Global structural stability and the role of cooperation in mutualistic systems

RESEARCH ARTICLE Global structural stability and the role of cooperation in mutualistic systems Jose ´R. PortilloID 1,4☯ *, Fernando Soler-ToscanoID 2☯ , Jose ´A. LangaID 3,4☯ 1Department of Applied Mathematics I, University of Seville, Seville, Spain, 2Department of Philosophy, Logic and Philosophy of Science, University of Seville, Seville, Spain, 3Department of Differential Equations and Numerical Analysis, University of Seville, Seville, Spain, 4Instituto de Matema ´ticas de la Universidad de Sevilla Antonio de Castro Brzezicki, Seville, Spain ☯These authors contributed equally to this work. *[email protected] Abstract Dynamical systems on graphs allow to describe multiple phenomena from different areas of Science. In particular, many complex systems in Ecology are studied by this approach. In this paper we analize the mathematical framework for the study of the structural stability of each stationary point, feasible or not, introducing a generalization for this concept, defined as Global Structural Stability. This approach would fit with the proper mathematical concept of structural stability, in which we find a full description of the complex dynamics on the phase space due to nonlinear dynamics. This fact can be analyzed as an informational field grounded in a global attractor whose structure can be completely characterized. These attractors are stable under perturbation and suppose the minimal structurally stable sets. We also study in detail, mathematically and computationally, the zones characterizing different levels of biodiversity in bipartite graphs describing mutualistic antagonistic systems of population dynamics. In particular, we investigate the dependence of the region of maximal biodiversity of a system on its connectivity matrix. On the other hand, as the network topology does not completely determine the robustness of the dynamics of a complex network, we study the correlation between structural stability and several graph measures. A systematic study on synthetic and biological graphs is presented, including 10 mutualistic networks of plants and seed-dispersal and 1000 random synthetic networks. We compare the role of centrality measures and modularity, concluding the importance of just cooperation strength among nodes when describing areas of maximal biodiversity. Indeed, we show that cooperation parameters are the central role for biodiversity while other measures act as secondary supporting functions. Introduction Phenomena from Natural and Social Sciences are usually modeled as complex networks for which a dynamic is defined among the nodes [1–4], sometimes associated to dynamical graphs [3,5–9], and where the study of stability is frequently a crucial fact [10,11]. From the keynote paper from Strogatz [9], many studies have focused on possible scenarios for the long time PLOS ONE PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 1 / 21 a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Portillo JR, Soler-Toscano F, Langa JA (2022) Global structural stability and the role of cooperation in mutualistic systems. PLoS ONE 17(4): e0267404. https://doi.org/10.1371/journal. pone.0267404 Editor: Pablo Martin Rodriguez, Federal University of Pernambuco: Universidade Federal de Pernambuco, BRAZIL Received: November 16, 2021 Accepted: April 7, 2022 Published: April 19, 2022 Peer Review History: PLOS recognizes the benefits of transparency in the peer review process; therefore, we enable the publication of all of the content of peer review and author responses alongside final, published articles. The editorial history of this article is available here: https://doi.org/10.1371/journal.pone.0267404 Copyright: ©2022 Portillo et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All real world seeddispersal databases files are available from the web-of-life database (accession number(s). M_SD_XX). https://www.web-of-life.es/map.php? dynamics of complex network with a given topology [12–15], being Population Dynamics [16–19], Economy [20–22] and Neuroscience [23–28] some of the areas where this important problem has been intensively studied. When the dynamics of the system is given by a set of differential equations, its behaviour generically depends on its global attractor [29–32], defined as information structure (IS) when its geometrical characterization is available [28,33]. An IS includes not only the information from the topology of the graph (structural network), but other key components that are crucial to understand all possible future scenarios. Indeed, an IS is the skeleton in the phase space describing topological and geometrical structural stability