Dynamics of two languages competing on a network: a case study
DDynamics of two languages competing on a network: a case study
Todd Kapitula ∗ Department of Mathematics and StatisticsCalvin UniversityGrand Rapids, MI 49546Panayotis G. Kevrekidis † Department of Mathematics and StatisticsUniversity of MassachusettsAmherst, MA 01003-4515
February 16, 2021
Abstract.
A language dynamics model on a square lattice, which is an extension of the one popularizedby Abrams and Strogatz [ ], is analyzed using ODE bifurcation theory. For this model we are interestedin the existence and spectral stability of structures such as stripes, which are realized through pulsesand/or the concatenation of fronts, and spots, which are a contiguous collection of sites in which onelanguage is dominant. Because the coupling between sites is nonlinear, the boundary between sitescontaining speaking two different languages is “sharp”; in particular, in a PDE approximation it allowsfor the existence of compactly supported pulses (compactons). The dynamics are considered as a functionof the prestige of a language. In particular, it is seen that as the prestige varies, it allows for a languageto spread through the network, or conversely for its demise. Keywords.
ODE bifurcation theory, language competition, prestige
Contents
1. Introduction
2. The model on a square lattice
3. Existence and spectral stability of stripes for the discrete model ∗ E-mail: [email protected] † E-mail: [email protected] a r X i v : . [ n li n . PS ] F e b . Spots: a case study anguage competition dynamics
1. Introduction
In their seminal paper Abrams and Strogatz [ ] developed a simple ODE model,˙ u = (1 − u ) u p − Au (1 − u ) p , (1.1)to help understand language competition and the decline in the number of people who speak such historiclanguages as Welsh, Quechua, and Scottish Gaelic. We will henceforth label (1.1) as the AS model (seeFigure 1 for a cartoon representation of this compartment model). The underlying assumptions for thismodel are that all speakers are monolingual, and the population is highly connected with no spatial or socialstructure. In equation (1.1) u represents the proportion of the population which speak language U . If v isthe proportion which speak language V , since all speakers are monolingual, v = 1 − u . The parameter p > volatility . The case p = 1 is a neutral situation, where transition probabilities from one languageto another depend linearly on local language densities. If p > p < p = 1 . ± .
25. The parameter
A > A can be used to represent the prestige associated with a particular language.Assume p >
1, so the volatility is low. The fixed points u = 0 (language V is preferred) and u = 1(language U is preferred) are stable, while u = B/ (1 + B ) with B = A / ( p − is unstable. If A < u (0) = 0 . u ( t ) → t → + ∞ , so the populationhas an affinity for language U . Or, language U has more prestige in the general population. On the otherhand, if A > u (0) = 0 .
5, then u ( t ) → t → + ∞ , so language V has more prestige in the wholepopulation. U Vu p Av p Figure 1: (color online) The compartment model associated with AS model (1.1). The variable u represents the proportion of the population which speaks language U , and v is the proportionwhich speaks language V . It is assumed u + v = 1. As pointed out by Mira and Paredes [ ], the monolingual assumption implies the two languages are sodissimilar that conversation is practically impossible between the two competing language groups. Theseauthors extend the AS model to allow for languages which are similar enough for there to be bilingualspeakers. The bilingual population subgroup satisfies, b = 1 − u − v , with 0 ≤ b ≤
1. The model becomes,˙ u = (1 − k )(1 − u )(1 − v ) p − Au (1 − u ) p ˙ v = (1 − k ) A (1 − v )(1 − u ) p − v (1 − v ) p , (1.2)where 0 < k < k = 0 means that conversation is notpossible between monolingual speakers, and k = 1 implies U = V . The larger the value of k , the moresimilar are the two languages. If k = b = 0, then model (1.2) reduces to model (1.1). An analysis of themodel (1.2) is provided in [ , ].An agent-based model associated with the AS model (1.1) when p = 1,˙ u = (1 − A ) u (1 − u ) , (1.3). Kapitula and P.G. Kevrekidis is considered by Stauffer et al. [ ]. In particular, on a lattice of dimension d an individual is assumed to feelthe influence of 2 d nearest neighbors. When A > A = 1. It is not clear how the two models compare when A <
1. An agent-based model is also consideredby Vazquez et al. [ ]. In the fully connected case the dynamics of the associated mean-field model areequivalent to those for the model (1.1).The AS model has been extended to networks. Each node of the network corresponds to a group whosedynamics are governed by the AS model, and then the dynamics between groups satisfy some other rule.Amano et al. [ ] collected and analyzed world-wide data taking into account such things as geographicalrange size, speaker population size, and speaker growth rate (i.e., changes in the number of speakers) ofthe world’s languages, and assessed interrelations among these three components to understand how theycontribute to shaping extinction risk in languages. The role of population density and how it effects theinteraction rates among groups is discussed by Juane et al. [ ] in the context of language shift in Galicia,which is a bilingual community in northwest Spain. They model the problem by looking at equations (1.2)on a network, with the strength of the interactions between nodes depending on the population density. Themodel for j = 1 , . . . , n is,˙ u j = (1 − k j )(1 − u j )(1 − v j ) p − A j u j (1 − u j ) p + K j ( u − u j )˙ v j = (1 − k j ) A j (1 − v j )(1 − u j ) p − v j (1 − v j ) p + K j ( v − v j ) . (1.4)Here y represents the average of the set, { y j } . The positive parameter K j is assumed to be a strictlyincreasing function of the population density. The authors Vidal-Franco et al. [ ] follow a similar strategy,except they assume the nonlinearities are of Lotka-Volterra type. Taking a different approach, Yun et al.[ ] assume a diffusion process to take into account spatial effects. Fujie et al. [ ] and Zhou et al. [ ]consider the problem of competition among more than two languages.In this paper we consider the language competition problem on a network under the assumption of lowvolatility, p >
