2.1 Afrikaans verbal deflexion

Germanic languages typically exhibit complex verbal morphology, with numerous forms found throughout the verbal paradigms, often without a transparent one-to-one mapping between form and meaning:

In this respect Afrikaans, which began as a contact variety of Dutch in the Cape Colony in the 17th and 18th centuries, behaves rather differently. Modern Afrikaans has only one form throughout the paradigm (corresponding to the stem of the Dutch verb):

This deflexion of verbal paradigms is just one manifestation of a larger-scale morphological regularization that separates Afrikaans from its ancestor (see Ponelis 1993).

The specific and rather unique setting in which Afrikaans arose has prompted scholars to discuss its origin extensively. There are three major competing theories: the superstratist hypothesis, according to which the structural features of Afrikaans arose from Dutch in a process of internal development; the interlectalist hypothesis, according to which the structure of Afrikaans is explained by competition between multiple different dialects of Dutch that were brought to the Cape; and the creolist hypothesis, according to which Afrikaans is a creole or semicreole that arose from the interaction between the colonizers, the native Khoekhoe population, and slaves brought by the Europeans from other parts of Africa and parts of Asia (Roberge 2002). It is clear, however, that contact must have played some role in the formation of the language; the specific classification of Afrikaans as a creole, semicreole or other kind of contact variety is less relevant here.

If the three main population constituents of the Dutch Cape Colony were in extensive linguistic contact, as seems likely (Ponelis 1993; Roberge 2002), then an amount of L2 learning will have taken place: both the native Khoekhoe and the imported slaves would have had to learn at least a limited amount of Dutch to communicate with the colonizers. The (adult) L2 Dutch spoken in the colony would form part of the input of the following generations of L1 learners, and through this mechanism of nativization of L2 output, changes could stabilize as part of the language that was in the process of formation (Trudgill 2011).¹ Complex meaning-to-form mappings are thought to be difficult for L2 learners generally (DeKeyser 2005), but experimental evidence lends an additional degree of credibility to this general idea in the specific case of the Dutch verb. Blom (2006) compared child L1, child L2 and adult L2 learners of Dutch for knowledge of Dutch verbal morphology by way of a sentence completion task; the L2 learners had either Turkish or Moroccan backgrounds and spoke Turkish, Moroccan Arabic or Tarifit (a Zenati Berber language of Morocco) as their L1. Compared to Dutch L1 learners, who exhibited an accuracy of 96% standard use of the verbal paradigm, child L2 learners showed 83% (Turkish) or 85% (Moroccan) accuracy. The adult L2 learners, however, attested only 57% (Turkish) or 56% (Moroccan) overall accuracy. This suggests that verbal morphology is an L2-difficult feature to acquire (irrespective of the learner’s L1) and may thus favour paradigmatic levelling (regularization) when a significant number of adult L2 learners are involved in the contact situation.

2.2 Erosion of null subjects in Afro-Peruvian Spanish

Consistent null subject (NS) languages such as standard forms of Spanish and Italian exhibit the omission of non-emphatic, non-contrastive referential subject pronouns in finite clauses (see Roberts & Holmberg 2010), illustrated here for Spanish:

In some contexts, such as with expletive subjects, a null pronoun is strictly obligatory:

An overt referential subject is, however, required under certain circumstances, and the use of null vs. overt pronouns in general is typically conditioned by complex semantic and pragmatic considerations. Consider the following examples (via Montrul 2004: 226):

With a null pronoun, the understood subject of the embedded clause in this example may or may not be coreferential with the subject of the main clause (7), whereas with an overt pronoun the referents of the subjects must differ: the interpretation in which the pronoun is bound is strictly ungrammatical if the pronoun is overt (8). Evidence now exists that features of this kind—ones that hinge on the syntax–semantics and syntax–pragmatics interfaces—are difficult for L2 learners to acquire (Sorace & Serratrice 2009; see also Walkden & Breitbarth 2019 for a theoretical account of the diachrony of null subjects as loss/gain of uninterpretable features).