in dynamical system [34]. Note that, for an autonomous system, an IS is just the detailed structure of the unique global attractor. In gradient systems, this IS induces a whole deformation of the phase space, drawing an informational landscape where the transient and asymptotic observed dynamics of the system hold [33]. This IS and informational landscape are fixed and attracting. As indicated above, they coincide with the global attractor. But, in non-autonomous systems, in which, for instance, parameters depend on time, this fixed structure and associated landscapes are also changing in time, loosing their invariance and attracting properties, but still being crucial for the description of the dynamics. This fact has been used, for instance, in Neuroscience to discriminate in detail subjects with disorders of consciousness [35]. Thus, and IS could not coincide with the standard definition of a global attractor as the object describing all the asymptotic behaviour of the system. This is why, even in an autonomous framework as we use in this paper, for attractors and IS is better if they are differentiated. In this paper we focus on N-dimensional Lotka-Volterra systems used in the study of population dynamics (see, for instance, [36,37]), but, by the Fundamental Theorem of Dynamical Systems [38], the results of this research can be extended to more general systems of differential equations. We show the dependence of dynamics on the topology of the graph, but, in addition, we claim that this fact it is only part of a more general principle: the dynamics on a graph is globally described by its associated IS, which is different from the structural base graph and whose nature is essentially informational. The IS for these systems is described as an hierarchical set of semi-stable stationary solutions linked by associated stable and unstable manifolds (see Fig 1), and informs not only on all the possible future scenarios of the system, but the way they are reached (metastability), the rate of convergence, and the zones describing phase transitions between different structures (bifurcation phenomena). This more complex scenario leads to define a generalization of the concept of structural stability introduced in [39], in line with [40,41], allowing for a more fine description of internal and transient dynamics in ecological systems. A mathematical model of differential equations describes the dynamics of nodes on a mutualistic system as follows: suppose Pis the total number of plants and Athe number of animals. Plants (and animals) are in competition among them and cooperation links are set from plants to aminals and viceversa. We introduce the following system of N=P+Adifferential equations for Spiand Saidescribing the population density for the i-th species: dSpi dt ¼SpiapiX P j¼1 bpij SpjþX A k¼1 gpik Sak ! dSai dt ¼SaiaaiX A j¼1 baij SajþX P k¼1 gaik Spk ! Spið0Þ ¼ Spi0 Saið0Þ ¼ Sai0 8 > > > > > > > > > > < > > > > > > > > > > : ð1Þ for each p i for 1 �i�Pand a i with 1 �i�A.apiand aai(α i in short) are the intrinsic growth PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 2 / 21 type=6 All synthetic computer-generated databases files are available from https://github.com/ DynamicGraphSystem/StructuralStability Code is also full available. Funding: This work was partially supported by FEDER Ministerio de Economı ´a, Industria y Competitividad grant PGC2018-096540-B-I00, and Proyectos Fondo Europeo de Desarrollo Regional (FEDER) and Consejerı ´a de Economı ´a, Conocimiento, Empresas y Universidad de la Junta de Andalucı ´a, by Programa Operativo FEDER 20142020 references US-1254251 and P20-00592. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. Competing interests: The authors have declared that no competing interests exist. rates in the absence of competition and cooperation for plants and animals, respectively, bpij �0,baij �0denote the competitive interactions and gpij �0and gaij �0the mutualistic strengths. Structural Stability focuses on the size of the region for the intrinsic parameters α i to reach optimal (maximal) biodiversity. Observe that (1) can be written as a general Lotka-Volterra model for nspecies as: _ ui¼uiaiþX n j¼1 aijuj !;i¼1;. . . ;N;ð2Þ or, equivalently, _ u¼uðaþAuÞ;ð3Þ with A= (α ij ) the interaction (or adjacency) matrix given by A¼B1G2 G1B2 " #ðPþAÞ�ðPþAÞ :ð4Þ Structural Stability for this model is introduced in [39] as a proper concept unifying the influences of network topology and parameter dependence in the system; it has been used in Theoretical Ecology to analyze robustness of biodiversity in these complex networks [42–48]. Essentially, structural stability of system (1) measures the region of intrinsic parameters