Fisher equation with turbulence in one dimension
We investigate the dynamics of the Fisher equation for the spreading of micro-organisms in one dimenison subject to both turbulent convection and diffusion. We show that for strong enough turbulence, bacteria , for example, track in a quasilocalized fashion (with remakably long persistance times) sinks in the turbulent field. An important consequence is a large reduction in the carrying capacity of the fluid medium. We determine analytically the regimes where this quasi-localized behavior occurs and test our predictions by numerical simulations.
keywords:Population dynamics, Turbulence, Localization.
Pacs:72.15.Rn, 05.70.Ln, 73.20.Jc, 74.60.Ge
where is a continuous variable describing the concentration of micro-organisms, is the diffusion coefficient and the growth rate.
In the last few years, a number of theoretical and experimental studies , , , ,  have been performed to understand the spreading and extinction of a population in an inhomogeneous environment. In this paper we study a particular time-dependent inhomogeneous enviroment, namely the case of the field subject to both convection and diffusion and satisfying the equation:
where is a turbulent velocity field. Upon specializing to one dimension, we have
Equation (3) is relevant for the case of compressible flows, where , and for the case when the field describes the population of inertial particles or biological species. For inertial particles, it is known  that for large Stokes number , i.e. the ratio between the characteristic particle response time and the smallest time scale due to the hydrodynamic viscosity, the flow advecting is effectively compressible, even if the particles move in an incompressible fluid. Let us remark that the case of compressible turbulence is also relevant in many astrophysical applications where (2) is used as a simplified prototype of combustion dynamics. By suitable rescaling of , we can always set . In the following, unless stated otherwise, we shall assume whenever and for . For a treatment of equation (3) with a spatially uniform but time-dependent random velocity, see 
The Fisher equation has travelling front solutions that propagate with velocity , . In Fig. (1) we show a numerical solution of Eq. (1) with , obtained by numerical integration on a space domain of size with periodic boundary conditions. The figure shows the space-time behaviour of , the color code representing the curves . With initial condition nonzero on only a few grid points centered at , spreads with a velocity and, after a time reaches the boundary.
A striking result, which motivated our investigation, is displayed in Fig. (2), showing the numerical solutions of Eq. (3) for a relatively ”strong” turbulent flow, where the average convection velocity vanishes and ”strong turbulence” means high Reynolds number ( a more precise definition of the Reynolds number and specification of the velocity field is given in the following sections). From the figure we see no trace of a propagating front: instead, a well-localized pattern of forms and stays more or less in a stationary position.
For us, Fig. (2) shows a counter intuitive result. One naive expectation might be that turbulence enhances mixing. The mixing effect due to turbulence is usually parametrized in the literature  by assuming an effective (eddy) diffusion coefficient . As a consequence, one naive guess for Eq. (3) is that the spreading of an initial population is qualitatively similar to the travelling Fisher wave with a more diffuse interface of width . As we have seen, this naive prediction is wrong for strong enough turbulence: the solution of equation (3) shows remarkable localized features which are preserved on time scales longer than the characteristic growth time or even the Fisher wave propagation time . An important consequence of the localization effect is that the global ”mass” (of growing microorganisms, say) , , behaves differently with and without turbulence. In Fig. (3), we show : the curve with red circles refers to the conditions shown in Fig. (1)), while the curve with green triangles to Fig. (2).
The behavior of for the Fisher equation without turbulence is a familiar S-shaped curve that reaches the maximum on a time scale . On the other hand, the effect of turbulence (because of localization) on the Fishe equation dynamics reduces significantly almost by one order of magnitude.