Rates of subject expression differ across varieties of Spanish, with American varieties exhibiting significantly higher rates of overt subjects compared to Peninsular Spanish: Martínez-Sanz (2011: 195) reviews the available literature and shows that these rates range from as low as 12% in Valladolid, Spain, to 60% in San Juan, Puerto Rico. In particular, it has been suggested that a number of Afro-Hispanic Languages of the Americas (AHLAs)—languages that emerged from contact between Spanish and African languages in the Americas in colonial settings—attest mixed² pro-drop systems (Sessarego & Gutiérrez-Rexach 2018). These languages not only employ overt subjects where Peninsular Spanish uses null subjects (9), they may also exhibit features of both NS and non-NS languages simultaneously (10):

One way of making sense of such data is to hypothesize that speakers have access to more than one grammar simultaneously (Kroch 1989; Yang 2002)—in this case, to both a NS and a non-NS version of Spanish—and employ these competing grammars variably and probabilistically (Toribio 2000; Sessarego & Gutiérrez-Rexach 2018; for an extensive variationist account of subject pronoun expression in Spanish–English bilinguals in New Mexico, see Torres Cacoullos & Travis 2018).

Empirical learning data are again relevant in suggesting explanations for the observed patterns. Margaza & Bel (2006) tested Greek adult L2 learners of Spanish on the expression of NS pronouns; the results demonstrate that L2 learners tend to overuse overt pronouns compared to native speakers, even when the L1 is a NS language. Specifically, in a cloze task in which participants had to either express or omit a subject pronoun, intermediate L2 learners employed null subjects 52% of the time in matrix clauses and 81.66% of the time in subordinate clauses, compared to 85.50% and 98.13% in advanced learners, and 96.00% and 100% in native controls (Margaza & Bel 2006: 92). These findings suggest that L2 learners—with the possible exception of those at an advanced proficiency level—struggle with the precise pragmatic conditioning of the null/overt contrast in a NS language (for converging evidence from different language pairings, see Bini 1993; Pérez-Leroux & Glass 1999). They thus support the notion that a contact situation involving (imperfect) L2 learning followed by L1 nativization may help to explain the diachronic erosion of null subjects.

In particular, such an explanation is plausible in the case of Afro-Peruvian Spanish, a variety spoken mainly in the coastal regions of modern Peru (Sessarego 2014; 2015; Sessarego & Gutiérrez-Rexach 2018). Afro-Peruvians—descendants of slaves who were brought from various parts of Africa and forced to work on plantations, in mines and as servants in cities from the 16th century onwards—amounted to 3.6% of Peru’s population in the most recent, 2017 census (INEI 2018: 222). Although Peru abolished slavery formally in 1854, most of the Afro-Peruvian population lived in relative poverty under a semi-feudal system until as recently as the second half of the 20th century (Sessarego 2015: 79). The probable linguistic consequences of these sociohistorical facts will be discussed in detail in Section 5.

2.3 Desiderata

Previous research thus suggests that the deflexion of the Afrikaans verbal paradigm and the erosion of null subjects in Afro-Peruvian Spanish may both be traced back to an earlier contact situation involving significant amounts of L2 learning. Apart from explicating what a “significant amount” means in this context, thereby providing a unified account of what is common to both cases, a mathematical modelling approach needs to be able to account for the differences between the two situations. In the case of Afrikaans, the loss of verbal morphology is complete: all verbs are reduced to one form throughout the paradigm. In the case of Afro-Peruvian Spanish, and several other AHLAs, the loss of null subjects is incomplete, in the sense that these languages attest features of both NS and non-NS languages. Moreover, in the case of Afro-Peruvian Spanish at least, there is evidence of the younger generations moving in the direction of the NS grammar again, suggesting that the mixed NS status may not be stable; fieldwork interviews suggest that the Afro-Peruvian variety could be lost in favour of standard Peruvian Spanish “in two generations, or maybe only one” (Sessarego 2014: 397). In the ideal situation, a modelling approach would predict in what circumstances—under what combinations of model parameters—each development is likely to unfold.

