António Abreu

L M Camarinha-Matos

Understanding Social Capital in Collaborative Networks

Keywords: Social Capital, Social Network, Collaborative Networks

The characterization and assessment of the social capital of a member in collaborative networks is an important element to help promoting the success of collaborative networks. However, models to measure the social capital are lacking. Applying some concepts from social networks theory, this paper discusses some perspectives and criteria to identify and measure the value of social capital of a member in the context of a Virtual organization Breeding Environment (VBE).

Introduction

According to various authors on Collaborative Networks (CNs), as well as reports from a growing number of practical case studies, the involvement in a collaborative network is commonly assumed to bring valuable (potential) benefits to the involved entities [START_REF] Beckett | Perceptions of value that sustain collaborative networks[END_REF][START_REF] Afsarmanesh | Challenges of collaborative networks in Europe[END_REF], [START_REF] Tenera | A TOC perspective to improve the management of collaborative networks[END_REF]. These benefits include an increase of the "survival capability" in a context of market turbulence, but also the possibility to better achieve common or compatible goals. On the basis of these expectations are, among others, the following factors: joining of complementary skills and capacities, access to new / wider markets and new knowledge, etc [START_REF] Camarinha-Matos | A contribution to understand collaboration benefits[END_REF].

However, it is important to realize that, when an enterprise is a member of a long-term networked structure, for instance a Virtual Breeding Environment (VBE), its value is not given only by its tangible assets -economic capital (such as: cash, resources, and goods). In this context, the existence of cooperation agreements, norms, reciprocal relationships, mutual trust, common infrastructures and common ontologies, allows members to operate more effectively in pursuit of their goals. In other words, there is an intuitive assumption that a VBE structure represents a group of organizational entities that have developed intangible assets of "social capital" that bring added value to its members. However, in spite of this assumption, it is, in fact, difficult to prove its relevance due to the lack of objective measurements, clearly showing the social capital value for each member [START_REF] Durlauf | Symposium on social capital: Introduction[END_REF], [START_REF] Adam | Social Capital: Recent Debates and Research Trends[END_REF], [START_REF] Sabatini | The Empirics of Social Capital and Economic Development: A Critical Perspective[END_REF].

Social capital metrics tailored to collaborative networks or even an adequate conceptual basis for social capital analysis is not available yet and might be an obstacle for a wider acceptance of this paradigm.

Nevertheless, in recent years some preliminary studies have explored the importance of social capital in the context of networked organizations [START_REF] Yu | A new idea for the research on cluster innovation network system model-based on the view of the use and creation of social capital[END_REF], [START_REF] Molina-Morales | Does homogeneity exist within industrial districts? A social capital-based approach[END_REF], [START_REF] Francis | Examining relational and resource influences on the performance of border region SMEs[END_REF] , however none of them proposed methodologies, approaches or support tools to help managers of enterprises to analyze and measure their social capital in a Virtual organizations Breeding Environments (VBE) context.

This paper introduces some discussion of the nature of social capital as a contribution to a future identification of a set of indicators that are suitable for collaborative networks. This work aims at contributing to answer the following main questions:

What are the main components that contribute to the formation of social capital of each member in a VBE context? How can social capital be measured in a VBE context?

Some Background

Although capital social is not new concept, its definition is not consensual among the various main disciplines that have addressed this topic (e.g. economy, sociology) and therefore several definitions can be found in the literature [START_REF] Bankston | Social Capital as a Process: The Meanings and Problems of a Theoretical Metaphor?[END_REF]. This problem results in part from the fact that the social capital needs to be analyzed from a multidimensional approach which has not occurred. For instance, the diversity of the perspectives on social capital varies according to this being focused on: the relationships that an actor has with other actors, the structure of relations among actors within a network, or both types of relationships [START_REF] Adler | Social Capital: Prospects for a new concept[END_REF].

As mentioned above, social capital has been defined from multiple and separated perspectives or for different purposes. From a sociological perspective, Bourdieu [START_REF] Bourdieu | Handbook of Theory and Research for Sociology of Education[END_REF] defined social capital as "the aggregate of the actual or potential resources which are linked to possession of a durable network of more or less institutionalized relationships of mutual acquaintance and recognition," while for Portes [START_REF] Portes | Social Capital its origins and applications in modern sociology[END_REF] capital social is "the ability of actors to secure benefits by virtue of membership in social networks or other social structures". On the other hand, from an economic perspective Nahapiet and Ghoshal [START_REF] Nahaiet | Social Capital, Intellectual Capital, and the organizational advantage Academy of Management Review[END_REF] defined social capital as "the sum of the actual and potential resources embedded within, available through, and derived from the network of relationships possessed by an individual or social unit. Social capital thus comprises both the network and the assets that may be mobilized through that network". For Fukuyama [START_REF] Fukuyama | Social Capital and the modern capitalist economy: Creating a high trust worplace[END_REF] "social capital can be defined simply as the existence of a certain set of informal values or norms shared among members of a group that permit cooperation among them".

Nevertheless, taking into account the above definitions, it is possible to conclude that social capital is a valuable asset which has some typical characteristics of economic capital such as: It can be invested with the expectation of future benefits [START_REF] Camarinha-Matos | Performance indicators based on colaborative benefits[END_REF] and, is convertible [START_REF] Bourdieu | Handbook of Theory and Research for Sociology of Education[END_REF]. But, on the other hand, it is different, since its existence lies in the relationships that are established within the network while economic capital can exist at the member level [START_REF] Robison | Is social capital really capital?[END_REF]. In other words, social capital cannot be traded by entities on an open market like as economic capital, but exclusively within a network [START_REF] Gant | Social capital and organisational change in high involvement and traditional work organisations[END_REF].

In order to categorize the components associated with the concept of social capital, several authors have proposed a multi-dimensional approach (see eg [START_REF] Nahaiet | Social Capital, Intellectual Capital, and the organizational advantage Academy of Management Review[END_REF]). However, the current limitations of existing methods and a poor understanding of the components that are the source of social capital have acted as a barrier to the development of this concept in other fields of application such as the collaborative networks. Therefore, identifying and characterizing the components that make up the social capital tailored to collaborative networks, is a necessary first step. Based on the literature, and taking into account the context of collaborative networks, the most relevant components, as a first approach, include: -Type of interaction among members -related to the forms of interrelationship that can occur between enterprises within a network. For instance, the following types of relationships can be identified: o Subordinate relationshipswhich characterize the authority and/or dependence structure within a network. o Peer relationshipswhich characterize the friendly relations within a network. -Frequency of contactrelated to the number of contacts between network members. -Intensity of contactrelated to the strength and nature of relationship in terms of potential benefits or losses. -Geographical dispersion of network membersrelated to the geographical distance among members. -Valuesrelated to the behaviour of members, such as: trust, solidarity, reciprocity, values systems alignment, rules and norms of governance. -Resourcesrelated to the number and type of resources that can be mobilized through the network. Such as: capital, raw-material, workforce, energy, goods, services, etc. -Knowledge -related to the type of knowledge that can be made available through the network, such as: skill, markets information, lobbying information, etc.

A Model to Analyze Social Capital

As mentioned above, one of the main weaknesses in this area in terms of social capital is the lack of appropriate formal methods to analyze social capital in the context of the networked organizations paradigm. We depart from the assumption that, the capital social of an enterprise is related to the "level of health" of the relationships that it establishes with other enterprises within a network. In order to analyze the social capital of each member in a collaborative network, it is necessary to develop a model that supports the analysis of the relationships among members and the assets that may be accessed through the network of contacts.

Therefore, as a first approach, these relationships are modelled using graphs, as illustrated in Figure 1. The main objective is to represent a network in symbolic terms from the perspective of analysis, abstracting reality through a set of connecting nodes.

Elementary-maps Enterprises Assets

Enterprises

Map of business contacts

Enterprise A Enterprise B

"Level of health" of the relationship

Map of enterprises' assets

Enterprise A Knowledge Y

Ownership relation

Enterprise A Market 1 Map of business contacts -A graph showing the network of contacts among members belonging to the network. In this case, the link's width represents the "level of health -LH" of the relationship between two enterprises, and its value is given by the following equation: However, the main difficulty is naturally the determination of each of the five components mentioned above. To collect and record those values without being intrusive in the network members' "life" requires further research and development.

Access relation Aggregate -maps Social Capital Resources Perspective

Social Capital Knowledge Perspective

Social Capital Market Perspective

ij ij ij ij ij ij VS w PRIC PRFC w SRIC SRFC w LH
these notions with concepts from the Social Network Analysis area, a useful tool to analyze in detail the map of business contacts can be obtained, as illustrated in Table 1. Assuming that the assets are classified into classes in accordance with their purposes, then, in this context it becomes possible to analyse the social capital of an enterprise according to different perspectives, such as:

-Capacity perspective -related to the ability of accessing to external resources.

-Innovation perspective -related to the ability of accessing to external knowledge.

-Market perspective -related to the ability of accessing to new markets.

Therefore, it is possible to define three notions of social capital that can be measured using a quantitative or a quantitative scale. However, in this discussion social capital is assumed as an abstract quantifiable value with the same meaning as utility concept.

Partial Social Capital (PSC) -Corresponds to the social capital of an enterprise under a single perspective. Its value is given by the following equation 1 :

j i v v v LH PSC Ani i A i A n j ij ik ) .... ( 2 1 1 (2)
Where: ik PSC -Partial social capital of enterprise i according to perspective k ij LH -"Level of health" of the relationship between enterprise i and enterprise j Ani v -Value assigned to asset An. The worth of the asset An is decided by enterprise i and depends on the relative importance/utility of each asset.

Effective Social Capital (ESC) -corresponds to the social capital that an enterprise uses in carrying out its activities. Its value is given by the following equation: 1 It shall be noted however that this equation does not properly capture all situations. For instance, it does not cover the cases in which an enterprise has relationships with two or more enterprises that hold identical assets. Total Social Capital (TSC) -corresponds to the maximum social capital that an enterprise can get from the network. Its value can be obtained through the sum of partial social capital. Its value is given by the following equation:

n k ik i PSC TSC 1 (4)
Where: i TSC -Total social capital of enterprise i ik PSC -Partial social capital of enterprise i according to perspective k

Potential Application

In order to analyze and measure the social capital of members in a collaborative network, the following example illustrates how the proposed approach can be used.

Let us consider a scenario, where we have a VO breeding environment (VBE) which contains seven organizations, as illustrated in Figure 2. Please note that the purpose of this example is only to illustrate the potential of this analysis, and in the network of business contacts, the "level of health" of the relationship among enterprises ( ij LH ) (see, Equation 1) is identical for all enterprises.

Fig. 2. Map of business contacts

Let us suppose the existence of three distinct classes of assets (Knowledge, Market, and Resources), as illustrated in Figure 3. Analyzing the graphs of enterprises' assets, it is possible to identify that in terms of knowledge (graph 3A) enterprise E4 does not have any asset while enterprise E7 has an exclusive asset K4. On the other hand, according to the analysis of resources (graph 3C), it is possible to verify that resource R2 is shared by both enterprises E4 and E7.

In order to analyze the social capital of an enterprise according to different perspectives, the following graphs are generated (Figure 4). From the perspective of innovation (graph 4A), it is possible to verify that enterprises E5 and E1 might have access to knowledge K4 (exclusive in this network) via the enterprise E7. On the other hand, from the perspective of market (graph 4B) enterprise E7, through enterprise E1, might access to market M1. From the perspective of resources (graph 4C) enterprise E4 might have access to all available resources within the network through enterprise E2 and E5. In order to illustrate how to estimate the social capital in a VBE context, let us make the following assumptions: There are records of the enterprises' past involvement in collaboration activities at the VBE management level, which makes possible to quantify the "level of health" of the relationship between enterprises ( ij LH ), such as: type of interaction among enterprises, number of contacts between members, value system alignment indicators, strength and nature of relationships in terms of potential benefits or losses. There are records of the type of assets that can be mobilized through the network at the VBE management level, and each enterprise decides on the relative importance/utility of each one of those assets.

