love dynamics between secure individuals: a modeling...

Nonlinear Dynamics, Psychology, and Life Sciences, Vol. 2, No. 4, 1998

Love Dynamics Between Secure Individuals: AModeling Approach

Sergio Rinaldi1,3 and Alessandra Gragnani2

A mathematical model composed of two differential equations, whichqualitatively describe the dynamics of love between secure individuals, ispresented in this paper with two goals. The general goal is to show howdynamic phenomena in the field of social psychology can be analyzed followingthe modelling approach traditionally used in all other fields of sciences. Thespecific goal is to derive, from very general assumptions on the behavior ofsecure individuals, a series of rather detailed properties of the dynamics oftheir feelings. The analysis shows, in particular, why couples can be partitionedinto fragile and robust couples, how romantic relationships are influenced bybehavioral parameters and in which sense individual appeal creates order ina community.

KEY WORDS: dyadic relationships; love dynamics: partner selection: community structure:stability; multiple equilibria.

INTRODUCTION

This paper deals with love dynamics, a subject that falls in the field ofsocial psychology, where interpersonal relationships are the topic of majorconcern. Romantic relationships are somehow the most simple case sincethey involve only two individuals. The analysis is performed following themodelling approach, which has recently received some attention in the sci-

1Centro Teoria dei Sistemi, CNR, Politecnico di Milano, Milano, Italy.-International Inst i tute for Applied Systems Analysis. Laxenburg, Austria.3Correspondence should be directed to Professor Sergio R ina ld i , Centro Teoria deiSistemi, CNR, Politecnico di M i l a n o , Via Ponzio 34/5. 2013? Milano. I t a l y ; e -mai l :rinaldi(a elet.polimi.it.

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284 Rinaldi and Gragnani

entific community (Butz, Chamberlain & McCown, 1996; Vallacher &Nowak, 1994).

Real-life observations tell us that very frequently love-stories developvery regularly and then level off for a very long time, as described by Lev-inger (1980) in his famous graph on the longitudinal course of partnersinvolvement. But there are also love-stories characterized by remarkabletemporal fluctuations. This spontaneously suggests the use of differentialequations for modelling close relationships, because differential equationscan explain stationary regimes as well as cyclic and chaotic regimes (Val-lacher & Nowak, 1994). Although Strogatz (1988) has indicated this pos-sibility in a one page pioneering paper, such an approach has never beenfollowed in social psychology. The only exception is a very recent study(Rinaldi, 1998) on the dynamics of love between Petrarch, an Italian poetof the 14-th century, and Laura, a beautiful and married lady. Their lovestory developed over 21 years and has been described in the Canzoniere,a collection of 366 poems addressed by the poet to his platonic mistress.In such a study, the main traits of Petrarch's and Laura's personalities areidentified by analyzing the Canzoniere; he is very sensitive and transformsemotions into poetic inspiration; she protects her marriage by reactingnegatively when he becomes more demanding and puts pressure on her,but at the same time, following her genuine Catholic ethic, she arrives atthe point of overcoming her antagonism by strong feelings of pity. Then,these traits are encapsulated in a model composed of differential equationswhere the variables are the emotions of the two individuals. Finally, themodel is analyzed and the result is that its solution tends toward a cyclicbehavior. In other words, the peculiarities of the personalities of the twolovers inevitably generate a love story characterized by recurrent periodsof ecstasy and despair. This is indeed what happened, as empirically ascer-tained by Jones (1995) through a detailed stylistic and linguistic analysisof the dated poems. It is interesting to note, however, that with the mod-elling approach the existence of the emotional cycle is fully understoodand proved to be inevitable, while with the empirical approach it is onlydiscovered.

We follow in this paper the same modelling approach to discuss thedynamics of the feelings between two persons. But instead of concentratingour attention on a specific and well documented case, we deal with themost generic situation, namely that of couples composed of secure indi-viduals (Bartholomew & Horowitz, 1991; Griffin & Bartholomew, 1994).For this we will first assume that secure individuals are characterized byspecific properties concerning their reactions to partner's love and appeal.The analysis of the corresponding model will show how the feelings be-tween two individuals evolve in time. Intentionally, we do not support these

Love Dynamics Between Secure Individuals: A Modeling Approach 285

properties with data, because we like only to highlight the power of themethod, namely the possibility of deriving the main characteristics of lovedynamics from individual behavior. It is worth anticipating, however, thatall the properties discussed in the paper are in agreement with commonwisdom on the argument.

