Step, dip, and bell-shape traveling waves in a (2 + 1)-chemotaxis model with traction and long-range diffusion

In this paper, a new (2 + 1)-dimensional chemotaxis model is introduced, the focus being the understanding of influences of cooperative mechanisms from traction forces, long-range diffusion to chemotaxis on the dynamical characteristics of waves and their transport. Applying the F-expansion method, three families of new traveling wave solutions of bacterial density and chemoattractant concentration are constructed, including step, dip, and bell-shape wave profiles. The dependence of the conditions of existence of our solutions with respect to the model parameters is fully clarified. We found that traction and long-range diffusion slow down the waves and entail the transport of a small number of particles. Surprisingly, the long-range diffusion increases the thickness of the wave but does not alter its magnitude. Among families of solutions constructed, dip waves travel faster may be used to explain fast coordination among particles. As they support the transport of large amounts of cells, step waves could explain the transport of particles in high dense media. Intensive numerical simulations corroborate with a pretty much good accuracy our theoretical analysis, confirming the robustness of our predictions. Traction and long-range diffusion deeply affect the wave dynamics, they must be taken into account for a better understanding of chemotaxis systems.


Introduction
Studies on the dispersal of active particles have shed light on different mechanisms associated with their response when placed under various mechanical and/or chemical conditions. An example of active particles is given by a distribution of bacteria in a uniform flow whose collective motion allows them to spread and colonize outward regions. Depending on their sensitivity and acquaintance with the local environment composition, such a collective motion has been characterized qualitatively and quantitatively [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. When placed in chemical fields with attractive properties, ciliated particles, for example, synchronize their flagella's rotation, the resulting quasi-linear run allows them to efficiently reach the core of higher chemical concentrations. Such a collective behavior termed chemotaxis has been widely studied since the pioneering work of Adler [1]. Besides the importance of chemotaxis in the cellular realm where it plays a major role in fertilization [2][3][4], intercellular communication [5,6], wound repair [2,3], and pattern formation [3,7], it has also been reported in animal and insect ecology [4], large-scale collective behavior [1,[8][9][10][11][12][13][14] and synchronization processes [5,8]. Though the literature is well furnished when it comes to chemotaxis, various issues associated with a combination of chemical and mechanical constraints cells might feature in their natural habitats that has just recently started receiving attention [10,[12][13][14][15][16][17][18][19][20][21][22]. It is known that the latter interactions may lead to interesting holistic dynamical behaviors that to the best of our knowledge remain to be fully understood.
The collective dynamic entailed by chemotaxis indicates that bacteria are social entities and are therefore prone to communicate with one another. In some extreme cases, communication is made possible through chemoattractant production. The latter is responsible for the attraction of other distant entities. This phenomenon significantly different from chemotaxis at the molecular level can be termed auto-chemotaxis and was recently observed through lenses of experimental settings [5]. Though the authors were not able to provide full and satisfying explanations of its specific contribution at the local level, they showed that it has the potential of biasing the primary chemotactic signal at the onset of bacterial motion. Depending on the strength of the bias, cells rearrange themselves and exhibit a different moving pattern. Independently, a theoretical approach to the question was addressed to demonstrate that auto-chemotaxis is a factor of instability [15,16], while friction and proliferation in the system have stabilizing effects. Such an antagonism is at the onset of pattern selection with a well-defined wave number [15]. On the other side, the fact that cells can become attractive by producing their own chemoattractant implies that in a given population, certain cells are more a e-mail: dkwilliam90@gmail.com (corresponding author) chemotactic than others, therefore may exert distant actions like traction forces. In cancer invasion for example, those distant actions allow cancerous melanocytic cells to escape tumorous regions and undergo metastasis [17]. In the dynamic of micro-swimmers assembly, distant traction forces were also reported. This means that the existence of traction forces might be a common feature in reactive systems, especially for higher active particles densities [16].
