Conservation laws and asymptotic behavior of a model of social dynamics

Nonlinear Analysis: Real World Applications 9 (2008) 183–196www.elsevier.com/locate/na

Conservation laws and asymptotic behavior of amodel of social dynamics

Maria Letizia Bertottia,∗, Marcello Delitalab

aDipartimento di Metodi e Modelli Matematici, Università di Palermo, ItalybDipartimento di Matematica, Politecnico di Torino, Italy

Received 13 February 2006; accepted 12 September 2006

Abstract

A conservative social dynamics model is developed within a discrete kinetic framework for active particles, which has beenproposed in [M.L. Bertotti, L. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in appliedsciences, Math. Mod. Meth. Appl. Sci. 14 (2004) 1061–1084]. The model concerns a society in which individuals, distinguished bya scalar variable (the activity) which expresses their social state, undergo competitive and/or cooperative interactions. The evolutionof the discrete probability distribution over the social state is described by a system of nonlinear ordinary differential equations.The asymptotic trend of their solutions is investigated both analytically and computationally. Existence, stability and attractivity ofcertain equilibria are proved.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Kinetic theory; Discretization; Boltzmann models; Population models; Nonlinearity; Asymptotic stability

1. Introduction

Several problems of interest in the applied sciences concern systems which are composed by a large number ofelements and have the following property: they depend, at the microscopic scale, not only on mechanical variables, suchas position and velocity, but also on some other quantity, the activity, which expresses a specific function of the elements,e.g. their social or biological state. Examples in this connection can be found in the literature on the competition betweentumor and immune cells, [2,4,5], the vehicular traffic flow, [3,12], the social dynamics, [10,11,14,17], the populationdynamics in biology, [13,16], and so on.

To investigate the behavior of similar systems, methods of the mathematical kinetic theory have been developed,somehow similarly to those for diluted gas. This mathematical approach is known as “generalized Boltzmann equation”,see e.g. [1,15], and the interacting entities are called “active particles”. The approach is based on the assumption thatthe microscopic state is a continuous variable defined on a bounded or unbounded domain. On the other hand, there aresituations in which the scalar social/biological state, rather than being naturally representable by a continuous variable,can attain only a finite number of values.

∗ Corresponding author. Tel.: +39 091 6657338; fax: +39 091 427258.E-mail addresses: [email protected] (M.L. Bertotti), [email protected] (M. Delitala).

1468-1218/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2006.09.012

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184 M.L. Bertotti, M. Delitala / Nonlinear Analysis: Real World Applications 9 (2008) 183–196

Mainly motivated by this fact, but also with the aim of reducing the computational complexity associated to theintegro-differential equation, we proposed in [6] a general “discrete” kinetic framework, suitable for the modelling oflarge complex systems and their microscopic interactions. This framework led to a new class of dynamical systemscontemplating stochastic interactions and expressed in the form of systems of partial differential equations or, inparticular cases, of ordinary differential equations. The first purpose in [6] was to initiate a program directed to establishwell posedness questions for this new class of differential equations. Some results in this direction were obtained forthe case of ordinary differential equations. Another aim was to have some insight on the qualitative behavior of thesolutions. Of course, this requires specializing the differential equations at hand. A specific example was worked out asan application of the general discrete setting, relative to a simple social dynamics interactions model. For that model,in [6] and later in [7], some analytical results were proved concerning properties of asymptotic stability of certainsolutions and several computational simulations were performed.

In the present paper, proceeding along this line of investigation, we focus on a new model, describing a closed systemfor which a suitable conserved quantity exists. The study of this model highlights aspects of the streamline of solutions,which provide a further step toward the knowledge of the dynamics of the new class of systems.

In detail, the contents of the paper are divided into four sections. The first section coincides with this introductionand is devoted to a discussion concerning the aims and the organization of the paper. In Section 2, we briefly recall aparticular case of the discrete kinetic framework developed in [6], suitable for the description of homogeneous in spacesystems, i.e. systems for which the distribution function only depends on the social/biological state. Then, starting fromthe model explored in [6,7], we construct the new social dynamics model. The study of the new model is containedin Section 3. For sake of concreteness and to be in the position to obtain detailed analytical proofs, we confine ourattention to a particular “low”-dimensional case. For that case we establish the existence of curves of equilibria andwe analyze the generic asymptotic trend. Then, we report on several computational simulations, relative to differentdimensions, and we integrate the experimental outputs with some comments and remarks. Finally, in the last sectionwe mention some future perspectives of work concerning both analytical questions and new modelling aspects, relativee.g. to the treatment of additional social phenomena (migration, taxation and other external actions).