The paper is organized as follows. In the next section we assume thatsecure individuals follow precise rules of behavior and we write two differ-ential equations encapsulating these rules. Then, we analyze these equa-tions and derive properties which formally describe the process of fallingin love, i.e., the transformation of the feelings, starting from complete in-difference and tending toward a plateau. Other results are concerned withthe influence that appeal and individual behavior have on romantic rela-tionships. Some of these properties are used to derive the consequenceson partner selection. Although the results are extreme, they explain to someextent facts observed in real life, such as the rarity of couples composedof individuals with very uneven appeal. Merits and weaknesses of the pre-sent analysis, as well as directions for further research, are briefly discussedat the end of the paper.

Although this is the first serious attempt of studying close relationshipsby means of differential equations, in a number of previous studies otherkinds of dynamical models have been used for predicting the evolution ofromantic relationships. In the most relevant of these studies (Huesmann& Levinger, 1976) a model, called RELATE, was proposed and extensivelyused for simulations. Such a model, discussed with other models in a recentsurvey (Baron, Amazeen & Beek, 1994), is a sort of computer model (basedon exchange theory) involving recursive manipulations of suitable "dyadicoutcome matrices" representing different interpersonal states. It is certainlya dynamic model, but it is much more complex and less elegant than apair of differential equations.

THE MODEL

The model proposed in this paper is a minimal model, in the sensethat it has the lowest possible number of state variables, namely one foreach member of the couple. Such variables, indicated by x1 and x2, are ameasure of the love of individual 1 and 2 for the partner. Positive valuesof x represent positive feelings, ranging from friendship to passion, whilenegative values are associated with antagonism and disdain. Complete in-difference is identified by x = 0. Details on how love can be measuredwith a single variable, how graphs of such a variable versus time can be


obtained through interviews, and what are the limitations of such proce-dures can be found in Levinger (1980).

The model is a crude simplification of reality. Firstly, because love isa complex mixture of different feelings (esteem, friendship, sexual satisfac-tion, . ..) and can be hardly captured by a single variable. Secondly, becausethe tensions and emotions involved in the social life of a person cannot beincluded in a minimal model. In other words, only the interactions betweenthe two individuals are taken into account, while the rest of the world iskept frozen and does not participate explicitly in the formation of love dy-namics. This means that rather than attempting to be complete, the aimis to check which part of the behaviors observed in real life can in theorybe explained by the few ingredients included in the model.

It is important to state that the time scale to which we like to referis an intermediate time scale (e.g., phases A, B, C in Levinger (1980)).More precisely, we are not interested in fast fluctuations of the feelings,like those controlled by the daily or weekly activities or by the sexualrhythms of the couple, nor in the long term dynamics induced by life ex-periences. Thus, the model can only be used for relatively short periods oftime (months/years), for example in predicting if a love story will be smoothor turbulent. This implies that the present study can only weakly relate toattachment theory (Bowlby, 1969, 1973, 1980), which has been a main in-vestigation tool in adult romantic relationships in the last decade (see, forinstance, Collins & Read, 1990; Feeney & Noller, 1990; Hazan & Shaver,1987; Kirkpatrick & Davis, 1994; Shaver & Brennan, 1992; Simpson, 1990).

