1982;Drazin and Reid, 1981;Ochoa and Murray, ' . 1983 , we consider the first onset of instability near C s 2 .…”

Section: Second-order Solutionsmentioning

confidence: 99%

Nonlinear Evolution of the Kelvin–Helmholtz Instability of Supersonic Tangential Velocity Discontinuities

Choudhury

1997

Journal of Mathematical Analysis and Applications

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A nonlinear stability analysis using a multiple-scales perturbation procedure is performed for the instability of two layers of immiscible, inviscid, arbitrarily compressible fluids in relative motion. Such configurations are of relevance in a variety of astrophysical and space configurations. For modes of all wavenumbers on, or in the stable neighborhood of, the linear neutral curve, the nonlinear evolution of the amplitude of the linear fields on the slow first-order scales is shown to be governed by a complicated nonlinear Klein᎐Gordon equation. Both the spatially dependent and space-independent versions of this equation are considered to obtain the regimes of physical parameter space where the linearly unstable solutions either evolve to final permanent envelope wave patterns resembling the ensembles of interacting vortices observed empirically, or are disrupted via nonlinear modulation instability.ᮊ 1997 Academic Press

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A non-linear analysis for spatial structure in a reaction-diffusion model

Cited by 7 publications

References 27 publications

Spatiotemporal pattern formation in a prey–predator model with generalist predator

Spatiotemporal pattern formation in a prey–predator model with generalist predator

Quasiperiodicity and chaos in the nonlinear evolution of the Kelvin-Helmholtz instability of supersonic anisotropic tangential velocity discontinuities

Nonlinear Evolution of the Kelvin–Helmholtz Instability of Supersonic Tangential Velocity Discontinuities

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