4.1 Motivation

It is conceptually useful to think of language change as an intertwined process of innovation and propagation. In the present model, L2 learners who acquire a lower probability of employing the L2-difficult grammar (as compared to L1 learners) are innovators. Whether and how these innovations propagate across the population depends, roughly, on how prevalent the L2 learner population is in the entire speech community. In general, interactions between individuals in a speech community are the result of a complex combination of factors involving properties of social networks, geographical distance and interaction frequency, to name but a few. Such factors can in principle be encapsulated in stochastic models of social dynamics inspired by techniques borrowed from statistical physics (Helbing 2010); however, inclusion of too many factors usually renders such models analytically intractable. Alternatively, when populations are large, stochastic fluctuations normally average out, and the system can be reduced to a simpler deterministic model, with correspondingly simpler analysis. This is the approach adopted here, developed in detail in Section 4.2. Section 4.3 provides simulation-based support for the deterministic approximation.

4.2 Deterministic approximation

Setting aside the complications arising from the full stochastic complexity of learning trajectories and interaction patterns, let us now focus on a simplified model in which learners behave according to the learning-theoretic expectations (12) and (14) and in which population size is so large that demographic noise cancels out. Technically, we consider an infinite well-mixed population of individuals in which the fraction of L2 speakers is σ and the fraction of L1 speakers is 1 – σ. Let p and q denote the probabilities of grammar G₁ in the L1 and L2 speaker populations, respectively (so that the probabilities of G₂ are 1–p and 1–q). Following Yang (2000), I assume a fraction α₁ of the output of grammar G₁ to be incompatible with G₂, and similarly a fraction α₂ of the output of G₂ to be incompatible with G₁. The parameters α₁ and α₂ will be called the (grammatical) advantages⁵ of G₁ and G₂ in what follows; numerical estimates will be given in Section 5. Assuming for simplicity that learners sample linguistic input from their environments at random, the penalty probabilities of the two grammars can then be expressed in the following simple forms:

To unpack this, note that the first term on the right hand side of the first equation, for instance, represents the event of the learner interacting with an L1 speaker (1–σ) who employs grammar G₂ (1–p) and utters a sentence which falls among those not compatible with G₁ (α₂). Similarly, the second term on the right hand side of the second equation represents the case of the learner interacting with an L2 speaker (σ) who employs grammar G₁ (q) and utters a sentence not compatible with G₂ (α₁), and similarly for the remaining two terms.

Following Yang (2000), we can now assume that the input to the language acquisition process of the (n+1)th generation of speakers is constituted by the linguistic output of the nth generation. Assuming learners reach the learning-theoretic asymptote as argued in Section 3, this implies setting

with the expectations given by (12) and (14). Expanding these with the help of the penalties (15), we arrive at the pair of equations

To reduce the number of parameters in these equations, it is useful to divide both the numerators and the denominators on the right hand side by α₂ (on the assumption that α₂ ≠ 0). This condenses the relation of the two advantage parameters α₁ and α₂ into a single number, the advantage ratio α = α₁/α₂:

The quantity D = d/α₂ represents the relative L2-difficulty of grammar G₁ scaled by the advantage of grammar G₂.

To recap, we have assumed grammatical competition between two options G₁ and G₂, the first of which is assumed to incur an amount of L2-difficulty, quantified by the ratio d = δ/ɣ of raw L2-difficulty δ to underlying learning rate ɣ. The two learning algorithms of Section 3 lead to a mixed population of L1 and L2 speakers whose dynamics are described by the pair of equations (18), on the assumption that speakers mix randomly. The dynamics depend on three model parameters:

• α = α₁/α₂ controls the ratio of grammatical advantages, indicating how much (plain) advantage the L2-difficult grammar G₁ has in relation to grammar G₂;

• D = d/α₂ represents the L2-difficulty of grammar G₁ in relation to the grammatical advantage of G₂;

• σ gives the proportion of L2 speakers in the population.