Three basic processes, namely return, instinct, and oblivion are assumedto be responsible of love dynamics. More precisely, the instantaneous rateof change dXi(t)ldt of individual's i love is assumed to be composed of threeterms, i.e.,

where Ri, Ii, and Oi must be further specified.The return Ri describes the reaction of individual i to the partner's

love and can therefore be assumed to depend upon xj , j = i. Loosely speak-ing, this term could be explained by saying that one "loves to be loved"and "hates to be hated." In order to deal with the most frequent situation,we restrict our attention to the class of secure individuals, who are knownto increase their reaction when the love of the partner increases. Moreprecisely, we assume that the return function Ri, is positive, increasing, con-cave and bounded for positive values of xj and negative, increasing, convexand bounded for negative values ofxj. Figure 1 shows the graph of a typical


Fig. 1. The return function Ri of a secure individ-ual i vs. the partner's love xj.

return function. The boundedness of the return function is a property thatholds also for non secure individuals: it interprets the psycho-physicalmechanisms that prevent people from reaching dangerously high stresses.By contrast, increasing return functions are typical of secure individuals,since non-secure individuals reduce their reaction when pressure and in-volvement are too high (Griffin & Bartholomew, 1994).

The second term Ii, describes the reaction of individual i to the part-ner's appeal Aj, Of course, it must be understood that appeal is not merephysical attractiveness, but, more properly and in accordance with evolu-tionary theory, a suitable combination of different attributes among whichage, education, earning potential and social position. Moreover, there mightbe gender differences in the relative weights of the combination (Feingold,1990; Sprecher, Quintin & Hatfield, 1994). For reasons similar to thosementioned above, we assume that instinct functions Ii(Aj) enjoy the sameproperties that hold for return functions. Nevertheless, for simplicity, wewill restrict our attention to the case of individuals with positive appeals.Notice that the reaction to the partner's appeal Ii(Aj) is greater than thereaction to the partner's love Ri(xj) when Xj is small (or negative). In par-ticular, at the beginning of the relationship (i.e., in the phase called "at-traction" by Levinger (1980)),x1 and x2 are small, so that the instinct termis the dominant term. In other words, the rate of change of the feeling isinitially determined by the appeal of the partner.

Finally, the third term Oi can be easily understood by looking at theparticular case in which Ri and Ii, are identically equal to zero, as in the


case in which an individual has lost the partner. If we assume that in suchextreme conditions Xj(t) vanishes exponentially at a rate ai, we must write

so that we can obtain

In the following, the parameter ai. is called forgetting coefficient and itsinverse 1/ai is a measure of the time needed to forget the partner afterseparation. Of course, a, is a behavioral parameter of the individual, butit is certainly influenced by culture and religion.

In conclusion, our model for secure individuals, is

where the functions Ri, and Ii annihilate for xj = 0 and Aj = 0 and satisfythe following properties (see Fig. 1)

Each individual i is identified in the model by two parameters (the appealA and the forgetting coefficient ai) and two functions (the return functionRi and the instinct function Ii). Such parameters and functions are assumedto be constant in time: this rules out aging, learning and adaptation proc-esses which are often important over a long range of time (Kobak & Hazan,1991; Scharfe & Bartholomew, 1994) and sometimes even over relativelyshort periods of time (Fuller & Fincham, 1995). The a-priori estimate ofthese parameters and functions is undoubtedly a difficult task, althoughsome studies on attachment styles (Bartholomew & Horowitz, 1991;


Carnelly & Janoff-Bulman, 1992; Griffin & Bartholomew, 1994) might sug-gest ways for identifying categories of individuals with high or low reac-tiveness to love and appeal. By contrast, the accurate estimate of theparameters appears forbiddingly difficult: it would require sitting down witha couple for weeks to obtain data from both partners on a long list ofquestions (Levinger, 1980). This identification problem will not be consid-ered in this paper, which is centered only on the derivation of the prop-erties of the model.

CONSEQUENCES AT INDIVIDUAL LEVEL

The analysis of model (1), taking into account properties (2), pointsout a series of properties concerning the dynamics of the feelings and theimpact that individual behavior has on the quality of romantic relationships.In the following, we simply state and interpret these properties, while theirformal derivations are reported in Appendix. As already stated, we consideronly the case of positive appeals.

Property 1

// the feelings are non-negative at a given time, then they are positive atany future time.

