Extended patchy ecosystems may increase their total biomass through selfreplication
Abstract
Patches of vegetation consist of dense clusters of shrubs, grass, or trees, often found to be circular characteristic size, defined by the properties of the vegetation and terrain. Therefore, vegetation patches can be interpreted as localized structures. Previous findings have shown that such localized structures can selfreplicate in a binary fashion, where a single vegetation patch elongates and divides into two new patches, in a process resembling cellular mitosis. Here, we extend these previous results by considering the more general case, where the plants interact nonlocally, this extension adds an extra level of complexity and shrinks the gap between the model and real ecosystems, where it is known that the planttoplant competition through roots and aboveground facilitating interactions have nonlocal effects, i.e. they extend further away than the nearest neighbor distance. Through numerical simulations, we show that for a moderate level of aridity, a transition from a single patch to periodic pattern occurs. Moreover, for large values of the hydric stress, we predict an opposing route to the formation of periodic patterns, where a homogeneous cover of vegetation may decay to spotlike patterns. The evolution of the biomass of vegetation patches can be used as an indicator of the state of an ecosystem, this allows to distinguish if a system is in a selfreplicating or decaying dynamics. In an attempt to relate the theoretical predictions to real ecosystems, we analyze landscapes in Zambia and Mozambique, where vegetation forms patches of tens of meters in diameter. We show that the properties of the patches together with their spatial distributions are consistent with the selforganization hypothesis. We argue that the characteristics of the observed landscapes may be a consequence of patch selfreplication, however, detailed field and temporal data is fundamental to assess the real state of the ecosystems.
1 Introduction
Spontaneous shift from a uniform cover of vegetation into a fragmented ecosystem constituted by a spatially periodic distribution of gaps, or patches is an irreversible process. This transition may occur either in water or nutrient limited territories. It is now widely admitted that facilitative and competitive interactions between individual plants can directly or indirectly account for the formation of vegetation patterns Lefever1997 (); LT (); Lejune_couteron (); Klausmeier1826 (); Rietkerk_2001 (); Meron2001 (); QUA:QUA10878 (); sherratt2005analysis (); d2006patterns (); vesipa2015noise (). The spatial distribution of such patterns can be modified the effect of climatic, terrain and anthropogenic influences Kefi_07 (); Dakos_11 (); Deblaw_12 (). Quantitative studies based on field observation have been made on: the Sahelian gapped patterns, constituted by Combretum micranthum trees JEC:JEC1126 (); ECY:ECY20088961521 (); Lefever2009194 (); patches of vegetation in arid high altitude environments in the tropical alpine ecosystems of the Andes formed by Festuca orthophylla (Poaceae); and for grasses or by Pycnophyllum tetrastichum cushions (Bolivia, Couteron20140102 ()).
Patterned vegetation landscapes are fragmented, i.e, the terrain is only partially covered by vegetation. Some of these landscapes are composed of vegetation patches, which may be sparsely or regularly distributed. These patches usually have a characteristic size and well defined circular shape. It has been shown recently that localized patches can be destabilized by a deformation of their circular shape, either leading to the formation of labyrinthine patterns bordeu2016laby (), or dividing into two new identical patches of smaller diameter bordeu2016self (). The latter is a phenomenon often called selfreplication and resembles mitotic cell division. It has been studied in the context of herbaceous populations in arid ecosystems bordeu2016self (). The selfreplication mechanism allows the transition from a single localized structure into a qualitatively different state, namely a hexagonal periodic pattern of vegetation. During the transition from localized to periodic pattern, the total biomass increases as newly formed patches contribute to the repopulation of the territory accessible to vegetation. From a theoretical point of view, selfreplication is a patterning phenomenon better known in physicochemical contexts rather than ecological systems. It is a generic mechanism of pattern formation, which has been observed and established in various nonequilibrium systems, such as fluids Magnetic (), liquid crystals refId0 (); Oswald200067 () chemical systems pearson:93 (); lee1994experimental (); PhysRevLett.79.1941 (); kaminaga2005black (); PhysRevLett.98.188303 (); DeKepper (); monine2002modeling (); Schaak1998386 (); PhysRevLett.81.1726 (); Tlidi_Gandica_16 (), in plant ecology meron2004vegetation (); bordeu2016self (), material sciences Ren_2003 (); Yasumasa (), granular fluid systems PhysRevLett.99.038001 (); sandnes2011patterns () and nonlinear optics PhysRevLett.89.233901 ().
In this contribution, we investigate the spacetime dynamics of vegetation under a selfreplication phenomenon by extending the previous work by Bordeu et al. bordeu2016self (); bordeu2016laby (), where a simple local model was used to illustrate that selfreplication was a possible mechanism for vegetation propagation. Here we consider general integrodifferential model instead of the simplified model of bordeu2016self (); bordeu2016laby (), here, nonlocal interactions are taken explicitly into account. The nonlocalities arise from the facilitating mechanisms, i.e. promoters of vegetation growth, competition mechanisms, which limit vegetation growth, and dispersion effects. This model corresponds to a variant of the theory of vegetation patterning established by R. Lefever Lefever1997 (), which focuses on the relationship between the structure of individual plants and the facilitationcompetition interactions existing within plant communities. It is now widely recognized that the existence of facilitation and competition interactions play an important role in the formation of selforganized vegetation patterns. Numerical simulations of our model show indeed a selfreplication process that leads moderately arid ecosystems to undergo a transition to higher biomass states, namely hexagonal patterns of vegetation patches. Moreover, we show that this kind of patterns may be obtained through the decay of a homogeneous vegetated landscape towards a less populated fragmented state, where hydric stress induces contraction of vegetated areas. Depending on the levels of aridity, the ecosystems may decay to a different type of patterned states or even become desert.