In this scenario, let us suppose that enterprise E4 needs to have access to some assets in order to accomplish two business opportunities, as illustrated in the table 2. Table 3 shows the relative importance of each asset regarding the increase of the "survival capability" according to enterprise E4 perspective.

Considering that the "level of health" of the relationships among enterprise is identical and equal to 1 ( ij LH =1), table 4 shows the effective social capital for each business opportunity and the total social capital for a radius of 1 around enterprise E4 (see, Figure 2). For instance, in order to achieve the Partial Social Capital (PSC) value for business opportunity 1, applying Equation 2, we get the following equation:

PSC ESC

, where: 41 ESC -Effective social capital of enterprise E4 to perform the business opportunity 1 and ikl PSC -Partial social capital of enterprise E4 according to the knowledge perspective to perform business opportunity l Table 4 shows that for business opportunity 1 the effective social capital that enterprise E4 can get from the network is greater than for business opportunity 2, despite the amount of assets involved being smaller (table 2). Furthermore, as the effective social capital is positive for both business opportunities (table 4) means that enterprise E4 can operate more effectively compared to a scenario of working alone out of this VBE. On the other hand, based on this model, it is possible to analyse the potential maximum of social capital that an enterprise may have access through the network. As an example, by applying the Equation 4, the Total Social Capital value (TSC) for a radius of 1 around enterprise E4, is given by the following equation: In this example, as the worth of effective social capital is lower than the total social capital, it means that E4 is not extracting all advantages of belonging to this community.

Conclusion

Reaching a better characterization and understanding of the role of social capital in collaborative processes is an important element for a better understanding of the behavioural aspects in the collaborative networks paradigm in its various manifestation forms. The definition of a set of indicators to capture and measure the capital social can be a useful instrument to the VBE manager, as a way to support the promotion of collaborative behaviours, and for a VBE member as a way to extract the advantages of belonging to a network. Using simple calculations as illustrated above, it is possible to extract some indicators. Some preliminary steps in this direction, inspired by the Social Networks analysis, were presented. The proposed model, although simplistic, has the advantage of providing a visual/graphical representation which is easy to understand.

Fig. 1 .

 1 Fig. 1. Simple model to analyze Social Capital in CN's.

  of health of the relationship between enterprise i and enterprise j ij SRFC -Frequency of contact between enterprise i and enterprise j, based on subordinate relation. ij SRIC -Intensity of contact between enterprise i and enterprise j, based on subordinate relation. ij PRFC -Frequency of contact between enterprise i and enterprise j, based on peer relation. ij PRIC -Intensity of contact between enterprise i and enterprise j, based on peer relation. ij VS -Value systems alignment between enterprise i and enterprise j.

  social capital of enterprise i to perform an activity l ikl PSC -Partial social capital of enterprise i according to perspective k to perform an activity l

Fig. 3 .Fig. 4 .

 34 Fig. 3. Map of enterprises' assets -(A) Knowledge (B) Market and (C)

4 -

 4 Partial social capital of enterprise E4 according the knowledge perspective, 42 LH -"Level of health" of the relationship between enterprise E4 and enterprise E2 and, 2 Ak v -Value assigned to asset k 2 (see,

4 TSC

 4 -Total Social Capital of enterprise E4, and x PSC 4 -Partial Social Capital of enterprise E4 according to perspective x, where x is knowledge, resources, and markets.

Table 1 .

 1 Mapping between SNA and Map of Business contact This graph shows the assets held by each enterprise and how they are shared. In this case, there are two sets of nodes: enterprises and assets. The nodes are connected by ownership/access relations.Aggregate-map -Graph showing how an enterprise may have access to assets held by another enterprise. It results from the aggregation between the map of business contacts and map of enterprises' assets. Based on this map, it is now possible to analyze the social capital of each enterprise through a visual representation of the components that make up the social capital. Therefore, this map will be composed of two sets of nodes (enterprises and assets), and two types of links.

		Social Network Analysis (SNA)	Map of Business Contact
						Key concepts
	Node -A social discrete entity such	Enterprises, organizations, people
	as: enterprises, actors, corporate or
	collective social units
	Relational tie -Type of ties or links	Subordinate relation
	between nodes			Informal relation
	Dyad -consists of a pair of actors and	Subordinate relation
	the possible ties between them		Peer relation
	Structural Variables -measure ties of	Frequency of contact
	a specific kind between pairs of actors.	Intensity of contact
						Value systems alignment
	Composition	variables	-	are	Geographical localization
	measurements of actors' attributes.	Number of assets
						Basic Analysis
	Nodal Degree -is a measure of the	Measures the ability to have access to
	activity of the actor.		others nodes through the network
	Define indicators in order to measure:
	-	Actor degree centrality
	-	Group degree centralization
	-	Actor closeness centrality
	-	Group closeness centralization
	-	Actor betweenness centrality
	-	Group betweenness centralization
	-	Degree of prestige
	-	Proximity prestige
	-	Status or Rank prestige
	Network Size			Number of members of the CN
	Density of network		Level of contacts
	Connectivity of network		Measures the concept of reachability
		-	Cutpoints		between pairs of nodes.
		-	Bridges
		-	Walks

Table 3

 3

	). Based on Equation 3, the Effective Social Capital (ESC) is worth 9 units in
	this case	(	41	4	k	1	)

Table 2 -

 2 Assets required Table -Assets utilityTable4-Social capital as an abstract value

	Business	Assets	Asset	Value assigned to	Business	Effective Social	Total Social
	Opportunity	required	(An)	asset An (Van)	Opportunity	Capital	Capital
	1	K2	R1	3	1	9	19
	2	K3, R3, R1	R3	2	2	7
			K2	9
			K3	2
			M1	3

However, the development of practical indicators and a software tool to analyze the social capital in collaborative networks still requires further work.

Acknowledgments -This work was supported in part by the FCT-MCTES -"Fundação para a Ciência e Tecnologia" (CTS multiannual funding) through the PIDDAC Program funds".


J A Carrillo

S Fagioli
email: simone.fagioli@univaq.it.

F Santambrogio §
email: filippo.santambrogio@math.u-psud.fr.

M Schmidtchen
email: m.schmidtchen15@imperial.ac.uk.

SPLITTING SCHEMES & SEGREGATION IN REACTION-(CROSS-)DIFFUSION SYSTEMS *

Keywords: cross-diffusion, reaction-diffusion, splitting schemes, variational schemes, segregation, pattern formation AMS subject classifications. 35K57, 35A15, 47N60

One of the most fascinating phenomena observed in reaction-diffusion systems is the emergence of segregated solutions, i.e. population densities with disjoint supports. We analyse such a reaction cross-diffusion system. In order to prove existence of weak solutions for a wide class of initial data without restriction about their supports or their positivity, we propose a variational splitting scheme combining ODEs with methods from optimal transport. In addition, this approach allows us to prove conservation of segregation for initially segregated data even in the presence of vacuum.

Introduction.

In this work we consider the following reaction-diffusion system for the evolution on an interval Ω of two species with population densities ρ, η ≥ 0 :

(1.1)

∂ t ρ = ∂ x (ρ∂ x χ (ρ + η)) + ρF 1 (ρ, η) + ηG 1 (ρ, η), ∂ t η = ∂ x (η∂ x χ (ρ + η)) + ηF 2 (ρ, η) + ρG 2 (ρ, η),
where χ : [0, ∞) → [0, ∞) is a C 1 super-linear function modelling nonlinear diffusion and F i and G i , i = 1, 2 model the reaction phenomena. Systems of this type appear naturally in mathematical biology. A fundamental biological phenomenon in interactions among different biological species is the inhibition or activation of growth whenever two populations occupy the same habitat. One species may promote or suppress the proliferation of the other species. In models involving cells or bacteria, the limited growth of different cell types can be attributed to volume or size constraints of the individual cells forming the different populations. The diffusive part in (1.1) was originally introduced in the seminal papers [START_REF] Gurtin | A note on interacting populations that disperse to avoid crowding[END_REF] and [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF][START_REF] Bertsch | On a degenerate diffusion equation of the form c(z)t = ϕ(zx)x with application to population dynamics[END_REF] and exhibits an intriguing phenomenon: segregated densities remain separated at all times. In fact, nonlinear diffusions are natural ways to include volume filling effects into mathematical biology models, see [START_REF] Painter | Volume-filling and quorum-sensing in models for chemosensitive movement[END_REF][START_REF] Calvez | Volume effects in the Keller-Segel model: energy estimates preventing blow-up[END_REF] in the case of the classical Keller-Segel system. They help to avoid blow-up in these aggregation models in a biologically meaningful way and lead generically to asymptotic stabilisation. In the absence of reactions, the system leads to the nonlinear diffusion equation

(1.2) ∂ t σ = ∂ x (σ∂ x χ (σ)) = ∂ 2 x β(σ).
with σ = ρ+η, in which χ (σ) models the resistance to compression of the whole group of individuals σ. The natural assumption on (1.2) in order to be a diffusion equation is β (s) > 0 for s > 0, possibly degenerating at s = 0, or equivalently χ (s) > 0 for s > 0. The particular relevant case of χ(s) = s 2 /2 can be understood as the mean-field limit of interacting particles with very localised repulsion, see [START_REF] Oelschläger | Large systems of interacting particles and the porous medium equation[END_REF][START_REF] Burger | On an aggregation model with long and short range interactions[END_REF][START_REF] Bodnar | Friction dominated dynamics of interacting particles locally close to a crystallographic lattice[END_REF][START_REF] Klar | A multiscale meshfree method for macroscopic approximations of interacting particle systems[END_REF]. Related reaction-diffusion models to (1.1) appear in tissue growth models where cell adhesion and volume effects are important factors determining cell sorting in heterogeneous cell populations, see [START_REF] Murakawa | Continuous models for cell-cell adhesion[END_REF][START_REF] Carrillo | Zoology of a non-local cross-diffusion model for two species[END_REF][START_REF] Burger | Sorting phenomena in a mathematical model for two mutually attracting/repelling species[END_REF][START_REF] Carrillo | Adhesion and volume constraints via nonlocal interactions lead to cell sorting[END_REF], and zebrafish lateral line patterning [START_REF] Volkening | Modelling stripe formation in zebrafish: an agent-based approach[END_REF]. They are also basic building bricks for a variety of cancer invasion models in the literature [START_REF] Chaplain | Mathematical modelling of cancer cell invasion of tissue. the role of the urokinase plasminogen activation system[END_REF][START_REF] Preziosi | Cancer modelling and simulation[END_REF][START_REF] Stinner | Global existence for a go-or-grow multiscale model for tumor invasion with therapy[END_REF][START_REF] Johnston | On the proportion of cancer stem cells in a tumour[END_REF][START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF][START_REF] Armstrong | A continuum approach to modelling cellcell adhesion[END_REF][START_REF] Armstrong | Adding adhesion to a chemical signaling model for somite formation[END_REF][START_REF] Armstrong | The impact of adhesion on cellular invasion processes in cancer and development[END_REF][START_REF] Gerisch | Mathematical modelling of cancer cell invasion of tissue: Local and non-local models and the effect of adhesion[END_REF][START_REF] Domschke | Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns[END_REF][START_REF] Sfakianakis | A multiscale approach to the migration of cancer stem cells: Mathematical modelling and simulations[END_REF] in which the coupling with other biologically meaningful modelling factors such as extracellular matrix, enzymatic activators and other substances are taken into account. These works usually involve drift terms due to long range attraction and/or repulsion between individuals leading to related mathematical difficulties with respect to (1.1), see for instance [START_REF] Kim | On nonlinear cross-diffusion systems: an optimal transport approach[END_REF].

