Structural stability of invasion graphs for Lotka–Volterra systems

Journal of Mathematical Biology (2024) 88:64 https://doi.org/10.1007/s00285-024-02087-8 Mathematical Biology Structural stability of invasion graphs for Lotka–Volterra systems Pablo Almaraz1,2 ·Piotr Kalita1,3 ·José A. Langa1· Fernando Soler–Toscano4 Received: 4 October 2022 / Revised: 11 March 2024 / Accepted: 19 March 2024 / Published online: 17 April 2024 © The Author(s) 2024 Abstract In this paper, we study in detail the structure of the global attractor for the Lotka– Volterra system with a Volterra–Lyapunov stable structural matrix. We consider the invasion graph as recently introduced in Hofbauer and Schreiber (J Math Biol 85:54, 2022) and prove that its edges represent all the heteroclinic connections between the equilibria of the system. We also study the stability of this structure with respect to the perturbation of the problem parameters. This allows us to introduce a definition of structural stability in ecology in coherence with the classical mathematical concept where there exists a detailed geometrical structure, robust under perturbation, that governs the transient and asymptotic dynamics. Mathematics Subject Classification 34C37 ·37C70 ·34D45 ·92D25 BPiotr Kalita piotr[email protected] Pablo Almaraz [email protected] José A. Langa [email protected] Fernando Soler–Toscano [email protected] 1Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Campus Reina Mercedes, 41012 Sevilla, Spain 2Grupo de Oceanografía de Ecosistemas, Instituto de Ciencias Marinas de Andalucía (ICMAN-CSIC), Campus Universitario de Puerto Real, Puerto Real 11519, Spain 3Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland 4Departamento de Filosofía, Lógica y Filosofía de la Ciencia, Universidad de Sevilla, C/ Camillo José Cela, s/n, 41018 Sevilla, Spain 123 64 Page 2 of 25 P. Almaraz et al. 1 Introduction. Invasion graphs and ecological assembly The relations between populations of interacting species in ecosystems can be described by structured networks, where nodes represent species, and the edges represent the fact that the presence of one species affects another one Bascompte and Jordano (2014). In order to understand the behavior of the ecosystem, however, it is necessary to study the dynamics of the interactions between species, i.e. how their quantities vary in time in relation to each other. The classical study of ecological dynamical models has been focused in their asymptotic behavior May (1973), but what is actually observed during the evolution in time of real systems is the presence of transient states Hastings et al. (2018). These transient states are known to last for hundreds of generations in many natural systems in which stochasticity is an integral part of their dynamics Hastings et al. (2021), so a major goal in current theoretical ecology is to evaluate the impact of transient dynamics on the persistence of communities in a constantly changing environment Hastings et al. (2018). The interplay between transient and asymptotic dynamics is particularly important when we want to analyze the way in which communities assemble, or the invasion of one or several species to a given state of the ecosystem. The description of the sequence of both the potential assemblies or invasions (bottom-up), and the disassemblies or extinctions (top-down) is usually described by a network structure whose nodes are subcommunities and edges represent the possibility of evolution from one subcommunity to another Hang-Kwang and Pimm (1993). Full knowledge of such structure allows us to draw a complete landscape of all possible states in all possible times for the associated ecosystem. As it encompasses the essential information on the ecosystem, following our earlier terminology, we call it the informational structure (IS), cf. Esteban et al. (2018), Portillo et al. (2022). The IS is the key object to investigate for a deeper understanding of the dynamics of the system, as it encapsulates both the transient states and the asymptotic dynamics. The complete characterization of the IS gives the information on the mechanics of ecological assembly. Indeed, given the close connection between the IS and the concept of assembly or community transition graph traditionally used in ecology (Hang-Kwang and Pimm 1993; Morton et al. 1996; Serván and Allesina 2021), the IS gives a picture of the pattern of possible developments of the ecological community containing the species present in the ecosystem. If a model is a dissipative system of autonomous Ordinary Differential Equations which has a finite number of equilibria, then the underlying IS is contained in the structure of its global attractor. The nodes of IS correspond to the equilibria of the system and the edges represent the heteroclinic connections between them. In this paper, we focus on the Lotka–Volterra system of ODEs. While we choose this relatively simple model, there may exist many other factors affecting the evolution of an ecosystem, so that the modelling approach can include a variety of functional forms, from very basic ones to highly nonlinear vector fields including stochastic delays, or even higher order terms. The system under consideration here has the form u i=ui⎛ ⎝bi+ n  j=1 aijuj⎞ ⎠for i∈{1,...,n}. 