With biological applications in mind, it is important to determine conditions such that the spatial distribution of microbial organisms and the carrying capacity of the medium are significantly altered by convective turbulence. Within the framework of the Fisher equation, localization effect has been studied for a constant convection velocity and quenched time-independent spatial dependence in the growth rate , , , . In our case, localization, when it happens, is a time-dependent feature and depends on the statistical properties of the compressible turbulent flows. As discussed in detail below, a better term for the phenomenon we study here might be ”quasilocalization”, in the sense that (1) spatial localization of the growing population sometimes occurs at more than one location; (2) these spatial locations drift slowly about and (3) localization is intermittent in time, as localized populations collapse and then reform elsewhere. For these reasons, the quasilocalization studied here is not quite the same phenomena as the Anderson localization of electrons in a disordered potential studied in . Nevertheless, the similarities are sufficiently strong that we shall use the terms ”quasilocalization” and ”localization” interchangeably in this paper. It is worth noting that the localized ”boom and bust” population cycles studied here may significantly effect ”gene surfing”  at the edge of a growing population, i.e. by changing the probability of gene mutation and fixation in the population.
For the case of bacterial populations subject to both turbulence and convections due to, say, an external force such as sedimentation under the action of gravity, we may think that the turbulent velocity can be decomposed into a constant ”wind” and a turbulent fluctuation with zero mean value, . We find that the localization shown in Fig. (2) can be significantly changed for large enough background convection .
We would like to understand why and how can change the statistical properties of in the presence of a random convecting velocity field. We wish to understand, in particular, whether spreads or localizes as a function of parameters such as the turbulence intensity and the mean ”wind” speed .
Our results are based on a number of numerical simulations of Eq. (3) performed using a particular model for the fluctuating velocity field . In Sec. we introduce the model and we describe some details of the numerical simulations. In Sec. we develop a simple ”phenomenological” theory of the physics of Eq. (3) based on our present understanding of turbulent dynamics. In Sec. we analyze the numerical results when the sedimentation velocity while in Sec. we describe our findings for . Conclusions follow in Sec. .
1 The model
To completely specify equation (3) we must define the dynamics of the ”turbulent” velocity field . For now, we set , neglect the uniform part and focus on . Although we consider a one dimensional case, we want to study the statistical properties of subjected to turbulent fluctuations which are close to thos generated by the three dimensional Navier-Stokes equations. Hence, the statistical properties of should be described characterized by intermittency both in space and in time. We build the turbulent field by appealing to a simplified shell model of fluid turbulence . The wavenumber space is divided into shells of scale , . For each shell with characteristic wavenumber , we describe turbulence by using the complex Fourier-like variable , satisfing the following equation of motion:
The model contains one free parameter, , and it conserves two quadratic invariants (when the force and the dissipation terms are absent) for all values of . The first is the total energy and the second is , where . In this note we fix . For this value of the model reproduces intermittency features of the real three dimensional Navier Stokes equation with surprising good accuracy . Using , we can build the real one dimensional velocity field as follows:
where is a free parameter to tune the strength of velocity fluctuations (given by ) relative to other parameters in the model (see next section). In all numerical simulations we use a forcing function , i.e. energy is supplied only to the largest scale corresponding to . With this choice, the input power in the shell model is simply given by , i.e. it is constant in time. To solve Eqs. (3) and (4) we use a finite difference scheme with periodic boundary conditions.
Theses model equations can be studied in detail without major computational efforts. One main point of this note is to explore the qualitative and quantitative dynamics of Eqs. (3),(4) and compare it against the phenomenological theory developed in the next section.
The free parameters of the model are the diffusion constant , the size of the periodic 1d spatial domain , the growth rate , the viscosity (which fixes the Reynolds number ), the mean constant velociy , the “strength’ of the turbulence and finally the power input in the shell model, namely . Note that according to the Kolmogorov theory , where is the mean square velocity. Since , we obtain that and are related as . By rescaling of space, we can always put . We fix and , corresponding to an equivalent . As we shall see in the following, most of our numerical results are independent of when is large enough. In the limit , the statistical properties of eq. (3) depend on the remaining free parameters, , , and . The important combinations of these parameters are discussed in the next section.
2 Theoretical considerations
We start our analysis by rewriting (3) in the form:
where and . Previous theoretical investigations  have shown that for , becomes localized in space for time-independent ”random” forcing (Anderson localization ). For large enough, a transition from localized to extended solutions has been predicted and observed in previous numerical and theoretical works . Here, we wish to understand whether something resembling localized solutions survives in equation (6) when both and depend on time as well as space.