The nonlinear diffusion equation (1.2) is well-studied, see [START_REF] Vázquez | The porous medium equation[END_REF], and it can be understood as a gradient flow, see [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF], of the energy functional

E(σ) =    Ω χ(σ) dx χ(σ) ∈ L 1 (Ω) +∞ otherwise
, in the metric space of probability measures endowed with a suitable topology induced by the L 2 -Monge-Kantorovich distance denoted by d 2 . The wellposedness of solutions to the nonlinear diffusion equation (1.2) was obtained in [START_REF] Otto | The geometry of dissipative evolution equations: the porous medium equation[END_REF] by means of the so called JKO-scheme, cf. [START_REF] Jordan | The variational formulation of the Fokker-Planck equation[END_REF], which is a particular case of the minimising movement scheme by De Giorgi, see [START_REF] Ambrosio | Minimizing movements[END_REF] and the references therein. The idea of such a scheme is to recursively construct a sequence by solving a minimisation problem in a certain metric space (X, d), corresponding to the space of probability measures endowed with d 2 in our case. Given some initial condition σ for Eq. (1.2) and a fixed time step 0 < τ < 1 we set σ 0 τ = σ, and then recursively define

(1.3) σ n+1 τ ∈ argmin σ∈X 1 2τ d 2 (σ, σ n τ ) + E(σ) ,
for n ∈ N. The seminal work of R. J. McCann [START_REF] Mccann | A convexity principle for interacting gases[END_REF] shows that E(σ) is displacement convex or geodesically convex on the metric space of probability measures on the line endowed with d 2 as soon as β is nondecreasing on (0, +∞) or equivalently χ convex, called the McCann's condition in short. We also refer to [2, Chap. 9], [24, p. 26], or [START_REF] Villani | Optimal transport : old and new[END_REF]Chap. 17] for this classical notion, and [START_REF] Bolley | Nonlinear diffusion: geodesic convexity is equivalent to Wasserstein contraction[END_REF] for related issues. Upon choosing a proper time interpolation σ τ , it can be proven that the sequence {σ τ } τ converges to a weak solution of Eq. (1.2), see [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Optimal transport : old and new[END_REF][START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF] and the references therein.

In this work, we propose a variational splitting scheme in order to construct weak solutions to the reaction-diffusion system (1.1). More precisely, we solve in an inner time step the diffusive part of the system by the JKO scheme related to the nonlinear diffusion equation (1.2), and then we transport both densities ρ, η through the flow generated by the equation for the total population σ. Note that in this step the total and individual masses of the populations are unchanged in time. In the second inner time step of the splitting scheme, we solve the system of ODEs parameterised by the spatial variable x ∈ Ω leading to the final approximation of our new population densities after a time step. This variational splitting scheme will be written in details in Section 2. The splitting between reaction and diffusion steps is natural from the numerical analysis view point as it has already been used for variations of Keller-Segel models where the diffusion step is solved by the JKO scheme [START_REF] Blanchet | Convergence of the mass-transport steepest descent scheme for the subcritical Patlak-Keller-Segel model[END_REF][START_REF] Carrillo | A hybrid mass transport finite element method for keller-segel type systems[END_REF] in the case of a single population density coupled with a system of reaction-diffusion equations.

Our main result shows the convergence of the splitting variational scheme towards weak solutions of the system (1.1). The main mathematical difficulty here arises from the cross-diffusion term allowing for segregation fronts to form in the solutions. This phenomenon was proven in [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF] in the case of initial data with separated supports for the populations. More precisely, while [START_REF] Gurtin | A note on interacting populations that disperse to avoid crowding[END_REF] constructs a source solution of the system without reactions similar to the well-known Barenblatt-Pattle profiles [START_REF] Vázquez | The porous medium equation[END_REF] for nonlinear diffusions, [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF] constructs a solution to the system without reactions by formulating it as a free boundary problem for a single effective equation, and by characterising the segregation front through this free boundary. This approach can only work in case the support of both populations are at a positive distance to each other initially. Later [START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF][START_REF] Bertsch | A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth[END_REF] combined both the nonlinear diffusion and the reaction to obtain a system similar to (1.1) showing similar segregation phenomena by regularisation techniques. However, their approach heavily relies on the absence of vacuum as they assume that σ 0 is bounded below by a positive constant.

These remarkable results have severe consequences -initially smooth solutions lose their regularity when both densities meet each other. In fact, they become discontinuous at the contact interface immediately. This phenomenon legitimises our functional space choice as bounded functions of bounded variation, see the precise notion of weak solution and assumptions on the initial data in the next section.

In contrast to [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF][START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF][START_REF] Bertsch | A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth[END_REF], we show the convergence of our variational splitting scheme for general initial data even in the presence of vacuum and for general nonlinearities. Moreover, we recover their result in [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF][START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF] about segregation fronts, even in the case of initial vacuum, by showing that initial data which are initially segregated remain segregated for all times. An important technical point in our proof relies on displacement convexity of an auxiliary functional that allows to obtain further regularity on the approximate solutions in order to pass to the limit in the nonlinear diffusion terms. This auxiliary functional imposes a slightly more restricted set of nonlinear diffusions satisfying some integrability condition at the origin, see the precise conditions in the next section.

The rest of the paper is organised as follows: In Section 2 we introduce the variational splitting scheme, present the main result, and explain the strategy of the proof. Section 3 is dedicated to deriving all estimates necessary for proving the existence theorem as well as the segregation theorem.

2. Preliminaries and main result. As already mentioned, our main aim is to study the existence of weak solutions for the following one-dimensional two species cross-diffusion and reaction system:

(2.1)                ∂ t ρ = ∂ x (ρ∂ x χ (ρ + η)) + ρF 1 (ρ, η) + ηG 1 (ρ, η), in [0, T ] × Ω, ∂ t η = ∂ x (η∂ x χ (ρ + η)) + ηF 2 (ρ, η) + ρG 2 (ρ, η), in [0, T ] × Ω, ρ(t, x)∂ x χ (ρ + η) (t, x) = 0, on [0, T ] × ∂Ω, η(t, x)∂ x χ (ρ + η) (t, x) = 0, on [0, T ] × ∂Ω, ρ(•, 0) = ρ 0 , η(•, 0) = η 0 , in Ω,
where Ω ⊂ R is an open bounded interval, T > 0. Moreover, χ denotes an internal energy density and F i and G i , i = 1, 2 model the reaction phenomena. Throughout we shall use the notation L 1 + (Ω) in order to denote the set of non-negative Lebesgue integrable functions. As mentioned before the space of bounded functions with bounded variation is a natural functional setting. Definition 2.1 (Space of functions of bounded variations). Let f : Ω → R. We define its variation with respect to a partition

P := {x 1 < x 2 < • • • < x |P | } ⊂ Ω by V P (f ) := |P |-1 i=1 |f (x i+1 ) -f (x i )|.
We call f a function of bounded variation if its total variation sup P V P (f ) < ∞ is finite. Here the supremum is taken over all partitions of Ω. We denote by BV (Ω) the set of functions whose variation is bounded. Equipped with the norm

f BV := sup P V P (f ),
the set BV (Ω) is a vector space.

We shall see in the remainder of this section that the vector space of bounded function with bounded variation is a good choice to construct solutions. In our analysis we will exploit the following property. Lemma 2.2 (BV (Ω)∩L ∞ (Ω) is an R-algebra.). The vector space BV (Ω)∩L ∞ (Ω) equipped with the pointwise multiplication

BV (Ω) ∩ L ∞ (Ω) 2 (f, g) → f g ∈ BV (Ω) ∩ L ∞ (Ω),
is a real algebra.

The proof of the previous result is standard. Notice that in one dimension, BVregularity implies boundedness. We prefer to write BV (Ω) ∩ L ∞ (Ω) for the sake of clarity and possible future generalisations.

P m (Ω) := µ ∈ M + (Ω) µ(Ω) = m ,
that is, the set of positive measures with mass m > 0. Consider a measure µ ∈ P m (Ω) and a Borel map T : R → R. We denote by ν = T # µ ∈ P m (Ω) the push-forward measure of µ through T , defined by

Ω f (y) dT # µ(y) = Ω f (T (x)) dµ(x),
for all Borel functions f on Ω. We call T a transport map pushing µ to ν. We endow the space P m (Ω) with the p-Wasserstein distance, p ≥ 1,

d p p (µ 1 , µ 2 ) = inf γ∈Π m (µ1,µ2) Ω×Ω |x -y| p dγ(x, y) .
Here, Π m (µ 1 , µ 2 ) is the set of all transport plans, γ, between µ 1 and µ 2 , that is the set of positive measures of fixed mass γ(Ω × Ω) = m 2 , defined on the product space such that π i # γ = µ i , for i = 1, 2, where π i denotes the projection operator on the i-th component of the product space. If µ 1 is absolutely continuous with respect to the Lebesgue measure the optimal transport plan, γ, is unique and can be written as γ = (id, T ) # µ. In addition there exists a Kantorovich potential, ϕ, that is linked to the transport map, T , in the following way

T (x) = (id -∂ x ϕ)(x). (2.2)
We refer to [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF][START_REF] Villani | Optimal transport : old and new[END_REF][START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF] and the references therein for a good account of the properties of transport distances and the state of the art in gradient flows/steepest descents of functionals in metric spaces of probability measures. While transport distances are an incredibly powerful tool for dealing with transport PDEs exhibiting a gradient flow structure, it is not applicable in the presence of source terms. This is owing to the fact that it is only defined for two measures of the same mass.

To resolve this shortcoming we will make use of the Bounded-Lipschitz distance d BL , classically used for the derivation of the Vlasov equation, see [START_REF] Neunzert | An introduction to the nonlinear Boltzmann-Vlasov equation[END_REF][START_REF] Spohn | Large scale dynamics of interacting particles[END_REF][START_REF] Cañizo | A well-posedness theory in measures for some kinetic models of collective motion[END_REF][START_REF] Cañizo | Measure solutions for some models in population dynamics[END_REF][START_REF] Gwiazda | Structured population equations in metric spaces[END_REF] and the references therein. The Bounded-Lipschitz distance d BL , also frequently called flat metric, is defined as follows

d BL (µ, ν) := sup Ω f d (µ -ν) | f L ∞ (Ω) , f L ∞ (Ω) ≤ 1 = µ -ν (W 1,∞ (Ω)) * .
(2.3) Note that the Bounded-Lipschitz distance may be defined in any space dimension with the properties below. Thus, we shall state all required properties in our setting. Since our problem is posed on the product space we extend the metric setting to the space M + × M + , the product space of nonnegative measures in the canonical way. For d ∈ {d BL , d p }, 1 ≤ p < ∞, we define the product metric (still denoted d) as

d(U, Ũ ) := d(ρ, ρ) + d(η, η) , (2.4) where U = (ρ, η), Ũ = (ρ, η) ∈ M + × M + . Proposition 2.3 (Properties of d BL ). Let µ, ν ∈ L 1
+ (Ω) be two densities. Then the following properties hold true:

(i.) d BL (µ, ν) ≤ µ -ν L 1 (Ω) , (ii.) d BL (µ, ν) ≤ d 1 (µ, ν), whenever µ(Ω) = ν(Ω). Proof. Let µ, ν ∈ L 1 + (Ω) be arbitrary and f ∈ W 1,∞ (Ω) with f W 1,∞ (Ω) ≤ 1. Hence we may write Ω f d(µ -ν) ≤ Ω |f | d|µ -ν| = µ -ν L 1 (Ω) .
Taking the supremum over all such functions f we get the first statement by using definition (2.3). For the second statement, we additionally assume that µ(Ω) = ν(Ω).