1. For ease we will primarily work with p = 2, but our experience is that other values of p > , , ]for some work in this area under the assumption of a single group). This may be an unrealistic assumptionin terms of language; however, it is less so if one assumes language U actually refers to those who have sometype of religious affiliation, and language V represents those who do not [ ]. We will assume the existenceof n distinct population groups, and let 0 ≤ u j ≤ j who speaklanguage U ( v j = 1 − u j speak language V ). For each j = 1 , . . . , n our model equation is a natural extensionof the compartment model illustrated in Figure 1,˙ u j = (cid:32) n (cid:88) k =1 I jk u pk (cid:33) · (1 − u j ) − A j (cid:32) n (cid:88) k =1 I jk (1 − u k ) p (cid:33) · u j , (1.5)where I jk ≥
0. We call the matrix I = ( I jk ) the influence matrix, and the term I jk represents the influencegroup k has on group j through the between-group reaction rate. If we think of the system (1.5) as beinga compartment model, then the term I jk u pk is the rate constant associated with the influence that the U speakers in group k have on the V speakers in group j , and I jk (1 − u k ) p is the rate constant associated withthe influence that the V speakers in group k have on the U speakers in group j . Clearly, when n = 1 thesystem (1.5) collapses to the AS model (1.1).We now compare the systems (1.4) and (1.5). In the case of no bilingual speakers the system (1.4)collapses to, ˙ u j = (1 − u j ) u pj − A j u j (1 − u j ) p + K j ( u − u j ) . (1.6)The systems (1.5) and (1.6) have the feature that the on-site dynamics are the same as those for the ASmodel. However, the coupling between groups is different; in particular, the model (1.6) assumes that group j is influenced by all of the other groups, whereas the model (1.5) allows for each group to be isolated fromsome of the other groups. Under the assumption that each external group has an equal influence on a givengroup, I jj = 1 and I jk = K j / ( n −
1) for all k (cid:54) = j , the system (1.5) becomes,˙ u j = (1 − u j ) (cid:104) u pj + K j u p (cid:54) = j (cid:105) − A j u j (cid:104) (1 − u j ) p + K j (1 − u ) p (cid:54) = j (cid:105) , (1.7) anguage competition dynamics where we use the notation, f (cid:54) = j = 1 n − (cid:88) k (cid:54) = j f k . The nonlinear coupling term for the model (1.7) is clearly very different than the linear coupling termassociated with the model (1.6). It is an open question as to whether this functional difference leads to aqualitative difference in the dynamics.A simple model such as (1.1) can also be used to model opinion propagation in a population in which itis assumed that people have either opinion U , or opinion V , where we think of V as being “not U ”. Marvelet al. [ ], hereafter referred to as MS, provide a model similar to (1.2) in which it is assumed there are threedistinct groups: those who hold opinion U , those who hold opinion V , and the remaining who are undecided,˙ u = (1 − u − v ) u − uv ˙ v = (1 − u − v ) v − uv. The underlying assumption in this model is that in order for one who initially holds opinion U to eventuallyhold opinion V (or vice-versa), the person first must become undecided. Wang et al. [ ] extended theMS model to allow for several competing opinions. The MS model was extended to networks by Bujalskiet al. [ ], and the extended model was studied using dynamical systems techniques. Tanabe and Masuda[ ] proposed and analyzed an interesting opinion formation model (hereafter labelled TM) in which it wasassumed that the population itself breaks down into two groups: congregators, and contrarians. In contrast,the MS model implicitly assumes the entire population is filled with congregators. One conclusion of theTM model is that if a large enough proportion of the population is contrarian, then no majority opinionwill be achieved. This is in contrast to the conclusion of those models in which it is assumed there are onlycongregators, as here a majority opinion is always obtained. The TM model was later refined by Eekhoff [ ],and the new model allowed for the effects of peer pressure, and incorporated the influence of zealots. Froma qualitative perspective the mean-field models used for opinion dynamics and language death have manysimilarities. Thus, although we frame our results using the formulation associated with language death, theyare also directly applicable to mean-field opinion formation models.In this paper we are primarily interested in the existence and stability of spatial structures for thenetwork system (1.5). We assume the groups have been arranged on a square lattice. The interactions onthis lattice are nearest-neighbor (NN) only. Our experience is that from a qualitative perspective the NNinteractions can be expanded without substantively changing the solution behavior as long as the interactionsare still somewhat spatially localized (the Implicit Function Theorem provides the theoretical justification).Moreover, there will be no preferential distinction in the reaction rates, I jk = I kj . This is a case study, sowe have not fully explored a large set of networks. That work will be left for a future paper. Our goal hereis not to do an exhaustive study for all types of influence matrices. Instead, we simply want to get a senseof what is possible for a given type of network.For this lattice configuration we start by considering the existence and stability of fronts and pulses forthe system (1.5). A front is a solution for which u jk = U j , and U j = 0 (or U j = 1) for 1 ≤ j ≤ n , and U j = 1(or U j = 0) for j ≥ n + (cid:96) and some (cid:96) ≥
1. In other words, to the left of n language V is spoken, and to theright of n + (cid:96) language U is spoken. A pulse is a solution for which U j = 0 for j ≤ n and j ≥ n + (cid:96) , and U j > n < j < n + (cid:96) . In other words, on the full lattice there is a stripe of language U speakers who aresurrounded by a group of V speakers. We will consider when fronts can travel, which implies that language U is invading language V , or vice-versa. We will also consider when pulses can grow or shrink. A growingpulse can be thought of as the concatenation of two fronts traveling in opposing directions, which impliesthat language U eventually takes over the entire network. A shrinking pulse eventually disappears, whichmeans that language U has gone extinct. As we will see, the prestige associated with speaking U ( A < V ( A >
1) plays a central role in the analysis. We will conclude with a case study for a spot, which is acontiguous group of sites with u jk > u jk = 0 - an island of U in a sea of V . Acknowledgements.