We study the characteristics of both selfreplication and fragmentation processes through the analytic and numerical analysis of a general integrodfferential model. We show, from a theoretical perspective, that depending on the levels of aridity localized patches can be more or less stable than the periodic pattern, a phenomenon previously studied in simpler local models PhysRevA.84.043848 (). In an attempt to conciliate the theoretical observations with real data, we consider two ecosystems, namely, from Zambia and Mozambique, these landscapes are composed of vegetation patches reaching large sizes, of the order of tens of meters in diameter (see Fig. 1). We perform statistical analysis of satellite images and find that patches have correlated characteristic patch sizes and interpatch distances along with other properties that support the hypothesis of the selforganization nature of these landscapes.
The article is organized as follows: in Section 2.1 the theoretical model is introduced, together with the phase diagram. A description of the methods used to analyze the satellite images are included in section 2.3. The results are presented in section 3. Theoretical results indicating the relationship with the wavelength and the range of the facilitative and competitive interactions are presented in the appendix. Finally, we present the Discussion, Conclusions and Perspectives of our work.
2 Methods
2.1 Mathematical model
The modeling of ecosystems is a challenging and complex problem. Here, we adopt the theory of vegetation patterns established by R. Lefever and coworkers twodecades ago to model the spatiotemporal dynamics of vegetation in which both space and in time are considered to be continuous variables Lefever1997 (); LT (); Lejune_couteron (). This theory incorporates the nonlocal facilitative and the competitive planttoplant interactions though kernels Lefever1997 (); Tlidi2008 (); Lefever2009194 (); LefeverTurner (). In the absence of these interactions, the resulting model is similar to the paradigmatic logistic equation introduced by Verhulst to study population dynamics Verhulst1845 (); mawhin2002heritiers (); mawhin2004legacy (). In what follows we consider vegetation of a single specie settled on a flat landscape under isotropic and homogeneous environmental conditions. To simplify further the description of the system, we assume that all plants are mature. Thus, we neglect age classes. This approximation can be justified by the fact that individual plants grow on much faster time scale comparing to the time scale of the formation of regular vegetation pattern. The only variable is the vegetation biomass density which is defined at the plant level. Let us introduce the biomass density, , that satisfies the following dynamical evolution Tlidi2008 (); Lefever2009194 ()
(1) 
where and are the spatial coordinates and time, respectively. The time derivative is represented by . The parameter , is the decaytogrowth rate ratio. It can be viewed as an indirect measure of resource scarcity or stress, that limits net biomass production and is what we refer to as aridity parameter. The first and the second terms on the righthandside of Eq. (1) account for the planttoplant facilitation and competition feedbacks, respectively. They describe the spatial extension of feedback effects in terms of the characteristic ranges and over which facilitative and competitive interactions operate, respectively. The facilitative interaction acts on the level of the aerial plant structure (crown) that involves sheltering, litter, water funneling or any other effect, such as seed production and germination that contribute to the biomass natural growth JEC:JEC1295 (). Let be the crown radius projected on the surface element centered on a point . Assuming that the length of facilitative interaction is equals to the radius of the crown, i.e., . On the other hand, in resourcelimited environments, plants should compete for their survival. In face of climate change and increasing drought periods, plants should adapt their root structures to overcome resource scarcity JEC:JEC682 (); JEC:JEC1126 (). Through their root structures (rhizosphere), each individual plant tends to deprive its neighbors of vital resources, such as water or nutrient uptake. Measurements of roots lateral spread indicate that they extend beyond the radius of the aerial structure (crown) by an order of magnitude JEC:JEC1126 (); ECY:ECY20088961521 (); Lefever2009194 (); Couteron20140102 ().
The competitive interaction between plants tends to oppose the facilitation mechanisms, by impeding vegetation growth. In arid landscapes, the length of facilitative interaction (crown) is much shorter than the length of the competitive interaction operating at the level of the rhizosphere which is the volume of soil around living roots. Let consider also that the range of the competition is such that , where, is the radius of the rhizosphere. The third term in Eq. (1) describes the spatial propagation of vegetation via seed dispersion. The parameter is the rate of propagation of the vegetation. The competitive planttoplan interaction is considered to be of the form
(2) 
where, is the strength of the competitive interaction, and is a normalization constant, that depends on the spatial dimension. In two dimensions, . The spatial propagation of vegetation via seed dispersion is assumed to have the form
(3) 
the parameter and are, respectively, the dispersion range of seeds and the strength of dispersive process. To simplify further the analysis we assume that the seed dispersion is described as diffusion . This can be obtained by considering a small dispersion range and a simple Taylor expansion, which leads to . To simplify further the problem, we consider that the facilitation is well described by a local process modelled by , where is the strength of the facilitative interaction.
The analysis we make in this work generalizes previous results. Here, the complete nonlocal integrodifferential model is analyzed, which accounts for an important step forward in the understanding of the behavior of this type of system. For a complete linear stability analysis of this model see Appendix 6.1.