We recall the dual definition of d 1 ,

d 1 (µ, ν) = sup Ω f d(µ -ν) | f ∈ Lip(Ω) s.t. f Lip(Ω) ≤ 1 ,
where Lip(Ω) is the set of Lipschitz-continuous functions, see [START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF]. Then property (ii.) is a consequence of this formulation, and this concludes the proof.

Corollary 2.4. Let U i = (µ i , ν i ) ∈ M + (Ω) 2 for i = 1, 2, 3. Furthermore as- sume µ i , ν i ∈ L 1 (Ω) for i = 1, 2 as well as µ 2 (Ω) = µ 3 (Ω) and ν 2 (Ω) = ν 3 (Ω). Then d BL U 1 , U 3 ≤ U 1 -U 2 L 1 (Ω) + d 1 U 2 , U 3 .
Throughout two quantities are crucial for our analysis -the sum of the two densities, σ, and the ratio, r, between one density, say ρ, and the sum

σ = ρ + η, and r = ρ σ ,
Note that, at this stage, the quotient can only be written formally. A straightforward computation shows these functions formally satisfy the following system of PDEs

     ∂ t σ = ∂ x (σ∂ x χ (σ)) + σ r( F1 + G2 ) + (1 -r) G1 + F2 , ∂ t r = ∂ x r∂ x χ (σ) + r(1 -r) F1 -F2 + (1 -r) 2 G1 -r 2 G2 , (2.5)
where we used Fi (σ, r) = F i (rσ, (1 -r)σ) and Gi (σ, r) = G i (rσ, (1 -r)σ), for i = 1, 2, to denote the reaction terms in the transformed variables. In order to simplify the analysis in Section 3, let us introduce the more concise notation A 1 (r, σ) := F1 + G2 , A 2 (r, σ) := G1 + F2 , and A 3 (r, σ) := F1 -F2 .

Note that these functions are Lipschitz and bounded as they are linear combinations of BV ∩ L ∞ functions. Thus the transformed system (2.5) can be rewritten in the more compact form

∂ t σ = ∂ x (σ∂ x χ (σ)) + σ rA 1 + (1 -r)A 2 , ∂ t r = ∂ x r∂ x χ (σ) + r(1 -r)A 3 + (1 -r) 2 G1 -r 2 G2 . (2.6)
Let us note here, that taking F i = G i = 0 in (1.1), yields the following nonlinear cross-diffusion system

∂ t ρ = ∂ x (ρ∂ x χ (σ)) , ∂ t η = ∂ x (η∂ x χ (σ)) , (2.7)
where the sum σ satisfies the nonlinear diffusion equation (1.2).

In order to be useful for our purposes, we need some properties on the internal energy density. The role of these properties will be clearer after the statement of our main result.

Definition 2.5 (Internal energy density). A function

χ : [0, ∞] → R is an internal energy density if (NL-i) χ ∈ C 0 ([0, ∞], R) ∩ C 2 ((0, ∞), R) with χ > 0, (NL-ii) lim h↓0 χ (h) = 0. (NL-iii) the integrals κ(x) := x 1 χ (s) s
ds, and K(σ) := σ 0 κ(x) dx exist. As mentioned in the introduction, the space of probability measures endowed with d 2 has proven to be an exceptional choice of a metric space. In this case the minimiser in Eq. (1.3) satisfies the optimality condition ϕ τ + χ (σ n+1 ) = const., (2.8) cf. [START_REF] Santambrogio | Optimal Transport for Applied Mathematicians[END_REF]Proposition 7.20], for instance. Here ϕ denotes the associated Kantorovich potential transporting σ n+1 to σ n in P m1+m2 , with m 1 and m 2 the masses of ρ and η, [44, Theorem 1.17]. Notice that this optimality is only obtained in those references for non-degenerate problems, where β (0 + ) > 0. However, similar approximation techniques as those developed in [41, p. 156] and [24, p. 27] allow to overcome the difficulties associated to degenerate diffusions, where β (0 + ) = 0. In the rest of the paper, we will proceed as if we were dealing with nonlinearities leading to nondegenerate diffusions since by this standard approximation procedure, the same result can be obtained for the degenerate ones. We are now ready to introduce our notion of weak solutions. > 0, and is therefore displacement convex. This fact will be crucial in Section 3.3 in order to prove Lemma 3.10. Let us just state here that it is necessary to obtain additional regularity from the dissipation of this functional on the nonlinear diffusion term, thus allowing us to pass to the limit in the approximating sequence.

Definition 2.6 (Notion of weak solutions

). A couple ρ, η ∈ C 0, T ; BV (Ω) ∩ L ∞ (Ω) is a weak solution to system (2.1) if σ = ρ + η ∈ L 2 (0, T ; H 1 (Ω)) and there holds Ω (ρ(t) -ρ(s)) ζ dx = t s Ω -ρ∂ x χ (ρ + η)∂ x ζ + (ρF 1 (ρ, η) + ηG 1 (ρ, η)) ζ dx dτ , Ω (η(t) -η(s)) ξ dx = t s Ω -η∂ x χ (ρ + η)∂ x ξ + (ηF 2 (ρ, η) + ρG 2 (ρ, η)) ξ dx dτ , for 0 ≤ s < t ≤ T any two test functions ζ, ξ ∈ C ∞ c (Ω). Remark 2.
2. Let us note that we can also allow for nonlocal reaction terms, i.e.

F i = W 1,i ρ + W 2,i η,
and similarly for G i , for i = 1, 2. The only assumption we need to impose on the kernels is that they are smooth and integrable. These models are found in modelling pattern formation, as for instance the kernel-based Turing pattern system [START_REF] Kondo | An updated kernel-based Turing model for studying the mechanisms of biological pattern formation[END_REF] or the nonlinear aggregation-diffusion system [START_REF] Volkening | Modelling stripe formation in zebrafish: an agent-based approach[END_REF].

3. Similarly, we can -at least formally -interpret the system (2.7) as a gradient flow of the functional

E(ρ, η) = Ω χ(ρ + η) dx,
in the product Wasserstein space.

2.2. Splitting scheme. We are now ready to introduce our splitting scheme for equation (2.1). Let some initial data ρ 0 , η 0 ∈ BV (Ω) ∩ L ∞ (Ω) be given such that there exists a function r 0 ∈ BV (Ω) such that σ 0 := ρ 0 + η 0 , and ρ 0 σ 0 = r 0 {σ0>0} , and 0 ≤ r 0 ≤ 1. Furthermore we assume F i and G i , i = 1, 2 are bounded and Lipschitz with respect to ρ and η and we impose G 1 (0, •) ≥ 0 and G 2 (•, 0) ≥ 0 to ensure positivity of solutions.

We fix 0 < τ < 1 and n ∈ {0, . . . , N }, with N ∈ N such that N τ = T . We then recursively construct the piecewise constant approximation to the system as follows. We impose (ρ 0 , η 0 ) = (ρ 0 , η 0 ), and then construct U n+1 = (ρ n+1 , η n+1 ) by the following scheme. We split the equation into a reaction step and a diffusion step on the time interval [t n , t n+1 ), with t n = nτ , for all 0 ≤ n ≤ N .

Reaction step.

The reaction phase consists of solving the system of ordinary differential equations

       ∂ t σ = Σ(σ, r) := σ rA 1 + (1 -r)A 2 , ∂ t r = R(σ, r) := r(1 -r)A 3 + (1 -r) 2 G1 -r 2 G2 , σ(t n ) = σ n , and r(t n ) = r n ,
in the time interval [t n , t n+1 ). We then set ρ n+1/2 := rσ t=(n+1)τ , and η n+1/2 := (1 -r)σ t=(n+1)τ . (2.9a) A straightforward computation reveals

∂ t (rσ) = rσ F1 + (1 -r)σ G1 , ∂ t ((1 -r)σ) = (1 -r)σ F2 + rσ G2 , (2.9b)
formally verifying that rσ and (1 -r)σ solve the reaction part of Eq. (2.1). Note that we shall solve the system rigorously in the subsequent section.

Diffusion step.

After the reaction phase we solve

(2.9c) U n+1 ∈ argmin U ∈P m 1 (Ω)×P m 2 (Ω) 1 2τ d 2 2 U, U n+1/2 + E(U ) ,
where m 1 = ρ n+1/2 (Ω) and m 2 = η n+1/2 (Ω). Let T denote the associated optimal transport map, i.e. T # U n+1 = U n+1/2 . We define

σ n+1 := ρ n+1 + η n+1 , and
r n+1 := r n+1/2 • T . (2.9d)
While the definition of σ n+1 is somewhat natural, the definition of r n+1 seems a bit surprising. Let us note here that, indeed, r n+1 = ρ n+1 /σ n+1 where σ n+1 > 0. For the precise argument we refer to Section 3.2, Eq. (3.3).

Combination of both steps -construction of a solution.

Throughout this paper we refer to Eq. (2.9a) as reaction step and to Eq. (2.9c) as diffusion step, respectively. Definition 2.8. [Piecewise constant interpolation] Let (r n , σ n ) n∈N be the sequence obtained from the splitting scheme. Then we define the piecewise constant interpolations by r τ (t, x) = r n (x), and σ τ (t, x) = σ n (x), as well as

ρ τ (t, x) = r n (x)σ n (x), and η τ (t, x) = (1 -r n (x))σ n (x), for all (x, t) ∈ Ω × [t n , t n+1
). Furthermore we write U τ := (ρ τ , η τ ).

We will say that two densities ρ, η ∈ BV (Ω) ∩ L ∞ (Ω) are segregated if the intersection of the interior of their supports is empty. We are now ready to state our main result.

Theorem 2.9 (Convergence to weak solutions). Let ρ 0 , η 0 ∈ BV (Ω) ∩ L ∞ (Ω) and assume there exists a function r 0 ∈ BV (Ω) such that r 0 = ρ 0 /(ρ 0 + η 0 ) on {ρ 0 +η 0 > 0} and 0 ≤ r 0 ≤ 1. Furthermore, we assume that the nonlinearity χ satisfies the assumptions in Definition 2.5 and the reaction terms F i and G i , i = 1, 2 are bounded and Lipschitz with respect to ρ and η with G 1 (0, •) ≥ 0 and G 2 (•, 0) ≥ 0. Then, upon the extraction of a subsequence, the piecewise constant interpolations (ρ τ ) τ >0 and (η τ ) τ >0 converge to a weak solution of system (2.1) in the sense of Definition 2.6. Moreover, if initially the two densities ρ 0 , η 0 are segregated, then, in the absence of cross-reactions, the limit densities ρ(t, •), η(t, •) remain segregated for all times.

3. Proof of the main result. This section is dedicated to proving the main result of the paper -the convergence of the approximation obtained by the splitting scheme to a solution of the system. It is organised as follows: in Section 3.1 and 3.2 we establish the crucial BV-estimates and L ∞ -bounds. In Section 3.3 we combine the estimates from the previous sections in order to get uniform estimates for a whole iteration. Finally, in Section 3.4 we show how to extract a convergent subsequence and identify its limit as a weak solution to system (2.1).

3.1.

Estimates for reaction step. Since the right-hand sides, Σ(σ, r), R(σ, r) are Lipschitz continuous in both components we note that the solution of the reaction system is unique. Proposition 3.1 (L ∞ estimates of the reaction step). Let (r n , σ n ) be given by our splitting scheme. Then there holds

0 ≤ σ n+1/2 ≤ σ n L ∞ exp(cτ ), and 0 ≤ r n+1/2 ≤ 1,
for some constant c > 0 independent of n and any x ∈ Ω.