This material is based upon work supported by the US National Science Foundationunder Grant No. DMS-1809074 (PGK).. Kapitula and P.G. Kevrekidis
2. The model on a square lattice
As already stated, we consider the dynamics of a square lattice with nearest-neighbor interactions only. Here u jk will represent the proportion of the population at site ( j, k ) who speak language U . We will henceforthassume that the prestige associated with language U is uniform throughout the lattice, A jk = A . It is aninteresting problem in its own right to allow for a spatially inhomogeneous distribution of the prestige andsee how it affects the prevalent dynamics. Moreover, we will assume p = 2. Our numerical experimentsindicate that from a qualitative perspective the results presented herein only need p > u jk = (cid:2) (cid:15) u jk + (cid:15) (cid:0) u j +1 ,k + u j − ,k + u j,k +1 + u j,k − (cid:1)(cid:3) (1 − u jk ) − A (cid:2) (cid:15) (1 − u jk ) + (cid:15) (cid:0) (1 − u j +1 ,k ) + (1 − u j − ,k ) + (1 − u j,k +1 ) + (1 − u j,k − ) (cid:1)(cid:3) u jk . Here 1 ≤ j, k ≤ n , and we assume in the model that at the edge of the square there are Neumann boundaryconditions, e.g., u n +1 ,k = u nk . The parameter (cid:15) > (cid:15) > dis f jk = f j +1 ,k + f j − ,k + f j,k +1 + f j,k − − f jk , the above ODE takes the more compact form,˙ u jk = ( (cid:15) + 4 (cid:15) ) u jk (1 − u jk ) [(1 + A ) u jk − A ] + 2 A(cid:15) u jk ∆ dis u jk + (cid:15) [1 − (1 + A ) u jk ] ∆ dis u jk . (2.1)If we assume that the interactions between neighbors are strong, i.e., (cid:15) (cid:29)
1, then upon setting R = (cid:15) + 4 (cid:15) (cid:29) ∂ t u = Ru (1 − u ) [(1 + A ) u − A ] + (1 + A ) u (1 − u )∆ u + [1 − (1 + A ) u ] |∇ u | . (2.2)Here ∆ represents the Laplacian, and ∇ is the gradient operator. The continuum model incorporates theexpected temporal dynamics associated with the original ODE model, but the coupling dynamics betweensites is dictated by an effective nonlinear diffusion. The PDE is physical in the following sense: u ( x, y, t ) = 0implies ∂ t u ( x, y, t ) ≥
0, and u ( x, y, t ) = 1 implies ∂ t u ( x, y, t ) ≤
0. Note the diffusion coefficient vanisheswhen the entire population supports one language, u = 0 or u = 1.When studying the solution structure to the ODE (2.1), or the accompanying PDE (2.2), we will firstfocus on the existence and spectral stability of time-independent patterns which vary in one direction only.For the ODE (2.1) we will set u j,k ( t ) = U j for all j, k , and U j will solve the 1D discrete model,0 = ( (cid:15) + 4 (cid:15) ) U j (1 − U j ) [(1 + A ) U j − A ] + 2 (cid:15) AU j ∆ j U j + (cid:15) [1 − (1 + A ) U j ] ∆ j U j , (2.3)where ∆ j f j = f j +1 + f j − − f j . For the PDE (2.2) we will set u ( x, y, t ) = U ( x ), and U ( x ) will solve thenonlinear ODE,0 = RU (1 − U ) [(1 + A ) U − A ] + (1 + A ) U (1 − U ) U (cid:48)(cid:48) + [1 − (1 + A ) U ] ( U (cid:48) ) , (cid:48) = dd x . (2.4)In both cases we will be looking for fronts/pulses, which for the full system will correspond to stripes. Thesesolutions act as transitions between regions where language U is dominant and language V is dominant. Remark 2.1.
Even though the derivation is dissimilar, the continuum model (2.2) is remarkably similar tothe mean-field model associated with the square lattice as provided for in [ , equation (48)]. The model(2.2) has the additional term, [1 − (1 + A ) u ] |∇ u | ; however, both models have the important feature thatthe diffusion coefficient is singular. Dynamically, both systems have the feature that small domains tend toshrink, and large domains tend to grow, and the domains tend to evolve in a way that reduces the curvatureof the boundary; see also further relevant discussion regarding the dynamics below. anguage competition dynamics
3. Existence and spectral stability of stripes for thediscrete model
A front solution to (2.3) satisfies U j = 0 (1) for j ≤ (cid:96) , and U j = 1 (0) for j ≥ k , where 1 < (cid:96) < k < n . Apulse solution will satisfy U j = 0 (1) for j ≤ (cid:96) and j ≥ k , and U j ∼ (cid:96) < j < k . The transitionbetween the states 0 and 1 will be monotone. A stripe solution to the full 2D model will be a pulse, or aconcatenation of two fronts. As we will see, the concatenation of two fronts provides for a “thicker” stripe.In the same spirit, we can also discuss multi-stripes, which are the concatenation of pulses and/or fronts. If (cid:15) = 0, the system uncouples, so a front can be constructed analytically. In this limit, for a front we set U j = 0 (1) for j = 1 , . . . , (cid:96) , and U j = 1 (0) for j = (cid:96) + 1 , . . . , n . We will refer to this front as the off-site front.Since each of the fixed points is stable for the scalar AS model, the front will be stable for the full system.By the Implicit Function Theorem the front will persist and be stable for 0 < (cid:15) (cid:28)
1. We can concatenatethese fronts when (cid:15) = 0 to form stable stripes, and then again apply the Implicit Function Theorem to showthe existence and stability for small (cid:15) .When (cid:15) = 0 we can construct another front by setting U j = 0 (1) for j = 1 , . . . , (cid:96), U (cid:96) +1 = A/ (1 + A ),and U j = 1 (0) for j = (cid:96) + 2 , . . . , n . Since all of the fixed points but the one at j = (cid:96) are stable for the scalarAS model, the front will be unstable for the full system with the linearization having one positive eigenvalue.By the Implicit Function Theorem the front will persist and be unstable with one positive eigenvalue for0 < (cid:15) (cid:28)
1. We will refer to this front as the on-site front.When (cid:15) = 0 the off-site and on-site fronts exist for any value of A . However, once there is nontrivialcoupling, we expect there will be an interval of A values which contains A = 1 for which the fronts willexist. In order to determine this interval we will do numerical continuation using the MATLAB package,Matcont [ ]. Using this package will also allow us to numerically continue bifurcation points in parameterspace. Setting R = (cid:15) (cid:15) , we will numerically explore the ( R , A )-parameter space. Since we analytically know what happens for R = 0, we are in a good position to use numerical continuation.For each fixed R > A . For a particularexample, consider the left figure in Figure 2. The horizontal axis is A , and the vertical axis is the L -normof the front. In this figure the solid (blue) curve corresponds to a stable front (which is off-site when A = 1),and the dashed (red) curve corresponds to an unstable front (which is on-site when A = 1). These twocurves meet at a saddle-node bifurcation point, which is denoted by an open black circle. We see there isan A − < < A + for which there are stable fronts for A − < A < A + , and no stationary fronts (at least asseen via numerical continuation) outside this interval. The values of A ± depend on R Each of the upwardshifts of the stable and unstable branches correspond to waveforms that are shifted by an integer number oflattice nodes to the left (hence the growth in norm). The right panel inFigure 2 shows the functions A ± asa function of R . While we do not show it here, even in the limit R → + ∞ the two curves do not convergeto 1; instead, we have A + (+ ∞ ) ∼ . A − (+ ∞ ) ∼ . R ,there is a stable stationary front. Outside the two curves, A ± ( R ), there is a traveling front. Traveling waves will be written as U ( x + ct ),so U j ( t ) = U ( j + ct ). Setting ξ = x + ct , the resulting forward-backward difference equation to which thetraveling wave is a solution is, cU (cid:48) = ( (cid:15) + 4 (cid:15) ) U (1 − U ) [(1 + A ) U − A ]+ 2 A(cid:15) U [ U ( ξ + 1) + U ( ξ − − U ( ξ )] + (cid:15) [1 − (1 + A ) U ] (cid:2) U ( ξ + 1) + U ( ξ − − U ( ξ ) (cid:3) . . Kapitula and P.G. Kevrekidis | U | R A traveling front, U->Vtraveling front, V->U Figure 2: (color online) Numerically generated existence curves for stationary V → U fronts, i.e., U j = 0 to the left, and U j = 1 to the right. The curve is given for R = 0 . (cid:96) -norm.Regarding the boundary in the right panel, inside the two curves there is a stable stationary front,and outside the curves the front travels. The invading language is provided in the figure. This system is solved using a variant of Newton’s method (see [ , , ] for the details).We consider in detail the case of R = 0 .