Apart from interspecies communication mechanisms, cells are also involved in hydrodynamic interactions which ensure an active transport of diffusive and non-active components over long distances. In aqueous media, swimming patterns of cells that consist of pulling, pushing, gliding, and scrawling imply that cells interactions with fluids constituents may lead to a variety of spatiotemporal patterns. Experimental and theoretical evidence showed that bacteria are collectively transported through wavy structures like solitary pulses [18][19][20][24][25][26][27][28][29][30]. Furthermore, other experimental findings suggest that hydrodynamic interactions coupled to the chemoattractant concentration gradient may guide traveling bands of bacteria at a constant speed [18][19][20]. Therefore, the inclusion of advection rate while modeling chemotactic behaviors is expected to substantially improve our knowledge of the system. Mathematical modeling in this view requires to couple the Keller-Segel chemotaxis model to Navier-Stokes equations [16,[21][22][23]. Another simple and realistic approach consists at collecting contributions emanating from stresses and traction cells exert upon each other in addition to the uniform flow coming from the background field within which cells are immersed in. In fact, recent evidences demonstrate that the collective motion of cells in a fluid generates flows mediated by the hydrodynamics interactions in between them [10,22,23]. It is therefore natural to wonder how both chemotaxis and traction forces contribute to producing structures like traveling waves observed from the Keller-Segel formulation for chemotaxis [24]. The latter model initially introduced to describe slime mold showed drawbacks and later experimental findings suggest that the issues may be circumvented when cells create their own chemoattractant gradient [25]. This implies that the collective motion of cells modifies the average flow of the medium, which in turn affects cellular response. In this way, cells' response to an excitatory source ceases to be proportional to the concentration gradient of the excitation. Another way of looking at the problem consists at considering the nonlocal character of bacterial response [3,26,31] which is responsible for the nucleation of long-range diffusion. In this work, we introduce a mathematical model that extends the existing ones, with the aim to fill in the above-mentioned gaps. To this end, we consider a chemotactic system where long-and short-range diffusion, chemotaxis, traction forces, cellular proliferation, and uniform advection are accounted for; dynamical behaviors of new traveling waves shall be investigated.
The rest of the manuscript is organized as follows. In Sect. 2 we derive the model and construct its analytical solutions. Secondly, we evaluate the effects of key parameters of the system on the proposed solutions and then make some predictions on the dynamical behavior of the system. In Sect. 3, numerical simulations are performed to ascertain our predictions. Some biological implications are discussed. Section 4 concludes the paper.

The model
In this description of collective behavior, we are interested in a bacterial population in a two-dimensional frame, immersed in a fluid experiencing a uniform flow. Besides the chemotactic velocity and the medium flow rate, the bulk bacterial motion also has a drift-velocity contribution emanating from the stress cells exert upon each other. Recent studies show that many energy sources simultaneously influence cells motion [30], but in the present paper, we consider the case where bacteria are in the presence of only one chemoattractant source. The timescale is large enough such that cells undergo proliferation, the nonlocal response ensures long-distance communications, and the cellular protrusion is also accounted for. The resulting model is an extended set of coupled partial differential equations that reads [3,24,[28][29][30][31] ∂c ∂t In Eq. (1), t is the time variable in seconds (s), ∂ y 2 , the two-dimensional Laplacian. n is the bacterial density, while c is the chemoattractant concentration. D 1 and D 3 are short-range diffusion while D 2 stands for the long-range diffusion of bacteria. χ 0 represents the chemotaxis strength, r is the proliferation rate of cells, σ , the medium's carrying capacity. The term k 0 n k 1 +k 2 n 2 describes the chemoattractant production. Numerical values of k 0 , k 1 , k 2 are not known to the best of our knowledge. β is the chemoattractant consumption's rate. δ = (δ x , δ y ) T is the two-dimensional bulk velocity field of the system. Recent studies of bacterial migration in a confined one-dimensional racetrack showed nucleation of strong flows emanating from cells closes to channel boundaries [14]. These flows were capable of swerving the bulk dynamic of cells placed far from channel walls. In this sense, the bulk velocity field in the system comprises an active part generated by the collective motion of cells and a passive part emanating from the perturbations of the medium within which cells are immersed. We propose a velocity field in the medium with the form δ = ∇ 0 + ∇ H (n, c), with δ 0 = δ 0 x + δ 0 y , and H (n, c) = τ 0 n 1 + λn . (2) δ 0 x , and δ 0 y are components of the uniform flow imposed by the medium within which cells and the chemoattractant are immersed.
