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Ecological Modelling 332 (2016) 1–7 Contents lists available at ScienceDirect Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel Interspecific interaction strength influences population density more than carrying capacity in more complex ecological networks Chuan Yan, Zhibin Zhang ∗ State Key Laboratory of Integrated Management of Pest Insects and Rodents in Agriculture, Institute of Zoology, Chinese Academy of Sciences, Beijing 100101, China a r t i c l e i n f o Article history: Received 18 December 2015 Received in revised form 26 March 2016 Accepted 29 March 2016 Keywords: Population density Species composition Population dynamics Species interaction Ecological network Communities a b s t r a c t Understanding the mechanisms determining population density of species in communities and ecological networks is an important task in ecological studies. Interactions and carrying capacity largely determine population density of species and then community structure. However, their impacts on population density have not been fully investigated in ecological networks. In this study, we examined the associations of interspecific interaction strength and carrying capacity with population density in three kinds of theoretical and empirical ecological networks with different complexity. We firstly demonstrated both the net direct and indirect interaction strength of a species received from the other species showed positive associations with population density of the species in all ecological networks (except for in predation networks), particularly in more complex ecological networks. Direct interaction was more important than indirect interaction in determining population density. Carrying capacity showed a positive association with population density, particularly in less complex ecological networks. Our results suggest that interspecific interaction strength is more important than carrying capacity in determining species dominance in more complex networks. © 2016 Elsevier B.V. All rights reserved. 1. Introduction Understanding species composition and structure of communities is a great challenge in ecology. Species dominance, which is measured by population density of each species, is an important indicator of community. It has been well known that population density of a species is determined by biotic interactions and environments (Krebs, 2008). Various theoretical models like Lotka–Volterra models and their derivatives were developed in two-species systems, and served well for ecologists in early literature for assessing the impacts of species interaction and carrying capacity on population density of interacting species (Lotka, 1925; Volterra, 1926; Pimm, 1982). In literature on food chains or webs, the bottom-up and top-down effects have been realized and debated over their relative importance for a long time, and a synthesis of both forces considering heterogeneity within or across trophic levels and flexibility is believed to an answer (Power, 1992). Three- or four-species models were developed to account for both bottom-up and top-down effects (Hastings and Powell, 1991; Abrams, 2005). In general, the effects on population ∗ Corresponding author. Tel.: +86 10 64807099; fax: +86 10 6480 7099. E-mail address: [email protected] (Z. Zhang). http://dx.doi.org/10.1016/j.ecolmodel.2016.03.023 0304-3800/© 2016 Elsevier B.V. All rights reserved. densities within a community are mostly studied in the systems of a few species, not in ecological networks. In natural ecosystems, species interact directly with each other in a variety of ways, which forms ecological networks. The network thinking is a prevalent systematic way of studying community and ecosystem (May, 1972; Patten, 1985). Network analysis contributed to identifications of some primary properties of ecosystems by examining the pathways, flows, storages, and net flows of static ecosystem models (Fath and Patten, 1999). Recently, there are increasing studies on food webs and mutualistic networks in literature (May, 1972; Bascompte, 2010; Thébault and Fontaine, 2010; Bascompte and Jordano, 2013), addressing various questions including properties of ecological networks, description of realistic webs, and stability of ecological networks, etc. Ecological network analysis, incorporating species composition and interaction, has provided a novel method to analyze community composition and dynamics (Proulx et al., 2005). However, previous studies mostly focused on network-level consequence like robustness and stability (May, 1972; Montoya et al., 2006; Bascompte, 2009; Thébault and Fontaine, 2010), whereas relations of population density of species with species interaction strength and carrying capacity within a network have not been investigated. It is necessary to investigate these relations so as to understand the role of abiotic and biotic factors in determining community composition. We hypothesized 2 C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7 that in more complex networks, the role of interspecific interaction strength would be more important than that of carrying capacity in explaining population densities within a community because species received more effects from other species. Besides direct species interactions, impacts of indirect interaction on community dynamics have also been noticed (Menge, 