Maximum Growth Rate Of Instability Research Articles

Epidemicity indices and reproduction numbers from infectious disease data in connected human populations

We focus on distinctive data-driven measures of the fate of ongoing epidemics. The relevance of our pursuit is suggested by recent results proving that the short-term temporal evolution of infection spread is described by an epidemicity index related to the maximum instantaneous growth rate of new infections, echoing concepts and tools developed to study the reactivity of ecosystems. Suitable epidemicity indices can showcase the dynamics of infections, together with commonly employed effective reproduction numbers, especially when the latter assume values less than 1. In particular, epidemicity evaluates the short-term reactivity to perturbations of a disease-free equilibrium. Here, we show that sufficient epidemicity thresholds to prevent transient epidemic outbreaks in a spatially connected setting can be estimated by generalizing existing analogues derived when spatial effects are neglected. We specifically account for the discrete nature, in both space and time, of surveillance data of the type typically employed to estimate effective reproduction numbers that formed the bulk of the communication of the state of the COVID-19 pandemic and its controls. After analyzing the effects of spatial heterogeneity on the considered prognostic indicators, we perform a short- and long-term analysis on the COVID-19 pandemic in Italy, showing that endemic conditions were maintained throughout the duration of our simulation despite stringent control measures. Our method provides a portfolio of prognostic indices that are essential to pinpoint the ongoing pandemic in both a qualitative and quantitative manner, as our results demonstrate. We base our conclusions on extended investigations of the effects of spatial fragmentation of communities of different sizes owing to connectivity by human mobility and contact scenarios, within real geographic contexts and synthetic setups designed to test our framework.

Growth assessment of the sea urchin Pseudechinus magellanicus (Temnopleuridae) in coastal habitats of central Patagonia, Argentina

Abstract Pseudechinus magellanicus is an ecologically important and small sea urchin in coastal and nearshore habitats off southern South America. We provide the first growth assessment for the species using tag (calcein) and recapture procedures in central Patagonia (Argentina). The individual growth rate of P. magellanicus ranged 0.05–1.3 mm year−1. The Brody–Bertalanffy and Richards growth models provided asymptotic maximum diameters of 29.89 and 26.01 mm, respectively. Both models yielded low values for the growth constant (k), with 0.046 (Brody–Bertalanffy) and 0.062 (Richards). Maximum instantaneous growth rate was estimated at 1.36 mm year−1 for the Brody–Bertalanffy model, and 2.69 mm year−1 for the Richards model. Model selection (corrected Akaike information criterion) showed a slight better fit for the Brody–Bertalanffy growth model compared to the Richards model. A significant variability in growth was observed within the studied population, which can be attributed to genetic factors and micro-environmental effects. P. magellanicus displays a combination of slow growth and small body size, with the lowest recorded growth performance index (θ = 3.72) recorded so far in sea urchins. The species has a long lifespan, with the most common adult sizes estimated to range from 15 to 21 years according to the Brody–Bertalanffy model. Due to the broad geographic distribution and occupation of contrasting habitats, further studies are necessary to explore growth of P. magellanicus under different environmental conditions and/or along a bathymetric gradient.

Modulational instability of two obliquely interacting interfacial waves in the presence of a basic current shear

Evolution equations are derived for two obliquely interacting interfacial waves in the presence of a basic current shear. Using these equations modulational instability analysis is carried out for a pair of obliquely interacting uniform wave trains considering both unidirectional and bidirectional perturbations. It is found that the region of instability reduces in the presence of a basic current shear. Variations in the perturbed wavenumber at marginal stability due to changes in wave steepness and vorticities in the two mediums are displayed graphically. The maximum growth rate of instability for unidirectional perturbations increases with the increase in vorticity in the lower layer, but the reverse effect is observed when the vorticity in upper layer increases. The maximum growth rate of instability decreases with the increase in the angle of interaction of the two wave packets when the angle is acute, but the effect is opposite when the angle is obtuse.