To motivate our subsequent analysis, consider first the case . In this limit, Eq. (3) is just the Fokker-Planck equation describing the probability distribution to find a particle in the range at time , whose dynamics is given by the stochastic differential equation:
where is a white noise with . Let us assume for the moment that is time independent. Then, the stationary solution of (3) is given by
where is a normalization constant and . It follows that is strongly peaked near the points where has a local minimum, i.e. and . Let us now consider the behaviour of near one particular point where . For close to we can write:
where . Equation (9) is the Langevin equation for an overdamped harmonic oscillator, and tells us that is spread around with a characteristic ”localizaiont length” of order . On the other hand, we can identify with , a typical gradient of the turbulent velocity field . In a turbulent flow, the velocity field is correlated over spatial scale of order where is the average kinetic energy of the flow. For to be localized near , despite spatial variation in the turbulent field, we must require that the localization length should be smaller than the turbulent correlation scale , i.e.
Condition (10) can be easily understood by considering the simple case of a periodic velocity field , i.e. . In this case, condition (10) states that should be small enough for the probability not to spread over all the minima of . For small or equivalently for large , the solution will be localized near the minima of , at least for the case of a frozen turbulent velocity field .
The above analysis can be extended for velocity field that depend on both space and time. The crucial observation is that, close to the minima of , we should have . Thus, although is a time dependent function, sharp peaks in move quite slowly, simply because near the maximum of . One can consider a Lagrangian path such that , where is one particular point where and . From direct numerical simulation of Lagrangian particles in fully developed turbulence, we know that the acceleration of Lagrangian particles is a strongly intermittent quantitiy, i.e. it is small most of the time with large (intermittent) bursts. Thus, we expect that the localized solution of follows for quite long times except for intermittent bursts in the turbulent flow. During such bursts, the position where changes abruptly, i.e. almost discontinuosly from one point, say , to another point . During the short time interval , will drift and spread, eventually reforming to become localized again near . The above discussion suggests that the probability will be localized most of the time in the Lagrangian frame, except for short time intervals during an intermittent burst.
We now revisit the condition (10). For the case of a time-dependent velocity field , we estimate as the characteristic gradient of the velocity field, i.e.
where stands for a time average. Now, should be considered as the mean kinetic energy of the turbulent fluctuations. In our model, both and are proportional to , the strength of the velocity fluctuations. Thus, we can rewrite the localization criteria (10) in the form:
where and are computed for . We conclude that for small values of , is spread out, while for large , should be a localized or sharply peaked function of most of the time. An abrupt transition, or at least a sharp crossover, from extended to sharply peaked functions , should be observed for increasingf .
It is relatively simple to extend the above analysis for a non zero growth rate . The requirement (10) is now only a necessary condition to observe localization in . For we must also require that the characteristic gradient on scale must be larger than , i.e. the effect of turbulence should act on a time scale smaller than . We estimate the gradient on scale as , where is the characteristic velocity difference on scale . We invoke the Kolmogorov theory, and set to obtain:
In (12), we interpret as the characteristic velocity gradient of the turbulent flow. Because and , it follows that the r.h.s of (12) goes as . Note also that on the average, which leads to the inequality:
a second necessary condition. Once again, we see that localization in a Lagrangian frame should be expected for strong enough turbulence.
One may wonder whether a non zero growth rate can change our previous conclusions about the temporal behavior, and in particular about its effect on the dynamics of the Lagrangian points where . Consider the solution of (3) at time , allow for a spatial domain of size , and introduce the average position
where . Upon assuming for simplicity a single localized solution, we can think of just as the position where most of the bacterial concentration is localized. Using Eq. (3), we can compute the time derivative . After a short computation, we obtain:
where and . Note that is independent of . Moreover, when is localized near , both terms on the r.h.s. of (16) are close to zero. Thus, can be significantly different from zero only if is no longer localized and the first integral on the r.h.s becomes relevant. We can now understand the effect of the non linear term in (3): when is localized, the non linear term is almost irrelevant simply because is close to . On the other hand, when is extended the non linear term drives the system to the state which is an exact solution in the absence of turbulent convection .
