BIOLOGICAL INVASION IN SOIL: COMPLEX NETWORK ANALYSIS

F Perez-Reche", S. N Taraskini", FM Neri$, C.A. Gilligan",L. da F Costa', M P Viana§, W Otten', D. Grinev'

£Department of Chemistry, University of Cambridge, Cambridge, UKtSt. Catharine's College, University of Cambridge, Cambridge, UK

$Department of Plant Sciences, University of Cambridge, Cambridge, UK§ Instituto de Fisica de Sao Carlos, Universidade de Sao Paulo, Sao Carlos, SP, Brazil

+SIMBIOS, University of Abertay Dundee, Dundee UK

ABSTRACT

A network model for soil pore space is developed and ap­plied to the analysis of biological invasion of microorgan­isms in soil. The model was parameterized for two soil sam­ples with different compaction (loosely and densely packed)from images derived from an X-ray micro-tomography sys­tem. The data were then processed using 3-D imaging tech­niques, to construct the networks ofpore structures with in thesoil samples. The network structure is characterized by themeasurement of features that are relevant for biological colo­nization through soil. These include the distribution of chan­nel lengths, node coordination numbers, location and size ofchannel bottlenecks, and the topology of the largest connectedcluster. The pore-space networks are then used to investi­gate the spread of a microorganism through soil, in which thetransmissibility between pores is defined as a function of thechannel characteristics. The same spreading process is in­vestigated in artificially constructed homogeneous networkswith the same average properties as the original ones. Thecomparison shows that the extent of invasion is lower in theoriginal networks than in the homogeneous ones: this provesthat inherent heterogeneity and correlations contribute to theresilience of the system to biological invasion.

Index Terms- Soil image analysis, complex network,biological invasion, percolation.

1. INTRODUCTION

The structure of soil and its transport properties are of signif­icant interest for various physical, geological, biological andagricultural reasons [1, 2, 3, 4, 5]. Usually soil is treated asa medium of pores of various sizes and shapes. It is a chal­lenging experimental task to obtain reliable information about

FPR, SNT, FMN and CAG thank BBSRC for funding (Grant No.RG46853), L. da F. Costa thanks FAPESP (05/00587-5) and CNPq(301303/06- 1) for sponsorship and M. P. Viana thanks FAPESP (07/50882­9) for financial support. WO and DG thank the Scottish Alliance for Geo­science, Environment and Society (SAGES) for support.

978-1-4244-3298-1/09/$25.00 ©2009 IEEE

soil structure and, in particular, about the spatial arrangementsof pores. Several techniques have been used for soil struc­ture analysis including serial sectioning [6], laser scanningconfocal microscopy [7], X-ray tomography [8] and nuclearmagnetic resonance imaging [9]. The structure of soil is ob­tained in the form of an image, which can then be processedby image analysis techniques and used for construction of anetwork model representing the soil structure [10, 11, 12].The network model can then be used for studying transportproperties of soil, e.g. water flow through the soil (see e.g.[13, 2]). Nowadays commercially available X-ray comput­erized tomography systems (also known as CT scanners)arecapable of resolving micron-sized pores in undisturbed soilsamples in three dimensions. Here we introduce the resultsfrom a X-ray computerized micro-tomography system thatachieves a resolution of 74JLm for contrasting samples of ap­proximately 2 x 2 x 4 cm representing loosely and denselypacked field soils. The resulting network structures are de­convoluted into a series probability distributions for chan­nel lengths, node coordination numbers, location and size ofchannel bottlenecks, and the topology ofthe largest connectedcluster defined. The invasion of a microorganism through thelargest connected cluster is then modelled using a set of rulesin which the transmissibility between each pair of connectednodes in the network is a function of channel properties link­ing the nodes. Under these rules, the transmission process isequivalent to an epidemiological process described by an SIR(susceptible-infected-removed) model. This, in tum, can bemapped onto isotropic bond percolation. According to thismapping, the system (i.e. the microorganism invading soil)exhibits a second-order phase transition from a non-invasiveto an invasive regime. The model enables us to compare artifi­cially constructed homogeneous networks with the same aver­age transmissibilities as the original ones. We show that thereare shifts in the invasion curves between the homogeneousand heterogeneous networks (as a function of transmissibil­ity. Those shifts are due to the existence of local correlationsin channel properties that reduce the extent of invasion in the

DSP 2009

Fig. 1. Images ofloosely (a) and densely (b) packed soil. Thegray colour represents the pore space in soil. The insets showthe middle slice (along the vertical axis) of the volumes withthe pores shown in yellow.