Age, Growth and Population Structure Analyses of the Purpleback Flying Squid Sthenoteuthis oualaniensis in the Northwest Indian Ocean by Beak Microstructure

To explore the feasibility of using beak microstructure information to estimate the age of Sthenoteuthis oualaniensis, the microstructures of the upper beaks of individual squid were applied in this work to analyze the ages and growth patterns of squid caught from February–May 2019 and from October–December 2020 in the northwest Indian Ocean. The results indicated that the squid samples in the two years were no older than 9 months, and the samples in 2019 were autumn population and 2020 were spring population. The linear growth model of the autumn population (2019) was the best model for describing the relationship between age and ML, while the power model of the spring population (2020) was the best for describing the relationship between age and ML. The maximum instantaneous growth rate (IGR) and absolute daily growth rate (AGR) values of the spring population were 0.24%/d and 1.09 mm/d, respectively, occurring in squid between 200 and 220 days of age. The maximum IGR and AGR values of the autumn population were 0.69%/d and 1.73 mm/d, respectively, occurring in squid between 200 and 240 days of age. The period from 141–260 days (5–8 months) was considered to correspond to the subadult stage in the whole life history of S. oualaniensis in the Northwest Indian Ocean. The beak microstructure information can be effectively applied to estimate the age of S. oualaniensis individuals.

Effect of momentum and heat losses on the hydrodynamic instability of a premixed equidiffusive flame in a Hele–Shaw cell

The linear stage of hydrodynamic instability of a laminar premixed flame propagating in a Hele–Shaw cell is investigated. Our theoretical model takes into account momentum and heat losses, temperature-dependent transport coefficients, and the continuous internal structure of the flame front. The dispersion relation is obtained numerically as a solution to an eigenvalue problem for the linearized governing equations. The obtained results are in good qualitative and quantitative agreement with previous studies. It is shown that the wall heat losses tend to weaken the hydrodynamic flame instability. On the contrary, momentum losses enhance the flame instability. It is demonstrated that for the adiabatic walls, an increase in the Hele–Shaw cell width results in a reduction of the instability growth rate. For the non-adiabatic walls, there is a competition between momentum and heat losses in narrow channels that may result in a non-monotonic dependence of the instability growth rate on the Hele–Shaw cell width. It is shown that the effects of the Prandtl number and the thermal expansion vary with the wall heat loss coefficient. A possibility of non-monotonic dependence of the maximum instability growth rate on the thermal expansion has been demonstrated.

Effects of nitrogen regulation and strip intercropping on faba bean biomass, nitrogen accumulation and distribution, and interspecific interactions

AbstractEffects of nitrogen fertilization on the accumulation and distribution of biomass and plant nitrogen content in intercropped faba beans, together with its impact on the interspecific interactions between faba bean and wheat has not been well studied. Here, we have used a logistic model to analyze the cropping systems (i.e., monocropped faba bean, monocropped wheat, and faba bean intercropped with wheat) by evaluating the effects of different levels of nitrogen application on the dynamic changes in biomass and nitrogen accumulation, along with the interspecific interaction. Using monocropped and intercropped faba beans in field experiments spanning two years, we found that nitrogen application increased both monocropped and intercropped faba bean yields. The yield of intercropped faba beans was greatest at the lower nitrogen input for two consecutive years. Nitrogen application increased the biomass and nitrogen accumulation in both monocropped and intercropped faba beans. The simulation parameters that increased were maximum nitrogen accumulation (A') and maximum accumulation rate (R'max) while the initial cumulative rate (r') decreased. Compared with monocropping, intercropped faba bean–wheat showed increased yield and biomass accumulation, along with a significantly increased initial growth rate (r) and a maximum instantaneous growth rate (Rmax). Intercropping reduced stem and leaf biomass and the nitrogen distribution ratio while increasing the pod content. In the early intercropped faba bean growth stage, there was no competition for nutrients or any kind of yield advantage, while in the later stages, the nutrient competitiveness of intercropping was stronger than the intraspecific competition, showing the advantages of intercropping on biomass and yield (i.e., relative crowding coefficient [K] >1, relative interspecific competitiveness [RC] >0, and relative competition intensity [RCI] <0). In summary, nitrogen application of 45 kg ha–1 could be the desired amount that maximizes the nutrient competition and interspecific advantages in the middle and late stages of faba bean growth when intercropped, thus contributing to the yield.