The function H takes into account the effects of traction forces generated by active particles present in the medium (here the bacterial cells). τ 0 is the maximum traction force ensuring a finite velocity at higher cellular density when the population increases in a quasi-dilute medium. λ measures the velocity decrease due to a higher density of cellular aggregation. In the thermodynamic limit, this choice of H ensures that the velocity remains finite both at low and high densities. The current model diverges from the classical Keller-Segel models [3,24,[29][30][31], since the chemotactic velocity drift, as well as wave profiles, remains bounded. This further sets the stepping stone to find relevant physical solutions for the model under consideration. In that endeavor, we assume a quasi-dilute medium, corresponding to λ → 0, meaning that hydrodynamic properties of bacteria are weak but non-null. Under this configuration, experimental studies revealed that one might still expect complicated patterns [7,32], and bacterial distribution per unit area may be very small but non-null. Hence, only a small amount of cells are expected to contribute to chemoattractant production, a situation entailing k 2 n 2 k 1 << 1. Considering lower power series of the saturating term from Eq. (1b) and plugging Eq. (2) into Eq. (1) yield the (2 + 1)-modified extended chemotaxis model are arbitrary real constants and are also not known as the k j s. Nevertheless, they will be chosen in accordance with biological relevance. β 1 , β 3 are referred to as the new chemoattractant production rate parameters. More than a simple reactiondiffusion-advection process, the system of Eq. (3) is dynamically interesting. It takes into consideration simple and nonlinear cross diffusion, in addition to incorporating two active transport mechanisms, namely chemotaxis and traction. Though traction has been the subject of intensive experimental studies [5,15,16,25], its contribution has not yet been mathematically formulated in chemotactic systems, to the best of our knowledge. Model Eq. (3) indicates that taking into account traction forces enhances both cellular and chemical displacement toward one another. In this sense, Eq. (3) can be used to explain a broad range of phenomena in systems made of composite materials. For example, it may describe the spreading behavior of two chemotactic subpopulations of bacteria moving toward each other with different velocities. Such a differential velocity may lead to rich phenomena like instabilities and new patterns formation just to name a few. We propose Eq. (3) as a viable generalized chemotaxis model that incorporates several aspects of bacterial collective motion not discussed in previous models [3,15,[28][29][30][31][33][34][35][36][37][38][39][40][41][42].