6. Our experience is that from a qualitative perspective thevalue of R is not particularly important. The numerical result is plotted in Figure 3. The points A ± aremarked with a (red) diamond. It should be the case that at these points c = 0; unfortunately, the fact thatthe linearization becomes singular at A = A ± precludes good convergence of the algorithm near these points.Away from these bifurcation points there is good convergence of the numerical algorithm. Assuming U j = 0to the left, and V j = 0 to the right, if c < V invades language U , whereas if c > U invades language V . We see here that if A > A + ∼ . V has more prestige, then language V invades language U . On the other hand, if A < A − ∼ . U has more prestige, thenlanguage U invades language V . Note that the speed increases as the preferred language becomes moreprestigious. Indeed, up to a small correction, and sufficiently far away from A ± , the wave speed follows theformal prediction of the continuum model, equation (4.1). The predicted curve, which is associated withthe limit R → + ∞ , is given by the black dashed line. This result has been numerically verified for severaldifferent values of R . One can observe the nontrivial effect of discreteness in establishing an interval wherethe fronts can be stationary. Indeed, the continuum model is found to possess vanishing speed at the isolatedpoint of prestige balance, namely at A = 1, while the discrete variant requires a detuning from this value inorder to enable such a depinning from the vanishing speed setting. Remark 3.1.
It is an interesting exercise to consider the scaling law for the wave speed as A → A ± ; however,we have not pursued this. The interested reader should consult Anderson et al. [ ], Kevrekidis et al. [ ]and the references therein for details as to how such a law may be derived. As is the case for fronts, if (cid:15) = 0 a pulse can be constructed analytically by setting U j = 0 (1) for 1 ≤ j ≤ (cid:96) and k ≤ j ≤ n , and U j = 1 (0) for (cid:96) < j < k . Since each of the fixed points is stable for the scalar ASmodel, the pulse will be stable for the full system. By the Implicit Function Theorem the pulse will persistand be stable for 0 < (cid:15) (cid:28)
1. We can concatenate these pulses when (cid:15) = 0 to form stable stripes, and thenagain apply the Implicit Function Theorem to show the existence and stability for small (cid:15) . If so desired, anguage competition dynamics c Figure 3: (color online) The numerically generated wave speed when R = 0 . A − ∼ . A + ∼ . we can also construct unstable pulses by setting U (cid:96) +1 = A/ (1 + A ) when (cid:15) = 0, and then using the ImplicitFunction Theorem for small (cid:15) . Assuming the background supports language V , the size of the pulse is thenumber of adjacent groups which support language U . For small (cid:15) the size is k − (cid:96) − V → U and a U → V stationary front. Consequently, the front dynamics completely determine the pulsedynamics. If the front is stationary, so is the pulse. If the front moves, so will the edge of the pulse. On theother hand, if the pulse is of size 1, 2, or 3, then the dynamics are not related to front dynamics. From adynamics perspective the pulse ceases to exist after a saddle-node bifurcation occurs.Using Matcont, the bifurcation point can be traced in ( R , A )-space. The results are presented in Figure 4.The pulse will exist inside the boundary curve. The cusp point is ( R , A ) ∼ (0 . , . R , A ) ∼ (0 . , . R > .
77 with 0 < − A (cid:28)
1, and is not shown in the figure. Note that the cusp point converges to A = 1as the size of the pulse increases, and satisfies A <
1. This is due to the fact that language U has moreprestige for A <
1. If the background was language U instead of language V , then the cusp point wouldsatisfy A > R is less than the cusp point value, and if A is small enough sothat ( R , A ) is below the bottom boundary curve, then the pulse will grow until it can be thought of asa concatenation of two fronts. Once this occurs the edges of the pulse will move according to the frontdynamics. The pulse grows because the prestige for language U is sufficiently large. On the other hand, if( AR , A ) is above the top boundary curve, then language V has sufficient prestige so that the backgroundlanguage prevails, and the pulse simply disappears in finite time. See Figure 5 for the corroborating resultsof a particular simulation. Remark 3.2.
If we assume a pulse of language V sits on a background of language U , then we will get thesame curves as in Figure 4. However, the dynamical interpretation leading to Figure 5 will be reversed. Inparticular, if A is too small the pulse will disappear, whereas if A is sufficiently large it will grow. We now consider the problem of concatenating individual pulses to form multi-pulses. For the sake ofconvenience and without loss of generality we assume that background consists of language V . As withthe single pulses, each of the multi-pulses will be stable when (cid:15) = 0, and they will persist as stable. Kapitula and P.G. Kevrekidis A j Figure 4: (color online) The left panel provides the numerically generated boundary of pulses ofsize 1 through 3. The boundary is given by a solid (blue) curve for the pulse of size 1, a (red)dashed curve for a pulse of size 2, and a (green) dashed-dotted curve for a pulse of size 3. For agiven pulse size, the pulse exists inside the two curves, and ceases to exist outside. The right panelgives an example of each pulse for R = 0 .
05 and A = 1. The pulse of size 1 is shown in the upperright panel, the pulse of size 2 in the middle right panel, and the pulse of size 3 in the lower rightpanel. structures for sufficiently small (cid:15) . Typically, the construction of multi-pulses would involve a discussionof tail-tail interactions between individual pulses, and an application of the Hale-Lin-Sandstede method(e.g., see [ , , , , – ] and the references therein). However, for the system under considerationthis is less relevant, as the nonlinear coupling between adjacent sites renders the transition from one stateto another to be super-exponential, instead of the exponential rates associated with linear coupling (seeFigure 6 for a representative demonstration of this phenomena). Consequently, to leading order one canthink of pulses as being compactons (a compactly supported structure), and fronts as being a compactlysupported transition between two states. In this light, to leading order, and as long as the individual pulsesare initially sufficiently separated, the dynamics associated with a concatenation of k pulses is really just thedynamics of k uncoupled pulses, each of which evolves according to the rules presented in Section 3.3.Since this is only a case study, we will focus on the example of the two-pulse, which at the (cid:15) = 0 limit welabel as j - k - (cid:96) . Here j and (cid:96) refer to the size of the pulse which supports language U , and k is the interveningpulse of size k which supports language V . For example, a 2-1-2 can be thought of when (cid:15) = 0 as thesequence of u -values, · · · · · · .First consider the 2-1-2 pulse. The boundary for which this solution exists is presented as a solid (blue)curve in Figure 7. The cusp point is ( R , A ) ∼ (0 . , . R , A ) values inside the curve thepulse will exist as a stationary solution and be stable, whereas outside the curve it does not exist. Froma dynamical perspective, if R < . A is chosen so that the point lies below the lower boundarycurve, then the solution will quickly become a single pulse of size 5 (i.e., the internal 0 becomes a 1), seethe center panel of Figure 8 with ( R , A ) = (0 . , . A is such that the point is also below the lowerboundary of the curve presented in the right panel of Figure 2, so that U invades V , then both fronts willtravel, i.e., expand until the entire lattice is overtaken by language U (see the left panel of Figure 8 with( R , A ) = (0 . , . R < . A is chosen so that the point lies above theupper boundary curve, then the solution will quickly decay to a pulse of size zero, i.e., language V is spokenover the entire lattice (see the right panel of Figure 8 with ( R , A ) = (0 . , . We will return to this aspect in more detail in the continuum limit analysis, see Section 4.1. anguage competition dynamics j t j t j t j t Figure 5: (color online) The results of a numerical simulation of the full ODE (2.1) where theinitial condition satisfies u jk (0) = u j(cid:96) (0) for all k, (cid:96) . The color white represents language V , and thecolor black represents language U . In the top two figures R = 0 .