Modulation instability of obliquely propagating ion acoustic waves in a collisionless magnetized plasma consisting of nonthermal and isothermal electrons

We have studied the modulation instability of obliquely propagating ion acoustic waves in a collisionless magnetized warm plasma consisting of warm adiabatic ions and two different species of electrons at different temperatures. We have derived a nonlinear Schr{\"o}dinger equation using the standard reductive perturbation method to describe the nonlinear amplitude modulation of ion acoustic wave satisfying the dispersion relation of ion acoustic wave propagating at an arbitrary angle to the direction of the external uniform static magnetic field. We have investigated the correspondence between two nonlinear Schr{\"o}dinger equations $-$ one describes the amplitude modulation of ion acoustic waves propagating along any arbitrary direction to the direction of the magnetic field and other describes the amplitude modulation of ion acoustic waves propagating along the direction of the magnetic field. We have derived the instability condition and the maximum growth rate of instability of the modulated ion acoustic wave. We have seen that the region of existence of maximum growth rate of instability decreases with increasing values of the magnetic field intensity whereas the region of existence of the maximum growth rate of instability increases with increasing $\cos \theta$, where $\theta$ is the angle of propagation of the ion acoustic wave with the external uniform static magnetic field. Again, the maximum growth rate of instability increases with increasing $\cos \theta$ and also this maximum growth rate of instability increases with increasing $\beta_{e}$ upto a critical value of the wave number, where $\beta_{e}$ is the parameter associated with the nonthermal distribution of hotter electron species.

Modulational Instability of Ion Acoustic Waves in a Magnetized Plasma Consisting of Isothermally Distributed Hot and Cold Electrons

Using the standard Reductive Perturbation Method a nonlinear Schr¨odinger equation is derived to study the modulational instability of small amplitude ion acoustic waves in a collisionless magnetized plasma composed of adiabatic warm ions, Maxwell-Boltzmann distribution of hot electrons as well as Maxwell-Boltzmann distribution of cold electrons, and the plasma system immersed in an external uniform static magnetic field (B0 = B0ˆz) propagating along the z-axis.The instability condition and the maximum growth rate of instability have been investigated analytically as well as numerically. We have studied the effect of each parameter of the present plasma system on the maximum growth rate of instability. In particular, it is found that the maximum growth rate of instability decreases with the increasing value of the ion cyclotron frequency with some set of values of the parameters associated with the present plasma system. Again, we have seen that the instability region decreases with the increasing value of the ion cyclotron frequency.

Instantaneous linear stability of plane Poiseuille flow forced by spanwise oscillations

In the present work, the stability of a plane Poiseuille flow forced by spanwise oscillations is studied via the instantaneous linear stability theory (LST). For streamwise Poiseuille flow and a spanwise Stokes layer, the superposition of these two linearly stable flows can lead to transient growth of perturbations. Periodic and aperiodic growth rates over time are found. Mode-crossing is observed in the aperiodic mode. Effects of oscillation amplitude W0 and frequency Ω are studied. The maximum instantaneous growth rate increases with increasing W0. At low W0, the stability of the flow is dominated by the corresponding Poiseuille flow or by the interactions between the two flows. At high W0, the stability characteristics of the Poiseuille–Stokes flow are very similar to those of the corresponding Stokes layer. It is demonstrated that oscillations have destabilizing effects in short-time intervals of one oscillation period. Oscillations of extremely low or high Ω have much weaker effects than those of medium Ω. The transient growth of the most unstable mode is traced by using direct numerical simulation (DNS). Instantaneous LST can capture the transient growth but fails to predict the accurate growth rate when the amplification ratio is higher than e9. The difference between LST and DNS is mainly due to the incorrect production of LST without nonlinear transportation. From the growing phase to the decay phase of the transient mode, the production term shifts from the destabilizing role to the stabilizing role because of the correlation-phase reversal of the perturbation velocities.

Age determination and growth pattern of temperate seabass Lateolabrax japonicus in Tango Bay and Sendai Bay, Japan

The temperate seabass Lateolabrax japonicus is a coastal and partial amphidromous fish that is commercially valuable in Japan. Fish age is an important and sensitive variable for stock structure assessment, and inaccurate aging can result in inappropriate management practices. However, there have been few studies on age determination in this species. By examining monthly changes in marginal increment growth, we validated that otolith increments are formed annually. This information was used to develop an age-determination method using otolith sections. Von Bertalanffy growth models were generated based on standard length and weight data for both sexes in Sendai Bay and Tango Bay, Japan. Differences in standard length- and weight-at-age and growth parameters were observed in both sexes and locations. Females exhibited larger standard lengths and weights at older ages than did males, in both locations. Individuals from Sendai Bay exhibited longer average asymptotic lengths, higher asymptotic weight, higher standard length-at-age and higher weight-at-age compared to their counterparts from Tango Bay. Decreases in growth rates in both Sendai Bay and Tango Bay occurred around the age at maturity, indicating that more energy was allocated to reproduction than growth after reaching maturation. Differences between Sendai Bay and Tango Bay in the age at which the maximum instantaneous growth rate and the asymptotic length are reached were thought to arise from differences in water temperature and biological productivity between the two bays.