2. A conservative model in social dynamics

Consider a population constituted by a large number of individuals interacting in pairs and assume that each individualis characterized by a scalar quantity, which we refer to as its “social state”, which can attain a finite number of values.One of the tasks of social dynamics analysis is to attempt to understand how the interactions between single individualsinfluence or determine the outcomes that can be observed at a macroscopic scale.

A mathematical framework to be used toward such a task has been developed in [6] as a particular structure within awider setting, suitable to treat more general situations. Specifically, it consists of the following mathematical structure:

dfi

dt=

n∑h=1

n∑k=1

�hkAihkfhfk − fi

n∑k=1

�ikfk, i = 1, . . . , n, (2.1)

where

• fi denotes the fraction, with respect to the overall number of individuals, with social state ui belonging to the set

Iu = {u1 = 0, . . . , ui, . . . , un = 1}(u = 0 and u = 1 are, respectively, the lowest and the highest possible values of the social state);

• the interaction rate

�hk = �(uh, uk) : Iu × Iu → R+

expresses the number of encounters per unit time between individuals with state uh and individuals with state uk;• the transition probability density

Aihk = A(uh, uk; ui) : Iu × Iu × Iu → R+,

M.L. Bertotti, M. Delitala / Nonlinear Analysis: Real World Applications 9 (2008) 183–196 185

with

n∑i=1

Aihk = 1 ∀h, k = 1 (2.2)

expresses the probability density for an individual with state uh to fall into the state ui after an interaction with anindividual with state uk .

With reference to this differential equations system, the following global existence and uniqueness theorem has beenproved in [6].

Theorem 2.1. Assume �hk �M for some positive constant M < + ∞. Then, for any given set {fi0} such that fi0 �0for i = 1, . . . , n and the set {fi0} is a discrete probability density:

n∑i=1

fi0 = 1,

the solution f (t)=(f1(t), . . . , fn(t)) of system (2.1), satisfying the Cauchy problem with initial conditions fi(t=0)=fi0for i = 1, . . . , n exists and is unique for all t ∈ [0, +∞). In particular, one has

∀t �0 : fi(t)�0 for any i = 1, . . . , n andn∑

i=0

fi(t) = 1. (2.3)

This result guarantees the well posedness of the model and establishes, for the flow of system (2.1), the positiveinvariance of the standard (n − 1)-simplex.

Formulating a specific social dynamics model amounts to determine the tables of the interaction rates and of thetransition probability densities. To keep models as simple as possible, one can start and take the encounter rate �hk tobe constantly equal to one. This was the case for the model developed in [6] and will be the case in the sequel too. Todetermine the transition probability densities, we move from the observation that the interactions between individuals,which typically modify the social state, may have the character of conflicts or collaborations. More specifically, somephenomenological observations suggest the following assumptions:

• when two individuals have close social state, then a competition occurs having as a result the fact that the individualplaced in the higher social position improves its situation, while the one in a lower position faces a further decrease(competitive behavior);

• when the social state of the individuals is sufficiently far away, the opposite behavior (altruistic behavior) occurs.

To translate these concepts into formulas, besides the number n of social classes, a parameter m can be introduced,possibly attaining any integer value between 0 and n − 1, which represents the distance between classes and whichdistinguishes the competitive and the altruistic behavior. This gives a tool which allows to assign, in correspondenceto any natural number n and to any integer m between 0 and n − 1, the value of the game table of the Ai

hk for everyi, h, k = 1, . . . , n.

There is also another assumption, which we want to take into account: if the social state of an individual is identifiedwith its wealth, a natural requirement is the conservation of the global wealth. Consider in this connection the case of acompetitive interaction between individuals belonging to two distinct classes. If one of these two classes is an extremeone, i.e. it is indexed by 1 or by n, some caution is needed in the definition of the transition probability densities. Anindividual of the class 1 cannot decrease his state, being already at the lowest possible level. A similar requirementoccurs when an individual of the class n is involved: he cannot increase his state, which is already at the highest possiblelevel. If the amount of wealth of the two individuals has not to be altered, we have to impose that the co-interactingindividual continues to stay in his own class too.

Besides, to possibly give some flexibility to the model, we might assume that not every individual belonging to theclass i, interacting with an individual of the class j, will change his state moving to the class i + 1 or i − 1, depending


Table 1The nonzero elements in the table of game of the conservative model

h = k : Ai=hhk = 1

h �= k :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

|h − k|�m

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

h = 1 or h = n : Ai=hhk = 1

h �= 1, h �= n :

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

h < k :

⎧⎪⎪⎨⎪⎪⎩

k �= n :{

Ai=h−1hk = �

Ai=hhk = 1 − �

k = n : Ai=hhk = 1

h > k :

⎧⎪⎨⎪⎩

k �= 1 :{

Ai=h+1hk = �

Ai=hhk = 1 − �

k = 1 : Ai=hhk = 1

|h − k| > m :

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

h < k :{

Ai=h+1hk = �

Ai=hhk = 1 − �

h > k :{

Ai=h−1hk = �

Ai=hhk = 1 − �

on the interaction kind the two are undergoing. Eventually, only a portion of individuals, identified by a coefficient �could change class, while a portion of individuals, identified by the coefficient 1 − � remain in the class i.