15. For the top left figure A = 0 . A = 0 . k is a small perturbation of a pulse of size 1. In the bottom two figures R = 0 .
5. For the bottomleft figure A = 1 . A = 0 . k is a small perturbation of a pulse of size 2. Remark 3.3. If R > . R , A ) ∼ (0 . , . R , A )points chosen outside of the domain bounded by the curve are exactly as that outlined above. For pointsbelow the curve the solution quickly becomes a single pulse of size 6, which again is the concatenation oftwo fronts. Each front will travel, and U will grow, if A is sufficiently small. For points above the curve thesolution again quickly decays to a pulse of size zero.Finally, consider the 2- k -2 pulse for any k ≥
3. Here we find this is a true concatenation of two pulsesof size 2, so the boundary curve is given by the dashed (red) curve in Figure 4. Moreover, the dynamics ofthis pulse is initially governed by the dynamics associated with a pulse of size 2 (see the bottom two panelsof Figure 5).While we do not present the corroborating details here, we now have the following rule-of-thumb. If westart with a two-pulse of size j - k - (cid:96) , and if k ≥
3, then the resulting dynamics will initially be independentlygoverned by those associated with the pulse of size j and pulse of size (cid:96) . The individual pulses “see” each other. Kapitula and P.G. Kevrekidis u
14 16 18 20 j -600-400-2000 l n ( u ) Figure 6: (color online) The top panel provides the numerically generated pulse of size 3, say u ,for ( R , A ) = (0 . , . u ). For j ≤
12 and j ≥
22 the numericallydetermined value of ln( u ) is −∞ . If the decay to u = 0 was exponential, the bottom panel wouldbe linear in j . Instead, it is concave down. only if the gap between the two is one or two adjacent sites. Indeed, this rule holds for any concatenationof pulses. As long as the distance between adjacent pulses is at least 3 sites, the existence boundary curveis exactly that associated with each individual pulse which makes up the entire multi-pulse. Moreover, thedynamics are governed by those associated with the single pulse until the distance between individual pulsesis reduced to one or two sites. We have proven stable fronts and pulses exists for small (cid:15) for the 1D model (2.1). We now remove theassumption that (cid:15) is small, and assume that a stable front/pulse exists for (2.1). The spectrum for theassociated linearized self-adjoint operator, L , is then strictly negative, so (cid:104)L v j , v j (cid:105) < . (3.1)We now consider the spectral stability for the original 2D model (2.1). The self-adjoint linearized operatorhas the form, L = L + 2(1 + A ) (cid:15) U j (1 − U j )∆ k . Using a Fourier decomposition for the eigenfunctions in the transverse direction, v jk (cid:55)→ v j e i ξk , − π ≤ ξ < π, we find, L v jk = [ L − A ) (cid:15) (1 − cos( ξ )) U j (1 − U j )] v j e i ξk . Since the second term in the sum is a nonpositive operator, by using the inequality (3.1) we can concludethat (cid:104)L v jk , v jk (cid:105) < . anguage competition dynamics A j Figure 7: (color online) The left panel provides the numerically generated boundary of the two-pulse 2-1-2 (solid (blue) curve) and 2-2-2 (dashed (red) curve). The two-pulse exists inside the twocurves, and ceases to exist outside. The right panel gives an example of each pulse when R = 0 . A = 1 .
0. The 2-1-2 pulse is upper right, and the 2-2-2 pulse is lower right.
Consequently, all the eigenvalues must be strictly negative, so the stable front/pulse for the 1D problem istransversely stable for the 2D problem.
4. Existence and spectral stability of stripes for thecontinuum model
We now consider the existence and spectral stability of solutions to the continuum model (2.2).
The existence problem is settled by finding solutions to the nonlinear ODE (2.4). Recalling R = (cid:15) + 4 (cid:15) ,under the assumption that neither language is more prestigious, A = 1, there exists the exact compactonsolution, U c ( x ) = 12 (cid:34) (cid:32)(cid:114) R x (cid:33)(cid:35) . In writing this solutions there is the implicit understanding that the compacton is continuous with U c ( x ) ≡ U c ( x ) ≡ u = 0 to u = 1. One front satisfies U c ( x ) = 0 for x ≤ − π (cid:112) /R , and U c ( x ) = 1 for x ≥ U c ( x ) = 1 for x ≤
0, and U c ( x ) = 0 for x ≥ π (cid:112) /R (again,this front can be translated). Note that the width of the front/pulse depends upon the reaction rate, R . Remark 4.1.
There is also an explicit compact solution when p = 3, U c ( x ) = 12 (cid:34) (cid:32)(cid:114) R x (cid:33)(cid:35) . Numerically, we see compactons for any p > j t j t j t Figure 8: (color online) The results of a numerical simulation of the full ODE (2.1) with R = 0 . u jk (0) = u j(cid:96) (0) for all k, (cid:96) . The color white represents language V , and the color black represents language U . In all three panels the initial condition is a smallperturbation of a 2-1-2 pulse. For the left panel A = 0 .
6, for the middle panel A = 1 .