123 Structural stability of invasion graphs... Page 3 of 25 64 where uiis the state variable for species i(e.g., population density or number of individuals); biis the intrinsic growth rate for species i; and aij is the direct effect of the average species jindividual on species i’s population growth rate Novak et al. (2016). We assume that the matrix A=(aij)n i,j=1is Volterra–Lyapunov stable (see Definition 3). For such a system, based on recent discoveries by Hofbauer and Schreiber (2022), we present an algorithm to construct the graph that represents the connections between the equilibria of the system, the IS, and we show that it is equivalent to the Invasion Graph (IG) as proposed in Hofbauer and Schreiber (2022). Thus, we complement the results of Hofbauer and Schreiber (2022) which states, in a more general framework, that the graph of connections is a subgraph of the IG, but the possibility that IG is essentially bigger is generally not excluded. We show that for a particular case of Lotka–Volterra system with Volterra–Lyapunov stable matrix the two structures coincide. In this way, we give a joint framework for the study of ecological assembly (Serván and Allesina 2021), Invasion Graphs (Hofbauer and Schreiber 2022) and Information Structures (Esteban et al. 2018; Portillo et al. 2022). We stress that our argument works only in the Volterra–Lyapunov stable case where the IG (and equivalently IS) is the directed graph and the results of Takeuchi (1996) allow to construct the unique minimal element, the globally asymptotically stable stationary point (GASS). While this assumption may be restrictive, the advantage is that we explicitly describe the structure of all connections between the equilibria. In the general case, the Lotka–Volterra systems may encompass many rich dynamic phenomena, such as limit cycles (Afraimovich et al. 2008), but the analytical algorithm to construct the whole dynamics for a general n-dimensional system is still unknown. The problem of structural stability is of a fundamental importance in biology: it concerns the question of whether the state of a system and its stability will survive upon a small perturbation of model parameters. Recently, Rohr et al. (2014) represented the structural stability of ecological networks as a problem of community persistence. Essentially, the aim is to provide a measure of the range of admissible perturbations to a system under which no interacting species become extinct, i.e. the community is feasible. Feasibility refers here to the existence of a saturated equilibrium vector, that is, given a particular combination of species interaction parameters and intrinsic growth rates (aij and biin (1), respectively) all of the abundances are strictly positive at the equilibrium. Thus, there is a connection between structural stability, as it is currently used in ecology, and the Modern Coexistence Theory (MCT) (Barabás et al. 2018), which aims at determining the number of species that can coexist in an ecosystem (Barabás et al. 2018). Invasion Graphs, as introduced by Hofbauer and Schreiber (2022), extends the concept of assembly graphs to the invasibility criteria of the MCT: the condition that a set of persisting interacting species should have positive per-capita growths rates when rare (Chesson 1994; Barabás et al. 2018). A novel contribution of our paper is to provide a link between Information Structures and Invasion Graphs through a measure of structural stability of global attractors that integrate both the transient and asymptotic dynamics. This achievement can be of paramount importance for a more detailed understanding of community coexistence and functioning in variable environments. 123 64 Page 4 of 25 P. Almaraz et al. Inspired by these considerations, and by the study on the stability of the global attractor structure (Bortolan et al. 2022), we show that, not only the stable equilibrium but also the whole assembly remains unchanged upon a small perturbation of model parameters. This result is interesting from a mathematical point of view as we get a result on structural stability for a problem which is not necessarily Morse–Smale, contrary to many classical structural stability results (see Bortolan et al. (2022) and references therein). On the other hand, its interest from the point of view of ecology is that it links the concept of structural stability from Rohr et al. (2014) with ecological assembly (Serván and Allesina 2021) and invasion dynamics (Hofbauer and Schreiber 2022). Indeed, the notion of stability of all the assembly can be viewed as the refinement of the notion of the stability of the persistent equilibrium (see Rohr et al. (2014)), as it induces the decomposition of the stability cones for the latter case into the smaller cones of the stability of assemblies. The structure of the paper is as follows: in Sect.2we formulate the problem and summarize its basic properties; in particular, we recall the result of Takeuchi (1996) on the existence and characterization of a globally asymptotically stable steady state. The next Sect.3is devoted to local properties of the system: we explicitly linearize it around the equilibria and study the properties of this linearization. The first main result of the paper, which states that the IS coincides with the IG is contained in Sect. 4. The following Sect.5contains the second main result, on the problem of structural stability, and on the stability cones for the assembly. Finally, in the appendices, we show that the considered problem is not necessarily Morse–Smale, and we formulate the open questions for the cases which are not Volterra–Lyapunov stable. 