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Examples of visualization of loosely and densely packedsoil samples are provided in Figs. Ia and Ib, respectively.The filled space represents the pores in the soil.

In order to construct the network representing the porevolume, all the pores were processed by a thinning algorithm[19,20,21], required in order to reduce each object (pore)to a respective l-voxel skeleton. The skeleton is a thin struc­ture located at the most central parts of the respective origi­nal shape. The skeleton retains all the topological features ofthe original shape (e.g. branching structure and cycles). Theskeletonized pores were then mapped onto a network as fol­lows. The skeleton is a set of intersecting curves with somedead ends (where the skeleton terminates). Each intersectionpoint and all dead ends were associated with the nodes of thenetwork [21]. The pore space around the skeleton betweentwo nodes is called a channel or link (edge) between twonodes. The axis of the channel thus coincides with the skele­ton. Figs. 2 (a) and (b) show the pore networks obtained forthe loosely and densely packed samples presented in Fig. I.

2. EXPERIMENTAL DETAILS

3. NETWORK ANALYSIS OF THE SOILSTRUCTURE

original networks , with greater effect of heterogeneity in thedensely packed soil.

The paper is organized as follows. In Sec. 2, the exper­imental technique used for soil sample analysis is described.Sec. 3 describes the details of the network models associatedwith the soil samples. Sec. 4 deals with biological invasion inthe in the network models. Concluding remarks are presentedin Sec. 5.

Soil aggregates (1 - 2 mm) of an arable sandy loam werepacked to attain bulk densities of 1.2 or 1.4 Mg/m', repre­senting two characteristic examples of a loosely and denselypacked field soil, respectively for this soil type. Full detailscan be found in Ref. [14]. We refer to these two soil samplesas loosely (1.2 Mg/rn") and densely (1.4 Mg/rn") packedsoils hereafter.

We scanned these samples with a Metris X-TEK Bench­top micro-tomography system [4, 15], using a molybdenumtarget, X-ray source settings of 155 kV and 25 ILA, and analuminum filter (0.25 mm) to reduce beam-hardening arte­facts.

2-D radiographs were collected at 1169 angular positionsand then reconstructed using a filtered back projection algo­rithm with a resolution of 74 ILm and isotropic voxel size).Each radiograph was averaged over 32 frames to improvesignal-to-noise ratio and ring artefacts were minimized dur­ing data acquisition. Numerical corrections were also ap­plied during the reconstruction to minimize remaining beam­hardening artefacts. Both 3-D volumes were then importedinto VGStudioMax v.I.2 .1 [16] and converted into 260 x 5258-bit TIFF image stacks with voxel-thick slices. Binary datasets were created by thresholding the grey-scale image stacksin Image] [17]. The choice of the threshold parameter wasbased on the 3-D statistical analysis of the histogram regioncorresponding to the pore-solid interface. In order to ob­tain an optimal threshold value this analysis took into accountvariation of the grey-scale values in pores of different shapesand sizes.

3.1. Network construction

After conversion into binary format, pore networks for bothsoil samples were reconstructed in 3-D by using the Visu­alization Toolkit (VTK) [18]. Isolated pores with volumesmaller than 104 voxels were discarded to aid data visualiza­tion and analysis (below we use I pixel = 74JLm as a unitof length). This has negligible impact on the topology of theresulting network on which we perform our analysis .

Fig. 2. The network models for the loosely (a) and densely (b)packed soil shown in Fig. I. The complete network (a) con­sists of Nnode = 10316 nodes and Nlink = 11502 links, whileNnode = 3612 and Nlink = 3846 for the complete network (b).Different colors correspond to distinct isolated clusters withthe largest connected component in blue.

Fig. 3. Probability density function of the arc-lengths of chan­nels (in pixels), p(L), in the LCC of the networks represent­ing loosely (solid circles) and densely (open squares) packedsoil. The upper right inset magnifies the region of small arc­lengths. The lower left inset shows the probability densityfunction of the node degrees, p(Z) for networks correspond­ing to loosely (circles) and densely (squares) packed soil sam­ples. Open and solid symbols are used to denote the completenetwork and the largest connected component in the network,respectively. Open and solid circles are not distinguishable atthis scale.

from the straight line connecting two nodes, so that be = 0in the case of zero deviations. The distribution of be for theLCC in both networks is shown in Fig. 4(a). It is evidentfrom Fig. 4(a) that the majority of channels (which are rela­tively short in length) do not deviate strongly from the shortestpath between two nodes. The deviations (the value of be) in­crease with the arc-length of the channels . This can be seenin Fig. 4(b) which demonstrates correlations between valuesof be and the arc-length .

