Sustainable network dynamics

Ecological Modelling 270 (2013) 43–53

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Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel

Sustainable network dynamics Arnaud Z. Dragicevic a,b,c,∗ , Bernard Sinclair-Desgagné d,e a

Agro ParisTech – Laboratoire d’Économie Forestière, 54042 Nancy Cedex, France INRA – Laboratoire d’Économie Forestière (UMR 356), 54000 Nancy, France c Chaire Forêts pour Demain (Agro ParisTech – Office National des Forêts), 54042 Nancy Cedex, France d HEC Montréal – Département des Affaires Internationales, Montréal, QC H3T 2A7, Canada e École Polytechnique ParisTech – Département d’Économie, 91128 Palaiseau Cedex, France b

a r t i c l e

i n f o

Article history: Received 10 June 2013 Received in revised form 10 September 2013 Accepted 12 September 2013 Available online 13 October 2013 Keywords: Bioeconomics Ecosystem-based management Forests Graph theory Connectedness

a b s t r a c t We propose a dynamic graph-theoretic model for ecosystem management as a control over networked system composed of target nodes and unmarked nodes. The network is represented by a complete graph, in which all vertices are connected by a unique edge. Target nodes are attracted by the objective function issued from the ecosystem-based management. They pull the network toward the objective position. The management policy is considered successful if the graph remains connected in time, that is, target nodes attain their objectives and unmarked nodes stay in the convex hull. At the time of the ecosystem network transfer, the model yields a separation theorem as well as a sustainability criterion to maintain full connectivity of the ecosystem. Our definition of sustainability can be easily linked to the general definition of sustainability as ecosystem integrity preservation. At last, we identify three management rules, dependent on the nature of connections, to ensure the maintenance of connectivity in time. The necessary and sufficient conditions for connectivity are (1) synchronous time-delays; (2) coequal magnitudes of increasing objective functions; and (3) nonnegativity of the objective function. © 2013 Elsevier B.V. All rights reserved.

1. Introduction “Ecosystem management is driven by explicit goals, executed by policies, protocols, and practices, and made adaptable by monitoring and research based on our best understanding of the ecological interactions and processes necessary to sustain ecosystem composition, structure, and function” (Christensen et al., 1996). It aims at maintaining the ecosystem health and integrity, given the numerous goods and services they provide. Health is the capacity of ecosystems to maintain their functions (Constanza et al., 1992). Integrity consists in maintaining ecosystem’s self-organizing structural complexity (Callicott, 1993). Ecosystems are connected networks (Patten, 1981; Higashi and Patten, 1989). They are regulated across many control variables in interactive networks. Studies of ecosystems as networked systems have recently been proposed (Strogatz, 2001; Fath, 2004; Jordan and Scheuring, 2004; Fath and Grant, 2007; Lanzen, 2007) but these studies do not focus on the matter of conserving connectivity between the ecosystem elements. And yet, the importance of interconnections within ecosystem networks is one of the most

∗ Corresponding author at: Agro ParisTech – Laboratoire d’Économie Forestière, 14 rue Girardet, 54042 Nancy Cedex, France. Tel.: +33 3 83 39 68 96; fax: +33 3 83 37 36 45. E-mail address: [email protected] (A.Z. Dragicevic). 0304-3800/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2013.09.003

important lessons learned from ecological research and natural resource management (Peterson, 1993). From the ecological perspective, high connectivity implies much interaction of animals, plants, energy, water, nutrients or other matter among elements (Cantwell and Forman, 1993). The network analysis enables to represent these organisms and exchanges as a collection of storages and flows (Fath and Patten, 1998). Ecological systems are also dynamic systems because their change is ubiquitous, so sustainability does not imply maintenance of the status quo. Besides, ecosystem status quo usually leads to failure in the long term (Connell and Sousa, 1983). Following the literature on controllability and stability of leaderfollower networks (Ji et al., 2008; Gustavi et al., 2010; Mesbahi and Egerstedt, 2010), we consider the ecosystem-based management as a control over networked system composed of target nodes and unmarked nodes. Target nodes can be biodiversity elements that need to be managed in order to achieve biodiversity goals. Unmarked nodes are all the other elements indirectly incorporated in the policy, for they are connected to target nodes. The network is represented by a complete graph, in which all vertices are connected by a unique edge. Target nodes are attracted by the objective function coming from the ecosystem-based management. They pull the network toward the objective position. The management policy must avoid disconnecting the ecosystem, which would cause its vulnerability. Therefore, the management policy is considered successful if the graph remains connected in

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time, that is, target nodes attain the objective and unmarked nodes remain within the graph. In graph-theoretic literature, Pichai et al. (1981) previously developed procedures to identify the minimal sets of connections which are essential for preserving the structural properties of the system. Given that a precise number of species to maintain key ecosystem processes is futile (Christensen et al., 1996), we cannot reduce the ecosystem networks to their minimal sets. Instead, we study their general connectivity. Since no formal model about the connectivity issue in the ecosystem-based management has been proposed so far, this work is a response to this need. We propose a dynamic graph-theoretic model, in which the evolution of the ecosystem is subject to time-delays. If time-delays occur and asynchrony in the evolution of different ecosystem elements arises, the biodiversity is maintained at several scales (Khanina, 1998). We find that the evolution of the ecosystem network excludes the possibility for the barycenter of the network and the objective coordinates to merge. We then propose a sustainability criterion to maintain full connectivity of the ecosystem in time, which enables to safeguard its utility domain. Our definition of sustainability can be linked to the general definition of sustainability as the conservation of the ecological integrity. Indeed, the structural aspects of ecosystems lead to a definition in which the typical rate of change of species and biological communities is impeded and implies the integrity loss, that is, the ecosystem is no longer complete (Leo and Levin, 1997). Our will is to uncover the properties of sustainable management toward natural systems when they are considered as dynamic networks, but also to make them applicable by displaying the constraints over the management rules. The model could respond to the needed theoretical formalization of decision-making in ecosystem-based management (Mills et al., 1993; Rooney, 2001; Reynolds, 2005; Jensen et al., 2009). Although theoretic models are often criticized because they simplify the factual complexity, they can be highly useful in identifying sensitive ecosystem components or in simulating alternatives (Lee, 1993). For example, the inability to monitor and manage all aspects of biodiversity has led to the development of paradigms that focus either on single species or whole ecosystems. Both have advocates and detractors (Payton et al., 2002). The keystone species concept can allow the ecosystem managers to combine the best features of both paradigms. Our model encompasses both paradigms in a sense that targeting specific nodes, which can be interpreted as keystone species (Power et al., 1996), and unmarking others, which become the subordinate or indicator species of the ecosystem, does not neglect the complexity and high connectedness of the ecological network. A keystone species either supports or alters the main pattern of the ecosystem; it is responsible for the existence of an ecosystem of a certain type (Khanina, 1998). The presence of such species supports stability and the maintenance of the ecosystem on its center of gravity (Ibim, 2010). In like manner, Conway (1989) suggested that keystone species permit to reestablish and sustain the ecosystem structure and stability. Although the model is meant to apply to different scenarios of ecosystem imbalance, it could also address the issue of reconnecting wildlife habitat and building ecological corridors. In this paper, we shall exemplify the analysis through the interactions between deer and forest communities to facilitate the understanding of the model. The increasing density of deer is inversely proportional to plant and animal species dependent on forest resources, as they cause extensive damage on forest ecosystems. Deer populations and habitat resources interact dynamically and such interactions vary over time (De Calesta and Stout, 1997; Augustine and Frelich, 1998). Management methods such as natural autoregulation or