of species for which we get maximal biodiversity. Note that a greater region for structural stability allows for lower values of individual intrinsic growth parameters but preserving a high level of biodiversity, pointing for robustness and resilience of species. The study of the size of the region for intrinsic growth parameters (in our case the α i parameters) for which a system reaches its optimal biodiversity (all the species present) is defined as Structural Stability in [39]. This is a crucial fact for the study of the robustness of biodiversity in an ecosystem, as it characterizes the borders for intrinsic growth to get maximal biodiversity. Fig 1. Information structure. Graph with six nodes (top left) where a dynamics is defined by means of a LotkaVolterra cooperative system with α i and γ ij parameters as shown in the tables below. The attractor associated to the system, the information structure (IS) is shown on the right. Its eight nodes correspond to non-negative stationary points in the dynamics of the system. These stationary points are characterised by the value of each node in the system. The nodes u i shown in white indicate that u i = 0 at the corresponding point of the IS. Black nodes indicate that u i >0. Links between nodes u i and u j are those in the system (top left) where both u i ,u j >0. The blue arrows linking different points of the IS represent transitions going from one stationary solution (limit with time approaching −1) to another (when time approaches + 1). For clarity, transitive arrows are not shown. https://doi.org/10.1371/journal.pone.0267404.g001 PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 3 / 21 Following the Linear Complementary Theory (LCP) associated to Lotka-Volterra systems [36,37,49], we introduce a partition of the phase space [50] for which we can estimate the area in which each stationary solution is globally stable, by measuring the intersection of its associated cone of biodiversity with the unit N-dimensional sphere [45]. However, specially in high dimensional systems, a stationary point with all its components strictly positive either does not exist, or, if this is the case, there also exists a big set of semistable stationary points. The presence of these stationary points is crucial for the description of the transient behaviour and metastability properties of the system, so that neglecting its study could lead to wrong conclusions. Moreover, the ways to reach a particular stationary solution are multiple, depending of the different (informational) landscapes [33] described in detail by its semistable solutions (see Fig 2). Thus, in this paper we study the structural stability for every possible future scenario of the system. We do it in two different ways. Firstly, we consider the whole set of stationary points (asymptotically stable, semistable, or even globally unstable), and not only the globally asymptotically stable point with all components positive (see [46,48] for a similar approach). For instance, the transition to one globally asymptotically stationary point to another by a bifurcation parameter is usually described as a sudden phenomenon, but, as we show in this paper, it is totally understandable by a careful study of the parameter region of stability for each stationary point and the way they intersect. Secondly, and maybe more important, we introduce the study of the internal dynamics for each level of biodiversity. The analytic calculation of the feasible region for any dimension has been established in Saavedra et al. [42] and in Song et al [51]. The region of maximal biodiversity is described by a cone [39,45,50] in the phase space for the intrinsic growth parameters αso that, for every αin this cone, the system will tend asymptotically to a stationary point with all components strictly positive. But there exists many ways to reach this global attracting state, each one defined by a different global attractor whose structure determines the transient behaviour. Indeed, in the interior of the cone of maximal biodiversity holds a rich set of different dynamical scenarios describing how species uses diverse strategies in order to reach the final stationary point. These distinct scenarios are described by different global attractors for which a topological description is available. On the other hand, in Theoretical Ecology the study of cooperative interactions between groups of plants and pollinators / seed-dispersal / ants and how they affect to biodiversity has received an intensive research in the last fifteen years [16–19,21]. Moreover, many studies conclude that the underlined topology of a complex network is somehow associated to the observed dynamics. Indeed, the dependence