1995; Křivan and Schmitz, 2004). Indirect effect requires the presence of intermediary species. Effects of indirect interactions included competition for common resource species (Levine, 1976), and predation cascade in food chains (Oksanen et al., 1981) on population density have been explored in theoretical models of a few species. Abrams et al. (1996) reviewed theoretical and empirical studies on indirect effects in food webs, suggesting that magnitude of indirect effects might be smaller than direct effects in most of cases because of density dependence and stochastic disturbance, but large enough in magnitude that they cannot be ignored. However, by applying economic input–output analysis in ecosystems (Network Environ Analysis) (Patten, 1978), studies have demonstrated that the effects of indirect interactions, usually quantified by flows of currency (energy or matter), often exceed the effects of direct connections on the overall throughflow in large ecosystems (Patten and Higashi, 1984; Patten, 1985; Fath and Patten, 1999; Borrett et al., 2010). The dominant indirect effect in ecosystem models raised the question whether it applies to population densities, making it crucial to test the performance of indirect effects in explaining population densities within the complex ecological networks, and more importantly, how the relative effects of indirect and direct interactions change with complexity of ecological networks. In this study, we examined the impacts of both direct and indirect interaction strength and carrying capacity on population density of species in three kinds of theoretical and empirical ecological networks through numerically solved multiple-species Lotka–Volterra equations that are widely used in the studies of population and community ecology. There are two kinds of indirect interactions in natural communities. The first one is that one species affects population density of another species through affecting population density of an intermediate species, i.e., density-mediated indirect interaction such as food chains; the second one is that one species affects another species’ abundance through affecting the interaction between the second and third species (Abrams et al., 1996). In this study, we mainly focused on the density-mediated indirect interaction. We have the following three predictions. First, in more complex networks, the role of interaction strength in determining population density will be more important than that of carrying capacity. Second, population density of a species is positively related to sum of all direct interaction strengths (net direct interaction strength, NDIS) which come from its directly connected partners, and also positively related to the net indirect interaction strength it received (net indirect interaction strength, NIIS). Third, NDIS would have a better performance in explaining population densities in networks than NIIS. It is notable that the “interaction strength” may be referred to different metrics in previous studies (Berlow et al., 2004) and here we defined it to the per capita effect of one species on another species in classic Lotka–Volterra equations, and it had “+” or “−” signs corresponding to any positive or negative effect caused by competition, amensalism, antagonism, neutralism, commensalism and mutualism. 2. Material and methods 2.1. Ecological networks We used two sets of systems: three types of simulated networks (random network, cascade prey–predator network and mutualistic network), and empirical mutualistic networks by Vázquez and Simberloff (2002) and food webs by Thompson and Townsend (2003), downloaded from an online Interaction Web DataBase (www.nceas.ucsb.edu/interactionweb/). The random networks, mutualistic networks and cascade prey–predator networks were constructed by following May (1972) and Allesina and Tang (2012), with randomly distributed coefficients carrying capacity (K) sampled from uniform distributions (100–200 limited) and interaction strength (IS) sampled from normal distributions (mean = 0, standard deviation (SD) = 0.01, 0.03 or 0.05). The normal distribution of interaction coefficients (with mean = 0) corresponds to right-skewed distributions of interaction magnitudes that are commonly discovered in empirical works (Wootton, 1997; McCann et al., 1998). The interaction coefficients in networks have signs corresponding to its positive or negative effects. For simulated networks, we set different species number (S = 30, 80 or 130), connectance (C = 0.05, 0.1, 0.3 or 0.5) and interaction strength level (for random and prey–predator networks, SD = 0.01, 0.03 or 0.05; for mutualistic networks SD = 0.01, 0.02 or 0.03). 50 networks were simulated for each combination of S, C, and SD. For more details, see Yan and Zhang (2014). For realistic networks, we only used their structures, and the interaction coefficients were similarly drawn from random distributions (see above). We used average links per species (ALPS = (S − 1) × C) and SD to indicate complexity of simulated networks. The larger the SD value, the larger the interaction coefficients (absolute value). 