Intuitively, one expects the (diachronic) loss of the L2-difficult grammar G₁ to be more likely for higher values of D and σ, as well as for lower values of α. These expectations are borne out by exact mathematical analysis, as will be described next.

In (18), the variables p and q represent the relative frequency of the L2-difficult grammar G₁ in the L1 and L2 speaker populations, respectively. We would now like to understand the range of behaviours this system is capable of. Each pair of values (p,q) with 0 ≤ p,q ≤ 1 is a possible population state, or in other words, the state space of the system consists of the unit square [0,1] × [0,1] = [0,1]². Of particular interest is the eventual fate of the population under the above modelling assumptions. In general, some points (p,q) of the state space may be expected to be attractors, drawing the population state to themselves over time, while other points may repel the population state. These configurations of the state space correspond to different empirical outcomes, as will be demonstrated next.

In the Appendix, it is proved that the system (18) can display one of two eventual outcomes for any fixed combination of model parameters α, D and σ. These are:

1. The L2-difficult grammar G₁ is used with some probability in one or both speaker groups. Mathematically, the system has two equilibria, the origin (p,q) = (0,0) and a further, non-origin state (p^*, q^*) ≠ (0,0). The former, however, is unstable, while the latter is asymptotically stable (a sink). The population will be attracted to (p^*, q^*) over time, with the consequence that the L2-difficult feature is retained in the language—with frequency p^* in the L1 speaker population, and with frequency q^* in the L2 speaker population.

2. The L2-difficult grammar G₁ vanishes from both groups. Mathematically, only one equilibrium exists, namely the origin (p,q) = (0,0), which is asymptotically stable. Consequently both the L1 and L2 populations eventually speak G₂ exclusively.

Of these outcomes, the latter corresponds to the proposed mechanism of L2 mutation followed by L1 nativization. The passage from phase I to phase II is governed by a complex interaction of the three parameters, one that however can be solved analytically. First, if α < 1, so that grammar G₁ has less advantage to begin with, phase II is predicted. This is unsurprising: if both grammatical fitness in the VL sense and L2-difficulty in the Trudgillian sense conspire against a grammatical option, no force exists to sustain it, and the option is predicted to disappear.⁶ The same outcome holds for the special case α = 1, in which the grammatical advantages are equal.

For α > 1, the situation is more complicated. If α ≥ D + 2, phase I is predicted for any combination of parameter values. In this case, the pure grammatical advantage enjoyed by the L2-difficult grammar G₁ is so high that no amount of L2 learning can suppress it entirely—the L1 speaker population will continue to speak G₁ at some non-zero frequency. For 1 < α < D + 2, however, the proportion of L2 speakers σ acts as a bifurcation parameter. For values of σ below a critical threshold

phase I is predicted. In this case, the proportion of L2 learners in the population is not high enough to completely suppress G₁. For values of σ exceeding this threshold, however, phase II is predicted: the non-origin equilibrium (p^*,q^*) vanishes and the population converges to (p,q)= (0,0), with both L1 and L2 speakers now employing grammar G₂ exclusively. In other words, as the proportion of L2 learners in the population grows, the speech community experiences a phase transition from phase I to phase II (Figure 1 and Table 1).⁷

Figure 1

Orbit diagram of the system showing the values of p (top row) and q (bottom row) at the stable equilibrium, for various values of advantage ratio α (columns), L2-difficulty D (horizontal axis) and proportion of L2 speakers σ (vertical axis). The dashed line supplies the bifurcation boundary σ_crit: the L2-difficult grammar G₁ is extinct in each speaker population above this line, but coexists with grammar G₂ below it.

4.3 Finite simulations

The above deterministic model turns on the assumption that learners are well-described by the learning-theoretic asymptotic expectations (12) and (14), as well as on the assumption that interactions between speakers are sufficiently random so that stochastic fluctuations in the population average out. These idealizing assumptions were made deliberately, to unleash the greater explanatory power of analytically soluble models, compared to simulations (see McElreath & Boyd 2007: 4–11). Nevertheless, the assumptions can be debated, and the question of how a finite, stochastic system would behave is certainly an interesting one. Although it is impossible to explore the entirety of the full stochastic model’s parameter space by way of simulations in this paper, I here report the outcome of proof-of-concept simulations which lend tentative support to the deterministic approximation analysed in Section 4.2.