0, and for theright panel A = 1 . If A (cid:54) = 1, numerical simulations indicate that the compacton fronts will travel at a constant speed whichdepends upon A . Moreover, the simulations suggest that the shape of the front at a fixed time is roughlythat of the compacton for A = 1. In order to derive an approximate analytic expression for the wavespeedwe plug U c ( x + ct ) into the PDE (4.2), multiply the resultant equation by ∂ x U c ( x + ct ), and then integrateover the domain where the front is nonconstant. Doing all this leads to the following predictions for thewave-speed, V → U, c = − √ Rπ ( A − U → V, c = √ Rπ ( A − . (4.1)The notation j → k corresponds to the front which has value j for x (cid:28) k for x (cid:29)
0. See Figure 9for the comparison of the theoretical prediction with the results of a numerical simulation of the PDE (4.2).Numerical simulations indicate that these are good predictions for a relatively large range of A for the 1DPDE model; recall the relevant discussion also in Figure 3. Moreover, we find that for R sufficiently large,and away from the saddle-node bifurcation points, these are also good predictions for the wave-speed for thediscrete model. Remark 4.2. If A <
1, so that language U is preferred, the front will move so that language U invadeslanguage V . On the other hand, if A >
1, so that V is preferred, V will invade U . The standing compactonwhich exists for A = 1 is then seen as a transition between these two invasion fronts. Let us now consider the spectral stability of these compactons. The 1D version of the PDE (2.2) is, ∂ t u = Ru (1 − u ) [(1 + A ) u − A ] + (1 + A ) u (1 − u ) ∂ x u + [1 − (1 + A ) u ] ( ∂ x u ) . (4.2)Writing u = U c + v , when A = 1 the linearized problem for v is, ∂ t v = 2 ∂ x [ U c (1 − U c ) ∂ x v ] + g ( U c ) v, (4.3)where, g ( U c ) = R ( − U + 6 U c −
1) + 2(1 − U c ) ∂ x U c − ∂ x U c ) . anguage competition dynamics A/(1+A) -1.5-1-0.500.5 c Figure 9: (color online) The numerically generated wave speed for the V → U front. The solid(red) curve corresponds to the analytic prediction, and the (blue) circles are the approximate wavespeed derived from a numerical simulation of the PDE (4.2) with R = 8 using the standard second-order finite difference schemes to approximate the spatial derivatives. Without loss of generality assume the solution in question is the V → U front, i.e., U c ( x ) = 0 for x ≤ − π (cid:112) /R , and U c ( x ) = 1 for x ≥
0. Outside the interval [ − π (cid:112) /R,
0] the linearized PDE (4.3) becomesan ODE, ∂ t v = − Rv.
The associated spectral problem is, λv = − Rv (cid:32) λ = − R, or v ≡ . Because of the degeneracy associated with the diffusion coefficient, the essential spectrum for the operatorcomprises a single point. On the other hand, if then upon using the expression for the compacton theassociated spectral problem is the singular Sturm-Liouville problem,12 ∂ x (cid:34) sin (cid:32)(cid:114) R x (cid:33) ∂ x v (cid:35) − R (cid:32) (cid:32)(cid:114) R x (cid:33) − (cid:33) v = λv. (4.4)If λ (cid:54) = − R , then for the sake of continuity we need Dirichlet boundary conditions at the endpoints, v (cid:32) − (cid:114) R π (cid:33) = v (0) = 0 . Regarding the interior problem, x ∈ [ − π (cid:112) /R, λ = 0 is v ( x ) = ∂ x U c . Since the front is monotone, this eigenfunction is of one sign. Consequently, by classicalSturmian theory λ = 0 is the largest eigenvalue, so the wave is spectrally stable.Now consider the concatenation of fronts. Since each front is a compacton, there will be no tail-tailinteraction leading to small eigenvalues. Consequently, each front will add another eigenvalue associatedwith the eigenvalue of the original front. The associated eigenfunction will simply be a spatial translationof the associated eigenfunction. In particular, if there are N fronts, then λ = 0 will be a semi-simpleeigenvalue with geometric multiplicity N . The multiplicity follows from the fact that each front can bespatially translated without affecting any of the other fronts.. Kapitula and P.G. Kevrekidis Suppose we have two fronts, so the solution is a flat-topped compacton. As the size of the top is nonzero,there will be two zero eigenvalues, and the rest of the spectrum will be negative. At the limit of a zero lengthtop we have the pulse compacton, U c ( x ) = 12 (cid:34) (cid:32)(cid:114) R x (cid:33)(cid:35) , − (cid:114) R π ≤ x ≤ (cid:114) R π.
Since the diffusion is zero at x = 0, so the eigenvalue problem is still degenerate, we can still think ofthis solution as the concatenation of two fronts, a left front and a right front. The eigenvalue at zero willhave geometric multiplicity two. One eigenfunction will be ∂ x U c of the left front, and zero elsewhere, whileanother will be ∂ x U c of the right front, and zero elsewhere. Using linearity, we note that one eigenfunction isthe sum of these two, which is precisely the expected spatial translation eigenfunction of the full compacton, ∂ x U c . A steady-state front solution to the 2D model (4.2) when A = 1 is the compacton, u ( x, y ) = U c ( x ). As wesaw in Section 4.3, for the 1D model (4.2) the original front is spectrally stable with a simple zero eigenvalue,and a concatenation of N fronts is spectrally stable with a semi-simple zero eigenvalue of multiplicity N .Let U ( x ) represent a spectrally stable concatenation of N fronts, which is a stripe pattern.Consider the spectral stability of the stripes for the full 2D problem. Denote the 1D self-adjoint lin-earization in (4.3) about the concatenation as L . The linearization about this striped pattern for (2.2)is, L = L + 2 U (1 − U ) ∂ y , which is also self-adjoint. Using the Fourier transform to write candidate eigenfunctions, w ( x, y ) = v ( x )e i ξy , we have, L w = (cid:0) L − ξ U (1 − U ) (cid:1) v e i ξy . We already know L is a nonpositive self-adjoint operator. Since ξ U (1 − U ) ≥
0, we can therefore conclude L is a nonpositive self-adjoint operator. Consequently, there are no positive eigenvalues, so the stripepattern inherits the spectral stability of the concatenation. In particular, it is spectrally stable.
5. Spots: a case study
We now consider the existence and spectral stability of spots. A spot is a contiguous set of sites on thelattice which all share language U (or V ). All other sites share language V (or U ). For example, a 2 × U . When (cid:15) = 0 a stable spot of any size and shapecan be formed. By the Implicit Function Theorem the spot willpersist and be spectrally stable for small (cid:15) . Our goal here is to construct a snaking diagram for this spot,and then briefly discuss the dynamics associated with small perturbations of a spot. First consider the snaking diagram associated with a steady-state solution. We will start with the configu-rations at R = 0 of a 1 × U sitting on a background of V . The results are plotted in Figure 10.The figure on the left gives the snaking diagram, and some stable solutions arising from the snaking aregiven on the right. For the snaking diagram stable solutions are marked with a (blue) square, and unstablesolutions are marked with a (red) dot. The initial 1 × U anguage competition dynamics | u | Figure 10: (color online) The numerically generated snaking diagram for a square lattice of size20 ×
20 when R = 0 . × | u | , represents the square of the (cid:96) -norm of the solution. The upper right panel has( A, | u | ) ∼ (0 . , . A, | u | ) ∼ (0 . , . A, | u | ) ∼ (1 . , . A, | u | ) ∼ (1 . , . U appears to have no upper bound. speakers for a stable solution is either a square or something that has roughly a circular geometry. Regardingthe transition from stable to unstable solutions, it is generally not a saddle-node bifurcation, e.g., at thetransition point the number of unstable eigenvalues will go from zero to two. Moreover, within the curve ofunstable solutions there are additional bifurcations where the number of positive eigenvalues either increasesor decreases. The solution structure is rich, but we leave a detailed look at it for a different paper. Remark 5.1.