of the forwards scenarios of a system on the topology of the underlying graph is usually pointed out [13,14,17–19,21,22,26,52–57]. A mathematical model by a system of differential equations for mutualistic networks in Ecology was introduced in Bastolla et al. [22]. Since then, many studies have been focused on this model class, as they provide a precise analysis for a global approach to these complex phenomena. They are represented by bipartite graphs representing two kind of species (classified into two sets, plants and animals) and the cooperative links between the groups [17–19,39,53]. These works studied how the architecture of the network relates to biodiversity. In particular, under some conditions is observed that the more nestedness of the network, the more probability for a richer biodiversity [58]; on the other hand, it also depends on other properties of the associated graph, and cannot be considered as the only marker for a higher biodiversity [59]. Moreover, several studies suggest that this index may not play the important role in shaping the network dynamics as it was previously believed. E.g., Payrato ´et al. show that nestedness is actually an entropic consequence of the degree sequence of the mutualistic networks, and not an irreducibly macroscopic feature [60]. PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 4 / 21 We study in detail the dependence of structural stability on several graph measures characterizing the underlying network described by adjacency matrix A(Results). Our findings, based on a deep computational analysis of biological and synthetic networks, conclude that cooperation parameters play the key role in biodiversity different to other graph measures such as modularity. Fig 2. The evolution of four different scenarios defined over the same graph in the same state. A three-nodes graph is considered. A system of differential equations as (2) is defined for the three nodes. Here, γ ji = 0.21 in all cases). Below, the evolution in time (red lines) from the state (0.2, 0.2, 0.2) of the system, depending on the value of the α i parameters which affect the behaviour of the nodes of the graph but not to its connectivity. The starting point of the red trajectories is always the same initial point but the trajectories are quite different. The changes in the trajectories are governed by the different information structures (figures delimited by the blue lines) in each of the dynamical systems which determine the future scenarios of the system. In each case, the trajectory goes to a special point which is the global stable solution in the phase space. https://doi.org/10.1371/journal.pone.0267404.g002 PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 5 / 21 Materials and methods Attractors as information structures A full mathematical study of systems like (1) is developed in [49]. In particular, sufficient conditions for existence and uniqueness of solutions are provided, so defining a dynamical system {T(t)} t�0 for (1) which possesses a global attractor A. The phase space X will be the space in which the dynamics takes place; in our case X¼IRN. We define adynamical system on Xas a family of non-linear operators fSðtÞgt2IRþ, SðtÞ:X!X u2X;SðtÞu2X; which describes the forwards dynamics of each u2X. In our case, S(t)u 0 =u(t;u 0 ), the solution represents the solution of (1) at time twith initial condition u(0) = u 0 . The global attractor is the central concept in dynamical system theory, since it describes all the future scenarios of the associated given phenomena. It is defined as follows [29–32,34,61, 62]: A set A�Xis a global attractor for {S(t):t�0} if it is 1. compact, 2. invariant under {S(t): t�0}, i.e. SðtÞA¼Afor all t�0, and 3. attracts bounded subsets of Xunder {S(t):t�0}; that is, for all B�Xbounded distHðSðtÞB;AÞ≔sup b2B inf a2AðSðtÞb;aÞ ! t!1 0: Suppose Ain (3) belongs to class S w or is Lyapunov-stable [63], i.e., A2S w , in the sense that there exists a diagonal positive matrix Wsuch that WA +A T Wis negative definite. In this case the whole structure of the global attractor for Lotka-Volterra systems can be characterized [49, 54]. Indeed, it is known that the dynamics of (3) generates an attractor, which is a structured finite set of stationary points (or equilibria) for the system, for which there exists a globally stable stationary point. The right part of Fig 1 represents the attractor corresponding to the graph on the left with the given α i and γ ij parameters. Due to the informational nature of a global attractor, this attractor characterization has been defined as information structure (IS) in [28]. The information structure for (1) not only informs on all the stationary points of the system, but