Formalizing the previous assumptions leads to a table of games, relative to fixed values of n, m, � (where n�3 is anatural number, 0�m�n − 1 and 0��1), for which the nonzero elements are given in Table 1.

We point out that for this model, in addition to (2.3), the following rule holds true:

n∑k=1

ukfk(t) = constant ∀t �0, (2.4)

which expresses the conservation of the global wealth of the population.In fact, the model refers to a closed system, while the model proposed in [6] corresponds to an open system, where

interactions with the outer environment allow dissipation/increase of the wealth through the interactions of the boundary(extreme) social classes.

3. Qualitative analysis and simulations

First of all, we notice that the study of the model seems to be more significant when the number n of social classesis odd. Indeed, it is in such a case that a middle class exists. Focusing then, on the case of n odd, we start recalling theresults obtained for the social dynamics model in [6,7].

For n = 3, nonisolated equilibria (all having some component equal to zero) were found to exist in the two extremecases corresponding to m = 0 and 2. In contrast, for m = 1, only an isolated equilibrium was proved to exist: ( 1

3 , 13 , 1

3 ).This equilibrium was shown to be “globally asymptotically stable”, namely stable and attractive with respect to allsolutions f (t) with initial data on the unitary 2-simplex, different from any of the equilibria coinciding with the verticesof the simplex, [6]. For n odd general and m = (n − 1)/2, an equilibrium configuration corresponding to the constantdistribution P = (f1, . . . , fn) with fi = 1/n for all i = 1, . . . , n was proved to exist, [7]. A great deal of computationalsimulations indicated this constant distribution equilibrium P as the asymptotic state for generic initial conditions onthe unitary (n − 1)-simplex. Moreover, all the performed simulations did confirm the scenario analytically establishedfor n = 3: i.e. they constantly indicated that, if m = 0 (totally cooperative systems) or m = n − 1 (totally competitivesystems), several asymptotic states appear, depending on the initial conditions. In contrast, if m �= 0 and m �= n−1, forgeneric initial conditions an unique asymptotic state appears, characterized by large concentrations on central valuesfor m relatively small, and exhibiting large concentrations on the extreme values u = 0 and 1 for m relatively large.


3.1. Analytical study of the case with n = 5 and m = 2

We come back now to the model described in Section 2. For sake of concreteness we fix from now on the value ofthe parameters n and m.

While the smallest odd dimension for which the model in [6] was analytically investigated is n = 3, we notice that,in the present situation, n = 3 is too low a value to be significant from the viewpoint of dynamics. Indeed, it turns outthat for n = 3, m = 0 and for n = 3, m = 1 as well no equilibrium solutions with all components different from zeroexist, while for n = 3, m = 2 every point (f10, f20, f30) of the unitary 2-simplex is an equilibrium. So, we choose todiscuss the case n = 5 and, in particular, we take m = (n − 1)/2 = 2.

Ifn=5 andm=2, system (2.1) consists of five evolution equations. However, the number of equations can be decreasedin view of (2.3): the dynamics takes place on the standard 4-simplex. Putting for example f3 = 1 − f1 − f2 − f4 − f5,we are left with the following system of four differential equations:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

df1

dt= −�(−f2 + f 2

2 + f1f2 + f1f4 + f1f5 + f2f5),

df2

dt= �(−f2 + f4 + f 2

2 − f 24 + f1f2 − f2f4 − f4f5 + f1f5),

df4

dt= �(−f4 + f2 + f 2

4 − f 22 + f5f4 − f4f2 − f2f1 + f5f1),

df5

dt= −�(−f4 + f 2

4 + f5f4 + f5f2 + f5f1 + f4f1).

(3.1)

The following result is immediate:

Proposition 3.1. The parameter � (0��1) does not influences the geometry of the solution curves of (3.1) in thephase space. It only plays a role in the temporal parametrization of the solutions of the evolution equations: as smalleris �, as smaller is the velocity of the solution curves,

df

dt=(

df1

dt,

df2

dt,

df4

dt,

df5

dt

).

The second result we get establishes the conservation of the global wealth. If the wealth of any individual of theclass i, with i = 1, . . . , n, is supposed to be equal to (i − 1)/(n − 1), the global wealth at time t is given by

n∑i=1

(i − 1)

(n − 1)fi(t).