We should point out that as in the case of single stripes being concatenated to form morecomplicated stripe patterns, we can concatenate single spots to form more complicated structures. All thatis required for each spot to essentially be an isolated structure is for the spots to be sufficiently separated.Our experience is that a minimal separation distance between two adjacent spots of three sites is enough.
Now let us consider the dynamical implications of the snaking diagram. In particular, we shall look at theeffect of varying A for fixed R = 0 .
1. Recall that for stripes we saw in Section 3.2 that outside the snakingdiagram traveling waves would appear; in particular, if
A < A − then language U would invade language V ,whereas if A > A + , then language V would invade language U . Consequently, we expect a similar behaviorfor spots; in particular, a spot will grow or die as a function of the prestige. For a particular example westart with a stable solution arising from the 1 × A = 0 . (cid:96) -norm of this solution is roughly 11. This solution is contained in the small stable branch shown inFigure 10 with A − ∼ . A + ∼ . A = 0 .
6. When looking at the snaking diagram, we see that there are no stable steady-state solutions with this value of A . The time evolution associated with this initial condition is providedin Figure 11. Of particular interest is the evolution of the square of the norm in the far right panel. Wesee that the norm is growing up to at least t = 50. While we do not show it here, the norm continues to. Kapitula and P.G. Kevrekidis | u | Figure 11: (color online) The time evolution of an A = 0 . A = 0 .
6. The panelon the far right shows the evolution of the square of the (cid:96) -norm of the solution. | u | Figure 12: (color online) The time evolution of an A = 0 . A = 0 .
75. The panelon the far right shows the evolution of the square of the (cid:96) -norm of the solution. grow until the all the nodes share the common language U . The growth in language U is manifested in thesquare becoming larger and larger as those nodes containing V at the boundary between U and V switch tolanguage U .Next suppose that A = 0 .
75. When looking at the snaking diagram, we see there is a (stable) steady-statesolution with this value of A and which also has a larger norm. The time evolution associated with thisinitial condition is provided in Figure 12. Of particular interest is the evolution of the square of the norm inthe far right panel, which in this case achieves a steady-state. The final state at t = 50 corresponds to thefirst stable solution on the snaking diagram where a = 0 .
75, and whose norm is greater than 11. Language U invades language V until a steady-state configuration is reached.For the next example suppose that A = 0 .
9. When looking at the snaking diagram, we see there is asteady-state solution with this value of A and which also has a smaller norm. The time evolution associatedwith this initial condition is provided in Figure 13. Of particular interest is the evolution of the square ofthe norm in the far right panel, which in this case also achieves a steady-state. The final state at t = 50corresponds to the first stable solution on the snaking diagram where A = 0 .
9, and whose norm is less than11. Language V invades language U until a steady-state configuration is reached. For the last examplesuppose that A = 1 .
1. When looking at the snaking diagram, we see there is no steady-state solution with anguage competition dynamics | u | Figure 13: (color online) The time evolution of an A = 0 . A = 0 .
9. The panelon the far right shows the evolution of the square of the (cid:96) -norm of the solution. | u | Figure 14: (color online) The time evolution of an A = 0 . A = 1 .
1. The panelon the far right shows the evolution of the square of the (cid:96) -norm of the solution. this value of A and which also has a smaller norm. The time evolution associated with this initial conditionis provided in Figure 14. Of particular interest is the evolution of the square of the norm in the far rightpanel, which in this case goes to zero. Language V invades language U until the entire lattice shares thecommon language V .In conclusion, we have the following rule-of-thumb if the initial configuration is near a steady statesolution. If the value of A is decreased, so that the prestige of language U increases, then a spot of U in a seaof V will grow until a stable steady-state associated with that value of A is achieved. If no such steady-stateexists, then eventually the entire lattice will share language U . On the other hand, if the value of A isincreased, so that the prestige of language V increases, then a spot of U in a sea of V will shrink in sizeuntil a stable steady-state associated with that value of A is achieved. If no such steady-state exists, theneventually the entire lattice will share language V . While we do not show it here, this rule was manifestedin every numerical simulation that we performed. It would be most interesting to translate this observationinto a precise mathematical statement. This is left as an interesting direction for future work.. Kapitula and P.G. Kevrekidis
6. Conclusions & Future Challenges
We have derived an ODE model of language dynamics on a square lattice which is a natural generalizationof the AS language model on one lattice site. The model can also be used to discuss, e.g., the spread of anopinion through the lattice, or the growth/decay of religious observance on the lattice. We also looked at thecontinuum limit of the ODE, which is a PDE which features a degenerate diffusion term. We numericallystudied the existence of special spatial structures on the lattice; primarily, stripes and spots. Through acombination of numerics and analysis we analyzed the dynamics associated with small perturbations of thesespatial structures. Finally, we provided rules-of-thumb to help understand how languages die and grow interms of their prestige, and interaction with neighboring communities.As is already evident from the discussion above, there are numerous directions in this emerging field thatare worthwhile of further study. Some are already concerning the model at hand. As highlighted earlier,features such as the bifurcation of traveling solutions from standing ones and their scaling laws, or the moreprecise identification of the discrete solutions and their tails from a mathematical analysis perspective wouldbe of interest. While it is unclear whether something analytical can be said about the bifurcation diagramof genuinely two-dimensional states such as spots, our numerical observations regarding the model dynamicsformulate a well-defined set of conjectures regarding the fate