Proof. We show first that there holds r n+1/2 ∈ [0, 1]. Assume the contrary, i.e. there exists an x ∈ Ω such that r n+1/2 (x) < 0 or r n+1/2 (x) > 1. If r n+1/2 (x) < 0 then, by continuity, there exists a time t (x) ∈ (t n , t n+1 ) such that r(t , x) = 0 and ∂ t r(t , x) < 0. However, this is absurd as

0 > ∂ t r(t , x) = R σ(t , x), r(t , x) = G1 σ(t , x), 0 ≥ 0.
Analogously, it can be shown that r n+1 (x) ≤ 1. Finally, for the positivity of σ we can use a similar argument. Let us assume σ n (x) ≥ 0 and σ n+1/2 (x) < 0 for some x ∈ Ω. Then, there exists another t such that 0 > ∂ t σ(t , x) = Σ σ(t , x), r(t , x) = 0, which clearly is a contradiction. For the L ∞ -bound we simply apply Gronwall's lemma and the fact that r ∈ [0, 1].

Since we control the L ∞ -norm of both r and σ the local solution provided by the Cauchy-Lipschitz Theorem is indeed global. Next we address the BV-estimates during the reaction step. Proposition 3.2 (Bounded variation of r n+1/2 and σ n+1/2 ). Let us consider (r n , σ n ) as initial data for our splitting scheme. Then, the reaction step is BV-stable in the following sense.

r n+1/2 BV + σ n+1/2 BV ≤ ( r n BV + σ n BV ) exp(cτ ),
for some positive c, depending only on the Lipschitz constants of F i , G i and the L ∞bounds on F i , G i , for i = 1, 2.

Proof. Using the transformed system, Eqs.(2.6), r and σ satisfy the following equations in the reaction step

∂ t σ = Σ(σ, r), and ∂ t r = R(σ, r).
Upon integrating in time we get σ(t) = σ(s) + t s Σ σ(τ ), r(τ ) dτ , and r(t) = r(s) + t s R σ(τ ), r(τ ) dτ . Now, let P ⊂ Ω be an arbitrary partition. We compute the variation of σ and r with respect to P and obtain

Q(t) := V P (σ(t)) + V P (r(t)) ≤ V P (σ(s)) + V P (r(s)) + t s V P (Σ(τ )) + V P (R(τ )) dτ , whence Q(t) ≤ Q(s) + c t s Q(τ ) dτ ,
where c only depends on the L ∞ -bounds and the Lipschitz-continuity of A i , for ∈ {1, 2, 3} and the L ∞ -bounds of σ and r. Applying Gronwall's lemma we finally obtain

Q(t) ≤ Q(s) exp c(t -s) .
Passing to the supremum on the right-hand side and then on the left-hand side yields the result.

3.2.

Estimates for diffusive step. This section is devoted to establishing BVestimates and L ∞ -bounds in the diffusive step. To this end we will make use of the following lemma. Lemma 3.3 (Same optimality conditions). (ρ n+1 , η n+1 ) be given by the JKO step for E, cf. (2.9c). Moreover, let

σ * ∈ argmin σ∈P m 1 +m 2 (Ω) 1 2τ d 2 2 (σ, σ n+1/2 ) + E(σ) ,
and T be the associated optimal transport map,

T # σ * = σ n+1/2 .
Then there holds σ * = ρ n+1 + η n+1 and T ρ = T η = T σ on supp(ρ n+1 ) ∪ supp(η n+1 ) for the associated optimal transport maps. Moreover the optimality conditions read

δE δρ + ϕ ρ τ = c 1 , δE δη + ϕ η τ = c 2 , and with δE δρ = δE δη = χ (σ n+1 ),
on supp(ρ n+1 ) ∪ supp(η n+1 ) as well as ϕ ρ = ϕ η = ϕ up to an additive constant.

Proof.

Let (ρ n+1 , η n+1 ) ∈ argmin 1 2τ d 2 2 (U, U n+1/2 ) + E(U ) ,
and let T ρ , T η be the associated transport maps, i.e. T ρ # ρ n+1 = ρ n+1/2 and T η # η n+1 = η n+1/2 . Note that the optimality conditions take the following form

χ (ρ n+1 + η n+1 ) + ϕ ρ τ = c ρ , on supp(ρ n+1 ), ≥ c ρ , elsewhere,
for ρ and

χ (ρ n+1 + η n+1 ) + ϕ η τ = c η , on supp(η n+1 ), ≥ c η , elsewhere,
for η, respectively. Thus the optimal maps are given by

T ρ = id + τ ∂ x χ (ρ n+1 + η n+1 ), on supp(ρ n+1 ), T η = id + τ ∂ x χ (ρ n+1 + η n+1 ), on supp(η n+1 ).
In particular, note that T ρ ≡ T η =: T , on supp(ρ n+1 ) ∩ supp(η n+1 ). We claim that

σ n+1 := ρ n+1 + η n+1 is a minimiser of the problem min 1 2τ d 2 2 (σ, σ n+1/2 ) + E(σ) . (3.1)
To this end suppose there exists σ * ∈ P m1+m2 (Ω) such that

1 2τ d 2 2 (σ * , σ n+1/2 ) + E(σ * ) < 1 2τ d 2 2 (σ n+1 , σ n+1/2 ) + E(σ n+1 ).
Then there holds

1 2τ d 2 2 (U n+1 , U n+1/2 ) + E(U n+1 ) = 1 2τ Ω |x -T ρ (x)| 2 ρ n+1 (x) dx + Ω |x -T η (x)| 2 η n+1 (x) dx + E(U n+1 ) = 1 2τ supp(ρ n+1 ) |x -T ρ (x)| 2 ρ n+1 (x) dx + supp(η n+1 ) |x -T η (x)| 2 η n+1 (x) dx + E(U n+1 ) = 1 2τ supp(ρ n+1 )\(supp(ρ n+1 )∩supp(η n+1 )) |x -T ρ (x)| 2 ρ n+1 (x) dx + supp(η n+1 )\(supp(ρ n+1 )∩supp(η n+1 )) |x -T η (x)| 2 η n+1 (x) dx + (supp(ρ n+1 )∩supp(η n+1 )) |x -T (x)| 2 σ n+1 (x) + E(U n+1 ) = 1 2τ supp(ρ n+1 )\(supp(ρ n+1 )∩supp(η n+1 )) |x -T ρ (x)| 2 σ n+1 (x) dx + supp(η n+1 )\(supp(ρ n+1 )∩supp(η n+1 )) |x -T η (x)| 2 σ n+1 (x) dx + (supp(ρ n+1 )∩supp(η n+1 )) |x -T (x)| 2 σ n+1 (x) + E(U n+1 ) ≥ 1 2τ d 2 2 (σ n+1 , σ n+1/2 ) + E(σ n+1 ) > 1 2τ d 2 2 (σ * , σ n+1/2 ) + E(σ * ) = 1 2τ Ω |x -S(x)| 2 σ * (x) dx + E(σ * ),
where S is the optimal transport map, such that S # σ * = σ n+1/2 . We may write σ * = ρ * + η * such that S # ρ * = ρ n+1/2 and S # η * = η n+1/2 . Then we may write

1 2τ d 2 2 (U n+1 , U n+1/2 ) + E(U n+1 ) > 1 2τ Ω |x -S(x)| 2 ρ * (x) dx + 1 2τ Ω |x -S(x)| 2 η * (x) dx + E(ρ * + η * ) ≥ 1 2τ d 2 2 (U * , U n+1/2 ) + E(U * ),
which is absurd for it implies U n+1 is not a minimiser. Hence we conclude that σ n+1 is a minimiser of problem (3.1) and, as a byproduct, T ρ ≡ T η on supp(ρ n+1 ) ∪ supp(η n+1 ) = supp(σ n+1 ). Finally let us address the construction of the partition σ * = ρ * + η * . To this end we set A := supp(σ * ) ⊂ Ω and recall that this set is compact. Therefore its complement in Ω is open, and we may write it as

A c = i∈N I i
i.e. as the countable union of open intervals. Taking the closure of these intervals, we see that

B := Ω \ i∈N cl(I i ),
satisfies A = B up to a set of measure zero. However, since σ n+1/2 is absolutely continuous the transport map cannot map intervals with positive mass onto a point. Thus, restricting S to B even makes the map injective and we can readily define its inverse. Setting ρ * = S -1

B # ρ n+1/2 and η * = S -1 B # η n+1/2
is a suitable decomposition of σ * , which concludes the proof. Proposition 3.4 (L ∞ stability of the diffusive step). Let (r n+1/2 , σ n+1/2 ) be given by the splitting scheme (2.9a). Then these quantities satisfy

0 ≤ σ n+1 ≤ σ n+1/2 L ∞ , and
0 ≤ r n+1 ≤ 1,
after the diffusion step for any 0 ≤ n ≤ N .

Proof. Choose x 0 ∈ argmax(σ n+1 ). Then, by the optimality condition, Eq. (2.8),

χ (σ n+1 ) + ϕ τ = c,
we have x 0 ∈ argmin(ϕ) where we used the fact that χ ≥ 0, cf. Definition 2.5, (NL -iii). Hence ϕ (x 0 ) ≥ 0 and consequently, by passing to the derivative in Eq.

(2.2) we get

T (x 0 ) = 1 -∂ xx ϕ(x 0 ) ≤ 1. (3.2)
where T is the transport map from σ n+1 to σ n+1/2 . After a change of variables we get

σ n+1 (x) ≤ σ n+1 L ∞ = σ n+1 (x 0 ) = T (x 0 )σ n+1/2 (T (x 0 )) (3.2)
≤ σ n+1/2 (T (x 0 ))

≤ σ n+1/2 L ∞ ,
for any x ∈ Ω. For the non-negativity we observe that T ≥ 0 by Brenier's theorem [START_REF] Brenier | Décomposition polaire et réarrangement monotone des champs de vecteurs[END_REF][START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Villani | Optimal transport : old and new[END_REF]. Note that we exploited here the fact that the problem is posed in one spatial dimension. Thus

σ n+1 (x) = σ n+1/2 • T (x)T (x) ≥ 0.
Finally the bounds for r n+1 follow from its definition, cf. Eq. (2.9d), as the composition with a monotone function does not change the infimum and the supremum of a function. This concludes the proof.

Proposition 3.5 (Bounded variation of r n+1 and σ n+1 ). Let (r n+1/2 , σ n+1/2 ) be given. After the diffusive step they satisfy the following estimate.

σ n+1 BV ≤ σ n+1/2 BV , and
r n+1 BV ≤ r n+1/2 BV .
Proof. The result for the BV-norm of the minimiser, σ n+1 , is shown analogously to the proof of Theorem 1.1., cf. [START_REF] De Philippis | BV estimates in optimal transportation and applications[END_REF]. Now we need to show that the BV-norm of the ratio r does not increase. Recall the definition of r n+1 , cf. Eq. (2.9d), as

r n+1 := r n+1/2 • T ,
where T is the transport map such that ρ n+1/2 = T # ρ n+1 and σ n+1/2 = T # σ n+1 . Note that it is indeed the same function and there holds T ≥ 0, by Brenier's theorem [START_REF] Brenier | Décomposition polaire et réarrangement monotone des champs de vecteurs[END_REF][START_REF] Brenier | Polar factorization and monotone rearrangement of vector-valued functions[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF][START_REF] Villani | Optimal transport : old and new[END_REF] . Now, let P ⊂ Ω be any partition of Ω. There holds

V P (r n+1 ) = V P (r n+1/2 • T ) = V P (r n+1/2 ) ≤ r n+1/2 BV ,
where P is another partition induced by the monotone map T . Taking the supremum over all partitions P , we get

r n+1 BV ≤ r n+1/2 BV .
Finally note that, indeed,

r n+1 (x) = ρ n+1 (x) σ n+1 (x) ,
on supp(σ n+1 ) as we shall see now. According to Lemma 3.3, the same transport map T pushes ρ n+1 onto ρ n+1/2 and σ n+1 onto σ n+1/2 . As a consequence the densities satisfy

ρ n+1 (x) = ρ n+1/2 T (x) T (x), and σ n+1 (x) = σ n+1/2 T (x) T (x), whence r n+1 (x) := r n+1/2 • T (x) = ρ n+1/2 σ n+1/2 • T (x) = ρ n+1 σ n+1 (x). (3.3)
3.3. Combined estimates for an entire splitting step. We have now garnered all information necessary to pass to the limit. Let us combine the estimates from the previous section in the following lemma. Proof. The uniform L ∞ -bound is a consequence of combining Propositions 3.1 and 3.4. We use these uniform L ∞ -bounds in the estimates for the BV -norm, cf. Propositions 3.2 and 3.5. Combining both for the reaction and diffusion step we also obtain the uniform BV -bounds.