the way they are connected, showing a precise hieralchical structure by levels of information (see Fig 1 and [28,33]). Global structural stability Under the hypotheses of Ain (3) to be Lyapunov-stable, it is known that there exists a unique global asymptotically stable stationary point [36]. But there also exists a huge set (at most 2 N ) of actual stationary points which are determining the transient dynamics, describing the closeness to phase transitions between different scenarios of biodiversity. This information is contained in the IS described above. Convex cone partition of IRN Let us introduce the precise definitions related to global structural stability: let D= {1, . . .,n}, I �Dand J=DnI. Let Aa Lypaunov-stable matrix and B .j =−A .j for j2Jand B .j =−I. j (negative PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 6 / 21 of identity matrix) for j2I, where B .j the j-column of matrix B. Define the convex cone as posðB:1;. . . ;B:NÞ ¼ fa2IRN:a¼r1B:1þ. . . þrnB:N;ri�0g:ð5Þ Then, for any α2pos(B. 1 ,. . .,B. N ), the unique globally stable stationary solution of (3) u� J¼ fu�ð1Þ;...;u�ðNÞgsatisfies [36]: fj2D=u�ðjÞ>0g ¼ J;and fi2D=u�ðiÞ ¼ 0g ¼ I: This is an important result as, given any possible stationary point of the system, there exists an associated convex cone as described in pos(B. 1 ,. . .,B. N ) such that, when α2pos(B. 1 ,. . ., B. N ), this stationary point is globally asymptotically stable [36]. For instance, if we consider a 4D Lotka-Volterra system, and J= {1, 2} (so that I= {3, 4},) the portion (i.e., the cone C J ) of the IR4space for parameter αassuring that the global asymptotic stationary point is of the form u� Jis given by CJ¼ fa2IR4:a¼r1ð A:1Þþr2ð A:2Þþr3ð I:3Þþr4ð I:4Þ;ri>0g: Even more interesting, the sixteen possible cones (2 4 )C J , for all possible J−choices, form a partition of IR4, i.e., the union of cones fills all the space and there is no intersection between their interiors. Fig 3 shows an example of the calculation for a graph of five nodes (n 1 ,n 2 at the left and n 3 , n 4 ,n 5 at the right). Competition (dashed lines) is assumed between every pair of nodes on the same side of the graph and cooperation (solid lines) exists when there is a link n i $n j joining nodes of different sides. Note that when considering cooperation relationships is a bipartite Fig 3. The cone of maximal biodiversity and structural stability. Left: Example (2 + 3)-bipartite graph with two nodes on the left and three nodes on the right. Competition (dashed lines) is between all elements on the same side, while cooperation occurs between elements on different sides for which there is an edge represented by a solid line. Cooperative relationships form a bipartite graph with two groups of nodes (left and right). Top right: Connectivity matrix −Mof the system. Competition parameters β ij >0 are set for all pairs of nodes n i ,n j in the same group (both at the left or the right). Cooperation parameters γ ij >0 exist for nodes n i ,n j in different groups (one in the left and the other in the right) only when there is an arrow n i $n j in the graph. In our experiments β ij =β ji and γ ij =γ ji . Bottom right: Equation to determine if a given point a �¼ ða1;...;a5Þ 2 IR5is in the maximal biodiversity cone. If there exist some r i �0, 1 �i�5, which verify (7), then a �is in the maximal biodiversity cone of M. To ensure the existence of a solution, the sum of the absolute values of each row or column of M(including the 1 in the diagonal) must be always lower than 2. This is equivalent to bound to 1 the sum of weights of all edges adjacent to each node of the graph (node degree bounded to 1). Given M,structural stability is defined as the proportion of points a �2IR5for which (7), has a solution. Since the cones are centred at 0, considered points can be limited to those on the surface of the IR5sphere of radius 1 centred at 0. https://doi.org/10.1371/journal.pone.0267404.g003 PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 7 / 21 graph. In general, β ij is not necessarily equal to β ji and the same for γ ij and γ ji . Matrix Min (6) (note that, in our case M=−A) contains all connectivity parameters. Observe that the diagonal is 1 and 0 represents no interaction between nodes. Given M, a certain a �2IRNis in the maximal biodiversity cone when there exist r 1 ,r 2 ,. . .,r N �0 verifying (7). Structural stability for M is defined as the proportion of a �2IRNin the maximal biodiversity cone; i.e., the structural stability of Mis equal to the proportion of points of the IRNsphere of radius 1 centred at 0 in the maximal biodiversity cone. Results Global structural stability In this section, we study the structural stability of each