2.2. Metrics of interaction The definition of interaction effects varies according to the purposes of different studies. Our current study focus on population densities in ecological networks, so we used classic Lotka–Volterra population dynamic models to quantify the interaction effects. Based on analytical results (Lawlor, 1979, and also see Appendix S1 and S2), the population density of a species at equilibrium was determined by the direct and indirect interact strength it receives and carrying capacity. We calculated two species-level metrics of interaction strength in networks. The net direct interaction strength was calculated by summing all the interaction strengths of species i received NDISi = J a / i ij j=1,j = , where J is the number of species interacting with species i. The net indirect interaction strength (NIIS) was calculated by summing all the indirect interaction strengths species i received. The indirect strength of one species on another species involves all potential indirect paths through the rest of community, and includes various interactions like mutualism, predation, competition, etc. The sign of indirect effect might be the same or opposite to that of direct effect. We derived the indirect strength by following Lawlor (1979), details in Appendix S1. Besides, we also calculated a first-order indirect interaction strength (see Appendix S2), which is the sum of interaction strengths multiplied by itsdirectly connected species’ net interaction strength respectively FIISi = J a NDISj / i ij j=1,j = . 2.3. Population dynamics models The population density change of one species can be represented by a combination of contributions from logistic growth and positive and negative contributions from other species (Eqs. (1)–(4)). In these equations, xi is the population density of species i, ri is the intrinsic increase rate, Ki is the carrying capacity, and aij is the per capita interspecific (i = / j) interaction strength. P is the number of species showing positive effects on xi , and N is the number of C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7 3 species showing negative effects on xi . Population change of xi = Logistic Growth + Positive contributions − Negative contributions Logistic growth = ri xi 1 − xi Ki (2) P Positive contributions = ri xi a x / i ij j j=1,j = (3) Ki N Negative contributions = ri xi (1) a x / i ij j j=1,j = Ki (4) If aij has either positive or negative sign, the Eqs. (2)–(4) can be combined as the Eq. (5). dxi = ri xi dt x 1− i + Ki J a x / i ij j j=1,j = Ki Fig. 1. The performance (R2 ) of explaining population densities by different metrics in the simulated random networks with different average links per species (ALPS) and standard deviation (SD) of interaction strength. Each dot represents the R2 from a linear mixed model with the one or two metrics as exploratory variables for networks with a particular parameter set. NDIS, net direct interaction strength; NIIS, net indirect interaction strength; NDIS + NIIS, both metrics of NDIS and NIIS; K, the carrying capacity. (5) J is the number of species interacting with species i. Analytical solution is often difficult to achieve for large networks. Thus, we numerically solved Eq. (5) by using the fourth-order Runge–Kutta method for 20,000 time units with a step of 0.01, implemented by deSolve package in R software (R Development Core Team, 2010). Initial population values for species in networks were randomly sampled from a uniform distribution between 5 and 10. The equilibrium abundance of each species was then used to infer the effects of metrics of interaction on population density within a network. Cycling and vanishing populations may occur in our results, but our additional analyses indicated that they did not affect our observations (for details, see Appendix S3). 2.4. Statistical analysis Linear mixed models (LMM) were applied to test the effects of K, NDIS, NIIS and FIIS on the population density of species at equilibriums within each simulated network. The response variable is the population densities within each network, the predictor variables are K, NDIS, NIIS, or both of NDIS and NIIS within each network, and the random variable is the networks (50 networks for each parameter set). For example, for simulated networks with a particular parameter set, the statistical model to calculate R2 for K is as population density = K + random (networks). In realistic webs, linear models (LM) were applied (26 food webs and 59 pollination webs). Nine pollination webs were not used when testing the effect of NDIS, because they were too large to calculate the inverse matrix for NDIS. The R2 of LMM or LM was used to make comparisons of variance explained between different metrics in different networks with different ALPS and SD, so as to infer the impact of direct and indirect interaction strength, and K on population density in ecological networks with different ALPS. Following the method provided by Nakagawa and Schielzeth (2013), the calculation of R2 for LMM was implemented by R package MuMIn, which is suggested to outperform other metrics to access model performance. 