This property implies that an undisturbed love story between secureindividuals can never enter a phase of antagonism. In fact when two indi-viduals first meet, say at time 0, they are completely indifferent one toeach other. Thus, x1 (0) = x2(0) = 0 and Property 1 implies that the feelingsimmediately become positive and remain positive forever. Actually it canbe noted that model (1) implies that at time 0

i.e., at the very beginning of the love story the instantaneous rates of changeof the feelings are determined only by the appeals and are therefore posi-tive if the appeals are such. The fact that antagonism can never be presentin a couple of secure individuals is obviously against observations. Indeed,in real life the feelings between two persons are also influenced by factsthat are not taken into account in the model. We can therefore imaginethat most of the times model (1) describes correctly the dynamics of the


feelings and this is what we have called "undisturbed" behavior. But fromtime to time unpredictable, and hence unmodelled, facts can act as distur-bances on the system. A typical example is a temporary infatuation foranother person giving rise to a sudden drop in interest for the partner. Ofcourse, heavy disturbances can imply negative values of the feelings. It istherefore of interest to know what happens in such cases after the distur-bance has ceased. The answer is given by the following property.

Property 2

Couples can be partitioned into robust and fragile couples. As time goeson, the feelings x1( t) and x2(t) of the individuals forming a robust couple tendtoward two constant positive values no matter what the initial conditions are.By contrast, in fragile couples, the feelings evolve toward two positive valuesonly if the initial conditions are not too negative and toward two negativevalues, otherwise.

Figure 2 illustrates Property 2 by showing in state space the trajectoriesof the system starting from various initial conditions. Each trajectory rep-resents the contemporary evolution of X 1 ( t ) and x2(t) and the arrows indi-cate the direction of evolution. Note that, in accordance with Property 1,no trajectory leaves the first quadrant. In Fig. 2a, corresponding to robustcouples, all trajectories tend toward point E+ = (x1 +, x2+), which is there-fore a globally attracting equilibrium point. By contrast, fragile couples(Fig. 2b) have two alternative attractors (points E+ and E-) with basins ofattraction delimited by the dotted line.

Figure 2b points out an interesting fact. Suppose that the couple is atthe positive equilibrium E+, and that for some reason individual 2 has a dropin interest for the partner. If the drop is not too large the couple recovers tothe positive equilibrium after the disturbance has ceased (see trajectory start-ing from point 1). But if the disturbance has brought the system into theother basin of attraction (see point 2), the couple will tend inevitably towardpoint E-, characterized by pronounced and reciprocal antagonism. This iswhy such kind of couples have been called fragile. In conclusion, robust cou-ples are capable of absorbing disturbances of any amplitude and sign, in thesense that they recover to a high quality romantic relationship after the dis-turbance has ceased. On the contrary, individuals forming a fragile couplecan become permanently antagonist after a heavy disturbance and never re-cover to their original high quality mode of behavior.

Of course, it would be interesting to have a magic formula that couldpredict if a couple is robust or fragile. Unfortunately, this is not possiblewithout specifying the functional form of the return functions, a choice that


Fig. 2. Evolution of the feelings in a robust cou-ple (a) and in a fragile couple (b) for differentinitial conditions.

we prefer to avoid, at least for the moment. Nevertheless, we can in partsatisfy our curiosity with the following property.

Property 3

If the reactions I1 and I2 to the partner's appeal are sufficiently high, thecouple is robust.


This property is rather intuitive: it simply says that very attractive in-dividuals find their way to reconciliate.

Figure 2 shows that the trajectory starting from the origin tends towardthe positive equilibrium point E+ without spiraling around it. This is truein general, as pointed out by the following property.

Property 4

Two individuals, completely indifferent one to each other when they firstmeet, develop a love story characterized by smoothly increasing feelings tendingtoward two positive values.

This property implies that turbulent relationships with more or lessregular and pronounced ups and downs are typical of couples composedof non secure individuals. Thus, we can, for example, immediately concludethat Laura and/or Petrarch were not secure individuals.

We can now focus on the influence of individual parameters and be-havior on the quality of romantic relationships at equilibrium. The firstproperty we point out specifies the role of the appeals.

Property 5

An increase of the appeal Ai of individual i gives rise to an increase ofthe feelings of both individuals at equilibrium. Moreover, the relative increaseis higher for the partner of individual i,

The first part of this property is rather obvious, while the second partis more subtle. Indeed it states that there is a touch of altruism in a woman[man] who tries to improve her [his] appeal.