2.2 Localized vegetation patches
The nonlocal equation (1) exhibits stable circular localized structures which are supported by one of the homogeneous steady state, (unpopulated state) or (homogeneously vegetated). In the context of vegetation dynamics, localized structures that emerge from the unpopulated state , correspond to circular patches of vegetation which are surrounded by bare terrain. As previously mentioned, periodic states and localized patches emerge as a selforganizing response of the system to changes in the parameters. Aridity (), is of particular interest, as it can be directly related to field measurements, and also relates to an everincreasing concern due to global warming. An increase in aridity (lower hydric resources available or other stress factors) may cause a contraction of savannas and woodlands, in a process called desertification, triggering the formation of vegetation patterns or even deserts. The necessary and sufficient condition for the formation of localized patches is the coexistence between a homogeneous cover and a periodic vegetation pattern. Implying the existence of an hysteresis loop. Inside this loop, there is a socalled pinning range of the aridity parameter where localized gaps or patches are stable LejeuneTlidiCouteron (); rietkerk2004self (); Tlidi2008 (). Similar pinning behavior occurs in many spatially extended systems where a homogeneous steady state coexists with a spatially periodic state POMEAU19863 (); PhysRevLett.73.640 (); PhysRevLett.84.3069 (). Pinning was first reported for front solutions by Y. Pomeau POMEAU19863 (), has also been applied to gap vegetation patterns PhysRevA.84.043848 (). Localized structures and localized patterns are a well documented phenomenon, concerning almost all fields of natural science including chemistry, biology, ecology, physics, fluid mechanics, and optics Tlidi_Focus_chaos_07 (); DSbook (); Rev18 (); Tlidi20140101 (); knobloch2015spatial (); Meron_book (); tlidi2015nonlinear (). It is worth mentioning that fairy circles are striking examples attributed to this category of localized vegetation patterns vanroyen (). Although, the mechanisms leading to their formation are still subject of debate among the scientific community. They have been interpreted as the result of a pinning front between a uniform plant distribution and a periodic hexagonal vegetation pattern Tlidi2008 (). Recent investigations support this interpretation tschinkel2012life (); cramer2013namibian (); tschinkel2012life (); getzin2015adopting (). On the other hand, fairy circles formation may result from front dynamics connecting a bare state and uniform plant distributions fernandez2014strong (); fernandez2013strong (); PhysRevE.91.022924 (). In all these works, the origin of fairy circle formation is intrinsic to the dynamics of the system. This means that the diameter of the fairy circle is determined rather by the parameters of the system and not by external effects, such as the presence of social insect or anisotropy. However, another theory based on external effects, such as termites or ants has been suggested Becker (); Picker (); Juergen (); Bonachela (). More recently, Tarnita and collaborators have shown that a combination of intrinsic and extrinsic effects could explain the origin of fairy circles Tarnita ().
In addition to analytically deriving predictions from Eq. 1, we will provide a case study for which the interpretation of vegetation patchiness based on localized structures and selfreplication is plausible.
2.3 Data analysis
Through the use of satellite images obtained from Google Earth, we have located regions in Zambia and Mozambique where vegetation patches dominate the landscape (see SI Table 1 for details).
Patch detection algorithm
In order to detect the vegetation, we developed a simple algorithm that segments the images, detects vegetation patches and extract properties of interest (Matlab vR2016b, see Supplementary Information Algorithm). The boundaries of every detected patch are used to extract the patch geometrical features, such as area, perimeter, equivalent diameter, and centroid positions, which are then used in the spatial analysis.
Equivalent radius and nearestneighbor distance
The equivalent radius of each structure is calculated as
(4) 
where corresponds to the area of the structure. The nearest neighbor distances is obtained by finding the minimum of the distance every patch and all the other patches.
Spatial distribution analysis
Lfunction
For the analysis of the spatial distribution of patches we make use of the modified Ripley’s function , for this, only the centroids of the patches are used. We define as Ripley_76 ()
(5) 
where is the distance from the reference point. and are the total number of points and the area, respectively. is the number of points that lay inside a circle of radius centered on a patch of reference chosen randomly, this is repeated for multiple patches. As control we compute the function for a set of randomly distributed point (null model), by repeating this 200 times, construct the confidence intervals. When the observed values of L go above (below) the confidence interval, we say the distribution of point is clustered (dispersed).
Radial distribution function
Similarly to the Lfunction, the radial distribution function , is computed using only the positions of the centroids of the detected objects, we define as Couteron_Kokou_97 ()
(6) 
where is the number of points that lie inside a ring of radius and width , centered on a given object. The numerator of this expression corresponds to the total number of points that lay inside the ring for the case of homogeneously distributed objects, is the overall number density. The radial distribution function, , measures the correlation of the position of the points, values of () indicate positive (negative) correlation. N(r) in numerically computed by measuring the distance between every two patches in the images, and then performing a histogram, where the binwidth corresponds to .
Fourier transform
The Fourier transform is used considering the binary image of the structures detected. It is defined as
(7) 
Where are the spatial coordinates of the image pixels, and are the spatial frequencies, usually is referred to as the spectrum of . Peaks in the spectrum indicate dominant wavelengths or typical sizes of the patches detected in the satellite images (see LT (); Lejune_couteron (); Debloaw_Couteron_2011 ().
3 Results
3.1 Selfreplication: From vegetation patches to extended pattern
Detailed mathematical analysis is provided for the first time for Eq. 1 (see the Appendix 6.1), this analysis shows that under a wide range of the aridity parameter, there is coexistence between homogeneous and pattern states. Thus, supporting stable localized vegetation patches. Moreover, as observed in models with only local interactions bordeu2016self (), vegetation patches in the nonlocal model may also be affected by a TuringPrigogine instability, where the patch elongates, increasing in size to finally split into two patches, through the decay of the central ”bridge” that connected them. This mitotic dynamics occurs repeatedly to each of the new patches, allowing the single initial patch to end up covering the whole system, through the formation of a periodic hexagonal pattern, as can be observed in Fig. 2.
It is through the selfreplication mechanism that a single vegetation patch can induce a increase the total biomass of a system although the aridity is too high to allow for an homogeneous cover of vegetation, Figure 6 shows the increase of the total biomass on the transition from a single to four patches through self replication.
To study the parameter zone where selfreplication is observed, we have performed a direct numerical approach for computing the rate at which the elongating unstable mode grows, see Fig. 4. We have found that given a sizescale, , for the nonlocal interactions in Eq. (1), there is a wide range of aridity, , for which an initial localized structure will destabilize through the selfreplicating mechanism (see Fig. 4). Lower values of , imply shorterrange nonlocal effects, this generates a leftshift in the selfreplicating window to lower values of aridity as can be seen in Fig. 4 when comparing the red (, ) and yellow (, ) curves to the blue curve (, ).