Gravitational Instability in the Dust Layer of a Protoplanetary Disk with Interaction between the Layer and the Surrounding Gas

We consider gravitational instability of the dust layer in the midplane of a protoplanetary disk with turbulence and shear stresses between the gas in the disk and that in the dust layer. We solve a linearized system of hydrodynamic equations for perturbations of dust (monodisperse) and gas phases in the incompressible gas approximation. We take into account the gas drag of solid particles (dust aggregates), turbulent diffusion and the velocity dispersion of particles, and the perturbation of the azimuthal velocity of gas in the layer upon the transfer of angular momentum from solid particles to it and from this gas to the surrounding gas in the disk. We obtain and solve the dispersion equation for the layer with the ratio of surface densities of the dust phase and gas being well above unity. The following parameters of gravitational instability in the dust layer are calculated: the critical surface density of solid matter and the Stokes number of particles corresponding to the onset of instability, the wavelength range in which instability occurs, and the rate of its growth as a function of the perturbation wavelength in the circumsolar disk at radial distances of 1 and 10 AU. We show that at 10 AU, the maximum instability growth rate increases due to the transfer of angular momentum of gas in the layer to gas outside it, a new maximum emerges at a longer wavelength, a long-wavelength instability “tail” forms, and the critical surface density initiating instability decreases relative to that determined without the transfer of angular momentum to gas outside the layer. None of these effects are observed at 1 AU, since instability in this region probably develops faster than the transfer of angular momentum to the surrounding gаs of a protoplanetary disk occurs.

Extraction of Photosynthesis Parameters from Time Series Measurements of In Situ Production: Bermuda Atlantic Time-Series Study

Computing the vertical structure of primary production in ocean ecosystem models requires information about the vertical distribution of available light, chlorophyll concentration and photosynthesis response parameters. Conversely, given information on vertical structure of chlorophyll and light, we can extract photosynthesis parameters from vertical profiles of primary production measured at sea, as we illustrate here for the Bermuda Atlantic Time-Series Study. The procedure is based on a model of the production profile, which itself depends on the underwater light field. To model the light field, attenuation coefficients were estimated from measured optical profiles using a simple model of exponential decay of photosynthetically-available irradiance with depth, which accounted for 97% of the variance in the measured optical data. With the underwater light climate known, an analytical solution for the production profile was employed to recover photosynthesis parameters by minimizing the residual model error. The recovered parameters were used to model normalized production profiles and normalized watercolumn production. The model explained 95% of the variance in the measured normalized production at depth and 97% of the variance in measured normalized watercolumn production. A shifted Gaussian function was used to model biomass profiles and accounted for 93% of the variance in measured biomass at depth. An analytical solution for watercolumn production with the shifted Gaussian biomass was also tested. With the recovered photosynthesis parameters, maximum instantaneous growth rates were estimated by using a literature value for the carbon-to-chlorophyll ratio in this region of the Atlantic. An exact relationship between the maximum instantaneous growth rate and the daily growth rate in the ocean was derived. It was shown that calculating the growth rate by dividing the production by the carbon-to-chlorophyll ratio is equivalent to calculating it from the ratio of the final to the initial biomass, even when production is time dependent. Finally, the seasonal cycle of the recovered assimilation number at the Bermuda Station was constructed and analysed. The presented approach enables the estimation of photosynthesis parameters and growth rates from measured production profiles with only a few model assumptions, and increases the utility of in situ primary production measurements. The retrieved parameters have direct applications in satellite-based estimates of primary production from ocean-colour data, of which we give an example.