We now allow a non zero mean flow . As before, we first set and consider a time -independent velocity field . Since the solution of (3) can still be interpreted as be the probability to find a particle in the interval at time , we can rewrite (9) for the case as follows:
The solution of (3) is localized near the point . Thus for small or large there is no major change in the arguments leading to (10). In general, we expect that will be localized near , provided the length is smaller than , i.e.
as a condition for localization, obtained in the study of localized/extended transition for steady flowsin an Eulerian context , . Here we remark that in a turbulent flow, Eq. (19) is only a necessary condition, because (10),(12) and (18) must all also be satisfied for to show quasi-localized states.
To study the change in the spatial behaviour of as a function of time, we need a measure of the degree of localization. to look for an some kind of order parameter. Although there may be a number of valuable solutions, an efficient measure should be related to the ”order”/”disorder” features of , where ”order” means quasi-localized and ”disorder” extended. As pointed out in the introduction, the total ”mass” of the organisms is strongly affected by a strongly peaked (or quasi-localized) , as opposed to a more extended concentration field. However, a more illuminating quantity, easily studied in simulations, is
where . Localized solutions of Eq. (3) correspond to small values of this entropy-like quantity while extended solutions correspond to large values of , which can be interpreted as the information contained in the probability distribution at time . In our numerical simulations, we consider a discretized form of (20), namely:
To understand how well describes whether is localized or extended, we consider the cases discussed in the Introduction in Fig.s (1) and (2). The numerical computations were done with for Fig. (1) and for Fig. (2), i.e. no turbulence and ”strong” turbulence (the attribute ”strong” refers to the conditions (10) and (12). In Fig. (4) we show corresponding to the two simulations, namely (red circles) and (blue triangles). The initial condition is the same for both simulations , i.e. a rather localized starting point. It is quite clear, from inspecting Fig. (4), that is a rather good indicator to detect whether remains localized or becomes extended. While for (a quiescent fluid), reaches its maximum value( for grid points) at (corresponding to uniform concentration ), for , is always close to its initial value , indicating that is localized, in agreement with Fig. (2).
Let us summarize our findings: when subjected to turbulence, we expect to be ”localized”, i.e. strongly peaked, most of the time for large enough and . Upon increasing the growth rate , the value of where shows Lagrangian localization should increase. Finally, for fixed and we should find a localized/extended crossover for large enough values of . Because our theoretical analysis is based on scaling arguments, we are not able to fix the critical values for which localized/extended transition should occour as a function of , and . However, we expect that the conditions (10),(12) and (18) capture the scaling properties in the parameter space of the model. Finally, we have introduced an entropy like quantity useful for analyzing the time dependence of of and for distinguishing between localized and extended solutions. In the following section, we compare our theoretical analysis against numerical simulations.
3 Numerical results for
We now discuss numerical results obtained by integrating equation Eq. (3). As discussed in Sec. , all numerical simulations have been done using periodic boundary conditions. Eq. (3) has been discretized on a regular grid of points. Changing the resolution , shifting to or , does not change the results discussed in the following. We use the same extended initial condition for all numerical simulations with the few exceptions which discussed in the introduction (the Fisher wave) and in the conclusions. For all simulations studied here, the diffussion constant has been kept fixed at .
We first discuss the case term and beginby understanding how well describes the localized/extended feature of the . In Fig. (5) we plot as a function of time for a case with . The behaviour of is quite chaotic, as expected. In Fig. (6) we show the functions for two particular times, namely (lower panel) and (upper panel). These two particular configurations correspond to extended () and quasi-localized () solutions. The corresponding values of are for and for . It is quite clear that for small strong localization characterizes while for increasing the behaviour of is more extended.
In Fig. (6) we also show (red circles) the instantaneous behavior of (multiply by a factor to make the figure readable). As one can see, the maximum of always corresponds to points where .
To understand whether the analysis of Sec. 2 captures the main features of the dynamics. we plot in FIg. (7), for (lower panel) and (upper panel), the quantity as computed from Eq. (8), i.e. by using the instantaneous velocity field . Although there is a rather poor agreement between and at , at time the is a rather good approximation of , i.e. when is localized. The spatial behavior of is dictated by the point where and the velocity gradient is large and negative. All the above results are in qualitative agreement with our analysis.