3. Results In random networks, the net direct interaction strength (NDIS) of a species mostly showed significant and positive impacts on population density of the species, while the net indirect interaction strength (NIIS) only showed significant and positive impacts at the higher SD level of 0.05 (Table S1, Fig. S1). The R2 of NDIS explained is higher than that of NIIS at all SD levels (Fig. 1). When taking both NDIS and NIIS into models (“NDIS + NIIS”), there was higher R2 than either NDIS or NIIS (Fig. 1). The R2 of NDIS, NIIS and “NDIS + NIIS” increased with the average links per species (ALPS) and SD of interaction strength (Fig. 1). The results of FIIS are similar to that of NIIS in random networks (Fig. S4). The carrying capacity showed significant and positive impacts on population density of the species, and its R2 decreased with increase of the average links per species (ALPS) and SD of interaction strength (Fig. 1). At the lower SD level of 0.01, K always showed higher R2 than interaction metrics, but at higher SD levels of 0.03 and 0.05, K showed only higher R2 than interaction metrics at low ALPS levels. In simulated cascade prey–predator networks, NDIS mostly showed significant positive impacts on population density s, while NIIS showed small negative impacts at lower SD levels but large negative impacts at higher SD levels (Table S2, Fig. S2). NDIS had a higher R2 than NIIS at all cases. Taking both NDIS and NIIS into models, they showed a similar R2 with NDIS at the SD level of 0.01, but showed a higher R2 than using NDIS or NIIS separately at higher ALPS and SD levels (Fig. 2a). R2 of NDIS and “NDIS + NIIS” in simulated cascade prey–predator networks increased with ALPS at the SD level of 0.01, but showed a dome-shaped relationship with ALPS at higher SD of interaction strength (Fig. 2a). In realistic food webs, NDIS mostly showed significant positive impacts on population density, while NIIS showed little impacts at the SD level of 0.01, but the negative impacts increased with increase of SD levels (Fig. 2b, Table S3). The R2 of NDIS is higher than that of NIIS at all SD levels, when taking both NDIS and NIIS into models, “NDIS + NIIS” showed a higher R2 than either NDIS or NIIS separately (Fig. 2b). R2 of NDIS and “NDIS + NIIS” in realistic food webs showed a consistent increase with increase of ALPS and SD levels of interaction strength (Fig. 2b). The results of FIIS are similar to that of NIIS in both networks (Fig. S5), but it showed positive effects on the population density. The results of carrying capacity on population density were similar to those in random networks (Fig. 2). In both simulated and realistic mutualistic networks, NDIS and NIIS both showed significant positive impacts on population density of both trophic levels (Fig. 3, Fig. S3, Tables S4 and S5). R2 of NDIS, NIIS, and “NDIS + NIIS” were similar in simulated mutualistic networks, with a little higher R2 in “NDIS + NIIS” at high ALPS and SD levels (Fig. 3). The R2 of NDIS, NIIS, and “NDIS + NIIS” increased with ALPS and SD levels of interaction strength at both trophic levels in simulated and realistic mutualistic networks (Fig. 3). The results of FIIS are similar to that of NIIS in both networks (Fig. S6). The results of carrying capacity were similar to those in random networks (Fig. 3). 4 C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7 Fig. 2. The performance (R2 ) of explaining population densities by different metrics in the simulated cascade prey–predator networks (a) and real food webs (b) with different average links per species (ALPS) and standard deviation (SD) of interaction strength. Each dot represents the R2 from a linear (mixed) model with the one or two metrics as exploratory variables for networks with a particular parameter set (a) or a real network (b). NDIS, net direct interaction strength; NIIS, net indirect interaction strength; NDIS + NIIS, both metrics of NDIS and NIIS; K, the carrying capacity. 4. Discussion In this study, we firstly demonstrated both net direct and indirect interaction strength of a species received could largely determine its population density, particularly in more complex ecological networks, while carrying capacity could largely determine its population density in less complex ecological networks. We found both the net direct and indirect interaction strength of a species received were positively associated with population density of the species in all three kinds of ecological networks except for in predator-prey networks, largely supporting the second prediction. We found net direct interaction strength explained more variance of population density than indirect interaction strength including both NIIS and FIIS in randomly and prey–predator networks, not in mutualistic networks (in which there was little difference), generally supporting our third prediction. Our results provide a new insight about the impacts of biotic and abiotic factors on structure and dynamics of ecological networks. The net direct and indirect interaction strength and the carrying capacity are important criteria in determining population density of a species, but their roles differ in networks with different complexity; the former is more important in more complex ecological networks, while the latter is more important in less complex ecological networks. 