Recalling now that, by (2.3), one has f3 = 1 − f1 − f2 − f4 − f5 during the evolution, we express the global wealthas a function Q of the four variables f1, f2, f4, f5:

Q : � → R, Q(f1, f2, f4, f5) = 2 − 2f1 − f2 + f4 + 2f5

4, (3.2)

where

� =⎧⎨⎩(x1, x2, x3, x4) ∈ R4 : xj �0 for any j = 1, . . . , n and

4∑j=1

xj �1

⎫⎬⎭ .

We are now ready to show the following proposition.

Proposition 3.2. The scalar function Q(f1, f2, f4, f5) is a first integral for the system (3.1).

Proof. The claim follows from the substitution of the r.h.s.’s of (3.1) into the formula

4dQ

dt= −2

df1

dt− df2

dt+ df4

dt+ 2

df5

dt. �


The first step toward a qualitative analysis of the system at hand consists in looking for equilibria. In this connectionwe find

Theorem 3.1. System (3.1) admits curves of degenerate equilibria belonging to the boundary �� and a curve of“internal” equilibria (i.e. equilibria with each component different from zero) as well. A parametrization of this curveis given by

r(�) = (�(�), �(�), �(�), �) for 0 < � < 1, (3.3)

where

�(�) = R(�)2(1 − � − R(�))/(�R(�) + R(�)2 + �2),

�(�) = R(�)�(1 − � − R(�))/(�R(�) + R(�)2 + �2),

�(�) = R(�), (3.4)

and in turn R(�) is the only root of the one parameter quartic equation in z:

z4 + z3� + z2�2 + z�3 + �4 − �3 = 0,

which, for the admissible values of the parameter �, is equal to a real number belonging to (0,1).

Proof. In view of Proposition 3.1, without loss of generality, we take �=1. Searching equilibria of (3.1) means lookingfor solutions of the system of algebraic equations⎧⎪⎨

⎪⎩f2 − f 2

2 − f1f2 − f1f4 − f1f5 − f2f5 = 0,

−f2 + f4 + f 22 − f 2

4 + f1f2 − f2f4 − f4f5 + f1f5 = 0,

−f4 + f2 + f 24 − f 2

2 + f5f4 − f4f2 − f2f1 + f5f1 = 0,

f4 − f 24 − f5f4 − f5f2 − f5f1 − f4f1 = 0.

which is easily seen to be equivalent to⎧⎪⎨⎪⎩

f2 − f 22 − f1f2 − f1f4 − f1f5 − f2f5 = 0,

−f2f4 + f1f5 = 0,

f4 − f2 − f 24 + f 2

2 + f1f2 − f4f5 = 0,

f4 − f2 − f 24 + f 2

2 − f4f5 + f1f2 = 0.

(3.5)

We observe at this point that two equations in this system coincide (this being related to the existence of the first integralQ in (3.2)). Consequently, nonisolated solutions may exist.

The proof can be divided into two parts. First, we look for internal solutions, namely solutions having all componentsdifferent from zero. Later on we will explore the possible existence of equilibria belonging to the boundary ��.

As for the first part, we confine our search to solutions (f1, f2, f4, f5) with fi > 0 for i = 1, 2, 4, 5 and f1 + f2 +f4 + f5 �1. Hence, in particular, we have f4 �= 0 and we can rewrite the system in the form⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

f1f5

f4

(1 − f1 − f5 − f1f5

f4

)− f1f4 − f1f5 = 0,

f2 = f1f5

f4,

f4 − f 24 − f4f5 + f1f5

f4

(−1 + f1 + f1f5

f4

)= 0.

(3.6)

The first and the third equation in (3.6) are obtained substituting to the unknown f2 appearing in the first and the thirdequation of (3.5) its expression in terms of f1, f4, f5 given by the second equation. Keeping now on a side the secondequation of (3.6), we see that the system composed by its first and third equation is equivalent to⎧⎪⎪⎨

⎪⎪⎩f4 − f 2

4 − f4f5 − f1f5 − f1f4 − f1f25

f4= 0,

f4 − f 24 − f4f5 − f1f5

f4+ f 2

1 f5

f4+ f 2

1 f 25

f 24

= 0.