The bottleneck diameter distribution is shown in Fig. 5(a). It follows from this figure that the number of channelswith large bottleneck diameters is reduced in the denselypacked soil in comparison with the loosely packed one.In order to characterize the position of the bottleneck ina channel {ij} it is convenient to introduce a parameter,bij = IL ib - Lbj II L ij, where Lib and Lbj are the arc-lengthsfrom node i to the bottleneck and from the bottleneck to thenode j , respectively. This parameter reflects the relative dis­tance from the bottleneck to the middle of the channel. If thebottleneck is in the middle of the channel then bij = 0 andif it is at one of the ends of the channel then bij = 1. Thedistribution of the values of parameter bij for both networksis shown in Fig. 5(b). The dip around b = 1 indicates that thebottlenecks mainly occur inside the channels but not close tothe nodes. This indicates that the crossing points of severalchannels (nodes) are characterized by relatively large pore

3.2. Network characteristics

The networks constructed according to the algorithm de­scribed in Sec. 3.1 represent the pore (or equivalently soil)topology of two soil samples. The networks are embeddedin 3D-space and each node is characterized by a positionvector R i . The nodes are connected to each other in a com­plicated way and, in general, the network consists of isolated(not connected to each other) clusters of different sizes. Thelargest connected component (LCC) (the blue clusters inFig. 2) which can be identified for each network is of partic­ular interest for biological invasions as described below. Inthe case of loosely packed soil, the LCC percolates throughthe sample and contain N;o~~ = 10183 nodes connected byNkn1c = 11369 channels. The LCC for densely packed soildoes not percolate through the sample volume and containsN LCC = 2613 nodes and N LCC = 2823 channels.node hnk

One of the standard characteristics of network connectiv­ity is the node degree (Z) distribution, p(Z), where Z is thenumber of links attached to a node (coordination number).These distributions are shown in Fig.3 (left lower inset) bothfor complete networks and LCe. The value of Z = 1 cor­responds to the number of dead ends in the networks. Byconstruction, there are no nodes with coordination numberZ = 2. As seen from Fig. 3, the probability density functionsreach a maximum value at Z = 3 and then quickly decaywith increasing Z. The complete networks are found rathersparsely connected with mean degrees ((Z ) = 2NlinkINnodc),

(Zl) ':::' 2.229 and (Z2) ':::' 2.130, for loosely and denselypacked soil samples, respectively. The LCC exhibit similarvalues, (Z l) ':::' 2.233 and (Z2) ':::' 2.161. Therefore, thedensely packed soil have smaller degree numbers (pores col­lapse under compaction).

The other important structural characteristics of the net­work are the arc-length, Lij, of the channel between nodes iand j , and bottleneck diameter of the channel, ¢Jij . The arc­length is defined as the length of the axis of the channel. Thebottleneck diameter, cPij, is defined as the minimal diameterof the maximally inscribed circle for crossection of the chan­nel. The values of cPij were obtained by calculating the maxi­mum distance transform value along the skeletons [22]. Boththese characteristics play an important role for biological in­vasions (see Sec. 4). In the case of limited image resolution,the value of cP* can also be associated with the resolution, i.e.all the pores of sizes less than the resolution are not repre­sented by the network model.

The distribution of the arc-lengths for both samples arepresented in Fig. 3. It can be seen from Fig. 3 that the numbersof short and very long channels are reduced in the denselypacked sample as compared with the loosely packed one.

The shape of the channels can be characterized by theirrelative arc-length as compared with the Euclidean distancebetween nodes, i.e. by the parameter be = 1- IRj -Ri IILij.The value of be describes the deviations of the channel axis

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4. BIOLOGICAL INVASION IN SOIL

The following transport processes can occur in the soil: (i)the spread of liquids such as water; (ii)the spread of gas (e.g.oxygen) and (iii) dispersal of microorganisms. The first twophenomena have many important applications and have at­tracted huge attention from the scientific community (see e.g.[11] and references therein) The last phenomenon, of greatimportan ce for biological processes including the spread ofpathogen, has attracted much less attention [23] and still ex­hibits several unsolved problems.