of a spot when the prestige is decreased orincreased that may be relevant to further explore mathematically. However, it would also be relevant toconsider variations of the model. Here, we selected as a first step of study to explore an ordered two-dimensional square lattice. However, the I jk may be relevant to generalize to more complex networks andmodified (influence or) “adjacency matrices” to explore their impact on the findings presented herein. Asindicated herein, the role of near-neighbor interactions is expected to maintain some of the key features weconsidered; yet in a progressively connected world, the consideration of nonlocal, long-range interactionsmay be of interest in its own right. Another possibility is to insert a spatially heterogeneous prestige A jk and examine how its spatial variation may influence standing and traveling structures. There are numerousvariants that can be considered thereafter, e.g., how does a local prestige variation interact with the travelingwave patterns explored herein? Such queries have been considered in other contexts where the interactionsbear a linear component recently, e.g., see Hoffman et al. [ ], but have yet to be considered in a fullynonlinear setting such as the one herein. Such studies, as applicable, will be reported in future publications. References [ ] D. Abrams and S. Strogatz. Modelling the dynamics of language death. Nature , 424:900, 2003.[ ] D. Abrams, H. Yaple, and R. Wiener. Dynamics of social group competition: modeling the decline of religiousaffiliation. Phys. Rev. Lett. , 107:088701, 2011.[ ] T. Amano, B. Sandel, H. Eager, E. Bulteau, J.-C. Svenning, B. Dalsgaard, C. Rahbek, R. Davies, and W. Suther-land. Global distribution and drivers of language extinction risk. Proc. Roy. Society B: Biological Sciences , 281:20141574, 2014.[ ] T. Anderson, G. Faye, A. Scheel, and D. Stauffer. Pinning and unpinning in nonlocal systems. J. Dyn. Diff.Eq. , 28:897–923, 2016.[ ] J. Bramburger and B. Sandstede. Localized patterns in planar bistable weakly coupled lattice systems. Nonlin-earity , 33:3500–3525, 2020.[ ] J. Bramburger and B. Sandstede. Spatially localized structures in lattice dynamical systems. J. NonlinearScience , 30:603–644, 2020.[ ] J. Bujalski, G. Dwyer, T. Kapitula, Q.-N. Le, H. Malvai, J. Rosental-Kay, and J. Ruiter. Consensus andclustering in opinion formation on networks. Phil. Trans. R. Soc. A , 376:20170186, 2018.[ ] R. Colucci, J. Mira, J. Nieto, and M. Otero-Espinar. Coexistence in exotic scenarios of a modified Abrams-Strogatz model. Complexity , 21(4):86–93, 2014. anguage competition dynamics [ ] A. Dhooge, W. Govaerts, and Y. Kuznetsov. Matcont: a MATLAB package for numerical bifurcation analysisof ODEs. ACM TOMS , 29:141–164, 2003.[ ] K. Eekhoff. Opinion formation dynamics with contrarians and zealots. SIAM J. Undergraduate Research Online ,12, 2019.[ ] C. Elmer and E. Van Vleck. A variant of Newton’s method for the computation of traveling waves of bistabledifferential-difference equations. J. Dyn. Diff. Eq. , 14(3):493–517, 2002.[ ] R. Fujie, K. Aihara, and N. Masuda. A model of competition among more than two languages. J. Stat. Phys. ,151:289–303, 2013.[ ] M. Haragus and A. Scheel. Corner defects in almost planar interface propagation. Ann. Inst. H. Poincare (C)Anal. Non Lineaire , 23:283–329, 2006.[ ] A. Hoffman, H. Hupkes, and E. Van Vleck. Entire Solutions for Bistable Lattice Differential Equations withObstacles , volume 250 of
Memoirs Am. Math. Soc.
Am. Math. Soc., 2017.[ ] H. Hupkes and S. Verdun Lunel. Analysis of Newton’s method to compute travelling waves in discrete media. J. Dyn. Diff. Eq. , 17(3):523–572, 2005.[ ] H. Hupkes and B. Sandstede. Stability of pulse solutions for the discrete fitzhugh-nagumo system. Trans. Amer.Math. Soc. , 365:251–301, 2013.[ ] H. Hupkes, D. Pelinovsky, and B. Sandstede. Propagation failure in the discrete Nagumo equation. Proc. Amer.Math. Soc. , 139(10):3537–3551, 2011.[ ] M. Juane, L. Seoane, A. Mu nuzuri, and J. Mira. Urbanity and the dynamics of language shift in Galicia. NatureComm. , 10:1680, 2019.[ ] T. Kapitula. Multidimensional stability of planar travelling waves. Trans. AMS , 349(1):257–269, 1997.[ ] P. Kevrekidis, I. Kevrekidis, and A. Bishop. Propagation failure, universal scalings and Goldstone modes. Phys.Lett. A , 279(5-6):361–369, 2001.[ ] S. Marvel, H. Hong, A. Papush, and S. Strogatz. Encouraging moderation: clues from a simple model ofideological conflict. Phys. Rev. Lett. , 109:118702, 2012.[ ] J. Mira and ´A. Paredes. Interlinguistic similarity and language death dynamics. Europhys. Lett. , 69(6):1031–1034, 2005.[ ] J. Mira, L. Seoane, and J. Nieto. The importance of interlinguistic similarity and stable bilingualism when twolanguages compete. New J. Physics , 13:033007, 2011.[ ] R. Moore and K. Promislow. Renormalization group reduction of pulse dynamics in thermally loaded opticalparametric oscillators. Physica D , 206:62–81, 2005.[ ] M. Otero-Espinar, L. Seoane, J. Nieto, and J. Mira. An analytic solution of a model of language competitionwith bilingualism and interlinguistic similarity. Physica D , 264:17–26, 2013.[ ] R. Parker, P. Kevrekidis, and B. Sandstede. Existence and spectral stability of multi-pulses in discrete Hamil-tonian lattice systems. Physica D , 408:132414, 2020.[ ] K. Promislow. A renormalization method for modulational stability of quasi-steady patterns in dispersivesystems. SIAM J. Math. Anal. , 33(6):1455–1482, 2002.[ ] B. Sandstede. Stability of multiple-pulse solutions. Trans. Amer. Math. Soc. , 350:429–472, 1998.[ ] B. Sandstede. Stability of travelling waves. In Handbook of Dynamical Systems , volume 2, chapter 18, pages983–1055. Elsevier Science, 2002.[ ] D. Stauffer, X. Castell´o, V. Egu´ıluz, and M. San Miguel. Microscopic Abrams-Strogatz model of languagecompetition. Physica A , 374:835–842, 2007.[ ] S. Tanabe and N. Masuda. Complex dynamics of a nonlinear voter model with contrarian agents. Chaos , 23:043136, 2013. . Kapitula and P.G. Kevrekidis [ ] F. Vazquez, X. Castell´o, and M. San Miguel. Agent based models of language competition: macroscopicdescriptions and order-disorder transitions. J. Stat. Mech. , page P04007, 2010.[ ] I. Vidal-Franco, J. Guiu-Souto, and A. Mu nuzuri. Social media enhances languages differentiation: a mathe-matical description. Royal Society Open Science , 4:170094, 2017.[ ] S. Wang, L. Rong, and J. Wu. Bistability and multistability in opinion dynamics models. Applied Math. Comp. ,289:388–395, 2016.[ ] J. Yun, S.-C. Shang, X.-D. Wei, S. Liu, and Z.-J. Li. The possibility of coexistence and co-development inlanguage competition: ecology-society computational model and simulation. SpringerPlus , 5:855, 2016.[36