maximum sustained yield management1 in limiting deer impacts on forest ecosystems have been unsuccessful, but an ecosystembased management approach is promising (Rooney, 2001; Latham et al., 2005; Clarke and Jupiter, 2010). The latter is based on management objectives, models of system dynamics and monitoring programs to assess the changes in the system (Waller and Alverson, 1997; Latham et al., 2005). After this starting section, we shall clarify the methodology of subset identification in Section 2. The dynamic behavior of the network is modeled in Section 3. Ensuing management prescriptions are given in Section 4. Section 5 is devoted to illustrating simulation examples. Section 6 concludes. 2. Subsets identification Mills et al. (1993) ask whether one can identify a set of species that are so important in determining the ecological functioning of a community that they warrant special conservation efforts. To address their question, we represent the interactions in the ecosystem by a complete undirected graph2  = {V, E} which defines the topology. The graph consists of vertices V = {1, . . ., N} indexed by the node members. At any given time t ∈ [0,∞), i and j represent two neighboring connected nodes. The graph also consists of the set of edges E = {(i, j) ∈ V × V} that represent inter-node interactions. Given the risk of edge distension and breakage, we assume that the set of edges E and the graph  are time-dependent. Ecosystem managers need to select a group of target nodes or identify a subset of target nodes by linking the nodes according to their interactions. Indeed, biologists have proposed that keystone species be special targets (Mills et al., 1993). If the ecosystem manager considers deer and trees as keystone species in the forest environment, that is, the target nodes of the objective function, then all other plant and animal species from the forest community represent the unmarked nodes. To identify the subset of target nodes, three types of interactions are considered:  t\t : target node-target node type of interaction  u\u : unmarked node-unmarked node type of interaction  t\u : target node-unmarked node type of interaction The target node-target node interaction can been illustrated as the existing interaction between deer and trees. For example, deer eat buds and saplings of trees, causing the adult recruitment failure (Rooney, 2001; Rooney and Waller, 2003). Trees are also considered as keystone structures, for they provide food and habitat to many forest species. In parallel, deer consume resources from the forest understory, such as forest herbs and shrubs (Augustine and Frelich, 1998; Rooney and Waller, 2003). This would be the target node-unmarked node interaction. Their consumption has numerous adverse effects on taxa that rely on the same plant species (McShea and Rappole, 1992; Rooney and Waller, 2003). Knowing that these taxa evolve in forest environments, which makes them tree-dependent, we fall on the unmarked node-unmarked node interaction and complete the graph. Let the ecosystem manager select a node from the graph and start to connect it to other nodes by any type of interaction.

1

See Rooney (2001) for a detailed description. A graph is connected if there exists a path between each pair of nodes, that is, if every node is reachable from some other node (Urban and Keitt, 2001). Although we looked at the case of incomplete graph, the aftereffects are related to the number of target nodes which must be lower than or equal to the number of unmarked nodes: a condition always verified in the ecosystem-based management approach. Further, the possible implementation of the predator-prey relationships would imply the use of directed graphs, whereas the ecosystem network we model has no orientation or has a dual orientation. 2

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By choosing the type, she can identify the target nodes and the unmarked nodes and their respective subsets needful for the dynamic graph-theoretic analysis. This method brings us to the rationale of Ramsey’s coloring of the edges in a graph in more than one color (Ramsey, 1930; Graham et al., 1990; Chomette, 2010). Our graph of n elements is composed of k-elements subgraph k , r=3

where  1 =  t\t ,  2 =  u\u and  3 =  t\u . The following identification theorem applies. Theorem 1 (Ramsey). ∀ (l )l ≤ r , ∃ n → (k)r , n ⊇ k .

The number of nodes in each subset is given by |t | = Nt and |u | = Nu respectively. Each node has a utility domain u∈(0, ], from which it gets utility from other nodes. This utility domain depends on the Euclidean distance d between the nodes. All nodes within the utility domain of a node form its utility node set. Definition 1. ∀i, j ∈ , and for node i’s utility node set i = {j ∈ i : 0 < |xi − xj | ≤ }, the utility domain uij is defined as uij = Ii (dij ) where



r

The proof is in the appendix. As a consequence, the subsets of target and unmarked nodes are identifiable. It now belongs to the ecosystem manager to decide on the ‘coloring’ of the ecosystem network. While in Ramsey’s works, the coloring of edges only implied the number of colors, we have different types of interactions which decide on specific attributes of the nodes at stake. Indeed, if the ecosystem manager decides to connect two nodes by  t\t , she specifies that both nodes are target nodes. The same reasoning applies with other two types of interactions. Therefore, some limitations on the coloring must be posited in order to achieve the identification of subgraphs and, by extension, the identification of subsets. Lemma 1.

E = {(i, j) ∈ V × V |i ∈ i ∀j}.

Lemma 1 states that adding an extra node to a subset of connected nodes must not modify the attributes of nodes already connected. Otherwise, the risk of multiple attributes of the same node can occur, i.e. a node becomes both a target node and an unmarked node for some management objective, making the identification of subsets unfeasible. In terms of graph construction, it means that the starting interaction of two identified nodes within the graph decides on the respective attributes of those nodes. All the remaining interactions in the network must be made from this starting subset and each additional interaction in the graph must be decided with respect to the existing attributes of nodes already connected. We can now make a general statement. Theorem 2.