Moreover, we have found that for aridity parameter values below the selfreplicating region, there is a different route for the selfreplicating mechanism (see ring instability inset, Fig. 4), here, an initial localized structure grows radially. After reaching a critical radius, the central portion of the structure decays, forming a doughnutlike shape. This structure is also unstable. By the consequent decay of two opposite sides of the doughnut, the structure ends up dividing into two new localized patches.
Following the simulations of a single initial localized patch (Fig. 2), we performed largescale simulations, containing hundreds of randomly distributed patches, the objective of this, was to assess how fundamental is the selfreplicating process in the emergence of the characteristic wavelength in the system. We proceeded as follows: (i) Considering a system of points (), we built and initial condition consisting of a spatial Poisson point process with rate , this generates on the order of 600 point randomly distributed in the twodimensional plane. Each of the points is considered as the center of a localized patch of points radius, the generated state can be observed in Fig. 5a. (ii) The random field is used as initial condition for simulating Eq. (1) for a parameter region inside the selfreplication windows (see Fig. 4, ). (iii) We let the system evolve, and obtain a transient state, that can be seen in Fig. 4c, although here we cannot observe new structures, because the evolution time was too short, it is enough time for the system to split the existing cluster into distinct structures.
When computing the Fourier transform of both the initial and the evolved state, we see that for the case of the initial state (Fig. 5b) the Fourier transform (apart from the central peak) exhibits a characteristic wavelength of the order of the size of the localized structures ( A.U), due to the almost monodispersed sizes of the computer generated structures (almost, because some of the structures merged to generate bigger clusters), the same phenomenon was observable in the analysis of Zambia’s landscape (Fig. 8c).
On the contrary, when analyzing the evolved state (Fig. 5d), we see immediately the emergence of a characteristic wavelength much larger than the size of the structures ( A.U.), this wavelength emerges as a result of the selforganizing nature of the system, where clusters split, rapidly arriving at a characteristic distribution.
3.2 Fragmentation: From homogeneous cover to patchy landscapes
We have show that through selfreplication a single or multiple localized patches may increase the total biomass of an system leading ultimately to a hexagonal patterned state. However this is not the only route by which a system can reach such patterned state, a second route is presented in Fig. 6, where an homogeneous vegetated state () is destabilized by an increase in aridity towards a labyrinthine pattern (). Further increasing aridity drives the system towards two possible final states, the first possibility is a reduction in aridity that takes the system to the region where localized patches are unstable and suffer from selfreplication (), here, the labyrinthine pattern breaks down into multiple localized patches that cover the system by generating a patterned state. Note that the bare state become stable when . On the other hand, when aridity is shifted from to , where localized patches are stable, then the system reaches a different state, consisting on a low number of isolated patches distributed in a disordered manner.
It is important to mention that a discontinuous transition from the stable homogeneous cover region () directly to any of the nonperiodic patch regions ( or ) generates an overall decay of the homogeneous state to the bare state. Thus, an intermediate level of aridity is necessary for allowing the destabilization of the homogeneous cover into a nonperiodic patch pattern.
3.3 Remotesensing observations
We have studied the spatial distribution of localized patches in eight distinct regions, four located in the Fombeni region, Mozambique, and other four located in the Mufumbwe District, NorthWestern Province of Zambia. The vegetation in Zambia is dominated be a medium size tree Brachystegia spiciformis. The vegetation patterns observed in these regions are not periodic, and are composed by vegetation patches, possibly composed by groups of trees forming compact clusters. Each patch can cover an area of up to thousands squaremeters, with an effective radius of tens of meters, as can be observed in Figs. 1a and 1f (and SI Figs. 11 and 12).
The maximal values of the radius of the structures extends over hundreds of meters, however, the typical value (mode) varies from region to region, ranging from 7 meters (zones 3, and zone 4 in Zambia) up to 36 meters (Zambia zone 4), most of the regions show a marked characteristic size of the structures (see Fig. 7a and 7f, and SI Figs. 11 and 12) and nearestneighbor distance (see Fig. 7b and 7g), indicating a preferred spatial patch distribution.
To study the spatial organization of the patterns, we perform two measurements, the first is a modified Ripley’s function L, which measures the level of clustering of a spatial point process (see formula 5), and the second one is the radial distribution function (6), used to measure the correlation in a point process. Both observables are closely related Couteron_Kokou_97 (), yet together they allow for further insight into the spatial organization. All regions Zambia and zones 1 and 4 in Mozambique exhibit scarcity of neighbors at close distance (observations below the 95% confidence interval) and then transition towards clustering is observed for larger distances (Fig. 7c and 7h). Similarly, when analyzing the correlation of the point distribution through the radial distribution function, we observe a positive correlation () regime in all the zone, in particular, regions with high clustering exhibits long range positive correlation, while zones 1 and 3 in Mozambique, which show no clustering exhibit a positive correlation only for a small distance (Fig. 7d and 7i). Because of the size of the regions considered (see SI Fig. 10) where the terrain is not homogeneous, and there is a qualitative change in the structures even within a single zone, this prevents the definition of a characteristic wavelength through a spatial Fourier transform on every region. Positive examples are zones 2 in Mozambique and Zambia, where an incipient characteristic wavelength can be defined (Fig. 7e and 7j).
To study if there is a relationship between the size of the patches and the distance between them, we make use of the results from the analysis of radius and nearestneighbor distance (NND) between structures (see SI Figs. 11 and 12) and study the correlation between them through a linear correlation analysis (see Fig. 8). We observe a strong correlation between the two quantities, indicating that there might be a underlying mechanism controlling both the structures size and the distance between them.
4 Discussion
Numerical observation of the nonlocal model studied suggest that selfreplication may be an important path for the propagation of vegetation, which allows for the propagation of the vegetation even in arid conditions where the homogeneous cover is not viable due to the lack of resource (nutrients and/or water). Moreover, it has been shown that a transition from a homogeneous vegetation cover to a patch patterned state may only occur if the transition occurs towards the selfreplicating region. However, in realecosystems processes occur at a slow time scale and there is a high number of uncontrolled parameters which prevent the assessment of the validity of the numerical observations. We however illustrate the plausibility of the process by referring to observations at a single point in time of patchy vegetation in Africa.