4.1. Effect of direct interaction Direct interaction is mostly concerned regarding its theoretical conception and empirical measure in literature. In theoretical models derived from classic Lotka–Volterra models, it is apparent that the interaction strength and K affect population density, because the equilibrium density is calculated as a sum of parameters including interaction strength and K values as the numerator part (see Appendix S1 and S2). The role of direct interaction strength in determining population density has long been recognized in Fig. 3. The performance (R2 ) of explaining population densities by different metrics in the simulated mutualistic networks (a) and real pollination webs (b) with different average links per species (ALPS) and standard deviation (SD) of interaction strength. Each dot represents the R2 from a linear (mixed) model with the one or two metrics as exploratory variables for networks with a particular parameter set (a) or a real network (b). Higher level indicates higher trophic level species, and lower level indicates lower trophic level species. NDIS, net direct interaction strength; NIIS, net indirect interaction strength; NDIS + NIIS, both metrics of NDIS and NIIS; K, the carrying capacity. models with a few species. Mostly studied at 2–4 species systems, classic Lotka–Volterra model and its derivatives provided basic explanation on how species interaction affects population density in mathematic ways (Lotka, 1925; Volterra, 1926). Applying Lotka–Volterra models in multiple-species systems is also common (Levine, 1976; Lawlor, 1979), but studies on the contribution of species interaction to population density in ecological networks have not been fully studied. Our results showed the simple metric of net direct interaction strength effect turned out to be a good predictor at most cases except when SD levels of interaction strength were too low, which implied that species that get more positive and less negative effects would have higher population density. It can be easily shown that many species are involved in multiple interactions including prey–predation and mutualism. From a network view, different biotic resources a species occupies are reflected in the positive interactions that other species interact with, such as prey, mutualism, and commensalism, while the negative interactions indicate predation, competition and amensalism. A common explanation of population density of species or species dominance in plants is the resources species can occupy C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7 (McKane et al., 2002). Thus previous studies on plants mostly focused on inter-specific competition, especially exploitative competition, while less concerned interactions are the biotic interactions such as the top-down effect and mutualistic interaction (e.g. pollination). The resources-partition hypothesis is also linked to the negative relationship between body size and population density in animals (White et al., 2007). It is apparently shown in our results that if species receive more positive effect and less negative effect from other species, species will be stabilized at a higher population density. The interaction strength used in this study is the per capita interaction strength in Lotka–Volterra models, which is not easy to measure in reality. In nature, variation of both time and space can affect estimation of interaction strength (Wootton and Emmerson, 2005). Experimental and observational approaches have been proposed in previous studies. Experimental approaches usually compare the absolute difference in the densities of target species to a manipulated species after some period of time (Paine, 1980). Observational approaches mainly include static regressions of population density s at different sites (Hallett and Pimm, 1979), measurements of energy flow by diets and biomass (Bascompte et al., 2005), direct observation (Wootton, 1997), modeling time series data (Ives et al., 2003), etc. In bipartite mutualistic networks, visiting frequency is a commonly used surrogate for interaction strength (Vázquez et al., 2005; Fang and Huang, 2016). The interaction strength is proportional to the frequency of interaction between the two species relative to all other interactions (Bascompte et al., 2006; Vázquez et al., 2007). It is apparent that abundance would determine this sort of interspecific interaction frequency or strength because abundant species should have more frequent encounters with individuals of many other species (Vázquez et al., 2005). Thus interaction frequency is not suitable to explain population density in communities or networks because of apparent autocorrelation problem. We suggest the observed interaction frequency should be corrected by using the observed population density data before it is used as proxy of per capita interaction strength for calculating the net direct or indirect interaction strength. The interaction strength in our study is not affected by population density or abundance (Berlow et al., 2004). We provided clear evidence that per capita interaction strength could well explain the population density of species in communities or networks, particularly in complex networks. 