As a result of Lemma 3.6 and Lemma 2.2 we obtain the following corollary.

Corollary 3.7 (BV -estimates and L ∞ -estimates for ρ, η). The sequences of approximated densities (ρ τ ) τ >0 , (η τ ) τ >0 are uniformly bounded, i.e.

ρ τ L ∞ (0,T ;L ∞ ∩BV ) ≤ C, and η τ L ∞ (0,T ;L ∞ ∩BV ) ≤ C,
where the constant C > 0 is independent of the parameter τ Finally, we need to prove an estimate on the cross-diffusion term to be able to pass to the limit later. This estimate is achieved in Lemma 3.10 which is preceded by two technical lemmas. We exploit the existence of an auxiliary functional guaranteed by Definition 2.5. Note that in the absence of the reaction part this would indeed be an entropy in the classical sense, i.e. that it is decayed along solutions. Since we are interested in a uniform estimate we shall begin by proving a control of this functional during each reaction phase. Lemma 3.8 (Control of the auxiliary functional in the reaction step). K increases at most at a constant rate independent of n. More precisely there holds

K(σ n+1/2 ) ≤ K(σ n ) + cτ, for any n ∈ N. Proof. A straight forward computation yields d dt Ω K(σ) dx = Ω κ(σ)∂ t σ dx = Ω κ(σ)σ(rA 1 + (1 -r)A 2 ) dx ≤ c,
using the uniform L ∞ -bound on σ and the fact that |Ω| < ∞. Hence,

K(σ n+1/2 ) ≤ K(σ n ) + cτ,
where c is independent of n.

Next, we address the diffusion step. As mentioned earlier the auxiliary functional, K, is an entropy for the diffusive part and from its dissipation we obtain the necessary regularity, as asserted in the following lemma. Lemma 3.9 (H 1 -bound for σ n+1 ). The minimiser of the JKO step satisfies the following estimate

τ ∂ x σ n+1 2 L 2 (Ω) ≤ K(σ n+1/2 ) -K(σ n+1 ) , for each 0 ≤ n ≤ N . Proof. Let (ρ n+1 , η n+1 ) ∈ argmin 1 2τ d 2 2 (•, U n+1/2 ) + E(U ) .
Let σ s = (T s ) # σ n+1 be the geodesic interpolation between σ s | s=0 = σ n+1 and σ s | s=1 = σ n+1/2 , given by

T s = (1 -s)id + sT , and
T = id -∂ x ϕ,
for the associated Kantorovich potential, ϕ, cf. Eq. (2.2). As a consequence the velocity field is given by

v s = (T -id) • T -1 s ,
satisfying the following continuity equation,

∂ s σ s = ∂ x (σ s v s ).
We differentiate the entropy along the geodesic and obtain

d ds Ω K(σ s ) dx = Ω κ(σ s )∂ s σ s dx = - Ω κ (σ s )∂ x σ s σ s v s dx = - Ω χ (σ s )∂ x σ s v s dx = - Ω ∂ x χ (σ s )v s dx.
Thus, at s = 0, the evolution of the entropy Eq. ( 9) becomes

d ds Ω K(σ s ) dx s=0 = Ω ∂ x χ (σ n+1 )∂ x ϕ dx.
Using the optimality condition, Eq. (2.8), we obtain

τ Ω ∂ x χ (σ n+1 ) 2 dx = - Ω ∂ x χ (σ n+1 )∂ x ϕ dx = d ds Ω K(σ s ) dx s=0 ≤ K(σ n+1/2 ) -K(σ n+1 ),
where the last inequality is a consequence of the geodesic convexity of the entropy, cf. Remark 2.7. This concludes the proof.

We combine the previous lemmas to obtain the desired estimate for a full iteration and finally for the piecewise constant interpolation, σ τ . Lemma 3.10 (Uniform L 2 ((0, T ) × Ω)-bound for ∂ x σ τ ). There holds

∂ x χ (σ τ ) L 2 ((0,T )×Ω) ≤ C,
for some positive constant depending only on T .

Proof. This statement is a consequence of combining Lemma 3.8 and Lemma 3.9 to get

τ ∂ x χ (σ n+1 ) 2 L 2 (Ω) ≤ K(σ n+1/2 ) -K(σ n+1 ) ≤ cτ + K(σ n ) -K(σ n+1 ).
Summing over n = 0 . . . N -1 gives

∂ x χ (σ τ ) 2 L 2 ((0,T )×Ω) ≤ cT + K(σ 0 ) -inf σ K(σ) ≤ C,
which yields the statement of the lemma.

Lemma 3.11 (Total-square 2-Wasserstein distance estimates). For every n ∈ {0, . . . , N } consider two consecutive steps for (2.9c), U n+1/2 = (ρ n+1/2 , η n+1/2 ) and U n+1 = (ρ n+1 , η n+1 ), then there exists a constant C such that

1 2τ N n=0 d 2 2 (U n+1/2 , U n+1 ) ≤ C.
Proof. Using the minimising property of U n+1 we have

1 2τ d 2 2 (U n+1/2 , U n+1 ) ≤ E(U n+1/2 ) -E(U n+1 ).
Adding and subtracting E(U n ) on the right-hand side, and considering that

E(U n+1/2 ) -E(U n ) = Ω χ(σ n+1/2 ) dx - Ω χ(σ n ) dx = Ω χ σ n + t n+1 t n σ(rA 1 + (1 -r)A 2 ) ds dx - Ω χ(σ n ) dx ≤ Ω χ (ξ(x)) t n+1 t n σ(rA 1 + (1 -r)A 2 ) ds dx
due to a Taylor expansion of χ around σ n , where ξ ∈ [σ n , σ n+1/2 ]. Owing to the uniform L ∞ -bounds on the σ and σ n+1/2 we may further simplify

E(U n+1/2 ) -E(U n ) ≤ χ L ∞ Ω t n+1 t n |σ(rA 1 + (1 -r)A 2 )| ds dx ≤ Cτ, (3.4)
where, in the last inequality, we used the L ∞ -bounds on σ, r as well as on F i , G i . Hence, we may summarise

1 2τ d 2 2 (U n+1/2 , U n+1 ) ≤ E(U n ) -E(U n+1 ) + Cτ,
for a full time step. Finally, summing over n gives the result:

1 2τ N n=1 d 2 2 (U n+1/2 , U n+1 ) ≤ N n=0 E(U n ) -E(U n+1 ) + Cτ = E(U 0 ) -E(U N τ ) + N (Cτ ) ≤ E(U 0 ) - inf U ∈M+×M+ E(U ) + CN τ ≤ C.
3.4. Convergence. We now prove the convergence result.

Proposition 3.12. The piecewise constant interpolations defined in Definition 2.8 admit subsequences converging uniformly to absolutely continuous curves, ρ, η, with respect to d BL with values in M + (Ω). Moreover, ρ and η are d BL -continuous functions on [0, T ].

Proof. The proof is based on the application of a generalised version of the Ascoli-Arzelà theorem, cf. Ref. [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], Section 3.

We begin by establishing 'almost continuity' of the approximation. To this end let 0 ≤ s < t ≤ T be two time instances. Then there exist two uniquely determined integers, m, k, m ≤ k -1, satisfying s ∈ ((m -1)τ, mτ ], and t ∈ ((k -1)τ, kτ ], such that

d BL (U τ (s), U τ (t)) ≤ k-1 n=m d BL (U n , U n+1 ) ≤ k-1 n=m d 2 BL (U n , U n+1 ) 1 2 |k -m| 1 2 , (3.5)
where U τ (t) = (ρ τ (t), η τ (t)) as defined in Definition 2.8. It becomes apparent that we need to address the bounded Lipschitz term next. To this end we use the triangulation established in Corollary 2.4 to estimate it by the L 1 -distance in the reaction step and the W 1 -distance in the diffusion step. Hence

k-1 n=m d 2 BL (U n , U n+1 ) ≤ 2 N n=0 U n -U n+1/2 2 L 1 (Ω) + 2 N n=0 d 2 1 (U n+1/2 , U n+1 ).
For the reaction step an argument similar to Eq. (3.4) yields

U n -U n+1/2 2 L 1 (Ω) = Ω |ρ n+1/2 -ρ n | + |η n+1/2 -η n | 2 = Ω (n+1)τ t n rσ F1 + (1 -r)σ G1 ds + (n+1)τ t n (1 -r)σ F2 + rσ G2 ds dx 2 ≤ Cτ 2 ,
by Eq. (2.9b), where C depends on the L ∞ -bounds of r, σ as well as F i , G i , for i = 1, 2. Using that the p-Wasserstein distances are ordered, we even have |k -m|

1 2 ≤ C √ τ |t -s| τ + 1 1 2 ≤ C( |t -s| + √ τ ).
Thus we get the 'almost 1 2 -Hölder continuity' for the curve U τ (t) and we obtain the uniform narrow compactness on compact time intervals by using the refined version of Ascoli-Arzelà's theorem, cf. [START_REF] Ambrosio | Gradient flows in metric spaces and in the space of probability measures[END_REF], Section 3. Corollary 3.13 (Strong convergence in L p (0, T ; L q (Ω))). Let 1 ≤ p, q < ∞ and (ρ τ ) τ >0 and (η τ ) τ >0 be the sequences of the piecewise constant interpolations as in Definition 2.8. Then there exist two functions ρ, η ∈ L p (0, T ; L q (Ω)) and subsequences, again denoted by (ρ τ ) τ >0 and (η τ ) τ >0 , such that ρ τ → ρ, and η τ → η, associated to the optimal map T , the integral above can be rewritten as (3.8)

We rewrite this equation in terms of the piecewise constant interpolation, Definition 2.8. For any 0 < s < t < T there are two uniquely determined integers m, k such that s ∈ (mτ, (m + 1)τ ] , and t ∈ (kτ, (k + 1)τ ] .

with r(t n ) = r n . By the uniqueness of the solution we have r(t) = r n on [t n , (n + 1)τ ] and therefore r n+1/2 = r n . Next we notice that supp(σ n+1/2 ) ⊂ supp(σ n ). This is due to the fact that the right-hand side of the equation for σ is premultiplied by σ ∂ t σ = σ(rA 1 + (1 -r)A 2 ), i.e. for any x ∈ Ω with σ(t n , x) = 0 there holds σ(t, x) = 0 for all t ∈ [t n , (n + 1)τ ] by the uniqueness of solutions, whence we deduce supp(σ n+1/2 ) ⊂ supp(σ n ). Hence we only need to prove that segregation is kept in the diffusion step. This is done by contradiction. Let us assume there exists an 1 ≤ n ≤ N such that Ω ρ n+1/2 (x)η n+1/2 (x) dx = 0, (3.9) while Ω ρ n+1 (x)η n+1 (x) dx > 0.

Then there exists δ > 0 and a set B ⊂ Ω with |B| > 0, such that |T (x)| < ∞ , ρ n+1 (x) > δ , and η n+1 (x) > δ,

for almost all x ∈ B. As both species have the same transport map T in common there exists a set A such that A = T (B) and 0 < δ < ρ n+1 (x) = ρ n+1/2 (T (x))T (x), 0 < δ < η n+1 (x) = η n+1/2 (T (x))T (x), which is absurd, for we assumed (3.9). Thus segregation is kept at each iteration which concludes the proof of the theorem.