∀ (i )i ≤ r , ∃ kr ⊆ n , E = {(i, j) ∈ V × V |i ∈ i ∀j}.

The proof is straightforward by Lemma 1. To see what this is about, consider the following four-node network.

x2

θt \t

x1

x3

θu \u

x4

θ t \u

Ii (dij ) =

dij

if

j ∈ i

0

if

j∈ / i



Definition 2. ∀ j ∈ , the network utility domain Ui () = j uij is the sum of utility domains contained in the network utility set. The utility can be interpreted as the supportable density of a species with respect to other species in the ecosystem. Indeed, the ecosystem manager seeks to maintain the right composition of carrying capacities or the maximum population sizes that the ecosystem can sustain. The distance characterizes the gap between the carrying capacities of species at stake. The overage of a species yields greater distances with other species’ proportions, which in fine jeopardizes the ecosystem balance. Taking into account the resources availability in a forest ecosystem, the utility set for a species represents the bearable carrying capacities of other species it interacts with, both directly and indirectly. For example, some plant species show negative responses to elevated deer density; this implies that the distance from the respective carrying capacities can be too high, compromising the ecosystem overall connectivity. The Euclidean norm distance between two random nodes i and j is dij = dji = |xi − xj | = [(xi − xj )T (xi − xj )] 1/2 ≥0

(2)

So the time derivative of the squared distance equals d˙ ij2

= 2dij d˙ ij = 2(xi − xj )T (x˙ i − x˙ j )

(3)

When the distance equals the utility domain or dij = uij , the condition for nodes i and j to evolve connected and, therefore, that the network is asymptotically stable in the sense of Lyapunov, is d˙ ij2 ≤ 0 : d˙ ij depends on the trajectories of nodes i and j. We consider network closed-loop systems in which the system outputs are used as the system inputs. We know that an ecosystem component has a dual role as receiver and transmitter of interactions (Patten, 1981). Let N be the number of nodes evolving in R2 . Let xi ∈ R2 denote the position of node i in the network. The set of all possible positions of the dynamical system is the configuration space. It is spanned by the stack vector of all the control inputs T

Nodes x1 and x2 are connected by  t\t which means that both are defined as target nodes. Since x2 connects with x3 by  t\u , x3 is thus an unmarked node. Let us now add x4 . The  u\u type of interaction between x3 and x4 implies that x4 is an unmarked node. In light of this information, the only possible interaction between x4 and x1 and x2 is the target node-unmarked node interaction. Otherwise, the subsets turn out unidentifiable. We finally have E = {(xi=1,2 , x4 ) ∈ V × V |i ∈ 2t , x4 }.

T ] which denotes the aggregated state of the network x = [x1T , . . ., xN as an involution. In our case, involution implies the preservation of the network during the dynamic transposition to an objective. The stack vector reflects the environmental policy space toward the network, for the control inputs are amassed and configured so as to arrive to the environmental objectives. According to Olfati-Saber and Murray (2004), the trajectory of a node obeys the model of a single-integrator dynamics

x˙ = ci , i ∈  = {1, . . ., N}

(4)

where ci denotes the control input for each node.3

3. Dynamic model The ecosystem manager identifies the subset of target nodes t and the subset of unmarked nodes u , such that t ∪ u =  and t ∩ u = ∅

45

(1)

3 The control can be seen as static, in the sense of structurally stable, that is, the potential perturbations do not affect the qualitative pattern of the trajectories (Boscain et al., 2006).

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The dynamics of unmarked nodes is given by the Laplacianbased control strategy (consensus) differential equation, meaning that each node moves in the direction of the average position of its utility nodes (Gustavi et al., 2010; Mesbahi and Egerstedt, 2010). A consensus problem is when spatially distributed elements of a network must reach overall utility without recourse to a central coordinator (Siljak, 1991; Spanos et al., 2005). The Laplacian dynamics has the advantage of converging to a steady-state, and we know that natural ecosystems operate near steady-states (Patten, 2010). Indeed, despite being often out of balance, ecosystems possess an innate tendency to develop and maintain steady-states. The interaction between nodes’ dynamics is realized through the control input for each node ci = −Lx where L is the graph Laplacian of the network. We introduce the utility updating time-delay parameter ε ∈ [0, 1], such that ε = 0 implies disconnection from vicinity and ε = 1 the absence of delay. When a node is transposed to some coordinates issued from the ecosystem management, it can update its utility from its utility nodes with delay. Time-delays can be problematic should the nodes update their utility coordinates too tardily to remain connected to other nodes; the rationale is that the timedelay jeopardizes the connectivity between the nodes.4 One way of interpreting delays is to think of belated responses of harmed species to a decrease in density of harmful species. For example, regardless of the decrease in abundance of deer, the population increase of the plant species threatened by its browsing can come off belatedly, making the rebalancing difficult, which may cause their disconnection from the network. Our way of incorporating the time-delay differs from that of Olfati-Saber and Murray (2004) but keeps the aim of studying consensus criteria when update responses are delayed. For a random unmarked node i ∈ u the dynamics is x˙ i = −Ni xi + εi



xk ∀i ∈ u

(5)

k ∈ i

If Ni = 0, the unmarked node will not move. As the graph stays connected, the consensus equation drives nodes to the same state value. If Ni > 0, by setting ˛ = 1/Ni in order to isolate the average value, we obtain ˛x˙ i = −xi + εi  where  =

(6)



x /Ni k ∈ i k

is the barycenter of i ⊆ . Its annulment

yields the equilibrium point under the unmarked node dynamics xi = εi   . Lemma 2.

The trajectory of the unmarked node converges to  .