Next we test the condition (11), which states that localization should become more pronounced for increasing values of . To test Eq. (11) we performed a number of numerical simulations with long enough time integration to reach statistical stationarity. In Fig. (8) we show as function of , where means a time average. In the insert of the same figure, we show to time dependence of for two different values of , namely and . The behavior of is decreasing as a function of , in agreement with (11). The temporal behavior of , for two individual realizations shown in the insert, reveals that, while on the average decreases for increasing , there are quite large oscillations in , i.e. the system shows both localized and extended states during its time evolution. However, for large localization is more pronounced and frequent. On the other hand, for small values of , localization is a ”rare” event. Overall, the qualitative picture emerging from Fig. (8) is in agreement with Eqs. (10) and (11)
In the previous section we argued that (10) and (11) apply also for a time-dependent function velocity . The basic idea was that is localized near some point which slowly changes in time, except for intermittent bursts. In sufficiently large systems, localization about multiple points is possible as well. During the intermittent burst, spreads and after the burst becomes localized around a new position . We have already shown, in Figs. (6) and (7), that our argument seems to be in agreement with the numerical computations using a time-dependent velocity field. To better understand this point, we measure defined in Eq. (16). We expect a small during localized epochs when is small. Each time interval when is localized, should end and start with an intermittent burst where may become large. Figs (9) illustrates the above dynamics. The solid red curve is multiplied by a factor while the blue dotted curve shows . The numerical simulation is for , i.e. to a case where localization is predominant in the system. Fig. (9) clearly shows the ”intermittent” bursts in the velocity . The stagnation point velocity, punctuated by large positive and negative excursions, typically wanders near . If we assume a single sharp maximum in , as in the upper panel of Fig. (6), the localized profile does not move or moves quite slowly. During an intermittent burst, grows significantly while spreads over the space. Soon after the intermittent burst (see for instance the snapshot at time in Fig. (9)), the velocity becomes small again and the corresponding value of decreases. Fig. (9) provides a concise summary of the dynamics: both localized and extended configurations of are observed as a function of time. During a era of localization, a bacterial concentration described by is in a kind of ”quasi-frozen” configuration.
Fig. (9) tells us that condition (10), which was derived initially for a frozen turbulent field , works as well for time dependent turbulent fluctuations. As increases, the system undergoes a sharp crossover and the dynamics of slows down in localized configurations. Additional features of this transition will be discussed later on when we focus on a quantity analogous to the specific heat.
Finally in Fig. (10) we show the probability distribution , obtained by the numerical simulations, for three different values of , namely and . As one can see, the maximum is shifted toward small values of for increasing , as we already know from Fig. (8). Fig. (10) shows that the fluctuations of about the mean are approximately independent of .
We now turn our attention to the case . We have performed numerical simulations for two growth rates, namely and . We start by analyzing the results for . In Fig. (11), we show the behaviour of as a function of , while in the insert we show the probability distribution for three values of . Upon comparing with Fig. (11) against Fig.s (8) and (10), we see that a nonzero growth rate does not change the qualitative behavior of the system, in agreement with our theoretical discussions in the previous section.
It is interesting to look at the time averaged bacterial mass as a function of . In Fig. (11) we show and as a function of . For large , when localization dominates the behavior of , is quite small, order of its maximum value, i.e. due to turbulence the population only saturates locally at a few isolated points. The reduction in tracks in , but is much more pronounced.
In Fig. (13) we show computed for the case and . As in Fig. (9), we plot and . The qualitative behaviour is quite close to what already discussed for the case . The whole picture for , as obtained by inspection of Fig.s (11) and (13), supports our previous conclusions that, as long as the systems is in a quasi-localized phase, the effect of in Eq. (3) is almost irrelevant. Note that for the system to be in the localized phase we must require that both conditions (11) and (12) must be satisfied.
According to our interpretation, we expect that for increasing the whole picture does not change provided is increased accordingly. More precisely, we expect that the relevant physical parameters are dictated by the ratios in Eqs. (11) and (12). To show that this is indeed the case, we show in Fig. (14) the results corresponding to those in Fig. (11) but now with instead of .