We conclude our analysis by considering perturbations of the behav-ioral characteristics of the individuals. A positive perturbation of the in-stinct function of individual i gives rise to a higher value of Ii(Aj) and istherefore equivalent to a suitable increase of the appeal Aj. Thus, one canrephrase Property 5 and state that an increase of the instinct function Ii,of individual i gives rise to an increase of the feelings of both individualsat equilibrium and that the relative improvement is higher for individuali. A similar property holds for perturbations of the return functions as in-dicated below.


Property 6

An increase of the return function Ri of individual i gives rise to an in-crease of both feelings at equilibrium, but in relative terms such an improve-ment is more rewarding for individual i.

This is the last relevant property we have been able to extract frommodel (1). Although some of them are rather intuitive, others are moreintriguing. But what is certainly surprising is that they are all mere logicalconsequences of our assumptions (2).

CONSEQUENCES AT COMMUNITY LEVEL

Now that we have identified the individual consequences of properties(2), we can use them to extend the study to a more aggregated level. Forthis we will develop a purely theoretical exercise, dealing with a hypotheti-cal community composed only of couples of secure individuals. Neverthe-less, since secure individuals are a relevant fraction of population, we canbe relatively confident and hope that our results retain, at least qualita-tively, some of the most significant features of real societies.

Consistently with model (1), an individual i is identified by appeal,forgetting coefficient, return function and instinct function, i.e., by thequadruplet (Ai, ai, Ri ,Ii,). Thus, a community composed of N women andN men structured in couples is identified by [A1 n, a1n, R1n,I1n, A2n, a2n,R2n, I2n], where the integer n = 1, 2, ..., N is the ordering number of thecouple. For simplicity, let us assume that individual 1 is a woman and 2 isa man and suppose that there are no women or men with the same appeal,i.e., Aih =Aik for all h = k. This means that the couples can be numbered,for example, in order of decreasing appeal of the women. Since we haveassumed that all individuals are secure, properties (2) hold for all individu-als of the community. Moreover, in order to maintain the mathematicaldifficulties within reasonable limits, we perform only an equilibrium analysisand assume that all fragile couples are at their positive equilibrium E+. Inconclusion, our idealized community is composed of couples of secure in-dividuals in a steady and high quality romantic relationship.

Such a community is called unstable if a woman and a man of twodistinct couples believe they could be personally advantaged by forming anew couple together. In the opposite case the community is called stable,Thus, practically speaking, unstable communities are those in which theseparation and the formation of couples are quite frequent. Obviously, thisdefinition must be further specified. The most natural way is to assumethat individual i would have a real advantage in changing the partner, ifthis change is accompanied by an increase of the quality of the romantic


relationship, i.e., by an increase of xi+. However, in order to forecast thevalue x1+[x2+] that a woman [man] will reach by forming a couple with anew partner, she [he] should know everything about him [her]. Generally,this is not the case and the forecast is performed with limited information.Here we assume that the only available information is the appeal of thepotential future partner and that the forecast is performed by imaginingthat the forgetting coefficient and the return and instinct functions of thefuture partner are the same as those of the present partner. Thus, the ac-tual quality x1+ of the romantic relationship for the woman of the h-thcouple is x1+( A1 a1 h , R f , /A A2h, o.2h, R2h, I2h) while the quality sheforecasts by imagining to form a new couple with the man of the k-th coupleis x1+(A1h, a1h, R1h, I1h, A2k, a2h, R2h, I2h). This choice of forecasting thequality of new couples obviously emphasizes the role of appeal. Quite rea-sonably, however, because appeal is the only easily identifiable parameterin real life.

The above discussion is formally summarized by the following definition.Definition 1A community [ A 1 n , a1n", R1n, I1n, A2n, a2n, R2n, I2n), n = 1, 2 , N is

unstable if and only if there exists at least one pair (h, k) of couples such that

A community which is not unstable is called stable.

We can now show that stable communities are characterized by thefollowing very simple but peculiar property involving only appeal.

Property 7

A community is stable if and only if the partner of the n-th most attractivewoman (n = 1, 2, ..., N) is the n-th most attractive man.