In the example we give, Lemma 2 implies that the density of some indicator species from the forest understory follows the movement of the ecosystem center of gravity, which preserves the species inside the ecosystem. The dynamics of target nodes is also based on the consensus equation. Moreover, it is based on the objective term which transposes the network to the objective position x = g.5 For node i assume di = |g − xi |. The dynamics for a random target node i ∈ t is given by x˙ i = −Ni xi + εi



k ∈ i

xk + F(xi , g) ∀i ∈ t

(7)

xk

xi

f ( di )

g

xk

xk

Fig. 1. An example of complete graph defined by the subsets of three unmarked nodes {3xk }, one target node {xi } subject to management function f(di ) and objective position g, and six edges {3xi xk , 3xk xk }.

where F(xi , g) is the management objective function such as



F(xi , g) =

f (|g − xi |)(g − xi )

if di > 0

0

if di = 0

(8)

The function F(xi , g) is continuous. Its magnitude is defined by a continuous scalar function f(di ) ≥ 0 which depends on node i’s distance to g. The objective function comes from the attractionrepulsion functions used in swarm models (Gazi and Passino, 2004). In these models, the motion dynamics depends on the distance between two agents and a function of attraction f(·) without the recourse of a central coordinator. If Ni = 0, the target node will, depending on f(·), either not move or head to g. If Ni > 0 and F(xi , g) = / 0, by setting ˇ = 1/[Ni + f(|g − xi |)] like in Gustavi et al. (2010), we obtain ˇx˙ i = −xi +

Ni εi  + f (|g − xi |)g Ni + f (|g − xi |)

(9)

Its annulment yields the equilibrium point under the target node dynamics xi = [Ni εi   + f(|g − xi |)g]/[Ni + f(|g − xi |)]. Lemma 3. The trajectory of the target node converges to an aggregate of  and g. In the example we give, Lemma 3 states that the density of keystone species such as deer is subjected to the combination of the ecosystem center of gravity and the objective decided by the ecosystem manager. Indeed, given the overall interconnections in the ecosystem, the target node is simultaneously subjected to two forces, one coming from the management objective function and the other coming from the populations’ weighted distribution in the ecosystem (Fig. 1). Lemmas 2 and 3 guarantee the boundedness of solutions of the system. This property is specified in Lemma 4. Lemma 4.

˝ = Co( ∪ g).

The proof is in the appendix (see Ji et al., 2008; Gustavi et al., 2010). Lemma 4 guarantees that the biological species composition remains unchanged in time and that their extinction is avoided, preserving the ecosystem initial configuration. Although the dynamics of target and unmarked nodes are distinctive, the objective of the ecosystem manager is to bring the network nodes to the objective coordinates. We thus have limt→∞ xi (t) = g ∀i ∈  whenever di > 0. Piecing the equilibrium points together gives ˇx˙ i = ˛x˙ i

(10)

or 4

The inertia in ecosystems has been widely documented in the Millennium Ecosystem Assessment reports. 5 Management objectives must be quantifiable and agreed upon by ad hoc agencies and participating stakeholders. The long-term objective of managing deer from an ecosystem perspective is the recovery and maintenance of forest structure and diversity, i.e. integrity, and ecological processes and forest ecosystem functions, i.e. health (Latham et al., 2005).

g = εi 

(11)

For F(xi , g) = / 0, g is equal to  weighted by the time-delay. Therefore, in the limit and absence of delay, the objective position and the barycenter overlap. And yet, this is impossible because of Lemmas 2 and 3. Given Lemma 4, the following separation theorem ensues.

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Theorem 3.

˝t = {Co( t ∪ g)|   = / g}.

The proof is in the appendix. According to Theorem 3, the nodes remain at best within in (t), as the barycenter cannot merge with g. Thereby, it is impossible for the ecosystem manager to transpose the ecosystem center of gravity to the objective coordinates. As a result, the consensus can only be a quasi-equilibrium. Such as shown by Patten (2010), the ecosystem shall at most operate near steady-states. Provided that the ecosystem falls only if it loses its center of gravity while swinging (Ibim, 2010), this impossibility is not fateful as long as the graph remains connected. In consequence, the only credible constraint that can be imposed upon the transposition of the network to the objective is that the network be connected. This brings us to the following sustainability criterion. Theorem 4. For limt→∞ xi (t) = g ˝t = {Co( t ∪ g)|   = / g} ,  (t) is connected.

∀i ∈ 

and

The proof is in the appendix (see Dattorro, 2005; Gustavi et al., 2010). Theorem 4 is the criterion that enables a network to reach the objective without compromising its connectivity, for disconnections induce the system vulnerability and jeopardize its stationarity. Provided that the graph remains connected, the Laplacian consensus constitutes a quasi-equilibrium. On the one hand, disconnected ecosystems are considered as sick and highly exposed to exogenous threats. On the other hand, we know that the value of a connected network is much higher than the sum of its parts, due to the superadditivity of synergies (WillsJohnson and Hornby, 2007). Besides, linkages are much more costly to form than to remove. Given that ecosystems work as a whole, and bearing in mind that the bailout of ecosystems needs public or private funds, the maintenance of connected ecosystems is a creation of both ecological and economic meanings of value. 4. Management rules The network properties previously defined allow us to put forward three management rules according to the type of interaction. For two random unmarked nodes i,j ∈ u , linked by  u\u , we have

47

When dij → , d˙ ij2 ≤ 0 if two conditions are met. The first is dij /(εi − εj ) ≥   which is verified for lim(εi − εj ) = 0. The second is Fi (xi , g) − Fj (xj , g) ≥ 0 and implies that fi (·) and fj (·) are two monotonically increasing functions such that fi (·) ≥ fj (·). Corollary 2. The efforts engaged to transpose the target nodes to the management objective positions and the utility-updating timedelays they entail must not break their utility domain. Otherwise, they disconnect. Necessary and sufficient conditions for connectivity are coequal magnitudes of increasing management objective functions and synchronous time-delays. In the example we give, Corollary 2 implies that the pressures from the environmental policy on both the deer and tree densities be proportional, gradually increasing in time, and, finally, that their responses to the overall management policy be synchronous, or else they will disconnect (Fig. 3). For two random nodes i∈ u and j∈ t , linked by  t\u , and for / 0, we have F(xj , g) = d˙ ij2 = −2Ndij [dij −  (εi − εj )] − 2dij F(xj , g)

(14)

When dij → , d˙ ij2 ≤ 0 if f(d) ≥ 0, which is verified by construction, and lim(εi − εj ) = 0. Corollary 3. The effort engaged to transpose the target node to the objective position and the utility-updating time-delays it entails must not break the utility domain between target and unmarked nodes. Otherwise, they disconnect. Necessary and sufficient conditions for connectivity are nonnegative management objective function and synchronous time-delays. In the example we give, Corollary 3 says that the management pressure on the deer population must not oscillate, and that responses of both the keystone species and understory plants to the overall management be synchronous, or else they will disconnect (Fig. 4). Overall, we see that the connectivity of a connected network built on the consensus protocol depends on rather simple constraints such as synchronous time-delays and standard criteria toward the management objective function. 5. Simulations

d˙ ij2 = −2Ndij [dij −  (εi − εj )]

(12)