Analytical solutions through the extended F-expansion method
The F-expansion method is a scheme used to solve nonlinear ordinary differential equations. In order to solve Eq. (3) using an extended F-expansion method, the traveling wave variable ξ = kx + ly − ωt is assumed. In the latter transformation, 1 k and 1 l are the wave widths along the x− and y− directions, respectively, ω the wave velocity taken in the laboratory frame. Thus, Eq. (3) become with Solutions of Eq. (4) are sought for a chemoattractant concentration of the form Inserting Eq. (5) into Eq. (4) leads to the fourth-order nonlinear ordinary differential equation provided that We restrict ourselves to the case where ω = kδ 0 x + lδ 0 y (which implies 0 = 0) to avoid nonphysical solutions that diverge at infinity. Such a choice is also consistent with recent numerical and theoretical results presented in [19,42,43] where the authors proved that the wave velocity in the Martiel-Golbeter chemotaxis model (for Dict yostelium discoideum dynamics) varies linearly with the imposed flow. On the other hand, Eq. (7) tells that variations of chemoattractant and bacterial waves are directly linked; hence, the corroborating results are obtained in the analysis of chemotactic Dict yostelium colonies [40]. The same study showed that by increasing background level, the directed propagation can be suppressed, due to memory inactivation. In other words, cells can be swept or their direction propagation swerved, enforcing the conclusion that medium flow rate has the potential of favoring the rise of progressive or regressive waves. In the present study, the transition from forward to backward propagation is attained when parameters are chosen such that the critical line δ 0x δ 0y = −l √ K −l 2 corresponding to stationary patterns is violated. In the latter configuration, our solutions may be either progressive (ω > 0) or regressive (ω < 0). The former is responsible for faster bacterial colonization of unoccupied regions, while the latter may be the signature of backward waves. In the flux limiting cases, the backward waves in a chemotactic system were shown to be responsible for a population saturation in a stable state accompanied by a transition toward unstable modes [44].
Physically acceptable solutions of Eq. (3) correspond to positive bacterial and chemoattractant wave amplitudes. In addition, both bacterial densities and chemoattractant concentrations must remain finite; hence, using Eq. (7) and assuming that n(ξ ) ≥ 0, the finiteness of the chemoattractant concentration implies the following restrictions Before proceeding further, Eq. (8) provides important characteristics about the solutions to be found below. The traction forces must be non-null and greater than the chemotaxis strength implying that the experimenter must choose an appropriate chemoattractant substance that is consistent with Eq. (8). Moreover, there exists a critical chemoattractant concentration that completely depends on system parameters (τ 0 , χ 0 , D 1 , D 3 ), below which the corresponding solutions are unphysical. The solutions will be viable only if the chemoattractant concentration is at least equal to c min , a situation which means that chemoattractant diffusion rate is always greater or equal than a minimal value The fact that D 1 > D 3 min implies bacteria diffuse faster than the minimum chemoattractant diffusion rate that is necessary to generate traveling chemical and bacterial waves. Furthermore, the minimal chemoattractant diffusion rate D 3 min broadly depends on the ratio between chemotaxis strength and traction forces, enforcing that the latter substantially modify the behavior of chemotactic systems. Taking into account the above considerations, Eqs. (6), (7) become where The next step of the F-expansion method consists at looking to solutions of Eq. (6') in a polynomial expansion [45][46][47][48][49][50]. We determine the order of the expansion by balancing the higher-order derivative with the higher-order nonlinear terms in Eq. (6'). Doing so permits a finite polynomial expansion where a 0 , a 1 , a 2 are real constants to be determined later. The function F(ξ ) is a solution of the auxiliary equation P 4 , P 3 , P 2 , P 1 , P 0 are real constants. Solutions of Eq. (10) may be found in [27,[45][46][47][48][49][50]. As explained in [45][46][47][48][49][50], solutions of Eq. (10) admit unbounded and localized structures. The latter ones are excellent candidates for describing real physical phenomena. We continue the methodology by plugging Eq. (9) into Eq. (6') and making use of Eq. (10). Further, we collect all the coefficients of power of F (F m (ξ ), m = 0, 1, 2, 3, 4), and setting each coefficients to zero yields a set of algebraic equations that we solve for the variables a 0 , a 1 , a 2 , K , β 3 and obtain different families of solutions presented below.

Existence and dynamical behavior of solutions Eqs. (11)-(17)
A straightforward analysis of Eqs. (11)- (17) shows that bacterial and chemoattractant solutions exist if D 2 = 0, σ 1 < 0, σ 2 > 0 which lead to the constraints β 1 < r σ + βc min = β 1 max , and r > β The first inequality of Eq. (18) proves that the linear chemoattractant production rate β 1 possesses a maximum value given by r σ (in the absence of chemoattractant production) and linearly increases with chemoattractant consumption rate β. Though the F-expansion method allows a large number of analytical solutions for the auxiliary equation Eq. (10), we just keep above families of solutions that are physically relevant as they satisfy the existence conditions Eqs. (8) and (18). The fact that the maximum chemoattractant production rate depends on the nominal chemoattractant consumption rate enforces to hypothesize the existence of a critical chemoattractant production-consumption line that may impede the wave generation process if violated. The second inequality of Eq. (18) shows that our system admits a minimal proliferation rate always greater than the threshold value r min .