stationary point in the system, independently of their stability properties. This is referred as Global Structural Stability. For (3), the zero solution is globally unstable, each stationary u� jpoint belongs to an informational level E i and possesses stable and unstable directions, and there exists just one stationary feasible point u�in the lower level which is globally stable (see Fig 1). To illustrate the description of global structural stability, consider a two-dimensional cooperative system given by _ u1¼u1ða1u1þau2Þ _ u2¼u2ða2u2þbu1Þ (ð8Þ with ai2IR and a,b>0. For a fixed network of connections in the system (given by values of the aand bparameters), the intrinsic growth rate of each species plays a crucial role. Indeed, a convex cone of αparameters in (8) is associated to these stationary points, and all of these convex cones form a partition of IR2[50], i.e., each cone has a nonvoid interior, the union of all cones is IR2and each pair of the interior of cones is disjoint (see Fig 4). This means that a given vector αof (8) belongs either to just the interior of one cone (determining the feasible stationary point, and so the future biodiversity of the system) or to the intersection of cones, made by rich manifolds showing a phase transition and a high sensibility to bifurcation scenarios in biodiversity. Fig 4. Description of cones for alpha parameters associated to 8with 2 ×2 matrices as indicated. A. Competitive case. B. Cooperative case. Note that there are four regions C ij ,i,j= 0, 1, each for the four possible stationary points. If α 2C ij , the globally asymptotically stable point u�for (8) has the positive components pointed by ij, i.e., α2C 11 means that in u� 1;u� 2>0:Note that borders of each cone are bifurcation lines, in the sense that a sudden attractor bifurcation occurs when passing through this border. Moreover, inside each cone, there exist interesting zones marking different attractor structures with the same globally asymptotically stable solution, which is also crucial for the study of the structural stability. https://doi.org/10.1371/journal.pone.0267404.g004 PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 8 / 21 The global structural stability of a system allows to study bifurcation and transitions between different biodiversity scenarios. Indeed, the borders, which are now mathematically well defined, of each cone are critical zones for the sudden transition to one biodiversity scenario to a different one. Moreover, we can also observe in a global way the dependence of the cone partition on the parameters of the system (see Fig 5). In Fig 6 we show the different cones for a three dimensional Lotka-Volterra system. Note, once more, that the union of cones forms a partition of IR3. Fig 5. Evolution of global structural stability on parameters on a 3D LV system. A. We observe the evolution of the size of the cones when changing parameter γ 23 from competition (0.4) to cooperation values (−0.4). We observe that cones of maximal biodiversity (u 111 ) and the cone associated to stationary point u 011 behaves monotonically increasing with γ 23 . B. The same result, now simultaneously changing γ 23 and γ 32 . Note that the cones for u 011 and u 111 now grow faster, while other cones with constant size now decreases (as those associated to u 010 and u 110 ). C It is shown the Global Structural Stability from a competitive system to a cooperative one, by changing all the parameters in matrix A. Note that, among all, it is the cone with maximal biodiversity the only one increasing with an income of cooperation in the system. https://doi.org/10.1371/journal.pone.0267404.g005 Fig 6. Two representations of the eight cones describing global structural stability. A. a 3D competitive LV system, with the cone of maximal biodiversity (dark blue). B. a 3D cooperative LV system, with a bigger cone of maximal biodiversity, pointing out the key role of cooperation in biodiversity. https://doi.org/10.1371/journal.pone.0267404.g006 PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 9 / 21 way they are joined. All this information is usually belittled in many research contributions. ISs are described in our Lotka-Volterra system with precision, so allowing for further studies on robustness, bifurcation phenomena or metastability of solutions, in which the role of the associated informational field may be crucial [28,33,62,72,73]. The concept of structural stability is used in [22], and defined as in the present paper in [39, 74]. In this work we have introduced a global framework to study the structural stability for every possible stationary point of a mutualistic system. It is very important to determine the robustness of each asymptotic