(3.7)


The first equation in (3.7) can be employed to express f1 in terms of f4 and f5. It gives

f1 = f 24 (1 − f4 − f5)

(f 24 + f4f5 + f 2

5 ), (3.8)

where the division by f 24 + f4f5 + f 2

5 is possible due to the fact that f4 and f5 are strictly positive. Substituting (3.8)into the second equation of (3.7) yields, after some minor passages,

(1 − f4 − f5)(f44 + f 3

4 f5 + f 24 f 2

5 + f4f35 + f 4

5 − f 35 ) = 0. (3.9)

Notice that in the case under consideration (1 − f4 − f5) is certainly different from zero. Indeed, (1 − f4 − f5) = 0would imply f1 = 0 and f2 = 0, and we are searching here internal points. So, Eq. (3.9) becomes

f 44 + f 3

4 f5 + f 24 f 2

5 + f4f35 + f 4

5 − f 35 = 0. (3.10)

Using Eq. (3.10) to get e.g. f4 in terms of f5, amounts to look for the roots of the one parameter polynomial in z,P�(z) = z4 + z3� + z2�2 + z�3 + �4 − �3.

We show next that for any 0 < � < 1 the polynomial P�(z) admits only one real root belonging to the interval (0, 1).It is immediate checking that

P�(0) < 0 for any 0 < � < 1, (3.11)

while

P�(1) > 0 for any 0 < � < 1. (3.12)

Also, the derivative of P�(z) with respect to the variable z is dP�/dz(z)= 4z3 + 3z2�+ 2z�2 + �3, and this implies that

dP�

dz(z) > 0 for any 0 < � < 1, for any 0 < z < 1. (3.13)

The inequalities (3.11)–(3.13) allow to deduce, in correspondence to any 0 < � < 1, the existence of a unique real rootR(�) of P�(z), belonging to (0, 1). More precisely, observing that in fact P�(1 − �) > 0 for any 0 < � < 1, we mayconclude that R(�) < 1 − �, namely R(�) + � < 1 for any 0 < � < 1. It is now an easy task verifying that, when f4 > 0,f5 > 0 and f4 + f5 < 1, also f2 given as in the second equation of (3.6) and f1 given as in (3.8) belong to (0, 1) andf1 + f2 + f4 + f5 �1 holds true (recall that � plays here the role of f5 and R(�) that one of f4). This completes thepart of proof relative to internal equilibria.

We pass now to consider the possible existence of equilibria belonging to the boundary ��. The starting point issystem (3.5).

We first complete the analysis of the case f4 �= 0. A curve of solutions in this context arises, consisting of points(f1, f2, f4, f5) for which f4 + f5 = 1. Indeed, in the first part of the proof, when passing from (3.9) to (3.10), only thecase f4 + f5 < 1 was treated. Allowing now the possibility that f4 + f5 = 1, we infer that the curve parametrized by

r1() = (0, 0, 1 − , ) for 0��1 (3.14)

consists of equilibria.Having examined all possible implications of system (3.5) in the case f4 �= 0, we next proceed assuming f4 = 0.

The second equation in (3.5) shows that two cases have to be explored: f1 = 0 and f5 = 0.If f4 = 0 and f1 = 0, solving system (3.5) leads to the conclusion that the point (0, 1, 0, 0) is an equilibrium and the

curve parametrized by

r2() = (0, 0, 0, ) for 0��1 (3.15)

consists of equilibria.


If f4 = 0 and f5 = 0, solving the system (3.5) leads to the conclusion that the curves parametrized, respectively, by

r3() = (, 0, 0, 0) for 0��1 (3.16)

and

r4() = (, 1 − , 0, 0) for 0��1 (3.17)

consist of equilibria.Of course, also the points (1, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (0, 0, 0, 0) are equilibria: it can be immediately seen

that they belong as a matter of fact to the curves of equilibria (3.14)–(3.17). �

Remark 3.1. Formulating the statement of Theorem 3.1 in terms of five components points (recalling that f3 =1 − f1 − f2 − f4 − f5), we have that the equilibria of system (2.1) for the conservative model under study when n = 5and m = 2 are precisely the points (f1, f2, f3, f4, f5) of the form

(0, 0, 0, 1 − , ) for 0��1,

(0, 0, 1 − , 0, ) for 0��1,

(, 0, 1 − , 0, 0) for 0��1,

(, 1 − , 0, 0, 0) for 0��1,

and

(�(�), �(�), 1 − �(�) − �(�) − �(�) − �, �(�), �) for 0 < � < 1, (3.18)

with �(�), �(�), �(�) as in Theorem 3.1. The equilibria expressed here in terms of the parameter � are of a majorinterest for us, because they correspond to a distribution of the individuals in the population, according to which allclasses, not only two, are represented.

Remark 3.2. Denoting for simplicity, for any 0 < � < 1, f [�] = (f1[�], f2[�], f3[�], f4[�], f5[�]) the equilibriumpoint in (3.18) corresponding to �, one can easily verify that the following relations between the components of theequilibrium itself hold true:

f2[�]f1[�] = f3[�]

f2[�] = f4[�]f3[�] = f5[�]

f4[�] = �

R(�)for 0 < � < 1.