When dij →, d˙ ij2 ≤ 0 if dij ≥   (εi − εj ) or dij /(εi − εj ) ≥   .6 This inequality is verified for lim(εi − εj ) = 0. If the distance between two unmarked nodes reported to the net utility time-delay is greater than or equal to the barycenter value, the nodes remain connected. Corollary 1. While being displaced toward the management objective position, the unmarked nodes must not be driven away by the utility-updating time-delays beyond their utility domain. Otherwise, they disconnect. Necessary and sufficient condition for connectivity is synchronous time-delays. In the example we give, Corollary 1 states that, once the management objective of decreasing the deer density has been initiated, the response of different understory species be synchronous or else they will disconnect (Fig. 2). For two random target nodes i,j ∈ t , linked by  t\t , and Fi (xi , / 0 and Fj (xj , gj ) = / 0, where gi = / gj or fi (di ) = / fj (dj ), we have gi ) = d˙ ij2 = −2Ndij [dij −  (εi − εj )] − 2dij [Fi (xi , gi ) − Fj (xj , gj )]

(13)

6 We can notice that the ecosystem barycenter is proportional to the distance between nodes and inversely proportional to their net utility time-delay.

Based on the properties and conditions previously obtained, the aim of this section is to illustrate the behavior of a simplified forest ecosystem through simulations. The stability conditions derived in Sections 3 and 4 guarantee the convergence to the objectives and the maintenance of connectivity. We shall consider three cases: one without time-delays, another one with synchronous time-delays, and a third one with asynchronous time-delays. To illustrate the behavior of a basic ecosystem of nine species initially on the point of disconnecting, we test the three cases defined above. Two of the species, meaning deer and trees, represent the target nodes with dynamics given by (7) and the remaining seven unmarked nodes, such as various understory flora and fauna, follow the dynamics given by (5). In all simulations, the term that pulls the target nodes toward the objective is defined as fi (|gi − xi |) = ai (gi − xi ). In the first case, the distances between carrying capacities of species have become such that the ecosystem is at the edge of imbalance. Put differently, the distances between population sizes of the keystone species that the environment can sustain reach critical values. The hypothetical population sizes are given by x(0) = [10 7 2 2 2 1 3 0.5 1], where Nt = {1,2} and Nu = {3, 4, 5, 6, 7, 8, 9}. As can be observed, the hypothetical distances between target nodes and between target nodes and unmarked nodes are perilous, compromising the subsistence of the convex hull. For example,

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A.Z. Dragicevic, B. Sinclair-Desgagné / Ecological Modelling 270 (2013) 43–53

xk

xi

xk

xi

g

xi

g

xk

xi

Fig. 2. An example of unmarked node-unmarked node type of interaction {xi xi } in a complete graph defined by the subsets of three unmarked nodes {3xi } and one target node {xk }.

xk

xk

f i ( di )

xi

gi gj

f j (d j)

xj

Fig. 3. An example of target node-target node type of interaction {xi xj } in a complete graph defined by the subsets of two unmarked nodes {2xk }, and two target nodes {xi , xj } subject to respective management functions fi (di ) and fj (dj ) and respective objective positions gi and gj .

the distance between deer and trees is of 3 (10 − 7 = 3) which, in instance, condemns the understory species to low levels (e.g. 10  2). The ecosystem manager sets the target nodes objective coordinates to g1 = 10 and g2 = 11. Indeed, her analysis shows that the reduction in the gap between the proportions of deer and trees (10−11 = abs(−1)) should bring the forest network to a sustainable state and should improve the capacities to grow of all nine species. As can be seen in Fig. 5, with ai = 1,2 = 2 so that f1 ≥ f2 , and εi = 1 for i∈{1, . . ., 9}, the connectivity condition (11) is respected. The target species converge toward the objectives g1 and g2 and the unmarked species remain close by. The distances between carrying capacities thus move away from the critical zone, and the graph remains connected for all t ≥ 0. In spite of the initial oscillatory movements in response to the environmental policy, the trajectories stabilize after a while. The terminal network barycenter is close to but not equal

f (d j)

xj

Fig. 4. An example of target node-unmarked node type of interaction {xi xj } in a complete graph defined by the subsets of three unmarked nodes {xi , 2xk }, and one target node {xj } subject to management function f(dj ) and objective position g.

to the objective coordinates, as had been put forward by Theorem 4. Indeed, at t = 50, the average objective coordinate is 10.5, while the network center of gravity stands at 10.1. In the second case, the initial positions and the objective function are identical, but, this time, synchronous updating time-delays are introduced. We assume that εi =/ 1,2 = 0.9. Fig. 6 shows that condition (11) is still satisfied. As a matter of fact, the trajectories demonstrate the ability of the ecosystem to connect in time as long as time-delays are synchronous. At t = 50, the network center of gravity of 9.1 is reasonably distant from the average objectives coordinates. In the third and final case, the initial positions and the objective function are identical, but, unlike the previous case, species-specific time-delays are introduced. Thereby, let ε1 = 1, ε2 = 0.8, ε3 = 0.8, ε4 = 0.7, ε5 = 0.5, ε6 = 0.9, ε7 = 0.1, ε8 = 0.7 and ε9 = 0.8. Fig. 7 shows that condition (11) is no longer satisfied. The figure illustrates the inability of the ecosystem to achieve full connectedness due to asynchrony in utility updating. Neither do the target nodes attain their objectives nor do the unmarked nodes catch up with the target nodes. All the species being interdependent, it turns out that the target nodes do not reach the objectives, even though the distance between the target nodes has shortened: their growth depends on the evolution of understory species and vice versa. At t = 50, the network barycenter value is 3.4, a value close to the early critical state. Again, the ecosystem is in danger of collapsing. As these elementary simulations show, when the graph converges to the objective without time-delay or in synchrony, its connectedness is preserved. In turn, this implies that the

10

8

L1 L2 F1

6

F2 F3 F4 F5

4

F6 F7 2

-

1

6

11

16

21

26

31

36

41

46

Fig. 5. Species coordinates (ordinates) as functions of time (abscissa) in an ecosystem where the graph remains complete and the management rules are satisfied.

A.Z. Dragicevic, B. Sinclair-Desgagné / Ecological Modelling 270 (2013) 43–53

49

10

8

L1 L2 F1

6

F2 F3 F4 F5

4

F6 F7 2

-

1

6

11

16

21

26

31

36

41

46

Fig. 6. Species coordinates (ordinates) as functions of time (abscissa) in an ecosystem with synchronous time-delays and satisfied management rules.