In this section, we address the probl em of invasion ofmi­croorganisms through the system of soil pores represented bya network (see Sec. 3). A question to answer is the following.Given a network of pores and dynamical rules for dispersal,can soil be invaded (occupied) by microorganisms and, if so,what are the conditions for such invasion? We approach thisproblem using standard methods of statistical physics for de­scription of non-equilibrium phenomena in complex systems.

4.1. Theoretical background

We consider invasion of a microorganism in structured soil,typified by hyphal (filamentous) growth through soil pores of

Fig. 5. Probability density function of (a) bottleneck diame­ters (in pixels) , p(cPij ), and (b) relative distances to the bottle­neck, p(0), in the LCC of the networks representing loosely(sol id circles) and densely (open squares) packed soil.

a fungal colony expanding from an initial site of introductionwith a fixed source ofnutrients at the site of introduction. Thedynamical rules are as follows: (i) We place an initial sourceof microorganisms on an arbitrary node i (as defined in Sec. 3)of the LCC of the network representing the pore space in thesoil. (ii) The organi sm can spread stochastically along any ofZ, channels linked to node i and thus , with given probability,Pij , can reach node j (j = 1, ... , Zi ), so that node j becomescoloni zed (occupied) by the microorganism. (iii) From nodej, the process of invasion continues in a similar manner withthe only exception that the organisms cannot move back topreviously colonized nodes.

The motion of the organism s along a channel is assumedto be a Poisson process, so that the probability, dPij (t), toreach node j for organisms moving from node i to node jalong the channel of length Lij in an infinitesimall y smalltime interval between t and t + dt is [24], dPij (t ) = (1 ­Pij(t ))f3ijdt , with straightforward solution , Pij(t) = 1 ­e -{3i j t , where the coefficient f3ij is the invasion rate along thechannel {ij} . The invasion rate can be estimated as f3i j =

Vij / L ij , with Vij being the typical velocity of motion of or­ganisms through the channel {ij} and L ij being the lengths(arc-length as defined in Sec. 3) of the channel between nodesi and j . According to the equation above the probability ofinvasion of node i . dPij (t ), approaches unity when t ----; 00 .

This leads to a scenario in which the whole LCC network isinvaded in the long-tim e limit, something that is unlikely tohappen in practice when nutrient is limited for the invadingmicroorganism. Accordingly, we introduce a time Tij , avail­able for exploration ofeach {ij} channel , i.e. the microorgan­ism can only move along the channel for t :::; Tij ' The finiteexploration time may be a consequence of a limited amountof nutrients inside the channel or a limited life-time of themicroorganisms. Rapid invasion is often of great importance

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We have investigated spread on two networks describing in­vasion of loosely and densely packed soil samples . In bothcases, the analysis has been performed on the LCC of the net­work (see Sec. 3). The topology of the largest connected com­ponent obviously depends on the value of ¢* , because all thechannel s with the bottleneck size less than ¢* are closed tospread.

4.2. Results

into the chann el. If the bottleneck diameter is greater than thebound value, ¢*, then the organisms can fit into the channel,so that invasion can occur with certain given probability alongthis channel. Therefore, the transmissibility of channel {ij}is defined as

Under these rules, the invasion process is identical to anepidemic process described by an SIR (susceptible-infected­removed) model which can be mapped onto isotropic perco­lation [25] by identifying the channel transmissibility withthe bond probability. In this mapping, the system (organ­isms invading soil) exhibits a second-order phase transitionfrom a non-invasive to an invasive regime with an increase ofthe control parameter, k. The order parameter, Any, invasionprobability in infinite system (probability that the initially in­fected site belongs to the spanning cluster), changes from zerovalue Any = 0 for k ~ kc to finite values for k > kc, wherekc is the critical value of the local invasion scale.

In bond-percolation, the bond probability plays the role ofthe control parameter [26]. For the sake of comparison withbond percolation, it is convenient to introduce an alternativecontrol parameter for invasion of soil, namely the mean trans­missibility, (T) = J(l - e -k/ L i j ) p (Lij ) d Lij, where p(Li j)

is an experimentally known probability density function ofthe channel lengths and averaging is taken only through thechannels with ¢ijin > ¢* which are accessible for invasion.

Invasion of soil is inherently heterogeneous because thechannel transmissibilities and coordination numbers varythrough the system. However, if the channel transmissi­bilities are independent random variables for each channel,then spread on a heterogeneous network characterized bythe distribution of T i j is equivalent to spread on the samenetwork with a homogeneous transmissibility, Ti j = (T)[27,28,29], meaning that the function Any ((T) ) is the samein the systems with heterogeneous and homogeneous (mean­field) transmissibilities. One of the aims of this study is toverify this property for the spreading process in soil.