regime, measured by the region that associated parameters reach. In this sense, we have introduced an IRN-partition for the α-parameters describing the different convex regions for each stationary solution, which is moreover globally asymptotically stable in these regions. To our knowledge, this is the first time structural stability is used to study all the possible future scenarios, and not only to determine maximal biodiversity. We are aware we have not taken all the important information from the existence of an information structure. Indeed, there are many possible configurations possessing the same global asymptotic stable stationary solution (see for instance Fig 1 in which the set of semistable stationary points above the last asymptotically stable one could be very different), and would deserve further research. Fig 11. Optimal modularity and structural stability. Left (top to bottom): Comparison of M_SD_20 with GB,GC and GD. Right (top to bottom): Comparison of M_SD_50 with EB,EC and ED. Spearman correlation coefficient is lower than −0.057 in all these cases except M_SD_20 (Table 2). It suggest that the formation of strongly interrelated communities within a biological system has a negative influence on maximal biological diversity. https://doi.org/10.1371/journal.pone.0267404.g011 PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 16 / 21 The level of the interdependence between structure and dynamics on complex networks is not always clear; sometimes it seems that this relation induces determination, and usually just correlation in most cases. In this paper we conclude that dynamics, although closely related, is mainly determined by the signed sum of the degrees of the net and not for other parameters of the topology of the network. To study the dependence of the topology and its associated dynamics we have introduced an N-dimensional Lotka-Volterra system of differential equations. Our results also suggest that optimal modularity has a negative impact on biological diversity (structural stability), but in a smoother way that the positive influence of the sum of cooperation γ ij values. That is, the formation of strongly interrelated communities within a biological system has a negative influence on maximal biological diversity. More conclusive results on modularity may require more extensive studies, maybe normalizing the connectivity matrix of the networks to a constant centrality degree or constant sum of cooperation coefficients. That way, the influence of modularity could be better estimated. Similar studies have been done in [39,43,60], and they have shown that some network properties, such as nestedness, are also a secondary process (not a core process) shaping coexistence. We conjecture that competition/antagonism β ij coefficients have negative influence on maximal biological diversity, but more extensive experiments could also be required to confirm this claim. Acknowledgments Authors thank Prof. Francisco J. Esteban, at the Faculty of Biology at Jaen University (Spain) for their useful suggestions to improve a previous version of this paper. We also want to thank the Computational Center at the Computer Engineering High Technical School at Seville University and Jeśus Cano for technical asistence. Author Contributions Conceptualization: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Data curation: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Formal analysis: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Funding acquisition: Jose ´A. Langa. Investigation: Jose ´R. Portillo, Fernando Soler-Toscano. Methodology: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Project administration: Jose ´A. Langa. Resources: Jose ´A. Langa. Software: Jose ´R. Portillo, Fernando Soler-Toscano. Supervision: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Validation: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Visualization: Jose ´R. Portillo, Fernando Soler-Toscano. Writing – original draft: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. Writing – review & editing: Jose ´R. Portillo, Fernando Soler-Toscano, Jose ´A. Langa. PLOS ONE Global structural stability and the role of cooperation in mutualistic systems PLOS ONE | https://doi.org/10.1371/journal.pone.0267404 April 19, 2022 17 / 21 References 1. Greene D, Doyle D, Cunningham P. Tracking the Evolution of Communities in Dynamic Social Networks. In: Proceedings of the 2010 International Conference on Advances in Social Networks Analysis and Mining. ASONAM’10. Washington, DC, USA: IEEE Computer Society; 2010. p. 176–183. Available from: http://dx.doi.org/10.1109/ASONAM.2010.17. 2. Markovitch O, Krasnogor N. 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