10

8

L1 L2 F1

6

F2 F3 F4 F5

4

F6 F7 2

-

1

6

11

16

21

26

31

36

41

46

Fig. 7. Species coordinates (ordinates) as functions of time (abscissa) in an ecosystem with asynchronous time-delays and unsatisfied management rules.

constraints on the objective function applied to the management rules are satisfied. 6. Conclusion This paper aimed at formalizing the criteria for a sustainable ecosystem-based management by borrowing tools from the leaderfollower networks. We first proved the existence of a subset of target nodes identified by the ecosystem manager and introduced the coherence principle in order to avoid the paradox of multiple attributes over nodes. We then proved the impossibility for a network to merge its barycenter with the objective position via the separation theorem. This result enabled us to reveal the minimal sustainability criterion which ensures that the ecosystem network elements remain

connected while being conveyed to the management objective position. The criterion guarantees that the ecosystem maintains its self-organizing structural complexity or integrity and preserves its utility domain. Our approach meets that of Smith et al. (2009) who assert that optimal policies must be determined over the subsets of connected systems. In light of the property of the objective function, the nature of connections and the utility-updating time delays between the nodes, we finally identified three management rules to ensure the maintenance of connectivity in time. A natural extension of this work would be to study the consensus dynamics on directed graphs, which would enable us to capture the relations of predator–prey or host–parasite. Despite our attempts to theorize an efficient ecosystem-based management, illustrated by the example of deer population management in forest environments, two of the issues remain unsolved.

50

A.Z. Dragicevic, B. Sinclair-Desgagné / Ecological Modelling 270 (2013) 43–53

First, management objectives must also be determined in the political arena in order to incorporate the socio-economic aspects of ecosystem management (Abildtrup and Jensen, 2013). Second, recovery of the structure, diversity and functions of forest ecosystems take a long time. Evaluating progress for such a period presents difficulties when goals need to be set in short time (Latham et al., 2005). Because graph theory is an idealized representation of network patterns, this paper ought to be considered as exploratory. And yet, it demonstrates that the analysis of network dynamics is needed to fathom the connectivity issues. Furthermore, ecosystem managers cannot afford to wait for better data and need rapid results. Such as pointed out by Urban and Keitt (2001), when research is conducted with lack of data, the graph-theoretic framework can play a role of heuristic in decision-making. Although graph theory does require minimum data of reasonable quality, it does not require long-term population data, so it can be used as a rapid assessment tool. Further research could apply the optimal control properties to network topology, so as to emphasize the graph-theoretic characterization of the ecosystem controllability. That way, we could reveal the ecosystem manager’s shadow values or willingness to abide by the management rules. Another avenue of research could consist in randomizing the ecosystem network, in which every edge would exist with some nonnull probability. Regardless of the appeal of such an approach in view of the stochastic processes in nature, it would involve the study of complete subgraphs on random graphs, thereby distorting the definition of convex hull. To conclude, various complementary works should be conducted to validate or invalidate the soundness of the field of study we have decided to tackle.

Let ( 1 ,  2 ) such that  1 +  2 = . Assume the result is true for some (ϕ1 , ϕ2 ) with ϕ1 + ϕ2 <  and thus for n1 and n2 such that n → ( 1 − 1,  2 ) and n → ( 1 ,  2 − 1). These integers exist only because of the induction method. We show that n = n1 + n2 verifies n → ( 1 ,  2 ). Consider a node s1 selected from n . Define subsets 1 and / sn , we have 2 of all remaining nodes, such that for all sn−1 = sn−1 ∈ 1 if and only if the interaction (s1 , sn−1 ) is of type  l , i where l = 1, 2.   1 2 Subsets  and  constitute a partition of the set of the n − 1 remaining nodes. Respective subgraphs 1 and 2 are obtained from nodes of these subsets. We have |1 | ≥n1 and |2 | ≥n2 . By assumption, we know that n1 → ( 1 − 1,  2 ). Hence, 1 ⊇ 1 −1 or 1 ⊇ 2 . In consequence, we have 2 ⊆ n . We add the node selected at the beginning, which is linked to 1 and thus to 1 −1 . We now have a complete graph of  1 − 1 +1 =  1 , in which all interactions are of the same type. By symmetry, we have the same case for |2 | ≥n2 . By induction, we conclude that for ( 1 ,  2 ) there exists n such that n → ( 1 ,  2 ). Let us now consider the case with more than two types. In this case, r = 3. The result is immediate by induction. Assume the result holds for  r types of interactions. Let n be integer such that n → (k)r , and m be integer such that m → (n)2 . Then integer m verifies m → (k)r +1 . For every interaction in m , we already have an interaction of one of two types: the interaction is of type  1 if and only if it was of type  r in the previous interaction. By induction, it means that the interaction of type  2 was of the type  r+1 in the previous interaction. The theorem being proved for two types, we know that m ⊇ n . If n ≡ 2 , we have m ⊇ 2 and consequently m ⊇ k as 1

n ≥ k. Since n → (k)r , we have m ⊇ k . r Eq. (5)

Acknowledgements This research was partly supported by a grant from the CIRANO Institute in Montréal (QC), Canada. The authors are indebted to John Galbraith (McGill University) and Serge Garcia (INRA) for their comments and feedback on different versions of the manuscript. The authors would also like to thank the two anonymous referees for their thorough reviews and suggestions.



=−

x˙

k ∈ u

(xi − εi xk ) −

= −Nu xi + εi



= −Nxi + εi

k ∈ t

(xi − εi xk )

− Nt xi + εi

x k ∈ u k

= −(Nu + Nt )xi + εi







x k ∈ t k

x k ∈ (u +t ) k



x k∈ k

Appendix. Proof of Theorem 1 (Ramsey) Let n denote a large number of nodes in a complete graph n . Let k be the number of nodes in subgraph k ⊆ n in which all nodes l

are connected by the same type of interaction  l where l = {l, . . ., r}, that is, there can be at most  r types of interactions. We call k a r monotype subgraph of k nodes. We note n → (k1 , k2 , . . ., kr ) if n is such that connecting nodes

Eq. (6)

˛x˙ i

r

r =3

−



k ∈ i

(xi − εi xk )



= − N1 Ni xi +

εi Ni

= −xi + εi

xk k ∈ i Ni



x k ∈ i k

= −xi + εi 

of (l )l ≤ r th type. Thereby, n nodes in n are contained in k subr graphs of k nodes. When k1 = k2 = ... = kr , n → (k)r . We have .