In order to prove this property, note first that Property 5 implies thata community is unstable if and only if there exists at least one pair (h, k)of couples such that


Condition (3) is illustrated in Fig. 3a in the appeal space, where eachcouple is represented by a point. Consider now a community in which thepartner of the n-th most attractive woman is the n-th most attractive man(n = 1, 2, ..., N). Such a community is represented in Fig. 3b, which clearlyshows that there is no pair (h, k) of couples satisfying inequalities (3). Thus,the community is stable. On the other hand, consider a stable communityand assume that the couples have been numbered in order of decreasingappeal of the women, i.e.

Then, connect the first point (A11, A21) to the second point (A12, A22) witha segment of a straight-line, and the second to the third, and so on untilthe last point (A1N, A2N) is reached. Obviously, all connecting segmentshave positive slopes because, otherwise, there would be a pair of couplessatisfying condition (3) and the community would be unstable (which wouldcontradict the assumption). Thus, A21 > A22 > ... > A2N. This, togetherwith (4), states that the partner of the n-th most attractive woman is then-th most attractive man.

On the basis of Property 7, higher tensions should be expected in com-munities with couples in relevant conflict with the appeal ranking. This re-sult, derived from purely theoretical arguments, is certainly in agreementwith empirical evidence. Indeed, partners with very uneven appeals arerarely observed. Of course, in making these observations one must keep inmind that appeal is an aggregated measure of many different factors and

Fig. 3. Community structures in the appeal space: (a) two points corresponding to twocouples (h, k) belonging to an unstable community (see (3)); (b) an example of astable community (each dot represents a couple).


that gender differences might be relevant. Thus, for example, the existenceof couples composed of a beautiful young lady and an old but rich mandoes not contradict the theory, but, instead, confirms a classical stereotype.

CONCLUDING REMARKS

A minimal model of love dynamics between secure individuals hasbeen presented and discussed in this paper with two distinct goals, onegeneric and one specific. The generic goal was to show how one couldpossibly deal with dynamic phenomena in social psychology by means ofthe modelling approach based on differential equations and traditionallyused in other fields of sciences. This approach is quite powerful for estab-lishing a hierarchy between different properties and distinguishing betweencauses and effects. The specific goal was to derive a series of propertiesconcerning the quality and the dynamics of romantic relationships in cou-ples composed of secure individuals.

The model equations take into account three mechanisms of love growthand decay: the pleasure of being loved, the reaction to the partner's appeal,and the forgetting process. For reasonable assumptions on the behavioral pa-rameters of the individuals, the model turns out to enjoy a number of re-markable properties. It predicts, for example, that the feelings of two partnersvary monotonically after they first meet, growing from zero (complete indif-ference) to a maximum. The value of this maximum, i.e., the quality of theromantic relationship at equilibrium, is higher if appeal and reactiveness tolove are higher. The model explains also why there are two kinds of couples,called robust and fragile. Robust couples are those that have only one sta-tionary mode of behavior characterized by attachment. By contrast, fragilecouples can also be trapped in an unpleasant mode of behavior characterizedby antagonism. All these properties are in agreement with common wisdomon the dynamics of love between two persons.

These properties have been used to derive the characteristics underwhich the couples of a given community have no tendency to separate (sta-bility). The main result along this line is that the driving force that createsorder in the community is the appeal of the individuals. In other words,couples with uneven appeals should be expected to have higher chancesto brake off.

As for any minimal model, the extensions one could propose are in-numerable. Aging, learning and adaptation processes could be taken intoaccount allowing for some behavioral parameters to slowly vary in time, inaccordance with the most recent developments of attachment theory. Par-ticular nonlinearities could be introduced in order to develop theories for


non secure individuals (Gragnani, Rinaldi & Feichtinger, 1997). Men andwomen could be distinguished by using two structurally different state equa-tions. The dimension of the model could also be enlarged in order to con-sider individuals with more complex personalities or the dynamics of lovein larger groups of individuals. Moreover, the process followed by eachindividual in forecasting the quality of the relationship with a potential newpartner could be modelled more realistically, in order to attenuate the roleof appeal, which has been somehow overemphasized in this paper. Thiscould be done quite naturally by formulating a suitable differential gameproblem. Undoubtedly, all these problems deserve further attention.