A generalized definition of reactivity for ecological systems and the problem of transient species dynamics

Summary Perturbations to an ecosystem's steady state can trigger transient responses of great ecological relevance. Asymptotic stability determines whether a generic perturbation will fade out in the long run, but falls short of characterizing the dynamics immediately after an equilibrium has been perturbed. Reactivity, traditionally defined as the maximum instantaneous growth rate of small perturbations to a stable steady state, is a simple yet powerful measure of the short‐term instability of a system as a whole. In many ecological applications, however, it could be important to focus on the reactivity properties of just some specific, problem‐dependent state variables, such as the abundance of a focal species engaged in interspecific competition, either predators or preys in a trophic community, or infectious individuals in disease transmission. We propose a generalized definition of reactivity (g‐reactivity) that allows to evaluate the differential contribution of the state space components to the transient behaviour of an ecological system following a perturbation. Our definition is based on the dynamic analysis of a system output, corresponding to an ecologically motivated linear transformation of the relevant state variables. We demonstrate that the g‐reactivity properties of an equilibrium are determined by the dominant eigenvalue of a Hermitian matrix that can be easily obtained from the Jacobian associated with the equilibrium and the system output transformation. As a testbed for our methodological framework, we analyse the g‐reactivity properties of simple spatially implicit metapopulation models of some prototypical ecological interactions, namely competition, predation and transmission of an infectious disease. We identify conditions for the temporary coexistence of an invader with a (possibly competitively superior) resident species, for transitory invasion of either prey or predator in otherwise predator‐ or prey‐dominated ecosystems, and for transient epidemic outbreaks. Through suitable examples, we show that characterizing the transient dynamics associated with an ecosystem's steady state can be, in some cases, as important as determining its asymptotic behaviour, from both theoretical and management perspective. Because g‐reactivity analysis can be performed for systems of any complexity in a relatively straightforward way, we conclude that it may represent a useful addition to the toolbox of quantitative ecologists.

Modulational instability of ion acoustic waves in a multi-species collisionless unmagnetized plasma consisting of nonthermal and isothermal electrons

A nonlinear Schrödinger equation is derived to study the modulational instability of finite amplitude ion acoustic waves in a collisionless unmagnetized plasma consisting of warm adiabatic ions and two distinct populations of electrons, one is due to distributed energetic electrons described by Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)] which generates the energetic electrons, and the other is the isothermal electrons. The instability condition and the maximum growth rate of instability have been investigated numerically. We have studied the effect of each parameter of the present plasma system on the maximum growth rate of instability. In particular, it is found that the maximum growth rate of instability increases with the increasing values of the wave number for any given set of values of the parameters associated with the present plasma system. It has also been shown that for any fixed value of the wave number, the maximum growth rate of instability increases with increasing values of the nonthermal parameter associated with the Cairns distributed energetic electrons.

Scalings for the Alfvén‐cyclotron instability: Linear dispersion theory and hybrid particle‐in‐cell simulations

AbstractThe Alfvén‐cyclotron instability is driven by an ion temperature anisotropy such that T⊥/T|| >1 where ⊥ and || denote directions perpendicular and parallel to a uniform background magnetic field Bo, respectively. The computations presented here consider a model of a magnetized, homogeneous, collisionless plasma. Two representations of the proton velocity distribution are considered: a single bi‐Maxwellian and a magnetospheric‐like configuration of two components, a more dense, relatively cool, isotropic component and a less dense, relatively hot, bi‐Maxwellian component which drives the instability. Only wave propagation parallel to Bo is considered. Using numerical solutions of the full kinetic linear dispersion equation, concise analytic expressions for the scaling of the dimensionless maximum instability growth rate and the corresponding dimensionless real frequency are derived as functions of three dimensionless variables: the hot proton temperature anisotropy, the relative hot proton density, and the hot proton β||. Furthermore, using one‐dimensional hybrid particle‐in‐cell simulations of this same instability, a third relation for the scaling of the maximum amplitude of the dimensionless fluctuating magnetic field energy density is derived.