Equations (8) and (18) describe the general conditions of the existence of bacterial and chemoattractant concentration waves but, do not provide any details on dynamical properties. We discuss below some dynamical properties of both bacterial and chemoattractant waves such as their profiles, velocities, amplitudes, and thicknesses, and we analyze effects of τ 0 , χ 0 , D 2 , D 1 , and β. To this end, we take β 1 = β 1max 10 , r = r min + r 0 , r 0 = 1.69 × 10 −9 being the proliferation rate given in [3]. Table 1 displays the parameters values used in our analyses.

Bell-shaped bacterial and chemoattractant: family A
The bell-shaped profile corresponds to Family A. Using parameters of Eq. (11a), explicit analytical expressions are given by  [3,19,29,30] χ 0 6.49 × 10 −5 cm 2 M −1 s −1 [3,19,29,30] δ 0 x , δ 0 y 10 −4 −10 −6 cm s −1 [18][19][20]29,30] where the associated velocity reads: From Eq. (20), one deduces the existence of minimal/maximal wave velocities where ω min = ω(l = l crit ) and ω max = ω(l = l 0 ). l crit is recovered by solving ω| l=l crit = 0, and the maximum velocity ω max is reached at l 0 . The latter is determined by solving dω dl | l=l 0 ; hence, we have l 0 and l crit depend on the system parameters, which means the experimenter has the potential of controlling fast or slow wave propagation if he accurately tunes the experimental setups. From Eq. (19) n ∞ bs = lim ξ →∞ n(x, y, t) → 0 and c ∞ bs = lim ξ →∞ c(x, y, t) → c min which mean that bacterial and chemoattractant wave amplitudes are finite as their associated velocity. This confirms that bellshaped solutions Eq. (19) are physical objects. The bell-shape profile Eq. (19) is displayed in Fig. 1 at the time t = 100. Bell-shape waves have been predicted analytically [24,[26][27][28][29][30][36][37][38]42], numerically [16,19,28,30,36,40,43,51], and experimentally observed [15,19,20,30] in one-dimensional chemotactic systems. Traveling waves in reactive systems are either matter carriers or information conveyors, the bell-shaped profile obtained here may explain collective bands of bacteria usually observed in reactive systems. Its velocity is depicted in panels (c)-(e) of Fig. 1. It is seen that the amplitude of ω bs decreases with increasing values of traction and long-range diffusion, respectively (see Fig. 1c-d), meaning that the traction and the long-range diffusion slow down the waves. Conversely, the amplitude of ω bs increases with increasing values of χ 0 implying that the chemotaxis strength accelerates the wave propagation. τ 0 , D 2 , and χ 0 have competing effects on the velocity of the wave. This property may be used in experiments to detect or characterize the waves. In all the cases, the velocity reduces with increasing values of l, thus thinner waves move faster than wider ones. Considering the influences of system parameters on the bell-shaped wave, one observes that when long-range diffusion increases (Fig. 2a), the wave thickness widens, hence the cellular distribution occupies a larger spatial domain. At high values of long-range diffusion D 2 , the coordination degree among units of the bacterial population decreases, resulting in a diminution of the global velocity of the aggregation. Such a result is in accordance with the idea that diffusion of particles should break up coordination and communication degree among particles. Wave thickness variations are a common feature in reactive systems as extensively discussed in [19,42,51], and its occurrence in the present study signifies that cells do not lose their active properties, but rather rearrange themselves to accommodate chemoattractant concentrations across the medium. In panels (b)-(e) of Fig. 2, wave thickness variations are accompanied by an amplitude variation. Amplitude and thickness increase are observed when traction Fig. 2b and short-range diffusion Fig. 2d increase, while chemotaxis Fig. 2c as well as the chemoattractant consumption rate Fig. 2e decreases. Wave thickness reduction or increase coupled with amplitude variations in systems with uniform flows have also been reported in autocatalytic fronts [52], the Fitz-Hugh-Nagumo model [53], and the Belousov-Zhabotinsky reaction [54,55]. We propose such a coupled dynamic between wave thickness and amplitude as a tool to slice spatial domains in intervals within which cells activity remains optimized, and above which cellular density drastically reduces.

