arXiv subjects

Andrey Morgulis

Publications and source records attributed to Andrey Morgulis.

11 recordsLinked to original sources

arXiv2026

Short-wave signal versus indirect prey-taxis

We address a short-wave asymptotic for one class of quasi-linear second-order PDE systems involving the cross-diffusion described by the so-called Patlak-Keller-Segel law. It is common to employ these equations for modeling the predator-prey community with the prey-taxis that means the interactions of two species of particles or cells or anything else through which the species called "predators" is capable of moving directionally while searching for the other species called "prey." However, we suppose the predators to be sensitive not to the prey density but to a driving signal produced by the prey. Additionally, the production of the driving signal is assumed to be sensitive to the intensity of an external field, which is independent from the community state. This is what we call the external signal. It can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. We assume that the external signal takes a general short-wave form and construct a complete asymptotic expansion for the short-wave solutions with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions. Further, we apply the short wave asymptotic to studying the stability or instability induced by the external signal following Kapitza's theory for the upside-down pendulum. Applying the general results to some special classes of external signals, we get examples of suppressing the taxical transport, examples of robustness of the species equilibrium to the signal up to a very strong stabilization or, oppositely, destabilization and somewhat like blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium. These results contribute to filling the gap in the literature, since the theory and techniques for the asymptotic integration of systems described above represent a weakly charted area.

math.AP↗

arXiv2025

Prey-taxis VS a Shortwave External Signal in Multiple Dimensions

We consider a model of the predator--prey community with prey-taxis. By that we mean the capability of the predators to get moving in a certain direction on the macroscopic level in response to the prey density gradients. Additionally, we suppose the same kind of sensitivity with respect to one more signal, called external, the production of which goes on independently of the community state. Such a signal can be due to the spatiotemporal inhomogeneity of the environment that results from the natural or artificial reasons. The model employs the Patlak--Keller--Segel law for responses to both ones. We assume that the external signal takes a general short-wave form, and we construct the complete asymptotic expansions of the short-wave solutions. This result generalizes the prior one by Morgulis \& Malal (2025) in two respects. First, we have addressed the case of multiple dimensions. Second, we have got rid of assuming the signal and corresponding solutions to take the form of a traveling wave, that makes our result novel even in one dimension. Further, we apply the short wave asymptotic to studying the stability or instability imposed by the external signal following Kapitza' theory for upside-down pendulum.

math.AP↗

arXiv2024

Prey-taxis vs an external signal: short-wave asymptotic and stability analysis

We consider two models of predator-prey community with prey-taxis, one relies on Patlak-Keller-Segel law (Lee et al, 2009), the other one employs the Cattaneo model of heat transfer following Dolak and Hillen (2003). Thus, the former one uses the prey density gradient for directing the predators flux, and the latter one -- for directing the vector density of sources of the predators flux. We assume the predators to be also capable of responding to an external signal in the same manner. Additionally, we assume that some scaling makes the dimensionless prey diffusivity and the wavelength of the external signal small quantities of the same order. In the case of Cattaneo's model, we additionally assume the resistivity to varying the predators flux to be as high as the reciprocal of the prey diffusivity. With these assumptions we construct the complete asymptotic expansion of the short-wave solution. We use this asymptotic to examine the effect of the short-wave signal on the formation of spatiotemporal patterns. We do so by comparing the stability of equilibria with no signal to that of the quasi-equilibria, by which we mean the simplest patterns directly forced by the external signal. Historically, such an approach goes back to Kapitza's theory for upside-down pendulum, and it had been already applied to examining other predator-prey systems and/or other limits (Morgulis and Ilin, 2020, Morgulis, 2023). This time the overall conclusion is that the external signal is likely not capable of creating the instability domain in the parametric space from nothing but it can substantially widen the one that is non-empty with no signal. The details, however, essentially depend on the speed at which the external signal propagates and on the system kinetics, at least when the signal amplitude is small

q-bio.PE↗

arXiv2021

The effect of boundary conditions on the stability of two-dimensional flows in an annulus with permeable boundary