2 Lotka–Volterra systems and their global attractors. In this section we introduce the Lotka–Volterra systems and, for Volterra–Lyapunov stable matrices in the governing equation, we formulate the results on the underlying dynamics. The key concept is the global attractor. This attractor contains the minimal invariant sets (in our case, the equilibria) and the complete trajectories joining them in a hierarchical way. In our case, each admissible equilibrium or stationary point describes a subcommunity of the system. If this admissible equilibrium has strictly positive components, it is also feasible. Equilibria are joined by complete trajectories, i.e., global solutions of the system defined for all t∈R.This structure encodes all possible stationary states of the system and the underlying backward and forward behavior of the dynamics via the heteroclinic connections. It is a directed graph, which has been defined as an information structure in Esteban et al. (2018), Kalita et al. (2019), Portillo et al. (2022), and it induces a landscape of the phase space defined as an informational field (Kalita et al. 2019). We start from definitions of classes of stable matrices. More information on them, as well as on the underlying dynamics of the associated Lotka–Volterra systems can be found in Hofbauer and Sigmund (1988,1998), Logofet (1993), Takeuchi (1996). Definition 1 A real matrix A∈Rn×nis stable if σ(A)⊂{λ∈C:Re λ<0}, where σ(A)is the spectrum of A. 123 Structural stability of invasion graphs... Page 5 of 25 64 Definition 2 A real matrix A∈Rn×nis D-stable if for every matrix D= diag{d1,...,dn}with di>0 for every ithe matrix DA is stable. Definition 3 A real matrix A∈Rn×nis Volterra–Lyapunov stable (VL-stable) if there exists a matrix H=diag{h1,...,hn}with hi>0 such that HA+ATHis negative definite (i.e. stable). Consider the following Lotka–Volterra system with Volterra–Lyapunov stable matrix A=(aij)n i,j=1and a vector b∈Rn. u i=Fi(u)=ui⎛ ⎝bi+ n  j=1 aijuj⎞ ⎠for i∈{1,...,n}.(1) Let n∈N. We denote C+={x=(x1,...,xn)∈Rn:xi≥0fori∈{1,...,n}}, and C+=int C+={x=(x1,...,xn)∈Rn:xi>0fori∈{1,...,n}}. Now let x=(x1,...,xn)∈C+.IfJ⊂{1,...,n}is a set of indices then we will use a notation CJ +={x∈C+xi>0fori∈J}. If x∈C+, then we denote J(x)={i∈{1,...,n}:xi>0}. Having such x∈C+, we have CJ(x) +={y∈C+yi>0fori∈J(x)}. We present a result on the system (1) from Takeuchi (1996). We will be first interested in its equilibria in C+. Clearly 0 =(0,...,0)∈Rnis one of them. If we choose the nonempty subset of indices J⊂{1,...,n},sayJ={i1,...,im}, then we will say that this set defines an admissible equilibrium if there exists a point x∈CJ +with xi=0fori/∈Jwhich is an equilibrium of (1). The statement will be made more precise with some auxiliary notation introduced with the next definition. Definition 4 Let A=(aij)n i,j=1∈Rn×nbe a matrix. If m<n, then m×mprincipal submatrix of Ais obtained by removing any n−mcolumns and n−mrows with the same indices from A, i.e. if 1 ≤i1<i2< ... < im≤n. The principal submatrix of Aassociated with the set J={i1,...,im}has a form A(J)=(A(J)jk)m j,k=1= (aijik)m j,k=1. 123 64 Page 6 of 25 P. Almaraz et al. Also, for a vector b∈Rnwe can associate with a set of indices J={i1,...,im} its subvector b(J)=(bij)m j=1. So, the set Jdefines a feasible equilibrium of the subsystem consisting only of the equations indexed by elements of Jand taking the variables outside Jas zero, if the solution of the system A(J)v =−b(J)has all coordinates strictly positive. We denote this solution by u∗(J). The associated admissible equilibrium of the original n-dimensional system is given by ui=0 for i/∈J, and uij=vjfor j∈{1,...,m}, i.e., ij∈J. We use the notation u∗=(u∗(J), 0i∈{1,...,n}\J). Since every subset of {1,...,n}can potentially define an admissible equilibrium, there may be maximally 2nof them (including zero), each of them determined uniquely by splitting {1,...,n}into the union of two disjoint subsets: the set Jon which the coordinates are strictly positive (this set defines the equilibrium) and the remainder on which they must be zero. It is not difficult to prove that for every subset of indices J⊂{1,...,n}the set {x∈C+xi=0forsomei∈J}=C+\CJ + is positively and negatively invariant with respect to the flow defined by (1). We recall the definition of the Linear Complementarity Problem (LCP). Given a matrix B∈Rn×nand a vector c∈Rnthe