Two clear features appear in Fig. (14). First the qualitative behavior of with increasing is similar for and . This similarity also applies to the probability distribution shown in the insert of Fig. (14). Second, there is a shift of the function towards large values of , i.e. the localized/extended transition occurs for larger values of with respect to the case . This trend is in qualitative agreement with the condition (12).
To make progress towards a quantitative understanding, we would like to use (10) and (12) to predict the shift in the localized/extended transition (or crossover) for increasing . For this purpose, we need a better indicator of this transition. So far, we used as a measure of localization: large values of mean extended states while small values of imply a more sharply peaked probability distribution. For , is the ”entropy” related to the probability distribution , solution of eq. (3). Thus for we can think of as the ”entropy” and of the diffusion constant as the ”temperature” of our system. This analogy suggests we define a ”specific heat” of our system in terms of and . After a simple computation we get using equations (8) and (20):
After allowing a statistically stationary state to develop, we then compute the time average to characterize the ”specific heat” of our system for a specific value of . It is now tempting to describe the localized/extended changeover associated with (3) in terms of the ”thermodynamical” function . In other words, we would like to understand whether a change in the specific heat can be used to ”measure” the extended/localized transition with increasing . The above analysis can be done also for , (when is no longer conserved) by using where the ”partition function” .
In Fig. (15) we show as a function of for (red curve with circles) and (green thin curve). Two major features emerge form this figure. First, is almost for small i.e. in the extended case. In the vicinity of a critical value , shows a rapid rise to large positive values and it stays more or less constant upon increasing . The large value of reflects enhanced fluctuations in (analogous to energy fluctuations in equilibrium statistical mechanics) when the population is localized. This behavior is in qualitative agreement with the notion of phase transition where (within mean field theory) the specific heat rises after a transition to an ”ordered state”. Here, the ”ordered state” corresponds to a quasilocalized, or sharply peaked probability distribution . Our numerics cannot, at present, distinguish between a rapid crossover and a sharp phase transition.
The second interesting feature emerging from Fig. (15) is that , the value of corresponding to the most rapid arise of , depends on , as predicted by our theoretical considerations. Indeed, as shown just below Eq. (12), we expect that . To check this prediction, we plot in Fig. (15) a third line (the blue line with triangles) which is just for plotted against . This rescaling is aimed at matching the position of the extended/localized changeover for the same independent of . The correspondence between the two curves in Fig. (15) confirms our prediction.
Fig. (15) shows that the statistical properties of can be interpreted in terms of thermodynamical quantities. How far this analogy goes, is left to future research. The quantity is in any case a sensitive measure of the extended/localized transition with increasing .
4 Numerical simulations for .
As discussed in Sec. , for fixed large , a mean background flow can eventually induce a transition from localized to extended configurations of . More precisely, for large , i.e. for large enough to satisfy (10) and (12), the system will spend most of its time in localized states provided the condition (18) is satisfied. Thus, for large enough we expect a transition from quasi localized (i.e. sharply peaked) to extended solutions. In this section we study this transition and check the delocalization condition in (18).
For this purpose we fix and which, according to our results in the previous section, correspond for to the case where localized states of dominate. As before, we use and of to characterize the statistical properties of for different values of . In Fig. (16) we show as a function of while in the insert we show the probability distribution for two particular values of . For we observe a quite strong increase of , a signature of a transition from predominately localized to predominately extended states. An interesting feature of for is the long tail towards small values of . This means that, occasionally, the system recovers a localized concentration distribution, as if .
The most striking feature appears near the critical value of where the transition a sharp rise in occurs. In Fig. (17) we show a two-peaked probability distribution for , where the slope of is the largest, and in the insert we show as a function of , where is computed using Eq.(22). Let us first discuss the result shown in the insert of Fig. (17). The specific-heat like quantity rises form at small , shows a bump where extended and localized states coexist, and then drops to for large. Note that the behavior of is different from what we observe in Fig. (15) suggesting a behavior reminiscent of a first order phase transition. We estimate as the critical value of where the behavior changes more rapidly. At the probability distribution is clearly bimodal, i.e. we can detect the two different phases of the system, one characterized by highly localized states and the other characterized by extended states. Turbulent fluctuations drive the system from one state to the other. The two maxima in are suggestive of two different statistical equilibria of the system. Note that is once again a good indicator of the transition from predominantly localized to predominantly extended states, as discussed in the previous section.