2. 1 .

 1 Metric structure. Consider Ω ⊆ R an open bounded interval, and denote by M + (Ω) the set of positive and finite measures. Throughout this paper we will make use of the following notation

7 . 1 .

 71 Notice that the functional K(σ) := Ω K(σ) dx, associated to χ, as in Definition 2.5, satisfies the McCann condition since K (s) = χ (s) s

Lemma 3 . 6 (

 36 BV -estimates and L ∞ -estimates). The sequence (r τ , σ τ ) τ >0 obtained by the splitting scheme is uniformly bounded in L ∞ (0, T ; L ∞ ∩ BV ). More precisely there holds sup t∈[0,T ] r τ L ∞ ≤ C, and sup t∈[0,T ] σ τ L ∞ ≤ C, and sup t∈[0,T ] r τ BV ≤ C, and sup t∈[0,T ]σ τ BV ≤ C,for some positive constant C < ∞ only depending on T and the initial data but not on τ .

  n+1/2 , U n+1 ) ≤ Cτ,where we used the total-square estimate, Lemma 3.11. Therefore Eq. (3.5) becomesd BL (U τ (s), U τ (t)) ≤

( 3 . 6 ) 1 = 1 =

 3611 1 τ Ω ρ n+1 (x) -ρ n+1/2 (x) ζ(x) dx = -1 τ Ω ρ n+1 (x) (T (x) -x) ∂ x ζ(x) dx + O d 2 2 (ρ n+1 , ρ n+1/2 ) = 1 τ Ω ρ n+1 (x)∂ x ϕ(x)∂ x ζ(x) dx + O d 2 2 (ρ n+1 , ρ n+1/2 ) .Thanks to the optimality condition, Lemma 3.3, we get1 τ Ω ρ n+1 -ρ n+1/2 ζ dx = -Ω ρ n+1 ∂ x χ (σ n+1 ) ∂ x ζ dx + O d 2 2 (ρ n+1 , ρ n+1/2 ) . Reaction part. Note that Ω ρ n+1/2 -ρ n τ ζ dx = Ω r n+1/2 σ n+1/2 -r n σ n r(τ )σ(τ ) ζ dτ dx, = Ω r n+1/2 σ n+1/2 F1 + (1 -r n+1/2 )σ n+1/2 G1 ζ dx + O(τ ) F 1 (ρ n+1/2 , η n+1/2 ) and G n+1/2 G 1 (ρ n+1/2 , η n+1/2), having used an approximation of Eq. (2.9b). Combination of both steps. Let us combine the reaction step and the diffusion step of the splitting scheme. Upon summing up Eqs.(3.6, 3.7) we obtain0 = 1 τ Ω ζ(x)(ρ n+1 (x) -ρ n (x)) dx + O(d 2 2 (ρ n+1 , ρ n+1/2 )) + Ω ρ n+1 ∂ x χ (ρ n+1 (x) + η n+1 (x))∂ x ζ(x) dx -Ω ρ n+1/2 F 1 (ρ n+1/2 , η n+1/2 ) + η n+1/2 G 1 (ρ n+1/2 , η n+1/2 ) ζ dx + O(τ ).

J. A. CARRILLO, S. FAGIOLI, F. SANTAMBROGIO, AND M. SCHMIDTCHEN

* JAC was supported by the Engineering and Physical Sciences Research Council (EPSRC) under grant no. EP/P031587/1. Most of the work was performed during a visit of FS to Imperial College thanks to a joint Imperial-CNRS Fellowship, the hospitality and support of both Imperial College and CNRS are warmly acknowledged. SF was partially supported by the local fund of the University of LAquila "DP-LAND" (Deterministic Particles for Local And Nonlocal Dynamics), and by the Erasmus Mundus programme "MathMods", www.mathmods.eu. SF thanks the Department of Mathematics at Imperial College in London for hosting him for a visiting period during which this work was carried out.

strongly in L p (0, T ; L q (Ω)) as τ → 0. Moreover the convergence holds pointwise in time, i.e. for all t ∈ [0, T ] there holds ρ τ (t) → ρ(t), and η τ (t) → η(t), strongly in L q (Ω).

Proof. Note that it suffices to show the result for (ρ τ ) τ >0 as the same argument applies for (η τ ) τ >0 . By Proposition 3.12 we can extract a subsequence, still denoted the same, such that

Furthermore, from the uniform BV ∩ L ∞ -bounds, for any t ∈ [0, T ] the sequence converges strongly in L 1 (Ω). Using the uniform L ∞ -bounds and the dominated convergence theorem, we obtain the pointwise-in-time convergence in any L q (Ω). Now let us apply the same argument on the whole domain, [0, T ] × Ω. The pointwise convergence and the uniform L ∞ (0, T ; L ∞ (Ω))-bound imply ρ τ → ρ, strongly in L p (0, T ; L q (Ω)) by the dominated convergence theorem. This concludes the proof. Lemma 3.14 (Identification of the limit). The sequence constructed in (2.9a), (2.9c) converges to a weak solution of Eqs. (1.1).

Proof. The proof consists of two parts -the diffusion part and the reaction part. We write them in their respective weak formulation and combine them to obtain the complete approximation of the weak formulation. Here we only show the argument for ρ as the corresponding result for η is shown analogously. We begin with the diffusion part.

Diffusion part. We consider the two steps before and after the application of the JKO scheme (2.9c),

and

For a given test function ζ ∈ C ∞ c (Ω), let T be the optimal transport map from ρ n+1 to ρ n+1/2 . Upon integration over Ω we get

Taylor expanding ζ(T (x)) around x yields

Moreover, using the fact that T (x) = x-∂ x ϕ(x), where ϕ is the Kantorovich potential Multiplying Eq. (3.8) by τ and summing from m to k -1, we obtain

-ρτ (τ , x)F 1 ρτ (τ , x), ητ (τ , x) + ητ (τ , x)G 1 ρτ (τ , x), ητ (τ , x) ζ(x) dxdτ .

We are now ready to pass to the limit. The strong convergence obtained in Corollary 3.13 allows us to pass to the limit in the nonlinear reaction terms. Moreover the crossdiffusion term converges due to the weak-strong L 2 (0, T ; L 2 (Ω))-duality, by Lemma 3.10 and Corollary 3.13. Passing to the limit τ → 0 we obtain the weak formulation for ρ. The same argument for η yields the statement.

We end the paper with a stunning result which can be seen as a generalisation of the result of Bertsch et al., cf. [START_REF] Bertsch | On interacting populations that disperse to avoid crowding: preservation of segregation[END_REF][START_REF] Bertsch | On a degenerate diffusion equation of the form c(z)t = ϕ(zx)x with application to population dynamics[END_REF]. In their papers they prove that initially segregated species stay segregated at all times. In these works only the cross-diffusion system is considered and the authors even allow for vacuum however imposing a rather restrictive assumption on the initial datum. We can drop their assumption that supp(ρ 0 ) < supp(η 0 ), i.e. that both species are ordered and prove the following, more general theorem. In the works [START_REF] Bertsch | A free boundary problem arising in a simplified tumour growth model of contact inhibition[END_REF][START_REF] Bertsch | A nonlinear parabolic-hyperbolic system for contact inhibition of cell-growth[END_REF] the authors study the system including reaction terms for initial data that are not necessarily ordered. However they impose boundedness away from zero which we are also able to drop. Theorem 3.15 (Segregation in the case of no cross-reaction.). Assume no crossreaction terms, i.e. G 1 ≡ G 2 ≡ 0 and that both species are initially segregated, i.e. Ω ρ 0 (x) η 0 (x) dx = 0.

Then, there exists a solution such that both species stay segregated at all times.

Proof. It suffices to show this property at the level of the discrete scheme since, once it is established, we use the strong L 2 -convergence of ρ τ and η τ to show segregation is also kept in the limit:

Thus let us now show the property at the level of the approximation. During the reaction step the segregation of ρ and η is kept as can be seen as follows. First recall that in the reaction step we solve Eq. (2.9a), with initial data σ n , r n such that

on the support of σ n . Thus we conclude that already r n (1 -r n ) = 0, on supp(σ n ). Using the fact that G1 = G2 ≡ 0 the equation for r reads


José Alberto Arauzo

Adolfo Lopez Paredes
email: adolfo@insisoc.org

Javier Pajares
email: pajares@insisoc.org

An Agent Based Information System for Project Portfolio Management

Keywords: information systems, multi-agent systems, project portfolio, scheduling

With the increase in globalization, integration of world markets, and rise in the concept of transnational corporations, the importance of project management has increased many folds. An efficient project management helps new enterprises to achieve competitiveness, speed of response, and closeness to customer demands by improving their flexibility and agility, while maintaining their productivity and quality. But for project management to be successful in the new competitive and dynamic environment management, systems should incorporate new characteristics. In these circumstances, the current challenge is to develop information and control systems for project management that exhibit intelligence, robustness and adaptation to the environment changes and disturbances. The introduction of multi-agent systems paradigms addresses these requirements, bringing the advantages of distribution, autonomy, scalability and re-usability. This paper proposes a distributed information and control system for project portfolio management that integrates strategic issues, planning and control into a community of software agents.

Introduction

Projects affect success rate of the organization and thus are vital for them. A project results in bringing something new. It can be a new product or service, increasing sales or capturing of new markets, reducing cost, or an innovative feature in the existing product. Project management helps organizations working in various places integrating their operations and achieving goals.

Project portfolio managers are involved in developing methodologies for valuing projects, selecting the best ones, balancing the portfolio in terms of risk, cost, etc, and coordinating the joint execution of individual projects, so that synergies can be achieved. The output of this process is a set of projects to be done, ranked in terms of strategic and financial importance to the firm.

Once the firm knows the "right projects to do", it is necessary to plan how "to do things right". Therefore, project schedules need to be developed, taking into account the availability of resources during each period.

Moreover, uncertainty overruns and dynamic changes make portfolio management more complex. Priority changes over time because of the addition of new interesting projects, changes in corporate strategy or simply, because of feedback information about individual project overruns affecting their expected returns.

In this paper, we propose an auction approach for online dynamic scheduling in multi-project environments, where projects can be rejected as a result of the scheduling tatônnement process. We use a multi-agent system where both the resources and the projects are artificial agents which play in real time an auction to maximize the portfolio efficiency.

This agent-based approach has two distinctive aspects with respect to other works: the integration of strategic decisions (accepting or rejecting new projects) and operative aspects (resource allocation), and the ability to manage resource flexibility.

Multi-agent systems for project planning and control

Projects are characterized by complexity (they include many components and dependencies), uncertainty (about the availability of resources, task durations), dynamic behavior (changes in the scope of the project, adding or removing unexpected tasks, re-scheduling processes) and are inherently distributed (each task may be completed by different resources or in different geographical locations). In the case of a multi-project environment, each one of these features is severely intensified.

In the last decade, many ideas of Artificial Intelligence (AI) have been proposed to solve operation management problems [START_REF] Kobbacy | AI and OR in management of operations: history and trends[END_REF]. Multi-agent systems are a distributed IA paradigm precisely designed to deal with problems of these characteristics. As Jennings and Wooldridge point out, multi-agent systems are suitable for problems having the following properties: complexity, openness (components of the system are not known in advance, can change over time, and are highly heterogeneous, dynamic in project management terms), with dynamical and unknown environments changing over time (uncertainty) and ubiquity (the activity is distributed over the complete structure) [START_REF] Jennings | Applying agent technology[END_REF] [START_REF] Wooldridge | An Introduction to Multiagent Systems[END_REF].