Our next goal is to prove that each one of the equilibrium points (3.18) is an attractor for all the solutionsf (t) = (f1(t), f2(t), f3(t), f4(t), f5(t)), evolving from initial data (f10, f20, f30, f40, f50) which, apart from sat-isfying fi0 > 0 for any i = 1, . . . , 5 and

∑5i=0fi0 = 1, have the same value of the wealth as the equilibrium. The route

to this goal will be the construction of a Lyapunov function. In this connection we start pointing out two properties ofthe coefficients Ai

hk appearing in Eq. (2.1) (Refs. [8,9] provided a hint at this point).

Lemma 3.1. For the conservative model under study, in the case n = 5, m = 2, the transition probability densitiesAi

h,k satisfy

Aih,k = Ah+k−i

k,h (3.19)

and

Aih,k = Ah

i,h+k−i (3.20)

for any (i, h, k) ∈ := {(i, h, k) ∈ NN3 : 1� i�5, 1�h�5, 1�k�5 and 1 − i�h + k�5 − i}.

The relations (3.19) simply express the fact that the wealth of any pair composed by a test individual and a fieldindividual undergoing an encounter has to be the same before and after the encounter. They hold true for the model


of Section 2, whatever n and m are. In contrast, the relations (3.20), which express microscopic reversibility (they sayindeed that the encounter (h, k) → (i, h + k − i) and the inverse encounter (i, h + k − i) → (h, k) occur with thesame probability), turn out not to be valid for general n and m. It is straightforward checking that they hold true forn = 5 and m = 2.

Lemma 3.2. For the conservative model under study, in the case n = 5, m = 2, the equations (2.1) may be written as

dfi

dt=∑h

∑k

Aihk(fhfk − fifh+k−i ), i = 1, . . . , 5, (3.21)

Moreover, for any � = (�1, . . . ,�5) ∈ R5 one has

5∑i=1

�i

dfi

dt= −1

4

∑i

∑h

∑k

(�i + �h+k−i − �h − �k)(fifh+k−i − fhfk)Aihk . (3.22)

Proof. By (2.2) and (2.3) Eq. (2.1) may be written as

dfi

dt= −

n∑h=1

n∑k=1

Ahikfifk +

n∑h=1

n∑k=1

Aihkfhfk, i = 1, . . . , n,

Here we are dealing with the case with n = 5. Using the microreversibility relations (3.20), we rewrite the first of thetwo terms on the r.h.s. so as to have (3.21). Exploiting now (3.21) as well as the conservativity relations (3.19) and themicroreversibility relations (3.20), we get

5∑i=1

�i

dfi

dt= −

∑i

∑h

∑k

Aihk�i (fifh+k−i − fhfk)

= 4

(−1

4

∑i

∑h

∑k


)

= − 1

4

∑i

∑h

∑k


− 1

4

∑i

∑h

∑k

Aihk�h+k−i (fifh+k−i − fhfk)

+ 1

4

∑i

∑h

∑k

Aihk�h(fifh+k−i − fhfk)

+ 1

4

∑i

∑h

∑k

Aihk�k(fifh+k−i − fhfk),

The claim is proved. �

We have now the ingredients for the proof of an H-Theorem. Indeed, for the conservative model under study, in thecase n = 5, m = 2 we have the following results.

Proposition 3.3. The function H =∑5i=1 fi log fi is not increasing along the solutions of (3.21). More specifically,

its Lie derivative vanishes at the equilibrium points and is negative elsewhere.


Proof. By the formula (3.22) with �i = log fi and since the function g : (0, +∞) → R, defined as g(x)=(x −1) log x

assumes the value 0 in x = 1 and is otherwise positive, we have

dH

dt=

5∑i=1

(log fi + 1)dfi

dt=

5∑i=1

log fi

dfi

dt

= − 1

4

∑i

∑h

∑k

logfifh+k−i

fhfk

(fifh+k−i − fhfk)Aihk �0.

In particular

dH

dt= 0 if and only if (fifh+k−i − fhfk)A

ihk = 0 for any i, h, k ∈ .

And by (3.21) this is a sufficient condition for a point to be an equilibrium. �

For any 0 < � < 1, call now C� the value of the wealth of the equilibrium f [�].

Proposition 3.4. For any 0 < � < 1, the function H=∑5i=1 fi log fi restricted to the points f = (f1, f2, f3, f4, f5),

which satisfy fi > 0 for any i = 1, . . . , 5 and∑5

i=0 fi = 1 and which have the same value C� of the wealth, attains itsminimum at the equilibrium f [�].