We prove the theorem for two types of interactions (r = 2) and then extend it to more than two types (r > 2). That is, for every interaction in n in type  l , where l = 1,2, there exists integer n such that 1 ⊆ n or 2 ⊆ n , and then extend the proof to l > 2. We first prove that the theorem holds for r = 2. Let n → ( 1 ,  2 ) be that integer n is true for ( 1 ,  2 ). This means that for every interaction in n in  1 or  2 , we have subgraph 1 or 2 . By induction, we show that for ( 1 ,  2 ) there exists n such that n → ( 1 ,  2 ), that is, the whole graph interacts via the two types of interactions.

 

i

in n induces a monotype subgraph k in which all interactions are

r = 3 and thus k

1 Ni

=

Eq. (7)

x˙ i



=−

k ∈ u

(xi − εi xk ) −

= −Nu xi + εi



x k ∈ u k

= −(Nu + Nt )xi + εi = −Nxi + εi



= −Ni xi + εi



x k∈ k



 k ∈ t

(xi − εi xk ) + F(xi , g)

− Nt xi + εi



x k ∈ (u +t ) k

x k ∈ t k

+ F(xi , g)

+ F(xi , g)

x k ∈ i k

+ f (|g − x|)(g − xi )

+ F(xi , g)

A.Z. Dragicevic, B. Sinclair-Desgagné / Ecological Modelling 270 (2013) 43–53

As we set (g − xi )T

Eq. (9)

=

ˇx˙ i

= = =

1 Ni +f (|g−x|) −Ni xi +εi







−Ni xi + εi

+ f (|g − x|)(g − xi )

x k ∈ i k



d˙ ij2

x +f (|g−xi |)g−f (|g−xi |)xi k ∈ i k

d˙ ij2

Ni +f (|g−xi |)

= −xi +

Ni εi





= −2Ndij2 + 2N[−di  (εi − εj ) + dj  (εi − εj )]

Ni +f (|g−xi |)

xk + f (|g Ni Ni +f (|g−xi |)

k ∈ i

= −2Ndij2 − 2Ndij [ (εi − εj )] = −2Ndij [dij −  (εi − εj )]

x +f (|g−xi |)g k ∈ i k

εi

Eq. (13) For two random target nodes i,j ∈ t , linked by  t\t , and for Fi (xi , / 0, where gi = / gj or fi (di ) = / fj (dj ) we have gi ) and Fj (xj , gj ) =

− xi |)g

d˙ ij2

Eq. (11)

T

= 2(xi − xj ) (x˙ i − x˙ j ) T

= 2(xi − xj ) [−Nxi + εi N



xk k∈ N

T

−xi +

Ni εi



xk + f (|g Ni Ni +f (|g−xi |)

− xi |)g

k ∈ i

Ni εi

εi



f (|g−xi |)g Ni +f (|g−xi |)

= εi



k ∈ i

g = εi

 g = εi





xk k ∈ i Ni

xk k∈ N

(εi − εj ) + Fi (xi , gi ) − Fj (xj , gj )]

(εi − εj ) + Fi (xi , gi ) − Fj (xj , gj )]

xi = / gi

xj = / gj

=0

= −2Ndij2 + 2 [−[(gi − xi ) − (gj − xj )]T [N

+

f (|g−xi |)g Ni +f (|g−xi |)

=0

= −2Ndij2 + 2 [−Ndi  (εi − εj ) − di2 fi (di )+(gi − xi )T (gj − xj )fj (dj )

xk Ni −Ni −f (|g−xi |) k ∈ i Ni Ni +f (|g−xi |)

+

f (|g−xi |)g Ni +f (|g−xi |)

=0

xk f (|g−xi |) k ∈ i Ni Ni +f (|g−xi |)

+

f (|g−xi |)g Ni +f (|g−xi |)

=0

+

f (|g−xi |)g Ni +f (|g−xi |)

Ni Ni +f (|g−xi |)

−

− εi



−1 +

Ni +f (|g−xi |) Ni +f (|g−xi |)









xk f (|g−xi |) k ∈ i Ni Ni +f (|g−xi |)

xk f (|g−xi |)] [Ni Ni Ni +f (|g−xi |) f (|g−xi |)

xk f (|g Ni f (|g−xi |)



+ Fi (xi , gi ) − Fj (xj , gj )]

f (|g−xi |)g Ni +f (|g−xi |)





k ∈ i

g = εi

T

= −2Ndij2 + 2(xi − xj ) [N

xk k∈ N

+Fj (xj , gj ))]

d˙ ij2 = −2Ndij2 + 2(xi − xj )T [N

Ni Ni +f (|g−xi |)

−εi

T

= 2(xi − xj ) [−N(xi − xj ) + N

=0

xk k∈ N

=0

k ∈ i

εi



− εj N



xk k∈ N

xk k ∈ i Ni

− εi



xk k ∈ i Ni



xk k ∈ i Ni



xk k∈ N



Due to Lemma 3, xi − xj = − [(gi − xi ) − (gj − xj )], so



xk k ∈ i Ni

− xi |)g





+Fi (xi , gi )−(−Nxj + εj N

=0

xk Ni Ni +f (|g−xi |)

εi



xk + f (|g k ∈ i Ni Ni +f (|g−xi |)

= 2(xi − xj ) [−N(xi − xj ) + εi N

ˇx˙ − ˛x˙ = 0

− −xi + εi



Ni εi

  we obtain

By (2), we know that di − dj ≡ dij = dji so



+

k∈ xk /N ≡ di

= −2Ndij2 − 2N[ (εi − εj )(di − dj )]

x +f (|g−xi |)(g−xi ) k ∈ i k

−xi [Ni +f (|g−xi |)] Ni +f (|g−xi |)



= −2Ndij2 − 2N[di  (εi − εj ) − dj  (εi − εj )]

Ni +f (|g−xi |)

−Ni xi +εi

51



xk (εi k∈ N

and

and

− εj ) + Fi (xi , gi ) − Fj (xj , gj )]



xk (εi k∈ N

− εj )

+ fi (|gi − xi |)(gi − xi ) − fj (|gj − xj |)(gj − xj )]]

+ Ndj  (εi − εj ) − dj2 fj (dj ) + (gj − xj )T (gi − xi )fi (di )]

In parallel, given that (gi − xi )T (gj − xj ) = di dj , the precedent reduces to d˙ ij2 = −2Ndij2 + 2 [N (εi − εj )(dj − di ) − di2 fi (di ) + fj (dj )di dj − dj2 fj (dj ) + fi (di )di dj ]

+ f (|g − xi |)]

− xi |)