The other aim of our analysis is to find how the order pa­rameter depends on both control parameters (k and (T) ) andthus identify the conditions for invasion of soil by microor­ganisms.

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as new niches are colonized predominantly by rapid invaders.Here we assume a local clock for colonisation ofeach succes­sive pore, which amounts to an assumption of a limited andequal amount of nutrient available for microorganism growthwithin each pore. Under these assumptions, the probability ofinvasion of node j from node i in the long-time limit (calledtransmissibility) is

Fig. 6. Dependence of the bottleneck diameter (in pixels) vsarc-length of the channel (in pixels) for loosely (solid circlesand solid line) and densely (open squares and dashed curve)packed soil, respectively. The straight lines represent the lin­ear fit by regression.

All the parameters affecting the transmissibility T i j dependon channel characteristics and vary from channel to channeldue to their dependence on e.g. channel topology. The param­eters Vij and Ti j can also depend on the node characteristics(such as amount of nutrient available at the node and the vol­ume of the pore associated with the node). This can result inmutual dependence of transmissibilities of the channels orig­inating from the same node. For simplicity, we assume thatthe values of Vij and Tij are the same for all channels withthe same local invasion scale, k = Vij Ti j , being a control pa­rameter in further analysis. The local invasion scale has themeaning of a typical channel length such that the channelswith L i j > k are more likely to be closed for invasion whilethe channels with L i j < k are more likely to be opened.

The bottleneck size plays the role of a geometrical cutofffor the channel, so that all the channels with bottleneck sizesless than a critical one, ¢*, are deterministically closed forbiological invasion. The value of ¢* can depend on the typeof microorganisms and soil properties. It is reasonable to as­sume, that if the minimal cross-section size along the channel(the diameter of the bottleneck or throat in the channel), ¢'I'/ ,is small enough, ¢ijin < ¢*, then the organisms do not fit

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Fig. 7(b) shows how the invasion probability Anv changeswith the mean transmissibility, (T), which is analogous tobond probability for a percolation problem. For small trans­missibilities (bond probabilities), the invasion probability (themass of the spanning cluster) is close to zero and it increaseswith (T) . The point where it starts significantly to depart fromzero is called the invasion threshold or critical transmissibil­ity, Te . For the percolation problem this value is known asa percolation threshold. It should be noticed that the valuesof the critical transmissibility (percolation threshold) are rela­tively high (Te ':::' 0.81 for loosely packed soil and T; ':::' 0.96for the densely packed one) , as compared with Te ':::' 0.25for bond percolation in simple cubic lattice and T; = 0.5 forbond percolation in square lattice [31], and approach T; = 1for bond percolation in a linear chain. This is a consequenceof the low node coordination in both networks with the meancoordination number (Z) ':::' 2.23 for loosely and (Z) ':::' 2.16for densely packed soil (see Fig. 3).

Fig. 7(b) also presents the data (open symbols) for Anv(k)obtained for the effective (mean-field) networks with homo­geneous transmissibility (T) . If the transmissibilities Ti j areindependent random variables then the invasion probabilityshould be the same both in heterogeneous and homogeneousmean-field networks characterized by the mean value of thetransmissibility [27, 28, 32, 33] , i.e. the open and solid sym­bol curves in Fig. 7(b) should coincide. However, as seenfrom Fig. 7(b), this is not the case for both networks and thusthe correlations in transmissibilities playa significant role forboth networks. They reduce the transmissibility for a givenvalue of (T) and thus make the real networks more resilientto invasion as compared to the homogeneous ones (the open­symbol curves are above the solid-symbol ones in Fig. 7(b)).

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First, we present the data obtained for the networks withall the channels included, i.e. with ¢* = O. Fig. 7(a) showsthe dependence of the invasion probability versus the localinvasion scale for loosely (solid circles) and densely (solidsquares) packed soil samples. The value of Anv increaseswith increasing k in such a way that Anv ':::' 0 for k ;S ke andAnv > 0 for k ~ ke , where ke is the critical value of the localinvasion scale (the absence of a sharp transition at k = ke isdue to finite-size effects). For loosely packed soil , k; ':::' 15,while ke ':::' 35 for densely packed soil reflecting the shift ofthe whole curve for Anv(k) to the higher values of k. Thismeans that the densely packed soil is more resilient with re­spect to invasion due to the fact that the pore sizes under com­pression decrease. Therefore, the local invasion scale shouldbe larger in order to activate more channels for the same levelof invasion as in the loosely packed soil. The open symbolsin Fig. 7(a) represent the invasion probability for topologi­cally the same networks but with homogeneous transmissibil­ity, Ti j = (T) . It is clear from Fig . 7(a) that the homogeneous(mean-field in transmissibility) networks are less resilient toinvasion (the open symbols are above the solid ones for thesame value ofk). This effect is much more pronounced for thedensely packed network. Therefore, heterogeneity in trans­missibilities makes the networks less acceptable for invasion.A similar effect has been observed in lattice models for spreadof epidemics [30].

The invasion probability, Finv, has been calculated nu­merically for several networks corresponding to different val­ues of ¢* as a function of the local invasion scale, k, andmean transmissibility, (T). The results of the analysis areshown in Fig. 8 for several values of ¢*. The LCC in thenetwork corresponding to the loosely packed soil becomessignificantly sparser with N;o~~ == 6903 and (Z) ~ 2.148for o; == 4 pixels and N;o~~ == 2526 and (Z) ~ 2.089 for¢* == 10 pixels. The LCC in the network for the denselypacked soil becomes even more sparse with N;o~~ == 2613and (Z) ~ 2.161 for o; == 1 pixel and N;o~~ == 1983 and(Z) ~ 2.134 for o; == 2 pixels so that strong finite-size ef­fects influence the invasion probability for greater values of¢* (not shown).

It follows from Fig. 8(a,b) that the removal of narrowchannels from soil reduces significantly the invasion proba­bility and thus results in the shift of the critical value of k; togreater values (cf. curves in (a) and (b) for different values of¢*) and makes the soil more resilient to biological invasion.The reduction in the channel number for the LCC with finitevalue of ¢* results in a decrease of the mean coordinationnumber and thus in an increase of the critical transmissibilityfor networks without narrow channels. This can be clearlyseen in Fig. 8(c,d) where the probability of invasion curvesare shifted to larger values of transmissibilities with increas­ing value of ¢*.

In the analysis above, several assumptions for the invasionprocess have been made. In particular, we assumed that theinvasion rate is inversely proportional to the arc-length of thechannels and depends on the channel area in a step-like fash­ion. Both these assumptions are rather simplistic and properanalysis based on the experimental data should be undertaken(which is the aim of our future work). It would also be desir­able to support our findings by analysis of several (rather thantwo) soil samples using different image resolutions.

5. CONCLUSIONS

To conclude, we have presented a network model based onreal data for soil structure and used this model to analyze a bi­ological invasion in soil. The main idea of the network modelis to reduce the pore space in the soil to a skeleton (network)which is topologically equivalent to the original pore space.The resulting networks corresponding to loosely and denselypacked soil exhibit the following topological properties: (i)the mean node coordination number is relatively small andclose to two; (ii) the network channels (links between nodes)are distributed in length and deviate from the straight lineconnecting two nodes with the deviation being proportionalto the length of the channel; (iii) the channel bottlenecks aremainly located inside the interior part of the channels ratherthan close to the nodes; (iv) the size of the channel bottleneckdecreases with increasing length of the channel.

An invasion of microorganisms is then defined on the

largest connected component of the network as a stochas­tic process on the network with transmission probabilitiesdepending on the properties of the network. Within this ap­proach, biological invasion is a critical phenomenon whicheither can or cannot occur in the system depending on thevalues of the control parameter (local invasion length scale).We have found that the networks associated with the porespace in soil are significantly heterogeneous both in topology(e.g. node connectivity and their spatial arrangement) andtransmissibility (invasion probability through the channel).The heterogeneity in transmissibilities makes the networkmore resilient to invasion in comparison with the same net­work all the channels of which are characterized by the same(mean) transmissibility. The topological heterogeneity isshown to result in correlation effects between transmissibili­ties of different channels and thus leads to further increase inresilience. It has also been demonstrated that the compactionof the soil changes the soil structure in such a way that thedensely packed soil is more resilient to biological invasion ascompared to the loosely packed soil.

6. REFERENCES

[1] W. Otten and C.A. Gilligan, "Soil structure andsoil-borne diseases: using epidemiological concepts toscale from fungal spread to plant epidemics," Euro­pean Journal of Soil Science, vol. 42, pp. 131-134,2006.

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