For , a straigthforward generalization of Eq. (16) leads to the following results for the velocity of a maximum in ,
One can wonder whether even for , the localized regime of small shown in Fig. (17) can be still characterized by , thus representing a pinning of the concentration profile despite the drift velocity . This question is relevant to understand whether the maxima for small in shown in Fig. (17) can be described using ideas developed for quasi localized probability distributions in Secs. and . To answer the above question, we performed a numerical simulation with a time-dependent uniform drift where . Thus changes periodically in time with an amplitude large enough to drive system from one regime to the other. If our ideas are reasonable, both and will become periodic functions of time. In particular, as l switches from small to large values, will go from in the localized regime to a large positive value in the extended phase.
Fig. (18) represents a numerical simulation for both and . In the upper panel we plot (red line) and the periodic function (we add an offset of in order to make the figure more readable). As one can see, indeed flattens out near periodically in time, and increases to large positive values in synchrony with the external time-dependent drift velocity. In the lower panel, we plot as a function of time; the graph clearly shows a periodic switching between the two statistical equilibria. A better understanding of the dynamics can be obtained from Fig. (19), where we show a contour plot of the normalized bacterial concentration (the horizontal axis is while the vertical axis is ). Localized states can be observed in the vicinity of and i.e when is near its smallest value, . Localized states are stationary or at most slowly moving whenever is small. During the period when is large, no localization effect can be observed. Fig. (20) we show four snapshots of taken from FIg. (19) at times . At and , when , the population is strongly localized while at and , when , is extended. The reason why even for a small is quite simple: according to our analysis in Sec. , localized states will form near shifted zero velocity points with negative slopes even for . When is large enough, there is no point where the whole velocity is close to . Every point in the fluid then moves in a particular direction, and the system develops extended states. Fig.s (18), (19) and (20) clearly support this interpretation.
In this paper we have studied the statistical properties of the solution of Eq.(3) for a given one dimnesional turbulent flow . Fig. (21) illustrates one of the main results discussed in this paper: the spatial behavior of the population subjected to a turbulent field. In particular, the figure shows four snapshots of taken from a numerical simulation (, , ) at times . The population shows strongly peaked concentration at time and , while at times and , is more extended. The popultion alternates strongly peaked solutions and more extended ones Our model is sufficiently simple to allow systematic investigation without major computational effort. From a physical point of view, the model can be interesting for compressible turbulent flows and whenever the field represents particles (such as the cells of microorganisms) whose numbers grow and saturate while diffusing and advecting. Our aim in this paper was to understand the statistical properties of as a function of the free parameters in the model. We developed in Sec. a simple theoretical framework. Based on three dimensionless parameters, we have identified three conditions which must be satisfied for quasi localized solutions of (3) to develop, given by Eqs. (10),(12) and (18).
All numerical simulations have been performed by using a grid resolution of points and a Reynolds number . Increasing the resolution will not change the numerical results provided the appropriate rescaling on conditions (10),(12) and (18) are performed, as shown in the following argument: let us define the grid spacing, i.e. , the Kolmogorov scale and the mean rate of energy dissipation, where . The turbulent field must be simulated numerically for scales smaller than the Kolmogorov scale. In the shell models, this implies that the largest value of is much larger than . The velocity gradient is of the order of . If the grid spacing is smaller than , no rescaling is needed in using the theoretical considerations derived in Sec. , namely equations (10), (12) and (18). On the other hand, if is larger than , as in our simulations, the velocity gradient goes as
Thus, by increasing the resolution, i.e. decreasing , we increase the velocity gradients and the condition (10) may not be satisfied unless we change or in an appropriate way. As an example of the above argument we show in Fig. (22) (insert) the value of computed for , and (red line with squares) and compared with the case, used in the main text, , and (green line with circles) already discussed in Sec. 3. For this particular case, we can superimpose the two curves by multiplying for the case by a factor which comes from equations (10) and (24). The final result agrees quite well with the case as shown in the same figure.