Step bacterial and chemoattractant waves: family F
Plugging the parameters Eq. (16a) into Eq. (7') and Eq. (8) allows us to construct step traveling waves whose analytical formula are The corresponding velocity associated with solutions Eq. (23) is Eur. Phys. J. Plus (2022) 137:353 Fig. 4 Step traveling waves for bacterial density (a) and chemoattractant concentration (b) (Eq. (23)). P 2 = 4, other parameters taken as in Fig. 1 Furthermore, Step waves obtained here translate the transition process happening between two different levels of bacteria as well as chemoattractant distribution. The gap between the levels depends on system parameters, which means that the experimenter has the potential of choosing how and the position at which the transition happens. Numerical values of short-range diffusion D 1 , D 3 , chemotaxis strength χ 0 , chemoattractant consumption rate β, and medium carrying capacity σ were chosen as in [3,19,29,30]. The fluids flow rate was taken according to experimental studies [18][19][20]29,30]. The other parameters τ 0 , D 2 , β 1 , β 3 to the best of our knowledge are not yet available. While the analytical formalism used allows to determine β 3 as given by Eqs. (11)-(17), the existence of solutions discussed yields a minimum value of β 1 . Though traction forces dominate the chemotaxis strength, the latter is shown to still have strong effects on wave characteristics namely the velocity, amplitude, and thickness. These observations are consistent with conclusions reported in [30]. More importantly, for the same set of parameter values, comparison of Eq. (20), Eqs. (22) and (24) yields ω dw > ω sw > ω bs : The dip waves travel faster than the step and the bell-shaped waves. In other words, dip waves are better candidates to achieve fast coordination of cells or to quickly convey a piece of information within a bacterial population. From Eqs. (19), (21) and (23), one derives the inequalities n sw > n dw > n bs .
Step waves carry a higher number of particles compared to dip and bell-shaped waves. Generally speaking, an optimal transport is expected for higher velocities and wave amplitudes, but the results obtained here draw the roadmap to characterizing an optimal transport as follows: while step waves ensure the transport of a higher number of particles, a faster transport is guaranteed through dip wave structures. In reactive systems without traction and long-range diffusion, it has been shown that optimal transport necessitates the coupled dynamic between short-range diffusion and feedback [56,57]. However, the present study stresses that traction and long-range diffusion deeply alter the optimal transport of waves, hence must be taken into account for a better description of waves propagation.