The properties of the landscapes observed in Mozambique and Zambia show some indications that support selforganization. These systems exhibit a characteristic distances between patches, and a characteristic patch size, together with clustering and long range correlations in the spatial distribution. This may indicate that patches are interacting with each other resulting in the spatial distributions observed. From our modeling perspective, localized patches which are too close to each other (high density) will compete for resources ending up with the disappearance of the one placed in the less favourable location. If the patches are too far apart (low density), the extra space will allow the growth of the patches resulting in selfreplication and the formation of new patches that will fill the previously unoccupied region. We must here make clear that we cannot corroborate the existence of these dynamical processes from the data analysis performed. The remote sensing data provides no temporal information and limited spacial resolution. Further onsite investigations are necessary to comprehend plantplant interaction in specific contexts, and to assess if there are local factors that contribute to the selforganization of the landscape.
Despite all of this, we have observed features that support the patch selfreplication hypothesis, for example, the observation of rings, which is a second route to selfreplication (see Fig. 4). Rings are usually transient state, leading to division into two or more patches bordeu2016self (); meron2004vegetation (). Examples of this can be observed in different regions of Zambia (for example, [S, E], and the surrounding areas). It is harder to observe the formation of rings in Mozambique, and we believe this relates to the facts that the structures in this location are smaller, preventing the ring instability to take place. Moreover the direct correlation between the size of the patches and the nearest neighbor distance, is also rendered by the model, which indicates that there is a dominant wavelength in the system that controls both the size of the structures and distance between them. This wavelength depend on the aridity parameter (see Appendix 6.1).
5 Conclusions and perspectives
By making use of a generic interaction redistribution model, we showed that self replication of localized patches is generically present even in the most general model involving nonlocal competition and facilitating interactions. We have shown, through this model, that a localized structure may undergo multiple selfreplications to finally cover the whole space available with a regular hexagonal pattern, and by doing so, let the total biomass in the system increase in time. Moreover, the selfreplication regime is may be an important stage in the decay of homogeneous covers of vegetation to hexagonal pattern states. It may be seen as an alternative to a more abrupt decay in biomass without this, the decay in the biomass is more abrupt resulting in a low number of sparse patches. We hypothesize that these processes are occurring in the observed landscapes of Mozambique and Zambia.
Through the analysis of satellite images, we have given evidence to support the selforganization hypothesis. Despite the limited amount of data, which prevents being assertive about the underlying processes, we believe that selfreplication could be considered as a possible mechanism for the regeneration dynamics of these landscapes. This phenomenon may occur for a moderate level of the aridity. Moreover, when considering a homogeneous cover, a gradual increasing of the level of the aridity results in a decay of the total biomass. The degradation of the ecosystems though fragmentation leads to either a periodic vegetation patterns or a random distribution of localized patches of vegetation. However, a periodic pattern is only obtained if the system relaxes in the region where selfreplication exists, and patches are unstable.
As mentioned in the Discussion section, more data, specially field measurements, long timelapse imaging and controlled experiments are necessary to confirm any claim on the nature of the observed patterns. However, measurements done here are a first step to understand the distribution of the observed patterns, and illustrate that selforganization is a strong candidate to explain them. Despite that there is no direct observation of selfreplication, the current state of some patches drive us to think that selfreplication might be an ongoing process, for example in the ”dividing” patches showed in Fig. 1.
The knowledge of the belowground structures of the type of vegetation considered in this contribution is rather limited. From the current literature only rooting depth data and a quantitative index of the vertical distribution of roots are available. which are irrelevant for the present study. Moreover, the full lateral roots extension has not yet been measured. For this reasons it was beyond the scope of the present contribution to perform a quantitative comparison between theoretical results and field observations. On the other hand, the model presented here has been previously confronted quantitatively to experimental measurement for two type of plants (Festuka orthophylla Couteron20140102 () and Combretum micranthum JEC:JEC1126 (); ECY:ECY20088961521 (); Lefever2009194 (). We believe that the same measurement could be realized for the type of vegetation of Zambia and Mozambique.
6 Appendix
6.1 Stability analysis of the integrodifferential model
The stationary homogeneous states of Eq. (1), describing uniform vegetation covers, are , and , given by the solution of
(8) 
where the feedback difference measures the community cooperativity. The first, , represents a territory totally devoid of vegetation, i.e., a bare state. The second, corresponds to uniform plant distributions that can be either monostable if or bistable when . In the monostable case, , the biomass density decreases monotonously with the aridity parameter and vanishes at (cf. Fig. 9). It exist only if . In the bistable case, , the uniform plant distributions extends up to the turning point sometimes also referred to as tipping point (saddlenode bifurcation point). The coordinates of this point are and .
To study the linear stability analysis of the stationary homogeneous states, , we introduce a small amplitude deviations from of the form
(9) 
where is a small parameter and the deviation from the is expressed in terms of Fourier modes in the space of wavevector . By replacing Eq. (9) and by linearizing with respect to , the dispersion relation obeyed by the eigenvalues reads
(10) 
The critical points associated with the TuringPrigogine instability are given by,
(11)  
(12) 
These two conditions provide the thresholds and the most unstable wavenumber associated with the TuringPrigogine instability. The critical wavenumber at the onset of the instability is given explicitly by
(13) 
The wavelength is . The above equation has two solutions, which we denote by and . The thresholds associated with the TuringPrigogine instability are solutions of
(14) 
The results of the linear stability analysis are summarized in Fig. 9. As the aridity level increases, the vegetation evolves toward extinction. For , the decrease of the phytomass density is monotonous. The bare state density is reached at the switching point , where the plant distribution state is destabilized through the TuringPrigogine instability turing1952chemical (); prigogine1968symmetry (). In the range , the uniform phytomass state is unstable. The corresponding levels of the aridity are obtained from the homogeneous steady states Eq. (8) (see Fig. 9).