The choice to abstract away from the individual-level stochastic dynamics of language acquisition was defended by way of an argument from the typically slow pace of language acquisition in Section 3; this entails assuming that learners receive a large amount of input during their learning period and also make only small adjustments at each presentation of input token, so that the learning rate parameter ɣ has a small value. In the absence of direct empirical estimates of ɣ, it is legitimate to worry about the effects that high learning rates may have on the ensuing dynamics. To explore this, ten Variational Learners were simulated for varying values of ɣ, exposed to a learning environment in which the relative frequency of grammar G₁ was 0.5 and the advantage parameters had the values α₁ = 0.25 and α₂ = 0.2. It was moreover assumed that grammar G₁ incurs an L2-difficulty of d = 2. Of the ten learners, five were randomly assigned to be L1 learners; the other five were L2 learners subject to the L2-difficulty of G₁. Each learner received 100,000 input tokens over their learning period and started from a randomly drawn initial value of p or q.

With these assumptions, the learning-theoretic expectations (12) and (14) predict an average probability of p = 0.56 for grammar G₁ in the L1 learner population, and an average of q = 0.06 in the L2 learner population. Figure 2A presents the simulation results. As expected, the predicted averages describe the population averages. On the other hand, for higher learning rates (larger values of ɣ), a larger residual variance about this expectation is observed across the population. It turns out, however, that this residual inter-learner variance, even when large, need not affect inter-generational, diachronic developments. Consider Figure 2B, which tracks the inter-generational trajectory of p and q across 15 generations of learners, each generation consisting of 100 learners (half L1, half L2), starting from p = q = 0.99 in generation 0 and subject to the same parameters as above. To channel input from generation n to generation n+1 in a reasonably realistic way, each learner in generation n+1 was randomly drawn two “parents” from generation n; each learner only received input from its parents. The consequence is that, particularly in the intermediate stages of the diachronic development, and particularly when the learning rate parameter ɣ has a high value, there is much variation across speakers in the community. However, since the ultimate force driving the evolution of p and q is deterministic in nature (a difference in the advantages of the competing grammars, combined with L2-difficulty of one of the options), this stochastic noise is not sufficient to disturb the development. With the above parameters, equation (19) implies a critical L2 proportion threshold of σ_crit = 0.3; since the actual proportion of L2 learners in the simulation is 0.5, we expect grammar G₁ to be driven to extinction over inter-generational time, i.e. the values of p and q to converge toward zero. This is exactly what is observed, even for noisy (high) values of the learning rate parameter. Moreover, the deterministic trajectories computed from (18), shown in Figure 2B as the connected curves, are broadly consistent with the simulated data in each generation. In fact, with higher learning rates, the predicted equilibrium (p,q) = (0,0) is attained quicker.

Figure 2

Finite simulations of the model. (A) Learning trajectories of 10 individual learners, half of them L1 and the other half L2 learners, for different learning rates ɣ. First 10,000 iterations shown only, sampled every 100 iterations for increased clarity; behaviour after this initial transient period is identical. (B) Diachronic trajectory of 15 generations of learners, 100 learners in each generation (half L1, half L2). The boxplots characterize the simulated data, i.e. the distribution of p (or q) across all learners at the end of their learning period. The solid curves give the deterministic prediction using equation (18). For simulation parameters, see text.

