The remarkable empirical finding that extensive fine-scale diversity is stable on long time scales, despite selective pressures, has spurred the study of several models, which can explain species formationand stability. However, the focus has been more on the possible mechanisms which sustain coexistence, rather than on their universality and robustness. We thus consider a model that includes generic forms of many-body non-linear interactions, which are non-local and depend on a set of parameters. Despite the complicated dynamics, we are able to aptly identify a region in the parameter space where the dynamics manifests universal and robust features which extend to the weakly non-linear regime. Large scale variations of patterns are well captured by a Ginzburg-Landau equation , whose form does not depend on the details of the underlying models. The mathematical framework we have introduced has much larger applicability than the one considered here, possibly including pattern formation in models with very different physical features . The analytical predictions yield a good agreement when compared to the numerical integration of a competitive Lotka-Volterra equation with non-local interactions .
Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this has spurred the study of many models, which can explain pattern formation. The modulations of patterns, occurring on long spatial and temporal scales, can not be captured by linear approximation analysis. Here, we show that, starting from a general model with long range couplings displaying patterns, the spatio-temporal evolution of large scale modulations at the onset of instability is ruled by the well-known Ginzburg-Landau equation, independently of the details of the dynamics. Hence, we demonstrate the validity of such equation in the description of the behavior of a wide class of systems. We introduce a novel mathematical framework that is also able to retrieve the analytical expressions of the coefficients appearing in the Ginzburg-Landau equation as functions of the model parameters. Such framework can include higher order nonlocal interactions and has much larger applicability than the model considered here, possibly including pattern formation in models with very different physical features .
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The remarkable empirical finding that extensive fine-scale diversity is stable on long time scales, despite selective pressures,
Regular spatial structures emerge in a wide range of different dynamics characterized by local and/or nonlocal coupling terms. In several research fields this
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several
many
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species formationand stability. However, the focus has been more on the possible mechanisms which sustain coexistence, rather than on their universality and robustness. We thus consider a model that includes generic forms of many-body non-linear interactions, which are non-local and depend on a set of parameters. Despite the complicated dynamics, we are able to aptly identify a region in the parameter space where the dynamics manifests universal and robust features which extend to
pattern formation. The modulations of patterns, occurring on long spatial and temporal scales, can not be captured by linear approximation analysis. Here, we show that, starting from a general model with long range couplings displaying patterns, the spatio-temporal evolution of large scale modulations at the onset of instability is ruled by the well-known Ginzburg-Landau equation, independently of
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weakly non-linear regime. Large scale variations of patterns are well captured by a
details of the dynamics. Hence, we demonstrate the validity of such equation in the description of the behavior of a wide class of systems. We introduce a novel mathematical framework that is also able to retrieve the analytical expressions of the coefficients appearing in the
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, whose form does not depend on the details of the underlying models. The mathematical framework we have introduced
as functions of the model parameters. Such framework can include higher order nonlocal interactions and
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one
model
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. The analytical predictions yield a good agreement when compared to the numerical integration of a competitive Lotka-Volterra equation with non-local interactions