4.2. Effect of indirect interaction Our results in large ecological networks are consistent to observations in some previous studies which were achieved in systems of a few species, that indirect interactions showed a relatively minor effect (Abrams et al., 1996). However, this result is inconsistent with the dominant indirect effect in ecosystem models (Fath and Patten, 1999). This difference could mostly come from the different study focuses and ways to quantify inter-species connections. Network analysis in ecosystem models focused on indirect energy flow and storage of the whole system, while in classic population and community models, the focuses are the population abundance of a targeted species and the indirect influence of another species on it through other species. There could be a large energy flow in the system, but this does not necessarily mean it will outweigh the direct effects on population density. How to integrate these two sets of models and contexts needs more crossing-field work in future studies. Previous studies have proposed several reasons for the minor effect of indirect interaction on population density. If interaction strength is small, then indirect interaction throughout the path way from one species to the targeted species would be small (Abrams et al., 1996). There are general mechanisms which 5 explain the reduction of indirect interaction effect: density dependence and stochastic disturbance (Wootton, 1994a; Abrams et al., 1996). Schoener (1993) and Menge (1995) both found that indirect effects were smaller than direct effects in many cases, but large enough not be ignored. However, some studies have documented important indirect interaction effects in systems with a few species (Dungan, 1986; Wootton, 1994b). Our results showed that the differences between effects of direct interactions and indirect interactions varied across network types. A big difference in the explained variances of population density between direct interactions and indirect interactions was found in simulated random and prey–predator networks and realistic food webs, but there was little such difference in mutualistic networks. Because the absolute value of interaction strength in our study is much smaller than 1.0, the value of indirect interaction strength multiplying direct interaction strength with the net strength its “partners” received would become smaller than direct interactions in magnitude, which may partially explain the relatively smaller impact of indirect interaction. For mutualistic networks, they had only positive interaction strengths, which made the direct interact strength and indirect interaction strengths correlated and then leaded to similar effects. The negative effect of NIIS on population density in prey–predator networks is inconsistent to our prediction, which implies that structure of network affects the effect of net indirect interaction strength on abundance. It is not surprising that the net indirect effects can show different signs with direct effects (Lawlor, 1979). This might explain why NIIS showed negative effects on population density in prey–predator networks. However, FIIS showed positive effects on population density. It is more easily calculated that NIIS, particularly when the network is large. Therefore, we propose FIIS might be a better metric as an indirect interaction in predicting population density or species dominance in communities or networks. Estimating indirect interaction effect in nature is much similar to estimation of direct interaction. Empirical studies to test indirect interaction effects by manipulating populations were proved to be applicable in a system of a few species (Wootton, 1992; Miller, 1994). It is not realistic to manipulate every species in a large community, and many studies included only short chains (Abrams et al., 1996). Another problem is the difficulty of distinguishing the direct and indirect effects when a change of targeted species is observed. Path analysis was also applied in estimating indirect effect (Johnson et al., 1991), but with shortcomings of a need for explicit causal assumption, large sample size in complex systems, and data variability (Wootton, 1994a). In our study, the FIIS would be an alternative method which only uses data of direct interaction strength. 4.3. Effect of carrying capacity (K) Carrying capacity (K) in Lotka–Volterra models is a mathematical abstraction of the organisms environment can maximally support, which is determined by many factors, such as climate, geographic gradients, and the traits of species themselves (Wangersky, 1978). In the Lotka–Volterra population models, K is usually included in the analytic solutions of population density (Appendix S1 and S2). In our study, the values of K were positively correlated to population densities at equilibrium in a network, which is consistent with the observations in models with a few species. Our novel finding is that under condition of weak interaction strength and smaller ALPS, population density is mainly determined by the carrying capacity, but as the complexity of ecological networks increases, the impact of K becomes less and less. In simple ecosystems with a few species or low and weak connectance, species seldom interact with each other, thus it is reasonable the 6 C. Yan, Z. Zhang / Ecological Modelling 332 (2016) 1–7 carrying capacity plays a more important role than species interaction in determining population density of the species; but in more complex ecosystems, resources are heavily overlapped, and species would face predation or parasitism or mutualism from other species, thus the contribution of the carrying capacity would become smaller. These results imply that carrying capacity should play more important roles in shaping structure and dynamics of more simple ecosystems (e.g. at early succession stages, desert ecosystems). 