ACKNOWLEDGMENTS

Preparation of this article was supported by the Italian Ministry ofScientific Research and Technology, contract MURST 40% Teoria deisistemi e del controllo.

We are grateful to Gustav Feichtinger, Lucia Carli, and Frederic Jonesfor their suggestions and encouragement.

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APPENDIX

Proof of Property 1

For x1 = 0 and x2 > 0 eq. (la) gives dx1/dt > 0. Symmetrically, forx1 > 0and x2 = 0 eq. (1b) gives dx2/dt > 0. Moreover, for x\ = x2 = 0, eqs. (1) givedx1/dt > 0 and dx2/dt > 0. Hence, trajectories starting from the boundary ofthe positive quadrant enter into the positive quadrant and remain there forever.

Proof of Property 2

The isoclines and of the system are given by (see

From properties (2) it follows that these isoclines intersect either at onepoint with positive coordinates (see Fig. 4a), or at three points, one with


positive coordinates and two with negative coordinates (see Fig. 4b). Thesetwo cases correspond, by definition, to robust and fragile couples. On thefirst isocline, trajectories are vertical, while on the second they are hori-zontal. Studying the signs of dx1/dt and dx2/dt in the regions delimited bythe isoclines, one can determine the direction of the trajectory of the systemat any point (x1,x2). These directions, indicated with arrows in Fig. 4, allowone to conclude that the equilibrium points E+ and E- are stable nodes,while the equilibrium point 5 is a saddle. Moreover, if the couple is robust(Fig. 4a) the equilibrium E+ is a global attractor, i.e., all trajectories tendtoward this point as time goes on. By contrast, if the couple is fragile (Fig.4b), there are two attractors, namely points E+ and E-, and their basinsof attraction are delimited by the stable separatrix of the saddle point 5(not shown in the figure). Since point 5 has negative coordinates and theseparatrix has a negative slope, it follows that the basin of attraction ofthe positive attractor (point E+) contains the positive quadrant as indicatedin Fig. 2.

Proof of Property 3

Assume that parameters and functions of the two individuals are suchthat the couple is fragile, i.e., the isoclines are like in Fig. 4b. Then, in-crease gradually I1(A2) and I2(A1). Since an increase of I\(AT) shifts thefirst isocline to the right, while an increase of I2(A1) shifts the secondisocline upward, after a while points S and E- collide and disappear(through a fold bifurcation). Thus, sufficiently high values of I1(A2) andI2(A1) guarantee that the isoclines are like in Fig. 4a, i.e., that the coupleis robust.

Proof of Property 4

Refer to Fig. 4 and to the region delimited by the two isoclines inwhich the origin falls. Inside this region dx1/dt and dx2/dt are positive andtrajectories cannot leave the region. This implies the property.

Proof of Property 5

Refer again to Fig. 4 and indicate with x1 + and x2+ the coordinatesof the positive equilibrium E+. Then, increase A2 of a small quantity AA2.This produces a rigid shift to the right of the first isocline while the seconddoes not move. Thus, point E+ shifts to the right along the second isocline.The consequence is that both components of the equilibrium increase, say


Fig. 4. The isoclines dx1ldt = 0 and dx2ldt - 0of the system and the directions of trajectoriesin robust (a) and fragile (b) couples. The equi-librium points E+ and E- are stable nodes, while5 is a saddle.

of Ax1+ and Ax2+, respectively. Obviously, Ax2+/AX1 + coincides with theslope of the second isocline and is therefore smaller than x2+/x1+ which isthe slope of the straight line passing through the origin and point E+.Hence, Ax2+/x2+ < Ax1+/x1+, which proves the property.


Proof of Property 6

Refer once more to Fig. 4 and note that an increase of the returnfunction of the first individual from R1(x2) to R1(x2)+SR1(x2) with 8R1 >0, moves the first isocline to the right of a quantity SR1(x2). This impliesthat point E+ moves to the right along the second isocline so that, usingthe same argument used in the proof of Property 5, one can conclude thatAx2+/x2+ < Ax1+/x1+.

love dynamics between secure individuals: a modeling...

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