Depending on the physical or organizational structures, many problems can be abstracted and managed as distributed systems using the multi-agent paradigm as metaphor. This decentralized approach in project management has been used since the last decade [START_REF] Yan | Application of Multi-Agent Systems in Project Management[END_REF], but it is in the last years when market based approaches [START_REF] Clearwater | Market-Based Control: A Paradigm for Distributed Resource Allocation[END_REF] are receiving a growing interest. Recently, Lee, Kumara and Chatterjee [8][9] have proposed a multi-agent based dynamic resource scheduling for distributed multiple projects using market mechanisms. Following the same research line, Confessore et al [START_REF] Confessore | A market-based multi-agent system model for decentralized multi-project scheduling[END_REF] propose an iterative combinatorial auction mechanism as coordination mechanism to resolve the same problem.

Other examples of agent based approaches in project management field can be found in the work of Kim and colleagues [START_REF] Kim | Distributed coordination of project schedule changes using agent-based compensatory negotiation methodology[END_REF], Wu and Kotak [START_REF] Wu | Agent-based collaborative project management system for distributed manufacturing[END_REF] or Cabac [START_REF] Cabac | Multi-agent system: A guiding metaphor for the organization of software development projects[END_REF]. As underlined in the introduction, our work makes use of a market metaphor and an auction mechanism to help project managers to take portfolio decisions about resources and about portfolio composition.

We define a multi-project scheduling environment in which the information system must operate.

At any instant t there are I projects in the system, each one denoted by i. Each project is characterized by a value V i , that can be interpreted as the revenue obtained for the project, a weight w i representing the strategic importance given to the specific project, a desirable delivery date D i , a limit delivery date D i * that cannot be exceeded, an arrival date of the project to the system, B i , and a limit answer date R i that represents the latest date to decide to reject the project.

The system is considered dynamic: while some projects are being developed other projects can be included or rejected in real time.

Each project i consists of J i activities, each one denoted by ij where i {1, 2,…, I} and j {1, 2,…, J i }. Let d ij the standard duration associated to activity ij and M the set of available resources. Each resource m {1, 2, 3…M} can just be assigned simultaneously to one activity (figure 1).

A set H of K competences H={h 1 , h 2 , ... h K } are necessary to complete the projects. Each resource is endowed with a given cost rate per unit of time, c m , and a subset H m of H of competences that can be performed. Each resource has a certain grade or ability to perform a competence. Therefore, the work capacity of resources can be symbolized by means of a vector of abilities per resource e m =(e m1 , e m2 ,…,e mk ), where e mf ≥ 0 shows the ability degree of resource m to perform the competence h f . If e mf = 0 then the resource m has not got the competence h f , if 0 <e mf < 1 the resource is able to perform inefficiently the competence h f , if e mf = 1 it has standard efficiency to perform the competence, and if e mf > 1 it will do it efficiently. Every activity j of project i is associated with a competence h(i,j). Any activity ij with a given h(i,j) can be performed by a resource m just if e m,h(i,j) ≥ 1. The duration of the activity ij depends on the resource assigned to perform it. We denote this duration as d ijm (duration of activity j of project i in resource m). It is calculated according to d ijm =d ij /e m,h(ij) , where d ij is the standard duration of activity j of project I (activity duration when is performed by a resource that has standard efficiency).

We include explicitly the option of reassigning resources in real time when a new project arrives to the system or when a disruption happens. The activities can be assigned to any resource that has the specific competences to perform it.

The overall efficiency (E) of the system will be evaluated by the average benefit obtained in a certain time interval T according to [START_REF] Cabac | Multi-agent system: A guiding metaphor for the organization of software development projects[END_REF]:

T TC V T B E i i i T ) ( (1)
where i are each one of the projects finished in T and TC i is the total cost to complete the i project. This cost has two components, the resource direct cost and the delay cost (expression 2).

* * 2 1 ) , ( ) ( 0 i i i i i i i i i i i J j ijm j i m i D F si D F D si D F w D F si where w d c Cost i (2)
The first addend corresponds to the resource direct cost to finish each activity j of project i. m(i,j) denotes the resource selected to comply with activity ij, and c m(i,j) the cost rate per unit of time of selected resource. The second addend is the delay cost associated to the project, where F i is the real conclusion date of the project. This cost is zero if the project is concluded before the desirable delivery date D i . If the project conclude between D i and the limit delivery date D i * , the cost will be proportional to the square of the delay over D i (F i -D i ). If the project is concluded after D i * the cost will be consider infinite. The problem considers the decision to reject projects if the benefit of perform the project don"t compensate the increment of total cost.

Multi-project environment definition

The system is mainly composed of two groups of agents: project and resource managers. A third type of agent is required in the system: The Monitoring-Auctioneer-Creator Agent (MAC). There is only one agent instance of this type which receives bids and asks from projects and resources and computes the exchange price to allocate resources to projects.

Project Manager Agents

Each project is represented by a Project Manager Agent (PMA) which is created and deleted in real time (new projects are evaluated to be accepted in the existing project portfolio, while the projects which finish are removed from the portfolio). Each one is characterized by its tasks, precedence relationships, due date, value, local programs and their execution state. The individual objective of each project agent is to look for "contracts" with resources that can perform the required activities and hence completing successfully the project. In order to achieve their goal, project agents make a plan that takes into account only their own activities (local schedule)

Resource Manager Agents

The system includes as many Resource Manager Agents as resources are considered. They are defined by their competences or skills and efficiencies. Each resource agent is aiming at performing tasks from project agents. The use of a resource is controlled by means of a Resource Manager Agent. Its goal is to increase it"s the level of occupation of the resources and the incomes. Project manager agents and resource manager agents make contracts to perform pending tasks of projects. The contracts will be based in the prices that emerge from an auction which is handled by the MAC Agent (Monitoring-Auctioneer-Creating Agent).

Monitoring-Auctioneer-Creating Agent (MAC)

The MAC agent plays three roles of centralized nature: (i) it creates manager Project Manager Agents when new projects are added to the system, (ii) it monitors the resource activities, and (iii) it plays as an auctioneer in a market procedure. This agent compiles real time data of resource states (i.e. availability, allocated activities, and current operations in resources) and shows a summary of this information. Moreover, as an auctioneer, it bids available time intervals of resources to projects, initiating an auction procedure that allocates activities to time intervals of resources.

Auction interactions

In our distributed multi-project system, the decision-making system is decentralized; it is distributed among every agent. Thus, each project creates its own schedule (local schedule) based on its own goal and information. In principle, this procedure entails some weaknesses, e.g. it can bring incompatible local schedules (several projects use simultaneously the same resource) and the local schedules can be globally inefficient, profitable projects may be rejected; most important projects may be delayed, etc. Accordingly, if we aim to design a functional decentralized scheduling process we need to resolve these difficulties. To do that, we propose a market mechanism (it was initially proposed by Davis R and Smith RG [START_REF] Davis | Negotiation as a Metaphor for Distributed Problem Solving[END_REF]) that ensures that local schedules are nearly compatible and globally efficient according to the expression (1). This market based multi-project scheduling approach is founded on Lagrangian Relaxation [START_REF] Luh | Scheduling of Manufacturing Systems Using the Lagrangian Relaxation Technique[END_REF], [START_REF] Zhao | Surrogate Gradient Algorithm for Lagrangian Relaxation[END_REF], a decomposition technique for mathematical programming problems.

In order to apply the market metaphor, the periods of time when resources are available are subdivided in a set small time intervals or time slots. Each time slot of every resource is modelled as a "good" that can be sold in an auction where the auctioneer acts as a seller. Thus, a local schedule is a bundle of time slots that has been allocated to a project.

The auctioneer (MAC Agent) proposes a price for each time slot from the current time to the end of the scheduling horizon. The scheduling horizon changes dynamically by coinciding with the latest time slot that some project has asked at any moment.

Project agents play the role of "bidders" in the auction mechanism. They bid for the required set of time slots needed to complete their pending tasks at the current time. Project agents try to find a set of time slots (Z i ) through the resource pool while incurring the minimum possible local cost (LC i ). This cost has two components (expression 3), the sum of the price of the selected time slots and the delay cost.

) (

i i i Z mt mt i F D w p LC i (3)
where p mt is the price of the time slot (t) of the resource (m).

If a project agent do not find a set of time slots that allows to schedule all the pending tasks before D i * with a cost smaller than the project value (V i ), it will not ask for any set of time slots. This indicates that the project is unprofitable at the correspondent round of bidding and must be rejected. Depending on the demand of bids, each resource agent modifies the price of its time slots to maximize its revenue, reducing at the same resource conflicts. In order to get this goal a subgradient optimization algorithm is used to adjust prices at each round of bidding. By means of this algorithm resource agents increase the price of the time slots where there is conflict (i.e. more than a project have asked for this time slot) and reduce the price of those time slots that have not been demanded. The process of price adjustment and bid calculation continues indefinitely decreasing the number of resource conflicts at each round.

Contract Interactions

The auction mechanism described above allows project agents to build compatible and globally efficient local schedules for their pending activities. Agents also interact through a complementary process to make firm agreements based on their local schedules. These agreements determine fixed programs for earliest scheduled tasks. When an agreement is closed, project agents remove the task involved in the agreement from the pending task queue.

Working example

To show the system performance we present a little example. We consider three different resources (R1, R2 and R3), endowed with the competences C1, C2 and C3 respectively. In table 1, we show a portfolio of five projects, and the tasks needed to complete the project. Each task is defined by means of the pertaining competence and expected standard time to be completed. The arrival date is the date when the project is included in the system. Projects can start in the starting date; otherwise, they should have been rejected before this date. Due Date 1 (DD1) is the most desirable duration whereas Due Date 2 (DD2) is the maximum allowed. All the projects have a weight of 1.

Figures 1 shows the system state at a given time (current time). In the upper area of the figures the relative duality gap evolution is presented. The prices of time slots are the solution of the dual problem and the duality gap is a measure of the difference between the primal and dual objective function, and hence it quantifies the quality of the solution [START_REF] Luh | Scheduling of Manufacturing Systems Using the Lagrangian Relaxation Technique[END_REF]. The relative duality gap is calculated as the duality gap divided by the dual solution. A small relative duality gap means that the prices are representative of the system state, thus, a good solution is achieved. In the lower part of the picture, charts of resources are presented.

In figure 1, we show the evolution of the tasks performed by each resource and the prices of the time slots after finishing the simulation. This picture is from the monitoring provide by the MAC Agent. We can see how the duality gap increases when the projects C and E arrive to the system. At this moment previous prices did not reflect the new system state (there are new projects in the system). After some time, the prices change to adapt themselves to the new system conditions, and the gap decreases again. The system not only gives us the dynamic schedule and the rejected projects, but the value of each resource as well.

Conclusions

Project portfolio management is a management process designed to help an organization to acquire and view information about all of its projects, then sort and prioritize each project according to certain criteria. Currently, in the global market, these activities have high influence in the enterprise competitiveness and.

Fig. 1 .

 1 Fig. 1. Tasks performed by resources. Tij denotes Task j of project Pi.

Table 1 .

 1 Example.

	Projects Task 1 Task 2 Task 3 Arrival	Starting	DD1 DD2	Value
					date	Date
	A	C1 35	C3 40	C1 25	0	0	120	180	30000
	B	C3 50	C1 25	C2 30	0	0	120	180	10000
	C	C2 30	C3 50	C1 10	50	90	150	270	15000
	D	C2 40	C3 45	C2 10	0	0	180	240	12000
	E	C3 45	C2 20	C3 50	50	90	150	270	30000

We propose a multi-agent system and an auction mechanism for online dynamic project selection, scheduling and control in multi-project environments. Projects have tasks to be completed, so they compete for the resources endowed with the capabilities required to do some pieces of work. The prices of resources emerge endogenously by means of an auction process.

We show some of the possibilities of our multi-agent approach to deal with some of the decisions that a manager needs to take within multi-project environments. The system allocates in real time resources to projects, and decides about project acceptance or rejection taking into account project value, profitability and (feedback) operational information. We also show how it is possible to discover which resources are the most valuable, so they should be preserved or added to the firm.