linear complementarity problem LCP(B,c) consists in finding a vector x∈Rnsuch that Bx +c≥0, x≥0, x(Bx +c)=0. If the matrix Ais Volterra–Lyapunov stable then the problem LCP(−A,−b)has a unique solution for every b∈Rn, cf. [Lemma 3.2.1 and Lemma 3.2.2] Takeuchi (1996). We cite the asymptotic stability result from Takeuchi (1996). Theorem 5 (Takeuchi (1996), Theorem 3.2.1) If A is Volterra–Lyapunov stable then for every b ∈Rnthere exists a unique equilibrium u∗∈C+of (1)which is globally asymptotically stable in the sense that for every u0∈CJ(u∗) +the solution u(t)of (1) with the initial data u0converges to u∗as time tends to infinity. This u∗is the unique solution of the linear complementarity problem LC P(−A,−b). In particular, if the solution u of the system Au=−b is positive, then u∗=u. We will denote this u∗as GASS (globally asymptotically stable stationary point). The following result is a straightforward consequence of the previous theorem. Corollary 6 If u∗is a GASS for the problem governed by (1), then for every set J = {j1,..., jk}⊂{1,...,n}, such that J(u∗)⊂J the point y ∈Rkdefined by yi=u∗ ji for i ∈{1,...,k}is a GASS for the k dimensional problem with A(J)and b(J). We present the definition of a global attractor (Hale 1988): 123 Structural stability of invasion graphs... Page 7 of 25 64 Definition 7 Let Xbe a metric space and let S(t):X→Xbe a semigroup of mappings parameterized by t≥0. The set A⊂Xis called a global attractor for {S(t)}t≥0if it is nonempty, compact, invariant (i.e. S(t)A=Afor every t≥0), and it attracts all bounded sets of X(i.e. if B⊂Xis nonempty and bounded then limt→∞ dist(S(t)B,A)=0, where dist(C,D)=supx∈Cinfy∈Dd(x,y)is the Hausdorff semidistance between sets C,D⊂X). If the mappings S(t):X→Xare continuous, for the global attractor existence we need two properties to hold: the dissipativity and asymptotic compactness Robinson (2001). As a consequence of Theorem 5we have the following result. Theorem 8 For every u0∈C+the problem governed by (1)has a unique solution which is a continuous function of time, and the initial data. Moreover, assuming the Volterra–Lyapunov stability of A, the problem has a global attractor. Proof The result follows the argument of Guerrero et al. (2017). We only need to prove the dissipativity, i.e. the existence of the bounded absorbing set; once we have it, the asymptotic compactness is trivial. To this end it is sufficient to prove that if n i=1uiwi≥Rfor Rlarge enough with some fixed weights wi>0, then d dt n  i=1 uiwi≤−D(R). Indeed defining |u|as n i=1uiwi, d dt |u|= d dt n  i=1 uiwi= n  i=1 uibiwi+ n  i,j=1 uiaijwiuj= n  i=1 uibiwi +1 2 n  i,j=1 ui(aijwi+ajiwj)uj≤c|u|−d|u|2. where c>0 and d>0 are some constants. Then if |u|≥ 2c d, then the right-hand side of the last expression is decreasing as a function of |u|, and d dt |u|≤2c2 d−d4c2 d2=− 2c2 d, which is enough for the global attractor existence.  3 Equilibria and the local dynamics While it is straightforward to find all the equilibria of (1) (it suffices to solve 2nlinear systems and determine the ones whose solutions are strictly positive, see also Lischke and Löffler (2017) for an efficient algorithm), finding the connections between them is a harder task. Our aim here is to give an algorithm that can be used to find exactly 123 64 Page 8 of 25 P. Almaraz et al. for which equilibria there exist heteroclinic connections, i.e. the solutions which tend to one equilibrium when time goes to minus infinity and another equilibrium when time goes to plus infinity. Before we move on to the study of the dynamics, we focus in this section on the local behavior in the neighborhood of the equilibria. 3.1 Linearization and its properties We construct the linearized system in the neighborhood of the equilibrium u∗of (1). Let u∗be an equilibrium and denote v=u−u∗. Then the system (1) can be rewritten as v i= n  j=1 ∂Fi(u∗) ∂uj vj+Gi(v), where Gi(v) =n j=1aijvjviis the quadratic remainder term. Assume that u∗is an equilibrium in which the variables are sorted in such a way that u∗ i= 0fori=1,...,k and u∗ i=0fori=k+1,...,n. Then for i=1,...,kthe equation of the above system is v i= k  j=1 vjaiju∗ i+ n  j=k+1 vjaiju∗ i+Gi(v), and, for i=k+1,...,n, v i=vi⎛ ⎝bi+ k  j=1 aiju∗ j⎞ ⎠+Gi(v), The linearized system has the following block diagonal form w=Bw=B11 B12 0B22w, (2) where the matrix B22 is diagonal and B22 ii =bi+k j=1aiju∗ j, while B11 ij =aiju∗ i, and B12 ij =aiju∗ i. We will name the subsets J⊂{1,...,n}corresponding to the admissible equilibria as admissible communities, according to the next definition. Definition 9 The set (community) I⊂{1,...,n}is admissible if there exists the nonnegative equilibrium u∗=(u∗ 1,...,u∗ n)of (1) with u∗ i>0 if and only if i∈I. The family of all admissible communities will be denoted by E⊂2{1,...,n}.The corresponding set of equilibria is denoted by E={u0,...,uK}.As0∈Ewe always denote u0=0. 