Viscous effects on the acoustics and stability of a shear layer over an impedance wall

The effect of viscosity and thermal conduction on the acoustics in a shear layer above an impedance wall is investigated numerically and asymptotically by solving the linearised compressible Navier–Stokes equations (LNSE). It is found that viscothermal effects can be as important as shear, and therefore including shear while neglecting viscothermal effects by solving the linearised Euler equations (LEE) is questionable. In particular, the damping rate of upstream-propagating waves is found to be under-predicted by the LEE, and dramatically so in certain instances. The effects of viscosity on stability are also found to be important. Short wavelength disturbances are stabilised by viscosity, greatly altering the characteristic wavelength and maximum growth rate of instability. For the parameters considered here (chosen to be typical of aeroacoustic situations), the Reynolds number below which the flow stabilises ranges from$10^{5}$to$10^{7}$. By assuming a thin but non-zero-thickness boundary layer, asymptotic analysis leads to a system of boundary layer governing equations for the acoustics. This system may be solved numerically to produce an effective impedance boundary condition, applicable at the wall of a uniform inviscid flow, that accounts for both the shear and viscosity within the boundary layer. An alternative asymptotic analysis in the high-frequency limit yields a different set of boundary layer equations, which are solved to yield analytic solutions. The acoustic mode shapes and axial wavenumbers from both asymptotic analyses compare well with numerical solutions of the full LNSE. A closed-form effective impedance boundary condition is derived from the high-frequency asymptotics, suitable for application in frequency domain numerical simulations. Finally, surface waves are considered, and it is shown that a viscous flow over an impedance lining supports a greater number of surface wave modes than an inviscid flow.

Magnetorotational instability of weakly ionized and magnetized electron-positron-ion plasma

The magnetorotational instability in a differential rotating weakly ionized and magnetized plasma consisting of electron, positron, ion, and neutral particles has been investigated by using the multi-fluid model. Satisfying the current neutrality and homogeneity of the system in the equilibrium state by assuming the same unperturbed angular velocity for charge species and neutrals, the general local dispersion relation (DR) has been derived by taking into account the collision effects. By analytical examination of the obtained DR in the arbitrary and high frequency regimes, the instability conditions have been found in which the presence of light positive species (positrons) plays an important role in the instability criteria. Moreover, numerical investigation shows the broadening of instability range as well as increasing the maximum growth rate of instability (especially for the small number density ratio of light to heavy positive species) in the presence of positrons. The obtained results of the present investigation will greatly contribute to the understanding of the particles' dynamics as well as dissipation mechanism in some astrophysical environments, such as the region of accretion disks surrounding the central of black holes and protoplanetary disks.

Numerical analysis of electrostatic ion cyclotron instability in an electron-positron-ion plasma

This paper presents a theoretical study of the effects of positron density on the electrostatic ion cyclotron instability in an electron-positron-ion plasma using the kinetic theory approach. It is supposed that positrons and electrons can drift parallel to the magnetic field either in the same or the opposite directions. The dispersion relation for the electrostatic ion cyclotron waves in an electron-positron-ion plasma is derived, and the numerical results are investigated. It is found that an increase in positron concentration increases the critical drift velocity for excitation of the instability in both configurations. It is also found that as the positron concentration increases the growth rate of instability decreases. In addition, it is shown that at low velocities the maximum instability growth rate for the unidirectional case is higher than the counter-streaming case; however, after a certain velocity, the maximum growth rate in the counter-streaming case dominates that of the unidirectional case.

Fourth Order Nonlinear Evolution Equation For Interfacial Gravity Waves In The Presence Of Air Flowing Over Water And A Basic Current Shear

A fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves as first pointed out by Dysthe (1979) is derived for gravity waves propagating at the interface of two superposed fluids of infinite depth in the presence of air flowing over water and a basic current shear. A stability analysis is then made for a uniform Stokes gravity wave train. Graphs are plotted for the maximum growth rate of instability and for wave number at marginal stability against wave steepness for different values of air flow velocity and basic current shears. Significant deviations are noticed from the results obtained from the third order evolution equation, which is the nonlinear Schrödinger equation.

Kinetic theory of the filamentation instability in a collisional current-driven plasma with nonextensive distribution

The evolution of filamentation instability in a weakly ionized current-carrying plasma with nonextensive distribution was studied in the diffusion frequency region, taking into account the effects of electron-neutral collisions. Using the kinetic theory, Lorentz transformation formulas, and Bhatnagar-Gross-Krook collision model, the generalized dielectric permittivity functions of this plasma system were achieved. By obtaining the dispersion relation of low-frequency waves, the possibility of filamentation instability and its growth rate were investigated. It was shown that collisions can increase the maximum growth rate of instability. The analysis of temporal evolution of filamentation instability revealed that the growth rate of instability increased by increasing the q-parameter and electron drift velocity. Finally, the results of Maxwellian and q-nonextensive velocity distributions were compared and discussed.