Since di dj = dj di we have d˙ ij2

xk k ∈ i Ni

= −2Ndij2 + 2 [N (εi − εj )(dj − di ) − di2 fi (di ) + fj (dj )di dj − dj2 fj (dj ) + fi (di )di dj ] = −2Ndij2 + 2 [N (εi − εj )(dj − di ) − fi (di )di (di − dj ) + fj (dj )dj (di − dj )]

g = εi 

= −2Ndij2 + 2 [N (εi − εj )(dj − di ) − fi (di )di (di − dj ) + fj (dj )dj (di − dj )] = −2Ndij2 + 2N (εi − εj )dji − 2dij [fi (di )di − fj (dj )dj ] = −2Ndij [dij −  (εi − εj )] − 2dij [fi (di )di − fj (dj )dj ]

Eq. (12)

= −2Ndij [dij −  (εi − εj )] − 2dij [Fi (xi , g) − Fj (xj , g)] d˙ ij2

T

= 2(xi − xj ) (x˙ i − x˙ j ) T



T



= 2(xi − xj ) [−Nxi + εi N = 2(xi − xj ) [−Nxi + εi N

xk k∈ N

− (−Nxj + εj N

xk k∈ N

+ Nxj − εj N

T

T

= 2(xi − xj ) [−N(xi − xj )] + 2(xi − xj ) [εi N =

−2Ndij2

T

+ 2(xi − xj ) [N



xk k∈ N

T

=

+ 2N[−(g − xi )

T







xk k∈ N

xk k∈ N

xk k∈ N

Eq. (14) For two random nodes i ∈ u and j ∈ t , linked by  t\u , and for F(xj , g) = / 0, we have

)]

]

− εj N



xk k∈ N

]

(εi − εj )]

= −2Ndij2 + 2 [−[(g − xi ) − (g − xj )] [N −2Ndij2



xk k∈ N



xk k∈ N

d˙ ij2

T

= 2(xi − xj ) (x˙ i − x˙ j ) T

= 2(xi − xj ) [−Nxi + εi N



xk k∈ N

T

= 2(xi − xj ) [−N(xi − xj ) + εi N (εi − εj )]

(εi − εj ) + (g − xj )]

T



xk k∈ N

T

= 2(xi − xj ) [−N(xi − xj ) + N (εi − εj )]

T

= −2Ndij2 + 2(xi − xj ) [N







− (−Nxj + εj N

xk k∈ N

xk k∈ N

xk k∈ N

− εj N





xk k∈ N

xk k∈ N

− F(xj , g)]

(εi − εj ) − F(xj , g)]

(εi − εj ) − F(xj , g)]

+ F(xj , g))]

52

A.Z. Dragicevic, B. Sinclair-Desgagné / Ecological Modelling 270 (2013) 43–53

References

Due to Lemma 3, we know that xj = / g. Henceforth T d˙ ij2 = −2Ndij2 + 2(xi − xj ) [N



xk k∈ N

(εi − εj ) − F(xj , g)] T

= −2Ndij2 + 2[−[(g − xi ) − (g − xj )] ] [N T

= −2Ndij2 + 2[−(g − xi ) N T

+(g − xj ) N



xk k∈ N



xk k∈ N



xk k∈ N

(εi − εj ) − F(xj , g)] T

(εi − εj ) + (g − xi ) F(xj , g) T

(εi − εj ) − (g − xj ) F(xj , g)

= −2Ndij2 + 2[−Ndi  (εi − εj ) + f (dj )di dj + Ndj  (εi − εj ) − dj2 f (dj )] = −2Ndij2 + 2[−Ndi  (εi − εj ) + Ndj  (εi − εj ) + f (dj )di dj − f (dj )dj dj ] = −2Ndij2 + 2Ndji [ (εi − εj )] − 2dji [f (dj )dj ] = −2Ndij [dij −  (εi − εj )] − 2dij f (dj )dj = −2Ndij [dij −  (εi − εj )] − 2dij F(xj , g)

Proof of Lemma 4 For i ∈ u , if Ni > 0, the trajectory of the unmarked node converges to the barycenter of the subgraph i ⊆  . Due to convexity, the barycenter lies within the convex hull. We conclude that the trajectory of the unmarked node i lies within . For i ∈ t , if Ni > 0, the trajectory of i converges to an aggregate of the barycenter of the subgraph i ⊆  and the objective position g. By convexity of

, node i remains in . As t→∞, the trajectories of nodes in  evolve within 0 . Since the trajectories of both the unmarked and the target nodes are on the boundary or in the interior of , both nodes stay inside the convex hull. 0 is thus an invariant set. Proof of Theorem 3 By Lemma 2 and Lemma 3, we know that the trajectories of the unmarked nodes follow the trajectories of the target nodes which converge to an aggregate of the barycenter and g. By (11), we know that the barycenter of the subgraph i ⊆  attains the objective position g ∀i ∈ , a result that contradicts the precedent. Finally, by Lemma 4, ˝ = Co( ∪ g). Proof of Theorem 4 By setting the dynamics via the matrix Kronecker product in stack vector form of the node’s positions, we have x˙

= (x − g)T (x − g)



=

[xT g T ]

I

−I

−I

I



x

g

= − [[x1 , ..., xN ]T (L ⊗ I2 ) + [F1 (x, g), ..., Fn (x, g)]T ] with L the Laplacian matrix of (t) and I the identity matrix of size 2. Let g = [g, ..., g]T . Taking V = (1/2)(x − g)T (x − g) as a graph-compatible Lyapunov function, which can be considered as a function of time, the time derivative yields ˙ = − [(x − g)T (L ⊗ I2 )(x − g)] − V˙ = (x − g)T (x˙ − g)

 i ∈ t

F(xi , g)

where V ≤ 0 due to the eigen-properties of L = LT ≥ 0, and −I and F being monotically increasing. As long as the graph is connected, then L ≥ 0 and, as such, −L ≤ 0. LaSalle’s invariance principle addresses the asymptotic stability of a system. Given the convergence to the objective, the principle gives x − g → 0 when t → ∞ and (x − g)T (L ⊗ I2 )(x − g) → 0. The system is then stable and x − g tends to the null-space of L asymptotically. This implies that all scalar positions of all the nodes tend to the same value. Given that x − g → 0 and given that g is in , which we know to be an invariant set by Lemma 4, the connectivity is preserved; this in turn implies that the dynamics should converge to a steady-state. Since the consensus is an equilibrium when x = g, and x = / g, the consensus is a quasi-equilibrium and the ecosystem operates near steady-states.

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