123 Structural stability of invasion graphs... Page 9 of 25 64 Whenever we speak about the multiple equilibria we will denote them by upper indices such as ui,uj. On the other hand, lower indices will denote coordinates of vectors u=(u1,...,un). The following lemmas summarize the properties of the matrix of the linearized system. Note that similar observations were made in different context in Lischke and Löffler (2017). Lemma 10 Assume that the matrix A of the system (1)is Volterra–Lyapunov stable. Consider the admissible community I and the corresponding equilibrium u∗. The system linearized around u∗has the form (2). The spectrum of the matrix B11 is contained in the open half-plane with the negative real part, i.e. σ(B11)⊂{z∈C: Re z<0}. Proof Denote u∗=(u∗ 1,...,u∗ n). The matrix {aij}i,j∈Ias a principal submatrix of A is Volterra–Lyapunov stable, cf. Cross (1978, Theorem 1 c). Hence is is also D-stable by Takeuchi (1996, Lemma 3.2.1). This means that the product diag((u∗ i)i∈I)(aij)i,j∈I is stable. This product is exactly B11. We are in a position to formulate a result of the properties of the linearized system (2). Lemma 11 Let A be Volterra–Lyapunov stable and let u∗be an equilibrium with the admissible community {1,...,k}. The spectrum of the matrix B, denoted by σ(B) is given by σ(B)=σ(B11)∪where λ∈if and only if λ=B22 ii for some i∈{k+1,...,n}, and σ(B11)⊂{Re λ<0}. So, if for some λ∈σ(B)we have Re λ≥0, then λis real and λ=B22 ii for some i ∈{k+1,...,n}. The eigenvector associated with the eigenvalue B22 ii is given by x =(x1,...,xk,0,...,0,1,0,...,0), where 1is on the position i and (xj)k j=1is some vector. Proof The assertion that σ(B11)⊂{Re λ<0}follows from Lemma 10. Now we prove that x=(x1,...,xk,0,...,0,1,0,...,0)is the eigenvector associated with eigenvalue B22 ii . We need to have B11 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ x1 . . . xk ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ +B12 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 . 1 . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ =B22 ii ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ x1 . . . xk ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ . Such (x1,...,xk)can be found because the matrix B11 −B22 ii Iis invertible as B22 ii ≥0 and it cannot be the eigenvalue of the matrix B11 as all its eigenvalues have negative real part. Moreover 0 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ x1 . . . xk ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ +B22 ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 . 1 . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ =B22 ii ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 0 . 1 . 0 ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ , 123 64 Page 16 of 25 P. Almaraz et al. Fig. 2 The Invasion Graph for the system (3) is the case, for example, if the matrix Ais symmetric, cf. Sect. 7.2. The following example demonstrates that this does not always has to be the case. Example 26 Consider the following system of three ODEs representing the May– Leonard problem, cf. Chi et al. (1998), May and Leonard (1975). u 1=u1(1−u1−1.5u2−0.05u3), u 2=u2(1−0.05u1−u2−1.5u3), (3) u 3=u3(1−1.5u1−0.05u2−u3). The matrix of the above system is Volterra–Lyapunov stable, cf. Kraaijevanger (1991). The system has five equilibria: zero, three one species equilibria and one three species equilibrium. The graph of connections (and, equivalently IG and IS) for the above system is presented in Fig. 2. The graph does not represent the full dynamics of the system because inside the global attractor there exists a solution which converges forward in time to the three species equilibrium (represented by the node 123) and backward in time to the heteroclinic cycle connecting the three nodes 1, 2, and 3. Notably, in the above example the graph has a 3-cycle consisting of three heteroclinic connections. We leave open the following question Question 27 Assume that the matrix A is Volterra–Lyapunov stable and that all equilibria are hyperbolic. Can we assert that if the IG is acyclic, then the global attractor consists only of the equilibria and their heteroclinic connections, and hence, this graph represents the whole dynamics of the system? 5 Structural stability of invasion graphs 5.1 Local structural stability If the system is Morse–Smale, then it is also structurally stable, i.e. C1small perturbation of its vector field produces a system whose global attractor has the same structure 123 Structural stability of invasion graphs... Page 17 of 25 64 [see Bortolan et al. (2022, Theorem 2)]. In this section we show that, although the system governed by (1) is not necessarily Morse–Smale, cf. Example 34, if all equilibria are hyperbolic, the small perturbation of Aand bproduces a system with the same graph of connections (and, if the global attractor consists only of the equilibria and their connections, with the same global attractor structure). In the next result B(A,ε) denotes the euclidean ball in Rn×nand B(b,ε)in the euclidean ball in Rn. Moreover, denote by E(A,b)the set of admissible communities for the problem with matrix A and vector b.ForI∈E(A,b)and i/∈Iwe will use the notation rA,b i(I)to denote the invasion rate corresponding to A,b. Theorem 28 Let A be a Volterra–Lyapunov stable matrix and let b∈Rnbe such that