When increasing the stress or aridity, the wavelength of the corresponding pattern increases, and simultaneously the morphology of the vegetation pattern changes, this is related to the observations made in Mozambique and Zambia, where an increase in the system wavelength is directly correlated with an increase in the size of the vegetation patches. The generic sequence of spatially periodic states obtained for increasing aridity is: spots of lower density forming an hexagonal lattice, alternating stripes of higher and lower density, and spots of higher density forming an hexagonal lattice LT (); QUA:QUA10878 (). This behavior has been also found for other mathematical models that include water transport by underground diffusion and/or above ground runoff Meron2001 (); Rietkerk2008169 (); Sherratt20138 (); Gowda ().
Acknowledgements
We acknowledge fruitful discussions with René Lefever and Pierre Couteron. We also appreciate the invitation from Pierre Couteron to submit this contribution to Ecological Indicators. M.T. received support from the Fonds National de la Recherche Scientifique (Belgium). I.B. was supported by CONICYT, Beca de Doctorado en el Extranjero No. 72160465. M.G.C. thanks the financial support of FONDECYT project 1150507. M.G.C. and M.T. acknowledge the support of CONICYT project REDES150046. This research was also supported by WallonieBruxelles International (WBI).
References
References
 (1) R. Lefever, O. Lejeune, On the origin of tiger bush, Bulletin of Mathematical Biology 59 (1997) 263.
 (2) O. Lejeune, M. Tlidi, A model for the explanation of vegetation stripes (tiger bush), Journal of Vegetation Science 10 (1999) 201.
 (3) P. Couteron, O. Lejeune, Journal of Ecology, Periodic spotted patterns in semiâarid vegetation explained by a propagationâinhibition model, Journal of Ecology 89 (4) (2001) 616.
 (4) C. Klausmeier, Regular and irregular patterns in semiarid vegetation, Science 284 (1999) 1826.
 (5) R. HilleRisLambers, M. Rietkerk, F. van den Bosch, H. Prins, H. de Kroon, Vegetation pattern formation in semiarid grazing systems, Ecology 8 (2001) 50.
 (6) J. von Hardenberg, E. Meron, M. Shachak, Y. Zarmi, Diversity of vegetation patterns and desertification, Phys. Rev. Lett. 87 (2001) 198101.
 (7) O. Lejeune, M. Tlidi, R. Lefever, Vegetation spots and stripes: Dissipative structures in arid landscapes, International Journal of Quantum Chemistry, 98 (2004) 261.
 (8) J. Sherratt, An analysis of vegetation stripe formation in semiarid landscapes, Journal of mathematical biology 51 (2005) 183.
 (9) P. D’Odorico, F. Laio, L. Ridolfi, Patterns as indicators of productivity enhancement by facilitation and competition in dryland vegetation, Journal of Geophysical Research: Biogeosciences 111 (G3) (2006).
 (10) R. Vesipa, C. Camporeale, L. Ridolfi, Noisedriven cooperative dynamics between vegetation and topography in riparian zones, Geophysical Research Letters 42 (2015) 8021.
 (11) S. Kéfi, M. Rietkerk, C.L. Alados, Y. Pueyo, V.P. Papanastasis, A. ElAich, R.C. De Ruitern, Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems, Nature 449 (2007) 213.
 (12) S. Kéfi, M. Rietkerk, M. van Baalen, M. Loreau, Local facilitation, bistability and transitions in arid ecosystems, Theoretical Population Biology 71 (2007) 367.
 (13) V. Dakos, S. Kéfi, M. Rietkerk, E.H. van Nes, M. Scheffer, Slowing down in spatially patterned ecosystems at the brink of collapse, Am. Nat. 177 (2011) E153.
 (14) V. Deblauwe, P. Couteron, J. Bogaert, N. Barbier, Determinants and dynamics of banded vegetation pattern migration in arid climates. Ecol. Monogr. 82 (2012) 3.
 (15) N. Barbier, P. Couteron, J. Lejoly, V. Deblauwe, O. Lejeune, Selforganized vegetation patterning as a fingerprint of climate and human impact on semiarid ecosystems, Journal of Ecology 94 (3) (2006) 537.
 (16) N. Barbier, P. Couteron, R. Lefever, V. Deblauwe, O. Lejeune, Spatial decoupling of facilitation and competition at the origin of gapped vegetation patterns, Ecology 89 (2008) 1521.
 (17) R. Lefever, N. Barbier, P. Couteron, O. Lejeune, Deeply gapped vegetation patterns: On crown/root allometry, criticality and desertification, Journal of Theoretical Biology 261 (2009) 194.
 (18) P. Couteron, F. Anthelme, M. Clerc, D. Escaff, C. FernandezOto, M. Tlidi, Plant clonal morphologies and spatial patterns as selforganized responses to resourcelimited environments, Phil. Trans. R. Soc. A 372 (2014) 20140102.
 (19) I. Bordeu, Labyrinthine dissipative patterns, Journal of Physics: Conference Series, 720 (2016) 012004.
 (20) I. Bordeu, M. Clerc, P. Couteron, R. Lefever, M. Tlidi, Selfreplication of localized vegetation patches in scarce environments, Scientific Reports 6 (2016) 33703.
 (21) A. Dickstein, S. Erramilli, R. Goldstein, D. Jackson, S. Langer, Labyrinthine pattern formation in magnetic fluids, Science 261 (1993) 1012.
 (22) P. Ribiere, P. Oswald, Nucleation and growth of cholesteric fingers under electric field, J. Phys. France 51 (1990) 1703.