4.4. Effect of complexity of ecological networks (ALPS) Ecological networks have different species richness, connectance, interaction types and strength magnitude in different ecosystems across geographic regions (Baiser et al., 2012) and temporal scales (Schoenly and Cohen, 1991). Traditionally, studies on ecologic networks were often focused on different network types such as food webs and mutualistic networks depending on the data availability, but recently studies combining different networks start to increase (Mougi and Kondoh, 2012; Pocock et al., 2012). Our results showed that the relative importance of direct and indirect interactions was different in different types of networks (see above). Regardless of network types, interaction strength of a species received from other species includes three parts: the number of interaction links, signs and the strength magnitude. When species richness and connectance of networks increase, each species has more interactions with other species on average. The indirect interactions also increase with web size (Abrams et al., 1996). However, at present it is still not clear how interaction strength per species affects population density of a species when species richness and connectance of networks increase. Our results firstly demonstrated that increase of average links per species (ALPS) increased the importance of direct and indirect interactions, but reduced the contribution of carrying capacity in determining population density. Nonetheless, in the cascade prey–predation networks, the dome-shaped relationship between ALPS and R2 of interaction strength indicates that impact of direct interaction strength decreased, while impact of indirect interaction strength increased, particularly at high SD levels of interaction strength. However, in realistic food webs, the relationship between ALPS and R2 of interaction strength is consistently positive. These results imply that species interaction should play more important roles in shaping structure and dynamics of more complex communities (e.g. at older succession stages, tropical ecosystems). 4.5. Effect of interaction strength magnitude (SD) Previous studies show that interaction strength has to be weak to stabilize communities (McCann, 2000; Montoya et al., 2006). It is known that distributions of observed interaction strengths in nature skewed toward many weak interactions and a few strong interactions (Wootton, 1997; McCann et al., 1998). Current theoretical and empirical studies tend to support theory of weak interaction strength (McCann, 2000). In theoretical studies, weak interaction strength is necessary to attain high persistence of complex networks. At present, it is not clear how interaction strength affects population density in large and well-connected ecological networks. In our study, interaction strengths were randomly drawn from normal distributions, and they were set to be weak to achieve stable equilibrium of ecological networks. We have clearly shown that increase of interaction strengths (larger SD) can increase the performance of both direct and indirect interactions in explaining population density, but decrease that of carrying capacity. Increased SD of interaction strengths would increase the effects of direct and indirect interaction on population density of the species. 5. Conclusions Our study firstly demonstrated that both direct and indirect interaction strengths are more important in determining population density or species dominance in more complex ecological networks than the carrying capacity. Thus, the interaction strength and carrying capacity are good indicators in reflecting the properties of ecological networks with different complexity. We predicted that in less complex ecosystems (e.g. in early stage of succession or in degenerated systems), species dominance would have a weak association with interaction strength but have a strong association with the carrying capacity; while in more complex ecosystems (e.g. in mature or diversified ecosystems), such relations would be reversed. With accelerated impact of human disturbance which has caused rapid loss of biodiversity, the impacts of interspecific interactions on species dominance or community composition will become weaker, while abiotic factors (e.g. climate warming, habitat change, etc.) would play more important roles in changing structure and dynamics of ecosystems. In future studies of ecological networks, methods of measuring per capita interaction strength or its indices need to be developed, so as to understand the mechanism of species dominance in ecological networks. Acknowledgements This work is supported by the grants of China Natural Science Foundation (Grant Nos. 31330013, 31500347), the “Strategic Priority Research Program” of the Chinese Academy of Sciences (Grant No. XDB11050300), and the Key Research Program of the Chinese Academy of Sciences (Grant No. KJZD-EW-TZ-L01). Appendix A. Supplementary data Supplementary material related to this article can be found, in the online version, at http://dx.doi.org/10.1016/j.ecolmodel.2016. 03.023. References Abrams, P.A., 2005. The consequences of predator and prey adaptations for topdown and bottom-up effects. In: Ecology of Predator–Prey Interactions. Oxford University Press, Oxford, UK, pp. 279–297. Abrams, P.A., Menge, B.A., Mittelbach, G.G., Spiller, D.A., Yodzis, P., 1996. The role of indirect effects in food webs. In: Food Webs. Springer, pp. 371–395. Allesina, S., Tang, S., 2012. Stability criteria for complex ecosystems. Nature 483, 205–208. 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