Proof. First of all, we explicitly point out that, in view of Remark 3.2, the components of each equilibrium point f [�]satisfy

fi[�] = (�(�))i−1�(�) for i = 2, 3, 4, 5 (3.23)

for some �(�) > 0, �(�) > 0.Then, we set hi = (fi − fi[�])/fi[�]. Together with the fact that the points f and f [�] belong to the unitary simplex

and share the same value of the wealth, this implies

5∑i=1

hifi[�] =5∑

i=1

fi −5∑

i=1

fi[�] = 0 (3.24)

and

5∑i=1

(i − 1)hifi[�] =5∑

i=1

(i − 1)fi −5∑

i=1

(i − 1)fi[�] = 0. (3.25)

Using now (3.24) and (3.25), we see that

H(f ) − H(f [�]) =5∑

i=1

fi[�](hi + 1) log(hi + 1) +5∑

i=1

fi[�]hi log fi[�].

The second addendum on the r.h.s. vanishes in view of (3.23)–(3.25). And, using again (3.24), we may write

H(f ) − H(f [�]) =5∑

i=1

fi[�][(hi + 1) log(hi + 1) − hi]�0,

where the last inequality follows from the fact that the function g : (−1, +∞) → R, defined as g(x)= (x +1) log(x +1) − x assumes the value 0 in x = 1 and is otherwise positive. �

As a consequence of Propositions 3.2. and 3.4, the function H plays the role of a Lyapunov function, allowing toestablish


Theorem 3.2. For any 0 < � < 1, the equilibrium point f [�] is stable and attractive with respect to the initial data(f10, f20, f30, f40, f50) which satisfy fi0 > 0 for any i =1, . . . , 5,

∑5i=0 fi0 =1 and have the same value of the wealth

as f [�].

3.2. Computational simulations for different values of n and m

We collect in this subsection some data and remarks relative to computational experiments.Computational simulations relative to the conservative model with n = 5 and m = 2 lead to the following list,

where, for some different values of � ∈ [0, 1], the components of the corresponding equilibrium points f [�] =(f1[�], f2[�], f3[�], f4[�], f5[�]) are given as well as the value of the wealth at f [�]. Notice that, when � is small, thewealth is small and the asymptotic equilibrium contemplates many “poor” individuals and a few “rich” individuals,while, when � is large, the wealth is large and the asymptotic equilibrium contemplates many “rich” individuals and a few“poor” individuals. This feature, also illustrated in Fig. 1, certainly looks compatible with what one would expect to be.

� f [�] Q

0.0 (1.000, 0.000, 0.000, 0.000, 0.000) 0.0000.1 (0.335, 0.247, 0.183, 0.135, 0.100) 0.3550.2 (0.200, 0.200, 0.200, 0.200, 0.200) 0.5000.3 (0.120, 0.151, 0.190, 0.239, 0.300) 0.6120.4 (0.069, 0.107, 0.166, 0.258, 0.400) 0.7030.5 (0.036, 0.070, 0.135, 0.259, 0.500) 0.7790.6 (0.016, 0.040, 0.099, 0.244, 0.600) 0.8430.7 (0.006, 0.019, 0.064, 0.211, 0.700) 0.8950.8 (0.001, 0.006, 0.032, 0.160, 0.800) 0.9380.9 (0.000, 0.001, 0.009, 0.090, 0.900) 0.9721.0 (0.000, 0.000, 0.000, 0.000, 1.000) 1.000

To have some more insight on the phase space picture for the conservative model with n=5 and m=2, one may lookat the four continua of equilibria belonging to the boundary of the unitary simplex described in Remark 3.1. Evaluatingcomputationally the eigenvalues of the linear part of the system of four differential equations (3.1) around each one ofthe equilibria of the form (f1, f2, f4, f5) = (0, 0, 1 − , ) or (0, 0, 0, ) or (, 0, 0, 0) or (, 1 − , 0, 0) for differentvalues of in (0, 1), e.g. for = 0.1, 0.2, 0.3, . . . , 0.9, one constantly finds that there are an eigenvalue equal to zero(a fact due to the degeneracy of the equilibria which appear in a continuum), an eigenvalue with positive real part andtwo eigenvalues with negative real part.

When searching equilibria for the conservative model with general n and m, one can immediately reduce the systemof n differential equations to a system of n − 1 equations in n − 1 variables by using the fact that

∑ni=1 fi = 1.

Furthermore, the conservation of the global wealth

n∑i=1

(i − 1)

(n − 1)fi(t)

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1 Q=0.25` l=0 .048`

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1 Q=0.5` l =0 .2`

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1 Q=0.75` l=0 .46`

Fig. 1. Asymptotic configurations in the case n = 5, m = 2, for values of the global wealth, respectively, equal to 0.25, 0.5, 0.75.