 (23) P. Oswald, J. Baudry, S. Pirkl, Static and dynamic properties of cholesteric fingers in electric field, Physics Reports 337 (2000) 67.
 (24) J. Pearson, Complex Patterns in a Simple System, Science 261 (1993) 189.
 (25) K. Lee, W. Mccormick, J. Pearson, H. Swinney, et al., Experimentalobservation of selfreplicating spots in a reactiondiffusion system, Nature 369 (1994) 215.
 (26) A. Muñuzuri, V. PérezVillar, M. Markus, Splitting of autowaves in an active medium, Phys. Rev. Lett. 79 (1997) 1941.
 (27) A. Kaminaga, V. Vanag, I. Epstein, “black spots”in a surfactantrich belousovzhabotinsky reaction dispersed in a waterinoil microemulsion system, Journal of Chemical Physics 122 (17) (2005) 4706.
 (28) T. Kolokolnikov, M. Tlidi, Spot deformation and replication in the twodimensional belousovzhabotinski reaction in a waterinoil microemulsion, Phys. Rev. Lett. 98 (2007) 188303.
 (29) P. Davies, P. Blanchedeau, E. Dulos, P. De Kepper, Dividing blobs, chemical flowers, and patterned islands in a reactionâdiffusion system, The Journal of Physical Chemistry A 102 (1998) 8236.
 (30) M. Monine, L. Pismen, M. Bär, M. OrGuil, Modeling triangular titration fronts in the O 2+ H 2 reaction on a catalytic Rh (111) surface, The Journal of chemical physics 117 (2002) 4473.
 (31) A. Schaak, R. Imbihl, Triangularshaped reaction fronts in a catalytic surface reaction, Chemical Physics Letters 283 (5â6) (1998) 386.
 (32) Y. Hayase, T. Ohta, Sierpinski gasket in a reactiondiffusion system, Phys. Rev. Lett. 81 (1998) 1726.
 (33) M. Tlidi, Y. Gandica, G. Sonnino, E. Averlant, and K. Panajotov, SelfReplicating spots in the brusselator model and extreme events in the onedimensional case with delay, Entropy, 18 (2016) 64.
 (34) E. Meron, E. Gilad, J. von Hardenberg, M. Shachak, Y. Zarmi, Vegetation patterns along a rainfall gradient, Chaos, Solitons & Fractals 19 (2004) 367–376.
 (35) X. Ren, J. Wei, On the spectra of threedimensional lamellar solutions of the diblock copolymer problem, SIAM Journal on Mathematical Analysis 35 (2003) 1.
 (36) Y. Nishiura, H. Suzuki, Higher dimensional slep equation and applications to morphological stability in polymer problems, SIAM Journal on Mathematical Analysis 36 (2005) 916.
 (37) B. Sandnes, H. Knudsen, K. Måløy, E. Flekkøy, Labyrinth patterns in confined granularfluid systems, Phys. Rev. Lett. 99 (2007) 038001.
 (38) B. Sandnes, E. Flekkøy, H. Knudsen, K. Måløy, H. See, Patterns and flow in frictional fluid dynamics, Nature Communications 2 (2011) 288.
 (39) M. Tlidi, A. Vladimirov, P. Mandel, Curvature instability in passive diffractive resonators, Phys. Rev. Lett. 89 (2002) 233901.
 (40) A. G. Vladimirov, R. Lefever, M. Tlidi, Relative stability of multipeak localized patterns of cavity solitons, Phys. Rev. A 84 (2011) 043848.
 (41) M. Tlidi, R. Lefever, A. Vladimirov, On Vegetation Clustering, Localized Bare Soil Spots and Fairy Circles, Lect. Notes Phys 751:381 (2008) 402
 (42) R. Lefever, J. Turner, A quantitative theory of vegetation patterns based on plant structure and the nonlocal FKPP equation, Comptes Rendus MÃ©canique, 340 (2012) 818.
 (43) P. Verhulst, Recherches mathématiques sur la loi d’accroissement de la population, Nouveaux mémoires de l’Académie Royale des Sciences et BellesLettres de Bruxelles, 18 (1845) 14.
 (44) J. Mawhin, Les héritiers de pierrefrançois verhulst: une population dynamique, Acad. Roy. Belg. Bull. Cl. Sci 13 (2002) 349.
 (45) J. Mawhin, The legacy of pierrefrançois verhulst and vito volterra in population dynamics, The first 60 (2004) 147.
 (46) R. Brooker, F. Maestre, R. Callaway, C. Lortie, L. Cavieres, G. Kunstler, P. Liancourt, K. TielbÃ¶rger, J. Travis, F. Anthelme, C. Armas, L. Coll, E. Corcket, S. Delzon, E. Forey, Z. Kikvidze, J. Olofsson, F. Pugnaire, C. Quiroz, P. Saccone, K. Schiffers, M. Seifan, B. Touzard, R. Michalet, Facilitation in plant communities: the past, the present, and the future, Journal of Ecology 96 (2008) 18.
 (47) H. Schenk, R. Jackson, Rooting depths, lateral root spreads and belowground/aboveground allometries of plants in waterlimited ecosystems, Journal of Ecology 90 (2002) 480.
 (48) O. Lejeune, M. Tlidi, P. Couteron, Localized vegetation patches: A selforganized response to resource scarcity, Phys. Rev. E 66 (2002) 010901.
 (49) M. Rietkerk, S. Dekker, P. de Ruiter, J. van de Koppel, Selforganized patchiness and catastrophic shifts in ecosystems, Science 305 (2004) 1926.
 (50) Y. Pomeau, Front motion, metastability and subcritical bifurcations in hydrodynamics, Physica D: Nonlinear Phenomena 23 (1986) 3.
 (51) M. Tlidi, P. Mandel, R. Lefever, Localized structures and localized patterns in optical bistability, Phys. Rev. Lett. 73 (1994) 640.