for all admissible communities I ∈E(A,b)the corresponding equilibria are hyperbolic. Then there exists ε>0such that for all matrices A ∈B(A,ε) and all vectors b ∈B(b,ε)we have E(A,b)=E(A,b). Moreover for every I ∈E(A,b)and every i /∈I we have rA,b i(I)>0⇒rA,b i(I)>0, rA,b i(I)<0⇒rA,b i(I)<0. Hence, the edges in both Invasion Graphs for A,b and A,b are the same. This implies that the graphs of connections for the problems with A,b and A,b coincide and the problem governed by the matrix A and the vector b is structurally stable in the class of Volterra–Lyapunov stable matrices. Proof The fact that all equilibria are hyperbolic means that ri(I)= 0 for every i/∈ Iand every I∈E(A,b). Note that since the eigenvalues depend continuously on the matrix, the set of Volterra–Lyapunov stable matrices is open and hence we can choose εsuch that every A∈B(A,ε)is Volterra–Lyapunov stable. Now, the fact that I∈E(A,b)means that A(I)u∗(I)=−b(I)has a positive solution u∗(I). From that fact that the mapping (A,b)→ u∗, which assigns to a nonsingular k×kmatrix A and vector b∈Rkthe solution u∗of the system Au∗=−b, which is continuous, we deduce that we can find ε>0 such that all admissible communities remain admissible. We prove that a nonadmissible community for (A,b)cannot produce an admissible one upon sufficiently small perturbation. Assume that I⊂{1,...,n}is not admissible. If at least one of the coordinates of the solution of the system A(I)u∗(I)=−b(I)is negative, then this negativity is preserved upon small perturbation of (A,b).Ifallare nonnegative, but at least one is zero, say u∗ j=0, then  i∈I,i= j akiu∗ i+bk=0 for every k∈I. In particular  i∈I\{ j} ajiu∗ i+bj=0. 123 64 Page 18 of 25 P. Almaraz et al. Denote by I0⊂Ithe (possibly empty) set of coordinates for which entries of u∗ are positive. Then I0corresponds to the admissible community for (A,b).Thelast equality means that rA,b j(I0)=0, which contradicts the assumption of hyperbolicity. We have proved that E(A,b)=E(A,b). The invasion rates ri(I)are the continuous functions of the vector b,matrixAand the equilibrium u∗related to the admissible community I. This means that if ri(I)is nonzero for the system governed by (A,b), it remains nonzero, and does not change sign, in a small neighbourhood of (A,b). This completes the proof.  5.2 The regions of structural stability. In this section we fix the matrix Aand we study the properties of the sets of vectors b∈Rnfor which the Invasion Graphs remain unchanged. Theorem 29 Let A be a Volterra–Lyapunov stable matrix and let b∈Rnbe such that for all admissible communities I ∈E(A,b)the corresponding equilibria are hyperbolic. Then there exists the unique maximal open neighbourhood Nof binRn such that for every b ∈Nwe have E(A,b)=E(A,b), all admissible equilibria corresponding to A,b are hyperbolic, and the Invasion Graphs coincide. Proof From Theorem 28 we know that there exists an open neighborhood of bsuch that the properties required by the theorem are satisfied. Let us denote by N(b)the family of all such neighbourhoods. It is nonempty. Then Nis the union of all elements of N(b). We continue by proving the lemma on convexity Lemma 30 Let A be Volterra–Lyapunov stable. Suppose that b1,b2∈Rnare such that •E(A,b1)=E(A,b2)=E, •for every I ∈Eand for every i /∈I we have r A,b1 i(I)= 0,rA,b2 i(I)= 0, and rA,b1 i(I)>0⇔rA,b2 i(I)>0. Then for every λ∈[0,1], denoting bλ=λb1+(1−λ)b2, we have •E(A,bλ)=E, •for every I ∈Eand for every i /∈I we have r A,bλ i(I)= 0, and r A,b1 i(I)>0⇔ rA,bλ i(I)>0. Proof Assume that I∈E. Then A(I)u∗ 1=−b1(I)and A(I)u∗ 2=−b2(I).This means that A(I)(λu∗ 1+(1−λ)u∗ 2)=−bλ(I)and Iis admissible for bλ.Onthe other hand, assume that Iis admissible for bλ, i.e. I∈E(A,bλ)but I/∈E. Restrict the system to those unknowns which correspond to the indices of I. The fact that I is admissible for the problem with bλmeans that this system has the strictly positive equilibrium, i.e. the solution vof A(I)v =−bλ(I)has all coordinates strictly positive, and by Theorem 5it has to be the GASS of the system with A(I), bλ(I), and the solution of LCP(−A(I), −bλ(I)). On the other hand, as I/∈E, the solutions u∗ 1and 123 Structural stability of invasion graphs... Page 19 of 25 64 u∗ 2of the problems LCP(−A(I), −b1(I)) and LCP(−A(I), −b2(I)), cannot have all coordinates strictly positive, and, because the invasion schemes IS for b1(I)and b2(I)coincide, the indices of zero and nonzero coordinates in both u∗ 1and u∗ 2are the same. Then λu∗ 1+(1−λ)u∗ 2must solve LCP(−A(I), −bλ(I)) and hence it must be a GASS for the problem with bλ(I), a contradiction with the fact that this GASS has all coordinates strictly positive. A straightforward calculation shows that for I∈E rA,bλ i(I)=λrA,b1 i(I)+(1−λ)rA,b2 i(I), which is sufficient to complete the proof of the Lemma.  