We consider the stability of two-dimensional viscous flows in an annulus with permeable boundary. In the basic flow, the velocity has nonzero azimuthal and radial components, and the direction of the radial flow can be from the inner cylinder to the outer one or vice versa. In most earlier studies, all components of the velocity were assumed to be given on the entire boundary of the flow domain. Our aim is to study the effect of different boundary conditions on the stability of such flows. We focus on the following boundary conditions: at the inflow part if the boundary (which may be either inner or outer cylinder) all components of the velocity are known; at the outflow part of the boundary (the other cylinder), the normal stress and either the tangential velocity or the tangential stress are prescribed. Both types of boundary conditions are relevant to certain real flows: the first one - to porous cylinders, the second - to flows, where the fluid leaves the flow domain to an ambient fluid which is at rest. It turns out that both sets of boundary conditions make the corresponding steady flows more unstable (compared with earlier works where all components of the velocity are prescribed on the entire boundary). In particular, it is demonstrated that even the classical (purely azimuthal) Couette-Taylor flow becomes unstable to two-dimensional perturbations if one of the cylinders is porous and the normal stress (rather than normal velocity) is prescribed on that cylinder.

physics.flu-dyn↗

arXiv2020

Drift, stabilizing and destabilizing for a Patlak-Keller-Segel system with the short-wavelength external signal

This article aims at exploring the short-wavelength stabilization and destabilization of the advection-diffusion systems formulated using the Patlak-Keller-Segel cross-diffusion. We study a model of the taxis partly driven by an external signal. We address the general short-wavelength signal using the homogenization technique, and then we give a detailed analysis of the signals emitted as the travelling waves. It turns out that homogenizing produces the drift of species, which is the main translator of the external signal effects, in particular, on the stability issues. We examine the stability of the quasi-equilibria - that is, the simplest short-wavelength patterns fully imposed by the external signal. Comparing the results to the case of switching the signal off allows us to estimate the effect of it. For instance, the effect of the travelling wave turns out to be not single-valued but depending on the wave speed. Namely, there is an independent threshold value such that increasing the amplitude of the wave destabilizes the quasi-equilibria provided that the wave speed is above this value. Otherwise, the same action exerts the opposite effect. It is worth to note that the effect is exponential in the amplitude of the wave in both cases.

q-bio.PE↗

arXiv2019

On the stability of the Couette-Taylor flow between rotating porous cylinders with radial flow

We study the stability of the Couette-Taylor flow between porous cylinders with radial throughflow. It had been shown earlier that this flow can be unstable with respect to non-axisymmetric (azimuthal or helical) waves provided that the radial Reynolds number, $R$ (constructed using the radial velocity at the inner cylinder and its radius), is high. In this paper, we present a very detailed and, in many respects, novel chart of critical curves in a region of moderate values of $R$, and we show that, starting from values of $R$, as low as $10$, the critical modes inherited from the inviscid instability gradually substitute the classical Taylor vortices. Also, we have looked more closely at the effect of a weak radial flow (relatively low $R$) on the Taylor instability and found that a radial flow directed from the inner cylinder to the outer one is capable of stabilizing the Couette-Taylor flow provided that the gap between the cylinders is wide enough. This observation is in a sharp contrast with the case of relatively narrow gaps for which the opposite effect is well-known.

physics.flu-dyn↗

arXiv2019

A remark on the disorienting of species due to the fluctuating environment

In this article we study the stabilizing of a primitive pattern of behaviour for the two-species community with chemotaxis due to the short-wavelength external signal. We use a system of Patlak-Keller-Segel type as a model of the community. It is well-known that such systems can produce complex unsteady patterns of behaviour which are usually explained mathematically by bifurcations of some basic solutions that describe simpler patterns. As far as we aware, all such bifurcations in the models of the Patlak-Keller-Segel type had been found for homogeneous (i.e. translationally invariant) systems where the basic solutions are equilibria with homogeneous distributions of all species. The model considered in the present paper does not possess the translational invariance: one of species (the predators) is assumed to be capable of moving in response to a signal produced externally in addition to the signal emitted by another species (the prey). For instance, the external signal may arise from the inhomogeneity of the distribution of an environmental characteristic such as temperature, salinity, terrain relief, etc. Our goal is to examine the effect of short-wavelength inhomogeneity. To do this, we employ a certain homogenization procedure. We separate the short-wavelength and smooth components of the system response and derive a slow system governing the latter one. Analysing the slow system and comparing it with the case of homogeneous environment shows that, generically, a short-wavelength inhomogeneity results in an exponential decrease in the motility of the predators. The loss of motility prevents, to a great extent, the occurrence of complex unsteady patterns and dramatically stabilizes the primitive basic solution. In some sense, the necessity of dealing with intensive small-scale changes of the environment makes the system unable to respond to other challenges.