Numerical experiments
In this section, we ascertain the ability of waves discussed above to propagate in a stable fashion way in model Eq. (3). To this end, direct numerical integrations of Eq. (3) on a square spatial domain of length L = 100 are performed. We take N = 512 points along each spatial direction with a time step is dt = 10 −2 . Integrations are performed through the pseudo-spectral method. We initially launched the simulations by introducing perturbed analytical solutions with a noise strength of ten percent of the initial amplitude of the wave. Simulations ran over a final time t fin = 200 s and results are displayed in Figs. 5, 6, 7. To be precise, panels (a), (b), (c) of Figs. 5, 6, 7 (resp. panels (d), (e), (f)) display snapshots of the bacterial density (resp. chemoattractant concentration) at t = 0 s, 100 s, 200 s, respectively, obtained with the analytical solutions Eqs. (19), (21), (23) as initial conditions. Though we inserted a random perturbation at the initial time, the results depicted in Figs. 5, 6, 7 show that initial solutions evolve without undergoing any collapses nor explosions. Our solutions are numerically stable ones. In addition, from Eqs. (19), (21), and (23), the analytical solutions found above predict waves whose profiles, widths, amplitudes, and velocities remain unchanged during their evolutions, but are slightly displaced due to small fluid velocities of magnitude about 10 −5 . Snapshots of waves displayed in Figs. 5, 6, 7 are in good agreement with analytical predictions. Our solutions are therefore stable ones, such that they are likely to be observed in experiments. Interestingly, analyzing Fig. 6 reveals that the heights of dip bacterial and chemical waves are slightly shifted by a magnitude of ∼ 10 −2 . The latter observation forces us to determine the absolute errors for bacterial and chemoattractant concentrations and found that |n ana (x, y, t) − n num (x, y, t)| ∼ 10 −2 , |c ana (x, y, t) − c num (x, y, t)| ∼ 10 −2 . Dip waves have not yet been recovered in chemotactic systems, to the best of our knowledge. The fact that their height slightly increases may be the signature of some biophysical features whose stability properties are beyond the scope of the present work. Our results are new ones and we do believe this work may motivate further two-dimensional experimental investigations in chemotaxis systems where traction forces, advection, and long-range diffusion are at play. To the best of our knowledge, such experimental investigations that take into account the latter effects are still missing.
In the above numerical simulations, we show that solutions constructed here are stable, even though families of solutions B, D, E, G are not presented. The solutions corresponding to Families B, D, E, and G exhibit periodic and triangular periodic waves. They   Fig. 1 actually represent important ways by which bacteria and chemoattractant are transported, in addition to the fact that they substantially match recent experimental results [18][19][20]. Periodic bacterial waves were proposed as a mechanism to inhibit the runaway fashions due to catalysis response of cells to chemoattractant they might produce [43,57]. In this sense, periodic waves corresponding to Families B, D, E, and G are stable solutions that could be used in the assessment of other aspects of the transport of chemotactic particles in fluids.

Conclusion
We discussed the existence and the dynamical behaviors of bacterial and chemical waves propagating in a (2 + 1)-dimensional chemotactic system. We introduced a new model that takes into consideration several processes including a uniform advection, Eur. Phys. J. Plus (2022) 137:353 long-range diffusion, chemotaxis, traction, cellular proliferation, and chemoattractant production degradation. The traveling wave variable is assumed and an extended F-expansion method is used to construct new traveling wave solutions like step, dip, and bell-shape waves. The quest for physically acceptable solutions enforces the existence of a minimum chemoattractant concentration which depends on both traction, chemotaxis strength, bacterial and chemical short-range diffusion. We showed that traction, shortrange diffusion, chemotaxis, and chemoattractant consumption rate have competing effects on the number of particles transported. The former parameters decrease the number of particles transported while the latter ones increase wave amplitudes. Increasing values of long-range diffusion decrease wave velocity but increase the wave width, the amplitude remaining unchanged. Besides, by comparing velocities and amplitudes of solutions presented, we observed that while dip waves travel faster hence are better candidates to explaining fast bacterial coordination, step waves have the potential of carrying a higher number of cells, hence may be considered as robust structures to perform transport in a highly dense system. The stability of solutions constructed is analyzed through direct numerical integration of the original model. Both numerical and analytical solutions remain very close hence ascertaining our theoretical predictions.

Data Availability Statement
This manuscript has associated data in a data repository. This manuscript has associated data in a data repository. [Authors' comment: The data that support the findings of this study are available in GitHub at https://github.com/wil-09/Codes/blob/master/Code.m.]

Declarations
Conflict of interest The authors declare that they have no conflict of interest.