Remark 31 Note that, by Lemma 11 the assumption that all invasion rates ri(I)are nonzero for i/∈Iis equivalent to saying that the admissible equilibrium corresponding to Iis hyperbolic. The next theorem states that the maximal neighbourhoods of Theorem 29 are convex cones and they group all points with a given invasion scheme IS, i.e. the given configuration of equilibria and signs of invasion rates. Theorem 32 The maximal neighbourhood Nof b given in Theorem 29 is an open and convex cone. Moreover, if for some point b ∈Rnwith all admissible equilibria being hyperbolic the invasion schemes for A,b and A,b are the same, then b ∈N. Proof We first prove the second assertion. Take b∈Rnsatisfying the assumptions of the theorem. By Lemma 30 the same assumptions are satisfied by every bλ∈ {λb+(1−λ)b:λ∈[0,1]}. By Theorem 28 for each λ∈[0,1]there exists an open neighborhood of bλon which the same assumptions also hold. The union of these neighborhoods is an open neighborhood of bwhich must be contained in Nand contains b. Now, convexity of Nfollows from Lemma 30. To prove that Nis a cone it is sufficient to see that A(I)(u∗)=−b(I)⇒A(I)(αu∗)=−αb(I)and rA,αb i(I)= αrA,b i(I). As a consequence of the above results we can represent the space Rnas a union of finite number of disjoint open convex cones Nk, with each cone corresponding to a given structure of the Invasion Graph, or equivalently, to a given IS.ThisIS is the same for every bin the cone. Note that the number of cones is finite as the number of possible invasion schemes is finite and any two vectors bwhich yield the same scheme must belong to the same cone. The points of nonhyperbolicity (that is, vectors bwhere at least one of the admissible equilibria is nonhyperbolic) constitute the residual set C. Rn= L  k=1 Nk∪C. 123 64 Page 20 of 25 P. Almaraz et al. The following statement holds b∈C⇔there exists I⊂{1,...,n}and i/∈I, such that ri(I)=0 and u∗(I)>0,where A(I)u∗(I)=−b(I). In other words, denoting by (A(I)−1)ij =aij(I)−1the entries of the inverse matrix to A(I). b∈C⇔there exists I⊂{1,...,n}and i/∈I,such that A(I)−1b(I)<0 and bi− k∈I j∈I aijajk(I)−1bk=0. This means that C⊂ I⊂{1,...,n} i∈{1,...,n}\I ⎧ ⎨ ⎩ b∈Rn:bi− k∈I j∈I aijajk(I)−1bk=0⎫ ⎬ ⎭ , i.e. the set of points of nonhyperbolicity is a subset of the union of a finite number of n−1 dimensional hyperspaces in Rn. In particular, Cis “small” compared to the sets Nk. 6 Appendix A. The dynamical system generated by (1)isnot Morse–Smale We begin this short section with a definition of a Morse–Smale system. We do not recall here all the necessary concepts: we refer, for example, to Bortolan et al. (2022, Section 2.1) for details on all notions presented in this chapter. Note that related definition in Bortolan et al. (2022) is more general: it allows for existence of periodic orbits. We present its simplified version only for gradient-like systems. Definition 33 Let Xbe a Banach space and let S(t):X→Xfor t≥0beaC1 reversible semigroup with a global attractor A⊂X. We denote the set of equilibria of {S(t)}t≥0as E, i.e. E={u∈X:S(t)u=u}for every t≥0. The semigroup is Morse–Smale if •The global attractor consists of the equilibria E, and nonconstant trajectories γ: R→Xsuch that limt→−∞ γ(t)=u∗ 1and limt→∞ γ(t)=u∗ 2where u∗ 1,u∗ 2∈E. •The set Eis finite and all equilibria in Eare hyperbolic. •If z∈Ais a nonequilibrium point such that limt→−∞ S(t)z=u∗ 1and limt→∞ S(t)z=u∗ 2, then the unstable manifold of u∗ 1and stable manifold of u∗ 2intersect transversally at every point zof intersection, that is the sum of their tangent spaces at zspan the whole space X:Tz(Wu(u∗ 1)) +Tz(Ws(u∗ 2)) =X. The Lotka–Volterra system (1) is defined on the closed positive coneC+. If a dynamical system is defined on a manifold M, then the transversality condition has the form 123 Structural stability of invasion graphs... Page 21 of 25 64 Fig. 3 Invasion graph for the example given by (34)foran ecological community with three species. The GASS represents a feasible community (all species are present), u∗= (0.2633778,0.1695335,0.377100). While the connections {3}→{1,3}and {1,3}→{1,2,3}are present in the graph, there is no connection {3}→{1,2,3} Tz(Wu(u∗ 1)) +Tz(Ws(u∗ 2)) =TzM, i.e. the tangent spaces of the stable and unstable manifolds span the tangent space of the whole M. It is possible to generalize the concept of Morse–Smale semigroups to manifolds with boundary, cf. Labarca and Pacifico (1990), Prishlyak et al. (2023), Robinson (1980). However, instead of taking this path, we extend the system to the whole Rnand we provide a simple example that the resultant system is not necessarily Morse–Smale. Hence, while the structural stability results are known to hold for Morse–Smale systems, cf., for example, Bortolan et al. (2022), our Theorem 28 is a structural stability result (in positive cone) beyond this class. Example 34 Consider the system u 1=u1(−u1+0.08u2−0.47u3+0.43), u 2=u2(0.66u1−u2+0.12u3−0.05), u 3=u3(0.56u1−0.28u2−u3+0.28). (4) Two of its admissible communities are {3}and {1,3}with the corresponding equilibria u∗ 1=(0,0,0.28)and u∗ 2=(0.2362255,0,0.4122863). The matrix A,asit is diagonally dominant, is Volterra–Lyapunov stable. The graph of connections is presented in Fig. 3, and contains the connection {3}→{1,3}. Both the unstable manifold of {3}and the stable manifold of {1,3}are contained in the {1,3}plane. In fact Wu({3})⊂Ws({1,3}), with Ws({1,3})being two-dimensional, and hence the sum of their tangent spaces cannot span the whole space. Also, note that if we set u2=0 and restrict the system to u1,u3variables only, for the resulting two dimensional system u 1=u1(−u1−0.47u3+0.43), u 3=u3(0.56u1−u3+0.28), (5) 123 64 Page 22 of 25 P. Almaraz et al. the same intersection, which was non-transversal in the 3D problem, becomes transversal, and the system is now Morse–Smale. 