5.1 Interim summary

To summarize the results of the foregoing sections, our mixed population of L1 and L2 speakers is capable of two qualitatively different kinds of long-term behaviour. The parameter α = α₁/α₂ expresses the ratio of the advantages of the two competing grammars. If α < 1, the state (p,q) = (0,0), in which both the L1 and L2 speaker populations use G₂ to the complete exclusion of the L2-difficult grammar G₁, is a stable equilibrium. If α > 1, this state is either stable or unstable depending on whether σ, the proportion of L2 speakers in the population, exceeds a critical threshold σ_crit. The value of σ_crit, in turn, depends on the magnitudes of α and D = d/α₂, the latter expressing the relative L2-difficulty of grammar G₁, scaled by the grammatical advantage of G₂. In this scenario, two evolutionary outcomes are possible: either total extinction of the L2-difficult grammar G₁ (when σ > σ_crit), or stable variation between G₁ and G₂ (when σ < σ_crit).

The three model parameters α, σ and D are all estimable in principle from empirical data: α from the frequencies of occurrence of different types of linguistic constructions, σ from population censuses, and D from L2 learning data. Suitable data of the last kind are presently lacking,⁸ but I will next attempt calibration of the first two parameters in the specific cases of Afrikaans deflexion and the erosion of null subjects in Afro-Peruvian Spanish. Here it should be borne in mind that all parameter estimates can be only approximate: some degree of uncertainty necessarily pertains to corpus estimates of grammatical advantages, and historical population data are wrought with problems well known to historians and demographers. I will therefore offer the parameter estimates as an approximate starting point, with the proviso that they come with an unknown degree of statistical uncertainty. This has important consequences on what we take the very goal of modelling to be. Even though exact quantitative statements about the equilibrium state of the linguistic system may be out of reach, it is still possible to infer something about the qualitative outcome of the contact situation in each empirical test case.

5.2 Calibrating grammatical advantages

In the case of Afrikaans verbal deflexion, the relevant competition is between a grammar that has person and number distinctions in the verbal paradigm (G₁, corresponding to Dutch) and a grammar that doesn’t (G₂, corresponding to Afrikaans). Parsing failure occurs when the learner–listener attaches the wrong interpretation (wrong person or number) to the surface form uttered by the speaker. Since Dutch is a non-NS language, person and number can always be inferred from the nominal domain, even if verbal inflection is eroded. On the face of it, this implies that the raw grammatical advantages of the two grammars are equal, so that α = 1. In practice, it may be argued that factors such as channel noise may lend the grammar with verbal inflection (G₁, i.e. Dutch) a slight advantage over the inflectionless grammar; on the other hand, redundant agreement has been suggested to present difficulty for L2 learners (Kusters 2008). However, the magnitudes of these effects are difficult to estimate outside the context of a strict laboratory study. In what follows, I will assume that such effects are negligible at the population level, and thus proceed with the maximum-parsimony assumption that α = 1, i.e. that neither grammar is more advantageous than its competitor.

The case of null subjects in Afro-Peruvian Spanish is markedly different. The NS grammar (G₁) is incompatible with overt expletive subjects (20), while the non-NS grammar (G₂) is incompatible with null thematic subjects (21):

As one might expect, thematic subjects far outnumber expletive subjects in discourse. Drawing on data on their frequencies of occurrence from multiple sources, Simonenko & Crabbé & Prévost (2019: 296, 299) estimate the grammatical advantage of a NS grammar to be about α₁ = 0.7 and that of the corresponding non-NS grammar to be only about α₂ = 0.05; that is to say, a difference of over an order of magnitude, implying an advantage ratio of α = 0.7/0.05 = 14 in favour of the NS grammar.⁹

This difference between the two case studies is important. In the case of Afrikaans, the fact that the two grammars are equiadvantageous means that, in the absence of any L2 learning, the population of speakers is stable (though not asymptotically) at any value of p: if σ = 0 and α = 1, then the value of p is always at equilibrium.¹⁰ Thus the only forces that could shift the state of the speech community (again, assuming the absence of L2 learners in the population) would have to be stochastic in nature. Once L2 learners facing an L2-difficulty with one of the grammars are introduced into the population, the equilibrium will shift: any amount of L2 learning implies that the origin (p,q) = (0,0) becomes the system’s only attractor (Table 1). Were those L2 learners to be removed from the population at a later stage, the speech community would then again assume its previous quasistable nature and come to rest at whichever value of p the L2 learning situation brought it to (this will correspond to p ≈ 0 if enough time has elapsed; see Section 5.4).