0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=11, m=5, Q=0.25‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=11, m=5, Q=0.5‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=11, m=5, Q=0.75‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=9, m=4, Q=0.25‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=9, m=4, Q=0.5‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=9, m=4, Q=0.75‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=7, m=3, Q=0.25‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=7, m=3, Q=0.5‘

0 0.25 0.5 0.75 1

0.2

0.4

0.6

0.8

1n=7, m=3, Q=0.75‘

Fig. 2. Asymptotic configurations for different values of n and m, for values of the global wealth, respectively, equal to 0.25, 0.5, 0.75.

implies that one of the n − 1 equations is in fact a linear combination of the remaining ones. In view of that, one mayexpect that the equilibria, when present, belong to continua.

It is tempting trying and looking whether it is a general fact that for the conservative model with odd values of n andm = (n − 1)/2, the point (f1, . . . , fn) with each component equal to 1/n is an equilibrium. This turns out not to bethe case: if one evaluates the r.h.s. of (2.1) at the point fi = 1/n for i = 1, . . . , n, one finds e.g. the following curiousresult:

Case n = 7, m = 3: (0, + 149 , − 1

49 , 0, − 149 , + 1

49 , 0).Case n = 9, m = 4: (0, + 1

81 , 0, − 181 , 0, − 1

81 , 0, + 181 , 0).

Case n = 11, m = 5: (0, + 1121 , 0, 0, − 1

121 , 0, − 1121 , 0, 0, + 1

121 , 0).Instead one finds, at least for each one of the values of n listed below, the existence in particular of an internal

equilibrium satisfying the symmetry condition fn+1−i =fi for i=1, . . . , (n−1)/2 and lying on the invariant hyperplaneQ = 0.5. A numerical approximation for the coordinates of this internal equilibrium follows:

Case n = 7, m = 3: f1 = f7 = 0.14480, f2 = f6 = 0.16631, f3 = f5 = 0.12282 and f4 = 0.13216.Case n= 9, m= 4: f1 =f9 = 0.11454, f2 =f8 = 0.12768, f3 =f7 = 0.11219, f4 =f6 = 0.09523 and f5 = 0.10071.Case n=11, m=5: f1 =f11 =0.09467, f2 =f10 =0.10309, f3 =f9 =0.09440, f4 =f8 =0.08851, f5 =f7 =0.07839

and f6 = 0.08189.Each one of these equilibria has in fact an oscillatory shape around the uniform shape in which each component is

equal to 1/n. Indeed, 17 ≈ 0.14286, 1

9 ≈ 0.11111, 111 ≈ 0.09091.

Always computationally one sees that for the linear part of the system of n − 1 differential equations around eachone of the (n − 1)-component equilibria corresponding to these equilibria there is an eigenvalue equal to zero, whileall remaining eigenvalues have negative real part. Accordingly, one could expect also these equilibria to belong to aone-dimensional continuum and to be asymptotically stable with respect to points of the hyperplane Q = 0.5. Thisseems to be confirmed by Fig. 2, where asymptotic configurations obtained computationally are shown, corresponding


to different values of the global wealth. Their shape replicate a structure characterized by the presence of many “poor”individuals when the wealth is small and many “rich” individuals when the wealth is small and characterized by a kindof oscillatory profile as well.

Finally, the computational evidence for the model under investigation, at least in the cases n odd and m= (n− 1)/2,is that the corresponding systems have negative divergence.

4. Perspectives

This paper is essentially devoted to the design of a new model of social dynamics and the study of its solutions.A qualitative analysis is carried out for a given choice of the model parameters n and m, leading to a detailed pictureof the equilibrium configurations and of their stability properties. In addition, simulations suggest that the qualitativebehavior described by Theorem 3.2 is confirmed for general values of n and m. Rigorous proofs for arbitrary n stillremain an open challenging objective. Some perspectives for future investigations follow.

Referring to analytical questions:

• a first goal to be pursued seems to be the recovering of the existence proofs of continua of equilibria, which aresomehow expected on the basis of the simulations described in the Section 3;

• the characterization of the shape of the equilibria needs further research;• another natural question, suggested by the computational evidence is how to eventually prove the asymptotic

stability of certain equilibrium configurations when the arguments employed in Lemma 3.1 do not work.

Referring to modelling aspects, the mathematical structure proposed in [6] can be possibly generalized in the followingdirections:

• enlarging the dimension of the activity variable to describe additional properties of the interacting individuals.For instance, their localization could be taken into account in view of the description of migration phenomena;

• including the effects of external actions, e.g. taxation or other political influences, with the aim of investigatingthe different possible asymptotic trends.

Finally, an explorative strategy can be adopted, directed to investigate which type of interactions and/or externalaction lead to a desired or undesired asymptotic behavior.

References

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Conservation laws and asymptotic behavior of a model of social dynamics