7 Appendix B. Questions and open problems 7.1 Wider classes of stable matrices The first question that we pose is related with the fact that in Rohr et al. (2014)the authors conjecture about stability of admissible equilibria for more general class of matrices A- D-stable ones and stable ones. As our result on the graph of connections relies on Theorem 5which uses the logarithmic Lyapunov function valid only in the class of Volterra–Lyapunov stable matrices A, it remains open to see if is holds in those wider classes. Question 35 Characterize the class of matrices for which the Invasion Graph (IG) corresponds to the graph of connections. Is this class essentially bigger then Volterra– Lyapunov stable ones? How does it relate to weaker notions of stable matrices such as, for instance, D-stable ones? 7.2 Symmetric case If the matrix Ais symmetric then the following function, as proposed by MacArthur MacArthur (1969), is Lyapunov: V(u)=− n  i=1 biui−1 2 n  i,j=1 aijuiuj. Indeed, after calculations we get, d(V(u(t))) dt =V(u)u(t)=− n  i=1 ⎛ ⎝bi+ n  j=1 aijuj⎞ ⎠ 2 ui.(6) If we assume that A, together with all its principal minors are nonsingular, then the problem (1) has the finite number of admissible equilibria which can all be explicitly calculated and one can construct the IG with the vertexes being exactly the equilibria. So for the case of symmetric and stable matrix A(for symmetric matrices stability and Volterra–Lyapunov stability are the same), the IG exactly corresponds to the structure of the global attractor. Again, the open question appears. Question 36 Does the IG correspond to the structure of a global attractor for a not necessarily stable but symmetric matrix A which, together with its all principal minors, is nonsingular? While we do not know how to answer this question we show that for symmetric case the Lyapunov function Vdrops along every edge in the IG. While this fact does 123 Structural stability of invasion graphs... Page 23 of 25 64 not guarantee the existence of the connection between the equilibria, this shows that the criterium associated with the Lyapunov function Vcannot exclude the edges in IG. Lemma 37 Let A be symmetric such that together with all its principal minors it is nonsingular and let u1and u2be admissible equilibria of (1) which correspond to the communities I1,I2. If there exists an edge I1→I2in IG then V (u1)>V(u2). Proof We may assume without loss of generality that I1∪I2=Rn. Otherwise we remove from the system the equations which correspond to the variables outside I1∪I2. We represent Rn=RI1\I2∪RI1∩I2∪RI2\I1, and we denote the projections on three subspaces as 1, 2, 3. Then, the matrix Aof the system can be written as A=⎛ ⎝ BCD CEF DFG ⎞ ⎠. Now as u1is equilibrium related with I1, hence 3u1=0, B1u1+C2u1=−1b, and C1u1+E2u1=−2b. Moreover, as the invasion rates at u1must be positive, it follows that D1u1+F2u1>−3b. Similar analysis at u2yields 1u2=0, E2u2+F3u2=−2b, and F2u2+G3u2=−3b. Finally as invasion rates at u2are negative we have C2u2+D3u2<−1b.Itfollowsthat (3u2)D1u1+(3u2)F2u1>−(3u2)3b, (1u1)C2u2+(1u1)D3u2<−(1u1)1b. Combining the two above inequalities we deduce (1u1)C2u2+(1u1)1b<( 3u2)F2u1+(3u2)3b. But (2u2)C1u1+(2u2)E2u1=−(2u2)2b, (2u1)E2u2+(2u1)F3u2=−(2u1)2b. Hence −(2u2)2b−(2u2)E2u1+(1u1)1b<−(2u1)2b −(2u1)E2u2+(3u2)3b. As Eis symmetric this means that −(u2)b<−(u1)b, which exactly implies the assertion as at equilibrium V(u)=− 1 2ub. 123 64 Page 24 of 25 P. Almaraz et al. Acknowledgements The authors have no Conflict of interest to declare that are relevant to the content of this article. We thank two anonymous referees for their valuable remarks which largely contributed to the final shape of the manuscript. The work of JAL, PA, and FST was partially supported by Proyectos Fondo Europeo de Desarrollo Regional (FEDER) and Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía, by Programa Operativo FEDER 2014-2020 reference P20-00592. The work of JAL was also partially supported by FEDER Ministerio de Economía, Industria y Competitividad grant PGC2018-096540-B-I00, and Proyectos Fondo Europeo de Desarrollo Regional (FEDER) and Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía, by Programa Operativo FEDER 2014-2020 reference US-1254251. The work of JAL and PK was supported by Ministerio de Ciencia e Innovación of Kingdom of Spain under project No. PID2021-122991NB-C21. The work of PK was also supported by Polish National Agency for Academic Exchange (NAWA) under Bekker Programme, project number PPN/BEK/2020/1/00265, as well as by National Science Centre (NCN) of Poland under Projects No. DEC-2017/25/B/ST1/00302 and UMO-2016/22/A/ST1/00077. 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