The case of null subjects is radically different. Here the original, L2-difficult grammar G₁ is much more advantageous than its competitor, G₂. Not only does this mean that the proportion of L2 learners in the population would have to be relatively high for G₁ to be completely replaced by G₂; the long-term prediction is also that if those L2 learners were to be removed, the population would drift back to the equilibrium p = 1 whose stability is guaranteed by the asymmetry in grammatical advantages in the absence of any L2 learning.

5.3 Calibrating demographics

Table 2, from Giliomee & Elphick (1979), gives an overview of the population of the Dutch Cape Colony from 1670 to 1820, the period relevant for the emergence of Afrikaans. Although it is evident from these figures that slaves, freed slaves and the indigenous peoples represented a significant fraction of the population at each stage, translating these data into figures of the likely proportion of L2 learners at the different time points is non-trivial. First, no data exist for the native population before 1798. Secondly, not all of the non-European population would be L2 learners: perhaps some of them would not have needed to learn Dutch, but even more importantly, at some point part of this population will have begun to acquire Dutch as an L1, or at least as a bilingual L2 in childhood. For these reasons, I have computed fairly wide interval estimates for σ, the proportion of (adult) L2 speakers in the colony, given in the rightmost column of Table 2. The left endpoint of each interval represents the conservative estimate that half of the non-European population would have spoken Dutch as an L2; the right endpoint gives the absolute maximum, on the (quite certainly unrealistic) assumption that the entire non-European population spoke Dutch as an L2.

Population figures for Colonial Peru are hard to come by; however, useful data exists for Lima in the early period, specifically the first half of the 17th century. Interpretation of the demographics in Table 3, from Bowser (1974), is complicated by the fact that the various sources from which the demographic data are drawn employ different classificatory principles, sometimes pooling Spaniards and people of mixed European–American background, or people of African and people of mixed African–European background, together. Sessarego (2015: 93) argues that people of mixed background were most probably bilingual in Spanish from childhood (as one of their parents would have been Spanish). They should therefore be exempted from any estimates of the proportion of adult L2 learners in the population. The interval estimates given in Table 3 were computed with the assumption that only the Black and Indigenous groups would contribute adult L2 learners to the population; the left endpoints of these intervals again correspond to the conservative assumption that 50% of these people spoke Spanish as L2, while the right endpoints correspond to the upper bound assumption that 100% of them did.¹¹ The resulting estimates of σ suggest a steadily, if slowly growing proportion of L2 speakers in the population, with figures roughly similar to those found in the Dutch Cape Colony.

I am not aware of useful demographic data for later stages of Colonial Peru; a census was carried out in 1725–1740, but it contains no information on slaves (Pearce 2001). Useful information is available, however, on the population dynamics of the slave population on individual plantations or haciendas in the later period; while such data can tell us little about the relative proportion of L2 speakers in the population, they are useful in illuminating further developments following the initial stages of colonization. Drawing upon data in Cushner (1980), Sessarego (2015: 107) reports an estimated yearly net growth of 1.4 slaves per hacienda for the period 1710–1767; yet at the same time, the yearly natural growth (estimated from records of slaves’ births and deaths, which were kept on haciendas run by Jesuits), is negative at –2.7 per hacienda. In other words, an average of 4.1 slaves must have been imported yearly, per hacienda, in this period to match the net growth rate. It follows that, as late as the second half of the 18th century, there must have been a steady influx of new slaves, and hence of potential adult L2 learners of Spanish, into these regions.

5.4 Predictions: Afrikaans

The main linguistic features of Afrikaans are estimated (based on extant written records) to have been in place by the end of the 18th century (Roberge 2002: 83). Since the Dutch founded their permanent colony in 1652 (Guelke 1979), this leaves around 150 years for the development of Afrikaans as a language separate from Dutch. Assuming one generation to correspond to roughly 30 years (Tremblay & Vézina 2000), the development is thus estimated to have occurred over about five generations of speakers. More precisely, five generations supplies an upper bound for the development; the linguistic change may have happened faster, too, but this is impossible to determine given the scant textual record in the earlier phases.