q-bio.PE↗

arXiv2016

Inviscid instability of an incompressible flow between rotating porous cylinders to three-dimensional perturbations

We study the stability of two-dimensional inviscid flows in an annulus between two porous cylinders with respect to three-dimensional perturbations. The basic flow is irrotational, and both radial and azimuthal components of the velocity are non-zero. The direction of the radial flow can be from the inner cylinder to the outer one (the diverging flow) or from the outer cylinder to the inner one (the converging flow). It had been shown earlier in Ref. \cite{IM2013a} that, independent of the direction of the radial flow, the basic flow can be unstable to small two-dimensional perturbations. In the present paper, we prove first that purely radial flow is stable and that flows with both radial and azimuthal components are always stable to axisymmetric perturbations. Then we show that both the diverging and converging flows are unstable with respect to non-axisymmetric three-dimensional perturbations provided that the ratio of the azimuthal component of the velocity to the radial one is sufficiently large. Neutral curves in the space of parameters of the problem are computed and it is demonstrated that for any ratio of the radii of the cylinders, the most unstable modes (corresponding to the smallest ratio of the azimuthal velocity to the radial one) are the two-dimensional ones. We also consider the corresponding viscous stability problem and construct an asymptotic expansion of its solutions for large radial Reynolds numbers. We compute the first-order viscous correction to inviscid eigenvalues and show that the asymptotic results give a good approximation to the viscous eigenvalues even for moderate values of radial Reynolds number, which indicates that the instability may be observed in real flows.

physics.flu-dyn↗

arXiv2013

Instability of a viscous flow between rotating porous cylinders with radial flow

The stability of a two-dimensional viscous flow between two rotating porous cylinders is studied. The basic steady flow is the most general rotationally-invariant solution of the Navier-Stokes equations in which the velocity has both radial and azimuthal components, and the azimuthal velocity profile depends on the Reynolds number. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. Neutral curves in the space of parameters of the problem are computed. Calculations show that the stability properties of this flow are determined by the azimuthal velocity at the inner cylinder when the direction of the radial flow is from the inner cylinder to the outer one and by the azimuthal velocity at the outer cylinder when the direction of the radial flow is reversed. This work is a continuation of our previous study of an inviscid instability in flows between rotating porous cylinders (see \cite{IM2013}).

physics.flu-dyn↗

arXiv2012

Instability of diverging and converging flows in an annulus

The stability of two-dimensional diverging and converging flows in an annulus between two permeable cylinders is examined. The basic flow is irrotational and has both the radial and azimuthal components. It is shown that for a wide range of the parameters of the problem, the basic flow is unstable to small two-dimensional perturbations. The instability is inviscid and oscillatory and persists if the viscosity of the fluid is taken into consideration.

physics.flu-dyn↗

arXiv2011

Steady streaming between two vibrating planes at high Reynolds numbers

We consider incompressible flows between two transversely vibrating solid walls and construct an asymptotic expansion of solutions of the Navier-Stokes equations in the limit when both the amplitude of vibrations and the thickness of the Stokes layer are small and have the same order of magnitude. Our asymptotic expansion is valid up to the flow boundary. In particular, we derive equations and boundary conditions, for the averaged flow. In the leading order, the averaged flow is described by the stationary Navier-Stokes equations with an additional term which contains the leading-order Stokes drift velocity. In a slightly different context (for a flow induced by an oscillating conservative body force), the same equations had been derived earlier by Riley (2001). The general theory is applied to two particular examples of steady streaming induced by transverse vibrations of the walls in the form of standing and travelling plane waves. In particular, in the case of waves travelling in the same direction, the induced flow is plane-parallel and the Lagrangian velocity profile can be computed analytically. This example may be viewed as an extension of the theory of peristaltic pumping to the case of high Reynolds numbers.

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