Similar considerations apply for a non zero mean flow , where the localized/extended transition should occours for larger values of according to (18). Finally, let us mention how we can predict the Reynolds number dependence of our analysis. According to the Kolmogorov theory, a typical velocity gradient is . Therefore, the transition from extended to localized solutions predicted by (10) can be observed provided either or . Hence, by increasing the Reynolds number, the extended/localized transition eventually disappears unless the diffusion term is properly rescaled.
Following the theoretical framework discussed in Sec. , we introduced a simple way to characterized how well is localized in space, namely using the entropy-like function (21) to illuminate the dynamics of the numerical solutions. The time average entropy was used to characterize the transition from extended to localized for increasing and from localized to extended solutions for increasing . We also found it useful to define a ”specific heat” by simply computing , where plays the role of temperature in the system. Notice from Eq. (10) that the physics is controlled by an effective temperature , i.e. rescaling is equivalent to changing .
The analogy between and some sort of effective temperature suggests that the rapid rise in the time average , observed in Fig. (15) near a characteristic value , might indicate a critical ”temperature” or diffusion constant . Fig. (15) highlights the rapid changes in from extended to localized states in the system. It will be interesting to study the behaviour observed in Fig. (15) from a thermodynamic point of view. As predicted by previous analytical studies  , with time-independent velocity field, a transition from localized to extended states has been observed by increasing . The interesting feature is that near this transition, the system shows a clear bimodality in its dynamics, at least in the probability distribution , more indicative of a first order transition.
We are not able at this stage, to predict the shape of the probability distribution of as a function of external parameters such as ,, and . It would be valuable to understand better when a quenched approximation (time-independent accumulation point in ) is reasonably good for our system, especially in regimes where microorganism populations are nearly localized. The reason why a quenched approximation may work is that the localized regime is quasi-static, in the sense that the solution follows the slow dynamics of accumulation points where with a large negative slope. A complete discussion of the validity of quenched approximation and analytic computations of is a matter for future research.
So far we have discussed the case of constant and positive. In some applications (both in biology and in physics) one may be interested to discuss with some non trivial space dependence. An interesting case, generalizing the work in ,  and  is provided by the equation,
with a turbulent convecting velocity field and where is positive on a small fraction of the whole domain and negative elsewhere. In this case, referred to as the ”oasis”, one would like to determine when can be significantly different from zero, i.e. when do the populations on an island or oasis survive when buffeted by the turbulent flows engendered by , say, a major storm (see  for a treatment of space-independent random convection). A qualitative prediction for results from the following argument: the extended and/or localization behaviour of depends on the ratio defined in Eq. (12). For small (i.e. small ) the solution must be extended and therefore one can predict that is significantly different from zero everywhere, wherever . On the other hand, for large , becomes localized. The probability for to be localized in one point or another is uniform on the whole domain. Thus, if the region where is significantly smaller then the region where , should approach to zero for long enough time.
In Fig. (23), we show a numerical simulation performed for a oasis centered on , performed with for and elsewhere for all . Thus the spatial average of the growth rate is . As a measure of , we plot its spatial integral as a function of time. The numerical simulations have been done with , and . In Fig.(23) we show three different values of . Let us recall that, when everywhere, as is now the case for the oasis, for , the system exhibits a transition from extended states to localized states, as illustrated in Fig. (15). As one can see, for the population tends to crash as predicted by our simple arguments. It is however interesting to observe that the dynamics of is not at all trivial. For and , seems to almost die and then recovers. Of course, our continuum equations neglect the discreteness of the population. At very low populations densities, a reference volume can contain a fractional number of organisms and extintion events are artificially supressed. Fig. (23) neverthelesst suggests an interesting feature of Eq. (25) with space dependent , worth investigating in the future.
Another interesting question tjhat deserves more detailed studies is the case two competing species with densities and . To illustrate the problem, consider the coupled equations: