Spatial effects in meta-food-webs
aa r X i v : . [ q - b i o . P E ] A ug Spatial effects in meta-foodwebs
Edmund Barter and Thilo Gross University of Bristol, Department of Engineering Mathematics, Bristol, UK * [email protected] ABSTRACT
In ecology it is widely recognised that many landscapes comprise a network of discrete patches of habitat. Thespecies that inhabit the patches interact with each other through a foodweb, the network of feeding interactions.The meta-foodweb model proposed by Pillai et al. combines the feeding relationships at each patch with thedispersal of species between patches, such that the whole system is represented by a network of networks.Previous work on meta-foodwebs has focussed on landscape networks that do not have an explicit spatialembedding, but in real landscapes the patches are usually distributed in space. Here we compare the dispersalof a meta-foodweb on Erd ˝os-R ´enyi networks, that do not have a spatial embedding, and random geometricnetworks, that do have a spatial embedding. We found that local structure and large network distances in spatiallyembedded networks, lead to meso-scale patterns of patch occupation by both specialist and omnivorous species.In particular, we found that spatial separations make the coexistence of competing species more likely. Ourresults highlight the effects of spatial embeddings for meta-foodweb models, and the need for new analyticalapproaches to them.
Foodwebs, the networks of trophic (feeding) interactions among a community of species, are among the paradigmaticexamples of complex networks. Their composition and dynamics have been studied extensively.
In nature,communities are often not isolated but are embedded in a complex structured environment that consists of distinctpatches of habitat.
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Depending on the system under consideration the patches may be lakes, islands, or actualpatches of forest left in an agricultural landscape. In typical environments, the communities at many similar patchesinteract through the dispersal of individuals between neighbouring patches. The aggregations of the foodwebs atpatches related by a complex spatial network are called meta-foodwebs.
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These systems, comprising localfoodwebs joined to each other by links between patches, can be represented by a network of networks (Fig 1b).The study of spatial interactions has a long history in Ecology. For instance in explaining the global coexistenceof similar competitors
13, 14 and the survival of multiple species on the same limiting resource.
Studies in thisarea often account for the presence or absence of the species at each patch as a binary variable.
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In these,so-called patch-dynamic, models the state of each patch changes in time due to local colonization and extinctionevents.Most patch-dynamic models focus on simple cases such as single populations or competitive interactionsbetween similar species. However, recently Pillai et al.
21, 22 set out a framework that incorporates trophicinteractions between species into patch-dynamic models. This meta-foodweb model considers complex foodwebsof many species with predator-prey and competitive interactions.The meta-foodweb model of Pillai et. al. has been used to demonstrate that general spatial heterogeneity canincrease stability of complex foodwebs.
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More specifically the number and distribution of links in the networkof patches have been shown to have non-trivial effects on the distribution of foodwebs among the patches.
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We have previously shown that species at different levels of a food chain may benefit from different distributionsof patch degrees (number of links to other patches). Other previous work addressed the prominent questionof whether, and under what conditions an omnivore predator can coexist with a specialist, who is a strongercompetitor. This demonstrated that an omnivore can persist only when the average number of links at eachpatch, the mean degree, is within a particular range. 1revious theoretical works investigated meta-foodwebs where the underlying patch network was assumed tobe an Erd˝os-R´enyi random graph or configuration model network.
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In such networks any given pair of patcheshas a fixed chance of interacting and therefore the networks typically have a small diameter , a measure of themaximum distance between nodes. In these networks the shortest path (series of links) between any given pairof nodes is small
27, 28 and they are subsequently termed small-worlds. By contrast, real world meta-foodwebsare constrained by geography and the individual’s ability to travel between patches. Such a spatial embeddingconstrains the possible networks as patches are only linked if they are close enough together for individuals todisperse between them. Geometric distances are translated in network distances and in the resulting network theshortest path between two patches can be relatively long.
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In a small world network, a species can quickly disperse from any node to every other node. By contrast, in alarge world pronounced geographical barriers, characterized by long network path lengths, may exists that impederapid colonization of distant parts. While we will provide a more detailed analysis below it is intuitively conceivablethat the large world nature of spatially embedded networks of patches creates spatial niches in which a species cansurvive with relatively little danger from competitors. Moreover, in small worlds the neighbours of any particularnode tend to be a representative sample from the network. A node in a small world is thus exposed to colonizationfrom the full range of communities that the system supports. By contrast, in large worlds the neighbours of a nodeare located in the same region of the network as the focal node, and most colonization will be from communitiesthat are very similar to the one established in the focal node. This reinforcement of communities may furtherpromote species persistence.Here we investigate the effects of spatial nature of a patchy environment (i.e. the large-worldishness) on thedispersal of foodwebs. We build our analysis on a comparison of non-spatial Erd˝os-R´enyi networks andexplicitly spatial, random geometric patch networks. We find two results: First, specialist consumers are lessabundant (occupy fewer patches) on the spatial patch networks. Second, when the landscape is also occupiedby a competitor, generalist consumers are more abundant on the spatial patch networks than on non-spatial patchnetworks. We conclude that these results are predominantly due to the larger distances between patches in thespatial networks.
We study a version of the model proposed by Pillai et al.. The model describes a set of species, each of whicheither occupies or is absent from each patch in a spatial network at each moment in time. Trophic interactionsbetween the species are represented by a global meta-foodweb (Fig. 1a). Following Pillai et al. we assume eachpatch contains only a subset of the species of the global foodweb, and these comprise a local food chain (Fig. 1b).The global meta-foodweb comprising all the species only becomes evident when an aggregation of the spatialsystem is considered.The trophic interactions of the meta-foodweb have implications on the ability of species to occupy each patch.A species must be able to feed at every patch it occupies. Primary producers can occupy an empty patch, but allother species can only occupy patches where their prey is present. Furthermore a species cannot share a patchwith another species competing for the same prey. Following Pillai et al. we assume that specialist consumersoutcompete their generalist counterparts for a particular prey all of the time. Hence, all local foodwebs are linearchains.A patch that contains at least one food source that can be utilized by a given species, and does not contain asuperior competitor to that species, can be occupied by it and we say the patch is available to that species.The species that occupy a particular patch change in time due to the colonization of the patch by species fromneighbouring patches and the extinction of the local populations at the patch. When established on a patch, species X colonises neighbouring patches that are available to it at the rate c X . When established at a patch, species X alsogoes extinct on that patch at the rate e X . In the following we set e X = c X = c for all species in the meta-foodweb.At all times the local foodwebs must satisfy the restrictions imposed on species by trophic interactions. Therefore, igure 1. Meta-foodwebs as networks of networks. Panel (a) shows a simple meta-foodweb of four species.Each node represents a species and the directed links are from prey to predators in a feeding relationship. Species A is a primary producer, while species B and C are specialist consumers. Species O is an omnivore, which canfeed on multiple trophic levels. Panel (b) shows a patch network. Rectangles represent each patch and links arebetween patches which species can disperse between. Each patch is occupied by a local food chain. The localnetworks change in time due to colonization and extinction events.when species X is established on a patch where its only prey is species Y , and species Y goes extinct on that patch,species X will also go extinct, we call this process indirect extinction . When a specialist predator, species X ,colonises a patch that is occupied by an omnivore species Y , and species X and species Y share a food source, thenspecies Y may no longer exist on that food source. This will lead to the extinction of species Y on that patch, unlessit can prey on species X , so has a food source it does not compete for. We call the extinction of a generalist by anarriving specialist driven extinction .To study the effects of space on species coexistence we consider dynamics on two different types of patchnetworks. The first type of network are those from the Erd˝os-R´enyi ( ER ) ensemble.
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These networks aregenerated by taking a set of nodes and assigning a link between each pair with a probability p = z / N , where N isthe number of nodes and z is the desired mean degree, i.e. the expectation value of the number of neighbours for arandomly chosen node. A central property of networks that determines how easy a particular network is to colonize is the degreedistribution p k , the probability distribution that a given node has k links.
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For ER networks the degree distributionis poisson distributed, so p k = z k e − z / k !. Because links are added randomly between nodes the ER is a non-spatialnetwork and every node can be reached from every other node in a small number of steps, which scales as log ( N ) . The second type of network considered here are from the ensemble of 2-dimensional random geometric ( RG )networks . These networks are generated by, first, randomly distributing N nodes in a unit square with uniformdistribution and then adding links between all pairs of nodes that are within a distance r of each other. The degreedistribution of RG networks is the same as that for ER networks, p k = z k e − z / k ! where the mean degree is given by z = π r N . The spatial distances between points in RG networks are translated into network distances and thereforethe shortest path between some pairs of nodes is large and scales with √ N / r . Although the ER and RG networks have the same degree distribution, they have different levels of degreecorrelations. In ER networks a link is equally likely between any pair of nodes, and so the degrees of neighbouringnodes are uncorrelated. In RG networks each node is at the centre of a circle with radius r that contains allits neighbours, we call this its spatial neighbourhood . A node’s degree is the number of nodes in its spatial igure 2. Spatial structure affects species abundance. The abundance (mean occupied fraction) of the species inthe meta-foodweb on Erd˝os-R´enyi (dashed lines) and random geometric (solid) networks. The colours of the linescorrespond to the colours of the species in the foodweb on the right. Most of the species are more abundant onErd˝os-R´enyi networks. The exception is the omnivore, which is more abundant on random geometric networkswhen coexisting with a specialist.neighbourhood. Two nodes that have a link between them in the network must have overlapping spatial neighbourhoods.As the number of nodes in the intersection of their spatial neighbourhoods is the same, the degree of a node in arandom geometric graph is correlated with the degree of its network neighbours.We simulate the dispersal of the species in the meta-foodweb on networks from each ensemble using a Gillespie-typealgorithm . By choosing ensembles with matching degree distributions we eliminate the effects on dispersalconsidered previously,
24, 25 and focus on the differences due to the difference in path lengths between the ensembles.For each species in the system there are two possible equilibrium states, either the species is extinct at themetapopulation level, i.e. does not occupy any patches, or the metapopulation is a constant size, i.e. it does. Afterthe simulation has reached this equilibrium for all species we record the time for which each patch was occupiedby each possible configuration of the species over the rest of the simulation. From these states we calculate thefraction of time each species occupies each patch (the patch occupation), the sum of which over all patches we termthe species abundance.
To begin we compare the dispersal of a single species on networks from ER and RG ensembles. We find that abovethe persistence threshold species abundance is higher on ER networks than RG networks, see Fig. 2. This meansthat populations of the species exist at more patches in ER networks than RG networks. Therefore, the globalextinction risk for a single species is comparatively greater on RG than ER networks.The difference in species abundance between the network ensembles is greatest close to the threshold, whilefor colonisation rates far above the threshold abundance is similar on both ensembles, see Fig. 2. In general therisk of global extinction is higher when abundance is lower. Therefore, the increase in extinction risk to a specieson RG , in comparison to ER networks, is greatest when the risk of global extinction is greatest.Let us now try to explain these findings with respect to the spatial structure of RG networks. Consideringparameters near the threshold, we find that the distribution of patch-wise occupations is broader on RG than ER networks, see Fig. 3a. Some patches in RG networks are occupied for a larger fraction of the time than any patchesin ER networks, while others are occupied for a smaller fraction of the time than any in ER networks. As extinction igure 3. The distribution of patch abundances for species A at different colonisation rates. The twocolonisation rates correspond to different regimes of overall species abundance. At c = . c = . c = . c = . RG networks are, when empty, colonised more quicklyby the species than others.To explain the variation in colonisation between patches we must consider which properties of a patch determinehow easy it is to colonise. It is well understood that a patch’s degree is correlated with its occupation.
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Thedistribution of patch occupations on ER networks has a similar shape to the degree distribution, reflecting theinfluence of degree on patch occupation. The broad distribution of patch occupation in RG networks reflects thegreater prominence non-degree effects in its determination. This indicates that in the large world network thedetailed spatial structure, rather than just the overall connectivity, is of greater importance.The probability of an unoccupied patch being colonised is affected by the occupation of its neighbours. Anunoccupied patch with frequently occupied neighbours will be colonised quickly, and so be unoccupied for arelatively short period. On the other hand, an unoccupied patch with infrequently occupied neighbours may not becolonised for a longer period. Therefore the properties of the neighbourhood of a patch can determine how often itis occupied.The observed differences between the distribution of patch occupations in ER and RG networks can thus beintuitively explained by the different average distances in these networks. In the ER networks the short distancesmean the neighbourhoods of all patches are similar. In the RG networks the long distances mean local neighbourhoodscan be more varied between nodes.We test the effect of differences between neighbourhoods explicitly by considering the occupation of nodeswith only the degree 10 in Fig. 4a. We find that the occupation of patches with degree 10 in ER networks has adistribution that is narrower than the distribution for all patches. For RG networks we find that the the occupationof patches with degree 10 has a relatively broad distribution, only a small amount narrower than the distributionof all patches. Therefore, patches of the same degree have more varied occupations in RG networks than in ER igure 4. Spatial embedding increases variation in occupation of patches with the same degree. Thedistributions of the fraction of time (patch occupation) that a primary producer occupies patches with degree k =
10, at different colonisation rates, c , in Erd˝os-R´enyi (dashed line) and random geometric networks (solid line).a) At a relatively low colonisation rate, when total abundance of the species is higher on Erd˝os-R´enyi networksthan random geometric networks. b) At a relatively high colonisation rate, when total abundance is similar on bothtypes of network. When overall abundance is low in random geometric networks, patches with the same degreehave a broad range of occupations.networks. This implies that, in spatial networks, the structure of a patch’s neighbourhood has a large influence onits occupation by the species.By observing the occupation of patches on example networks, such as in Fig. 5a, we find meso-scale patternsof patch occupation. There are distinct regions of high occupation patches separate from regions of low occupationpatches. In the large world network the species is distributed unevenly across the landscape, as regions withdifferent local structures are separated by large spatial distances. In RG networks there are often more lowoccupation regions than high occupation regions, and as such the species abundance is lower on these than equivalent ER networks. Spatial distances reduce the number of patches the species can easily colonise and so abundance islower in the large world networks.We now consider the situation at higher values of c , for which the species has similar abundance on both typesof network. At these parameter values the distribution of patch occupations is similar on both types of network,see Fig. 3b. Furthermore, the distributions are similar when considering only patches of degree 10, see Fig. 4b,suggesting that at these parameter values degree is a good indicator of occupation on RG networks as well as ER networks. When overall abundance is high the neighbourhoods of nodes in the large world are more similar and sohave a lesser impact on the occupation of individual patches.Observing example networks shows that spatial variations do exist in networks at high overall abundance, eventhough degree is a good indicator of patch occupation. This is because the RG networks have degree correlations.In RG networks high degree nodes have more high degree neighbours and low degree nodes have more low degreeneighbours. Therefore spatial regions with many high degree nodes are more highly occupied that spatial regionswith many low degree nodes. In the RG the degree correlations translate to spatial correlations so the speciesabundance varies between different regions.Furthermore, we find that the mean occupation of patches in RG networks with relatively low degree is lowerthan for those in ER networks, while the mean occupation of patches in RG networks with relatively high degrees igure 5. Species abundance on the patches of an example random geometric network with 10 nodes and h k i =
10 with c = .
8. a) Occupation by the specialist is highest in the regions most dense with patches, and theregions close to them. b) Occupation by the omnivore is highest in regions of medium density which are spatiallyseparated from regions with high specialist occupation. The regions with high omnivore occupation complementthe regions of high specialist occupation.is higher than for those in ER networks, see Fig. 6. The effects of degree on occupation are enhanced on RG networks due to degree correlations. Typical high degree patches in RG networks also have high degree neighboursand are colonised more easily than typical high degree patches in ER networks. Therefore, as well as the regionalvariations due to large network distances, in spatial networks the species distribution is uneven due to the structuralcorrelations of nearby nodes.To summarise, both long network distances and degree correlations cause the emergence of meso-scale patternsin the occupation of patches in the random geometric network. At low overall abundance the spatial separationshave the dominant effect, and the results is large variations between the occupation of different patches. At highoverall abundance spatial patterns are predominantly due to degree correlations, and these result in smaller spatialvariations in patch-wise occupation. We now extend our study to the meta-foodweb shown in Fig. 1a, which includes both predator-prey feeding andspecialist-omnivore competition interactions.Species A , a primary producer, that can survive on abiotic resources; species B , a secondary consumer, thatfeeds upon the primary producer; species C , a top predator, that feeds upon the secondary consumer; and species O , an omnivore that preys upon the primary producer and secondary consumer.Species B , C and O can only occupy patches alongside their prey. In addition, species O is out competed by thespecialist predators and therefore cannot survive on patches where species C is established. When both species B and species O occupy a patch, species B out-competes species O for feeding on species A , however species O canstill survive by feeding on species B instead.For each species, X , there is a range of values of c X / e X for which that species is unable to persist in a network.For specialist species the range is characterized by a critical value of µ X . When c X / e X < µ X the species cannotpersist in the network and will become globally extinct, but when c X / e X > µ X the species will persist and beexpected to occupy a quasi-steady fraction of the patches.
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For an omnivore species Y which is a competitor igure 6. Degree correlations increase variation between patches with different degrees. The distributions of thefraction of time (patch occupation) that a primary producer occupies patches with different degrees, k , inErd˝os-R´enyi (dashed line) and random geometric networks (solid line). a) Patches with the relatively low degree, k = k =
15, both at a colonisation rate, c = .
0, when totalabundance of the species is similar on both types of network. Degree correlations in random geometric networkslead to lower occupation of patches with low degree and higher occupation of patches with high degree, comparedto patches in the Erd˝os-R´enyi networks.to species X there are two critical values. Species Y can only persist in the network when both c Y / e Y > µ Y , and c X / e X < ν Y , where the limit ν Y is itself dependent on c Y / e Y . When c X / e X > ν Y the omnivore’s competitor occupiesa large fraction of the patches and the omnivore is unable to persist anywhere in the network, becoming globallyextinct. Though we have set c X / e X = c for all species, due to indirect extinctions the threshold value, µ X , is not thesame for all species.Using the foodweb, we can establish how the spatial embedding affects species that interact with feeding orcompetitive interactions. We investigate: a) how the distribution of a predator is affected by the uneven distributionof its prey, and b) how the distribution of an omnivore is affected by the uneven distribution of its superiorcompetitor.We start by studying the food chain of species A , B and C that contains only predator-prey relationships. Wefind that all these specialist species have similar behaviour to a single species, and are less abundant in RG than ER networks with the same degree distributions under the same dispersal conditions, see Fig. 2.The set of patches occupied by a predator is a subset of the set of patches occupied by its prey. Furthermore theeffective extinction rate of the predator is larger than that of its prey. Therefore it is harder for a predator to disperseand the abundance of a predator cannot be greater than the abundance of the prey species it consumes. We find thatthe abundance of predators is lower on RG networks even at parameter values where the abundance of their preyis similar in both ensembles. This suggests that the spatial effects of the underlying network experienced by thesingle species are experienced by all species in the chain.For the single species on RG networks, we found that structural properties lead to spatial variations in patchabundance. We find that a predator utilising a species as its food source has an even more uneven distribution thanits prey, see Fig. 7. The variation in the in habitability of patches in the underlying network is exaggerated by thedispersal of intermediate species in the chain. For species A , its prey occupies all patches equally, but structuralproperties of the network mean it will find some patches are more hospitable than others. For species B , preying on igure 7. Variations in patch occupation are larger for predators than prey. Comparison of the occupation of a)species B and its predator b) species C on patches from Erd˝os-R´enyi networks (dashed lines) and randomgeometric networks (solid lines) at c = .
0. The variation of patch occupation for the predator, species C , onrandom geometric graphs is larger than for the prey, species B , due to the combination of direct spatial effects, andthe uneven distribution of the prey.species A , the structural properties have the same effects, and in addition its prey is unevenly distributed. As species A and B undergo similar dispersal processes the uneven distribution of prey aligns with the structural variations.Therefore, species B experiences greater variation in habitability, and therefore occupation, across the patches inthe network. The local structures in large world networks have a greater impact on the distribution of species higherin the food chain.Now let us focus on the effects on the distribution of the omnivore species O due to coexisting with themeta-population of the species C . The two species coexist only for colonisation rates in the range µ C < c < ν O .When c < µ C , species C cannot persist in the meta community and so the patches available to species O are thesame patches available to species B , and the omnivore behaves identically to the specialist species B . When c > ν O the omnivore becomes extinct due to driven extinction from competition with the specialist.When the specialist and omnivore coexist, the omnivores abundance is greater on RG networks than on ER networks, see Fig. 2. We previously saw that the large distances in spatial networks hinder the occupation ofpatches by the specialists, by contrast these distances aid the occupation of patches by the omnivore.One reason the omnivore is more abundant on RG than ER networks, in the coexistence regime, is the loweroverall abundance of species C . More of the patches are available to the omnivore for more of the time in the RG networks. Further, colonisation of a patch that is occupied by species O by species C , causes the local extinctionof species O . Therefore when a patch which is occupied by species O and species B neighbours a patch occupiedby species C the colonisation rate of species C acts to increase the effective extinction rate of species O . In spatialnetworks omnivore populations encounter specialist populations less frequently, and competition has a smallereffect on the occupation of patches by the omnivores.We have seen that, on both ER and RG networks occupation by specialist species increases with node degree,see Fig. 8. However the variation is greater in RG networks, with high degree nodes more likely to be colonisedthan lower degree nodes. We now investigate whether the greater variation in patch occupation by the specialist on RG networks, affects the distribution of the omnivore.On both types of network, at c = .
8, occupation by the omnivore is largest on nodes with an intermediate igure 8.
The mean fraction of time patches of different degrees are occupied by each species. The bars areshaded in three sections, these show the fraction of time patches are occupied by: bottom) the omnivore, this canbe with or without species B ; middle) species C , which must also be with species B ; and top) species B but neitherspecies C or the omnivore. In the random geometric networks the intermediate degree patches have lowabundance of species C , and high abundance of the omnivore.degree. Low degree nodes are hard for all species to occupy, while the large amount of time species C occupieshigh degree nodes means that they are often unavailable to the omnivore.The effect of competitor occupation on omnivore occupation of high degree nodes is greater on RG than on ER networks. Figure 8b for RG networks, has a pronounced peak in omnivore occupation at intermediate degrees. Thehigh occupation by the specialist of high degree patches makes them almost always unavailable to the omnivoreand the omnivore rarely occupies them. In spatial networks some patches are occupied by the specialist so oftenthat a population of omnivores is almost never established on them.Patches with intermediate degrees are relatively unlikely to be occupied by the specialist on RG networkscompared to ER networks. Subsequently, the occupation of these patches by the omnivore is high, almost as highas occupation by the omnivore in the absence of competition, which is equivalent to occupation by species B . Thelarge network distances mean that the omnivore is able to colonise these nodes and establish populations that rarelyencounter competition from specialist populations.Both RG and ER networks have poisson degree distributions, and many more patches with intermediate degreethan with the highest degrees. Subsequently, the overall abundance of the omnivore is greater on the RG than the ER networks. The spatial distances in large worlds make more patches more hospitable to the omnivore than theymake less hospitable, and therefore decrease the chance of global omnivore extinction.To summarise, the large path lengths in RG networks mean that some regions of the graph are hospitable to theomnivore but inhospitable to the specialist. Therefore, the omnivore can colonise these regions without sufferingregular competitive extinctions, and consequentially, occupy them a large fraction of the time. In other words, theomnivore can fill in gaps in the specialists dispersal, see Fig. 5b. We used agent-based simulations to investigate the dispersal of a foodweb, comprising predator-prey and competitiveinteractions, on spatially embedded patch networks.The abundance of primary producers and specialist consumers, which are only affected by predator-prey nteractions, is lower on spatially embedded random geometric patch networks than non-spatial Erd˝os-R´enyi patchnetworks. Furthermore, in spatially embedded networks there is a greater variation between the fraction of timedifferent patches are occupied by any species. By visualising patch abundance of specialists, we found that thesevariations correspond meso-scale patters of patch occupation. Regions dense with patches of high occupation arespatially separated from regions of patches with low occupation.A specialist consumer only occupies patches that its prey also occupies. In general its prey is also a specialistconsumer. The local factors, such as high degree, that make a patch easy for the prey species to occupy it also makeit easy for its predator to occupy. Further, spatial variations in prey occupations reduce the availability of patchesfor its predator in some regions of spatial networks. Consequently, the difference in the abundance of a predatorbetween Erd˝os-R´enyi and random geometric networks is larger than the difference in the abundance of its prey.We assume that generalist species, such as omnivores, are weaker competitors than specialist species withthe same prey¸ such that the generalist is driven to local extinction in any interaction. Subsequently, reducedoverall specialist abundance in random geometric networks aids generalist persistence. Further, we identify thatthe meso-scale structure in spatially embedded networks aids the persistence of generalist species in three ways.Firstly, patches can have relatively high degrees but low specialist abundance due to network separation from themost occupied regions of the network and patches with higher degree are easier for the generalist to colonise.Secondly, patches with low specialist occupation are often grouped together and the generalist can persist in such aregion without stochastic extinction. Thirdly, regions with low specialist occupation are separated by long distancesfrom the regions of high specialist occupation and so the generalist populations there are rarely threatened by drivenextinction. The combination of these factors leads to greater abundance of generalist species on random geometricnetworks, compared to Erd˝os-R´enyi networks.Two properties of the spatially embedded networks cause the significant regional variations in patch occupation:long distances between nodes and degree correlations. Previous work has indicated that long distances have agreater effect than degree correlations on dispersal processes, and our results also support this conclusion. Whenregional variations are large, degree is a poor indicator of patch occupation, but when regional variations are smallit is a good indicator. This suggests that when regional variations are larger, they are being primarily caused by longnetwork distances and when regional variations are smaller we are seeing the effect only of degree correlations. Inother words, long distances have a greater effect than degree correlations on species dispersal over spatial networks.Real patch landscapes are usually inherently spatial and these results suggest that structures resulting fromthis have many implications for environmental management. For example, our results demonstrate the contrastingresults possible from measuring diversity at different spatial scales. Further, the ability of a species to inhabit apatch is dependent on the structure of surrounding patches and long range effects. In situations, such as deforestation,where a managed removal of patches attempts to preserve habitat by preserving a single, well populated patch, theseattempts may fail if the neighbourhood of that patch is destroyed such that recolonisation becomes more difficult.Alternatively, well intentioned environmental management to maintain a specialist species can destroy the spatialseparation that allows omnivores to persist, sheltered from specialist populations.The spatial embedding of agents is a property of many networks found in ecology and elsewhere. However,dynamic process on networks incorporating spatial structure are relatively poorly understood. Most existinganalytical approaches for dispersal processes on networks focus on local structure, such as degree correlations.Our results show that local properties in spatial networks often have a smaller impact than long network distances.Though there has been some recent development of methods for considering spatial separation of nodes in networks, further advances are required to characterise space in network models, and understand its effects on dynamicprocesses.One particular challenge for analytical approaches to spatial networks is how to characterise their spatialstructure. The most popular model for generating spatial networks, using random geometric graphs, results innetworks with several notable properties that make them different from random networks. For instance higherclustering coefficients, larger diameters, larger average path length and positive degree correlations. Theseproperties are not independent but are jointly caused by the spatial embedding. For example, nodes near each otherare neighbours and the radius containing their neighbourhoods largely intersect, resulting in clustering and also egree correlations. Our results show that both long separations and local properties are simultaneously importantin explaining the dynamical behaviour of systems on spatial networks. Hence, methods to better understand thesesystems should be capable of accounting for the combination of these properties.We finish by suggesting how this may be accomplished. It may be possible to approximate the regions ofa spatial graph of homogeneous patches by a graph of heterogeneous meta-patches. The properties of each ofthese meta-patches would be constructed to reflect the region of the underlying spatial graph that the particularpatch represents. A random geometric graph in this model would have meta-patches with properties drawn froma distribution determined by the formation processes of a random geometric graph. For example, each meta-patchcould have a local extinction probability, based on the number of patches included and the number of links betweenthem. The graph of meta-patches contains many fewer patches, and could be potentially be analysed using availablenumerical methods. The sensitivity of the model to meta-patch properties may be analysed to determine the systemssensitivity to the underlying properties of spatial patch networks that inform them. ata Availability
This study did not involve any underlying data.
References May, R. M. Will a Large Complex System be Stable?
Nature , 413–414 (1972). McCann, K., Hastings, A. & Huxel, G. R. Weak trophic interactions and the balance of nature.
Nature ,794–798 (1998). McCann, K. S. The diversity-stability debate.
Nature , 228–233 (2000). Neutel, A.-M., Heesterbeek, J. A. P. & de Ruiter, P. C. Stability in Real Food Webs: Weak Links in LongLoops.
Science , 1120–1123 (2002). Montoya, J. M., Pimm, S. L. & Sol´e, R. V. Ecological networks and their fragility.
Nature , 259–264(2006). Gross, T., Rudolf, L., Levin, S. A. & Dieckmann, U. Generalized models reveal stabilizing factors in foodwebs.
Science (New York, N.Y.) , 747–50 (2009). Lafferty, K. D. et al.
A general consumer-resource population model.
Science , 854–857 (2015). Levins, R. Some Demographic and Genetic Consequences of Environmental Heterogeneity for BiologicalControl.
Bulletin of the Entomological Society of America , 237–240 (1969). Hanski, I.
Metapopulation Ecology (Oxford University Press, 1999).
Warfe, D. M. et al.
Productivity, disturbance and ecosystem size have no influence on food chain length inseasonally connected rivers.
PLOS ONE , 1–11 (2013). Gramlich, P., Plitzko, S., Rudolf, L., Drossel, B. & Gross, T. The influence of dispersal on a predator–preysystem with two habitats.
Journal of Theoretical Biology , 150 – 161 (2016).
Kivela, M. et al.
Multilayer networks.
Journal of Complex Networks , 203–271 (2014). MacArthur, R. H. & Wilson, E. O.
Theory of Island Biogeography (Princeton University Press, 1967).
Harrison, S. Local extinction in a metapopulation context: an empirical evaluation.
Biological journal of theLinnean Society , 73–88 (1991). Levins, R. & Culver, D. Regional Coexistence of Species and Competition between Rare Species ,1246–1248 (1971). Levin, S. A. Dispersion and Population Interactions.
The American Naturalist , 207–228 (1974).
Nee, S. & May, R. M. Dynamics of Metapopulations: Habitat Destruction and Competitive Coexistence.
Journal of Animal Ecology , 37–40 (1992). Hanski, I. & Gilpin, M. Metapopulation dynamics: brief history and conceptual domain.
Biological Journalof the Linnean Society , 3–16 (1991). Hanski, I. Metapopulation dynamics.
Nature , 41–49 (1998).
Hanski, I. The Levins model and its variants. In
Metapopulation Ecology , chap. 4, 55–75 (Oxford UniversityPress, 1999).
Pillai, P., Loreau, M. & Gonzalez, A. A patch-dynamic framework for food web metacommunities.
TheoreticalEcology , 223–237 (2009). Pillai, P., Gonzalez, A. & Loreau, M. Metacommunity theory explains the emergence of food web complexity.
Proceedings of the National Academy of Sciences of the United States of America , 19293–8 (2011). Gravel, D., Canard, E., Guichard, F. & Mouquet, N. Persistence increases with diversity and connectance introphic metacommunities.
PloS one , e19374 (2011). B ¨ohme, G. A. & Gross, T. Persistence of complex food webs in metacommunities. arXiv:1212.5025 (2012).
Barter, E. & Gross, T. Meta-food-chains as a many-layer epidemic process on networks.
Phys. Rev. E ,22303 (2016). Newman, M.
Networks: An Introduction (Oxford University Press, 2010).
Klee, V. & Larman, D. Diameters of random graphs.
Canad. J. Math , 618–640 (1981). Bollob´as, B. The Diameter of Random Graphs.
Transactions of the American Mathematical Society ,41–52 (1981).
Watts, D. J. & Strogatz, S. H. Collective dynamics of ’small-world’ networks.
Nature , 440–442 (1998).
Banavar, J. R., Maritan, A. & Rinaldo, A. Size and form in efficient transportation networks.
Nature ,130–132 (1999).
Dall, J. & Christensen, M. Random geometric graphs.
Physical review. E, Statistical, nonlinear, and softmatter physics , 016121 (2002). Erd˝os, P. & R´enyi, A. On random graphs I.
Publ. Math. Debrecen , 290–297 (1959). Gilbert, A. N. Random Graphs.
Annals of Mathematical statistics , 1141–1144 (1959). Erd˝os, P. & R´enyi, A. On the evolution of random graphs.
Bull. Inst. Internat. Statist , 343–347 (1961). Pastor-Satorras, R. & Vespignani, A. Epidemic spreading in scale-free networks.
Physical Review Letters ,3200–3203 (2001). Penrose, M.
Random Geometric Graphs (Oxford University Press, 2003), 1 edn.
Friedrich, T., Sauerwald, T. & Stauffer, A. Diameter and Broadcast Time of Random Geometric Graphs inArbitrary Dimensions.
Algorithmica , 65–88 (2013). Gillespie, D. T. A general method for numerically simulating the stochastic time evolution of coupled chemicalreactions.
Journal of Computational Physics , 403–434 (1976). Isham, V., Kaczmarska, J. & Nekovee, M. Spread of information and infection on finite random networks.
Phys. Rev. E , 046128 (2011). Whittaker, R. H. Evolution and measurement of species diversity.
Taxon , 213–251 (1972). Pilosof, S., Porter, M. A., Pascual, M. & K´efi, S. The multilayer nature of ecological networks.
Nature Ecology& Evolution , 0101 (2017). Riley, S., Eames, K., Isham, V., Mollison, D. & Trapman, P. Five challenges for spatial epidemic models.
Epidemics , 68 – 71 (2015). Challenges in Modelling Infectious { DIsease } Dynamics.
Pastor-satorras, R. et al.
Epidemic processes in complex networks.
Reviews of Modern Physics , 1–62(2015). Estrada, E., Meloni, S., Sheerin, M. & Moreno, Y. Epidemic spreading in random rectangular networks.
Phys.Rev. E , 052316 (2016). Acknowledgements
This work was supported by the EPSRC under grant codes EP/N034384/1 and EP/I013717/1.
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There is no competing interest for any of the authors. uthor contribution