 (52) P. Coullet, C. Riera, C. Tresser, Stable static localized structures in one dimension, Phys. Rev. Lett. 84 (2000) 3069.
 (53) A. Champneys, Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics, Physica D: Nonlinear Phenomena 112 (1998) 158.
 (54) M. Tlidi, M. Taki, T. Kolokolnikov, Introduction: Dissipative localized structures in extended systems, Chaos: An Interdisciplinary Journal of Nonlinear Science 17 (2007) 037101.
 (55) N. Akhmediev, A. Ankiewicz, Dissipative Solitons: From Optics to Biology and Medicine, Lecture Notes in Physics, Springer, 2008.
 (56) O. Descalzi, M. Clerc, S. Residori, Localized States in Physics: Solitons and Patterns, Springer, 2011.
 (57) M. Tlidi, K. Staliunas, K. Panajotov, A. Vladimirov, M. Clerc, Localized structures in dissipative media: from optics to plant ecology, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 372 (2014) 20140101.
 (58) E. Knobloch, Spatial localization in dissipative systems, conmatphys 6 (1) (2015) 325.
 (59) E. Meron, Nonlinear physics of ecosystems, CRC Press Boca Raton (2015).
 (60) M. Tlidi, M. Clerc, Nonlinear Dynamics: Materials, Theory and Experiments, 173 Springer 2015.
 (61) M.W. van Rooyen, G. K. Theron, N. van Rooyen, W.J. Jankowitz, and W.S. Matthews, Mysterious circles in the Namib Desert: review of hypotheses on their origin, J. Arid. Environ. 57, (2004) 467.
 (62) W. Tschinkel, The life cycle and life span of namibian fairy circles, PloS one 7 (2012) e38056.
 (63) M. Cramer, N. Barger, Are namibian âfairy circlesâ the consequence of selforganizing spatial vegetation patterning?, PloS one 8 (2013) e70876.
 (64) S. Getzin, K. Wiegand, T. Wiegand, H. Yizhaq, J. von Hardenberg, E. Meron, Adopting a spatially explicit perspective to study the mysterious fairy circles of namibia, Ecography 38 (2015) 1–11.
 (65) C. FernandezOto, M. Tlidi, D. Escaff, M. Clerc, Strong interaction between plants induces circular barren patches: fairy circles, Phil. Trans. R. Soc. A 372 (2014) 20140009.
 (66) C. FernandezOto, M. G. Clerc, D. Escaff, and M. Tlidi, Strong Nonlocal Coupling Stabilizes Localized Structures: An Analysis Based on Front Dynamics, Phys. Rev. Lett. 110 (2013) 174101.
 (67) D. Escaff, C. FernandezOto, M. G. Clerc, M. Tlidi, Localized vegetation patterns, fairy circles, and localized patches in arid landscapes, Phys. Rev. E 91 (2015) 022924.
 (68) T. Becker, S. Getzin, The fairy circles of Kaokoland (northwest Namibia) origin, distribution, and characteristics, Basic Appl. Ecol. 1 (2000) 149.
 (69) M.D. Picker, V. RossGillespie, K. Vlieghe, E. Moll, Ants and enigmatic Namibian fairy circlescause and effect, Ecol. Entomol. 37 (2012) 33.
 (70) N. Juergens, The biological underpinnings of namib desert fairy circles, Science 339 (2013) 1618.
 (71) J.A. Bonachela, R.M. Pringle, E. Sheffer, T.C. Coverdale, J.A. Guyton, K.K. Caylor et al., Termite mounds can increase the robustness of dryland ecosystems to climatic change, Science 347 (2015) 651.
 (72) C.E. Tarnita, J.A. Bonachela, E. Sheffer, J.A. Guyton, T.C. Coverdale, R.A. Long, R. M Pringle, A theoretical foundation for multiscale regular vegetation patterns, Nature 541 (7637) (2017) 398.
 (73) I.P.D.J. Harris, P.D. Jones, T.J. Osborn, D.H. Lister, Updated highâresolution grids of monthly climatic observationsâthe CRU TS3. 10 Dataset. International Journal of Climatology 34(3) (2014) 623642.
 (74) B.D. Ripley, The secondorder analysis of stationary point processes. Journal of Applied Probability. 13, (1976) 255â266
 (75) P. Couteron and K. Kokou, Woody vegetation spatial patterns in a semiarid savanna of Burkina Faso, West Africa, Plant Ecology 132 (1997) 211.
 (76) V. Deblauwe, P. Couteron, O. Lejeune, J. Bogaert, N. Barbier, Environmental modulation of selforganized periodic vegetation patterns in Sudan, Ecography 34 (2011) 990.
 (77) A. M. Turing, The chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London B: Biological Sciences 237 (1952) 37.
 (78) I. Prigogine, R. Lefever, Symmetry breaking instabilities in dissipative systems. ii, The Journal of Chemical Physics 48 (1968) 1695.
 (79) M. Rietkerk, J. van de Koppel, Regular pattern formation in real ecosystems, Trends in Ecology and Evolution 23 (2008) 169.
 (80) J. Sherratt, Historydependent patterns of whole ecosystems, Ecological Complexity 14 (2013) 8.
 (81) K. Gowda, H. Riecke, M. Silber, Transitions between patterned states in vegetation models for semiarid ecosystems, Phys. Rev. E 89 (2014) 022701.
Supplementary information
Areas of study
Location 





Zambia zone 1  903  
Zambia zone 2  903  
Zambia zone 3  903  
Zambia zone 4  903  
Mozambique zone 1  1348  24.89  
Mozambique zone 2  1348  24.89  
Mozambique zone 3  1348  24.89  
Mozambique zone 4  1348  24.89 
Measured properties
Algorithm
Here we present the algorithm written for the detection of vegetation patches in the eight regions selected, the parameters used for each region analyzed are indicated within the code.