http://www.diva-portal.org This is the published version of a paper published in Population Ecology. Citation for the original published paper (version of record): Landi, P., Minoarivelo, H O., Brännström, Å., Hui, C., Dieckmann, U. (2018) Complexity and stability of ecological networks: a review of the theory Population Ecology, 60(4): 319-345 https://doi.org/10.1007/s10144-018-0628-3 Access to the published version may require subscription. N.B. When citing this work, cite the original published paper. Permanent link to this version: http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-154081 Population Ecology (2018) 60:319–345 https://doi.org/10.1007/s10144-018-0628-3 REVIEW Complexity and stability of ecological networks: a review of the theory Pietro Landi1,2 · Henintsoa O. Minoarivelo1,3 · Åke Brännström2,4 · Cang Hui1,5 · Ulf Dieckmann2 Received: 20 April 2017 / Accepted: 29 June 2018 / Published online: 6 July 2018 © The Author(s) 2018 Abstract Our planet is changing at paces never observed before. Species extinction is happening at faster rates than ever, greatly exceeding the five mass extinctions in the fossil record. Nevertheless, our lives are strongly based on services provided by ecosystems, thus the responses to global change of our natural heritage are of immediate concern. Understanding the rela- tionship between complexity and stability of ecosystems is of key importance for the maintenance of the balance of human growth and the conservation of all the natural services that ecosystems provide. Mathematical network models can be used to simplify the vast complexity of the real world, to formally describe and investigate ecological phenomena, and to understand ecosystems propensity of returning to its functioning regime after a stress or a perturbation. The use of ecological-network models to study the relationship between complexity and stability of natural ecosystems is the focus of this review. The concept of ecological networks and their characteristics are first introduced, followed by central and occasionally contrasting definitions of complexity and stability. The literature on the relationship between complexity and stability in different types of models and in real ecosystems is then reviewed, highlighting the theoretical debate and the lack of consensual agreement. The summary of the importance of this line of research for the successful management and conservation of biodiversity and ecosystem services concludes the review. Keywords Biodiversity · Community · Complex networks · Ecosystem · Resilience Introduction if sometimes we do not realize it, our lives are strongly based on services provided by ecosystems, thus the responses to In the geological era of the Anthropocene, our planet is global change of our natural heritage are of immediate changing at paces never observed before (Millennium concern for policy makers. As ecosystems are composed Ecosystem Assessment 2005). Pollution, natural resources by thousands of interlinked species that interact directly or exploitation, habitat fragmentation, and climate change are through their shared environment, such as nutrients, light, only some of the threats our biosphere is facing. Species or space, a holistic perspective on the system as a whole is extinction is happening at faster rates than ever, greatly normally required to predict ecosystem responses to global exceeding the five mass extinctions in the fossil record. Even changes (Wolanski and McLusky 2011). A systems-analysis approach is thus often crucial for acquiring an understand- ing of all the dynamical feedbacks at the ecosystem level * Pietro Landi and for accurately managing the biodiversity that we rely on [email protected] in terms of ecosystem services. In particular, mathematical 1 Department of Mathematical Sciences, Stellenbosch network models can be used to simplify the vast complex- University, Stellenbosch, South Africa ity of the real world, to formally describe and investigate 2 Evolution and Ecology Program, International Institute ecological phenomena, and to understand how ecosystems for Applied Systems Analysis, Laxenburg, Austria react to stress and perturbations (Dunne 2006). 3 Centre of Excellence in Mathematical and Statistical Complex-networks models are composed of a set of com- Sciences, Wits University, Johannesburg, South Africa partments, describing either species or coarser functional 4 Department of Mathematics and Mathematical Statistics, groups, and a set of links that represent interactions or Umeå University, Umeå, Sweden energy or biomass flows among compartments. Thus, such 5 Mathematical and Physical Biosciences, African Institute models can describe both biotic and abiotic interactions for Mathematical Sciences, Muizenberg, South Africa Vol.:(0123456789)1 3 320 Population Ecology (2018) 60:319–345 among species, i.e., both interactions among the species management and conservation of biodiversity and ecosystem themselves and interactions with their external environ- services in the current era of the Anthropocene. ment, and consequently they can often successfully be used to assess ecosystems stability to perturbations. Stability of an ecosystem can be understood as its propensity of return- Ecological networks defined ing to its functioning regime after a stress or a perturbation in its biotic components (e.g., decline in species abundances, An ecological network describes interactions among species introduction of alien species, and species extinction) or abi- in a community (Pascual and Dunne 2006). There are dif- otic components (e.g., exploitation, habitat fragmentation, ferent types of interactions, e.g., trophic interactions (feed- and climate change). A challenging and central question ing), mutualistic interactions (pollination, seed dispersal, that has interested ecologists and systems analysts alike for etc.), and competitive interactions (interference for common decades is how the stability of an ecosystems depend on its resources). Ecological networks can be represented as a set complexity, as roughly measured by the ecosystems’ diver- of S nodes, characterizing the species, connected by a set sity in species and their interactions (Johnson et al. 1996; of L links, characterizing possible interactions among each Worm and Duffy 2003; Dunne et al. 2005; Hooper et al. ordered pair of species (Newman 2010; Estrada 2012). Links 2005; Kondoh 2005; Loreau and De Mazancourt 2013). can be described by either a binary variable (0 or 1, absence To appreciate the importance of this question, we first rec- or presence of interaction) or by a real number characterizing ollect and differentiate between the major different functions the weight (or strength) of the interaction. In the first case that ecosystems continuously provide. Natural ecosystems the network is called unweighted, while in the second case sustain life and provide services that can be divided into four it is called weighted. Moreover, interactions can be undi- areas (Millennium Ecosystem Assessment 2005): provision- rected (or symmetric), meaning that species i affects species ing, such as the production of food and water; regulating, j to a certain amount and equally vice versa, or directed such as the control of climate and disease; supporting, such (or asymmetric), meaning that species i can affect species as nutrient cycles and crop pollination; and cultural, such as j differently from how species j affects species i (Fig. 1). spiritual and recreational benefits. For the management and Moreover, interactions can be described by their sign (+ or conservation of ecosystems services it is important to know −). For example, trophic networks (food webs) are charac- how the complexity of an ecosystem is related to its stability, terized by the fact that one species is feeding on the other, thus how the diversity of species in the ecosystem and the thus the coefficients aij (describing the effect of species j on network of their interactions can contribute to maintaining a species i) and aji (describing the effect of speciesi on species stable supply of services. This is especially important in an j) will obviously have opposite signs (thus their product will era in which the pressure exerted on natural ecosystems is be negative, aijaji < 0), i.e., one species is benefiting while becoming stronger and stronger, influencing their structure the other is suffering from the interaction. In mutualistic and functioning, while the services they provide are vital for networks both species are benefiting from the interaction, a continuously increasing number of people. In particular, thus both coefficientsa ij and aji will be positive (and so their human activities, directly or indirectly, tend to simplify the product, aijaji > 0), while in competitive networks both spe- composition and the structure of natural ecosystems. There- cies are suffering from the interaction, thus both coefficients fore, understanding the relationship between complexity and aij and aji will be negative (thus their product will be again stability of ecosystems is of key importance for the mainte- positive, aijaji > 0) (Fig. 2). Notice therefore that trophic nance of the balance of human growth and the conservation networks cannot be undirected (symmetric), since the two of all the natural services that ecosystems provide. Using coefficients describing the interaction always have opposite ecological-network models to study the relationship between sign (and typically also different absolute values). complexity and stability