In the well-mixing, infinite-population setup adopted in Section 4.2, the fact that the grammatical advantages are in balance in this case (α = 1) means that, in theory, even one adult L2 learner would suffice to drive the L2-difficult grammar to extinction, given enough time, irrespective of the absolute size of the population. The crucial question then concerns the time scale of the development: for fixed grammatical advantage ratio α and L2-difficulty d, the proportion of L2 speakers σ directly controls the time to extinction of G₁, i.e. how long it takes for the population to converge to the attractor (p,q) = (0,0) from some initial state (p₀, q₀) = (1, q₀). (The initial probability in the L2 population, q₀, is of course an unknown.) Since the strength of L2-difficulty d is also unknown, the best one can do is to compute these times-to-extinction for several combinations of the parameter values, thereby hopefully establishing at least a subset of the parameter space that predicts the empirically observed facts.

Iterating the pair of equations (18) repeatedly, one can find the number of generations it takes for the population to converge to the attractor (p, q) = (0,0) from various initial conditions (p₀,q₀) = (1, q₀), for various selections of d and for σ = 0.2 and σ = 0.6 (roughly corresponding to the endpoints of the interval estimates in Table 2). Convergence to the attractor is here defined, somewhat arbitrarily, as the values of both p and q being below 0.001 (0.1% use of G₁).

Figure 3 plots the passage times found using this procedure. From these results, it is clear that convergence to the attractor in 5 generations is possible. For proportions of L2 speakers on the order of σ = 0.6, a strength of L2-difficulty greater than about d = 0.5 is sufficient; for lower proportions on the order of σ = 0.2, L2-difficulties in excess of about d = 5 guarantee convergence in up to 5 generations. In future work, it would be important to estimate plausible values of d using independent evidence, so that model fit can be evaluated in a more principled manner (subject to fewer researcher degrees of freedom). Having said that, the above demonstration shows that the model makes the development of Afrikaans possible (even if not necessary).

Figure 3

Time of convergence to attractor in the Afrikaans case study, for various values of L2-difficulty d, proportion of L2 speakers σ and initial probability of L2-difficult grammar in the L2 speaker population q₀.

5.5 Predictions: Afro-Peruvian Spanish

In the case of Afro-Peruvian Spanish, the situation is radically different: in this case, it needs to be explained why the null subject grammar was never completely overthrown by the corresponding grammar without null subjects, which would be favoured by L2 learning. Recall that the critical value of the bifurcation parameter σ, the proportion of adult L2 learners in the population, was found in Section 4 to be

where α = α₁/α₂ is the ratio of the advantages of the two grammars and D = d/α₂ represents the relative L2-difficulty of G₁, scaled by the advantage of G₂. The lower bound of σ_crit occurs as D → ∞, namely

On the other hand, in Section 5.2 the value α = 14 was estimated in this particular case. This corresponds to a of 13/14 ≈ 0.93. In other words, the proportion of L2 speakers in the population would need to be at least about 0.93—and possibly higher, if D turned out to have a small value—for the grammatical advantage enjoyed by the L2-difficult grammar G₁ to be overcome and for the origin (p,q) = (0,0) to be an attractor. Since based on the available demographic data such a high proportion of L2 speakers never obtained in Colonial Peru (Section 5.3), we infer that null subjects were never about to be completely lost in the Afro-Peruvian variety of Spanish.

For any given non-zero σ, however, a stable, attracting state is implied in the interior of the state space (see again Section 4.2). This would appear to correspond, qualitatively, to the linguistic classification of Afro-Peruvian Spanish as a mixed NS language (Section 2.2)—that is, as a variety sometimes employing, sometimes not employing, null subjects. It is worth pointing out, however, that as the proportion of L2 speakers decreases over time, this interior equilibrium tends to a stable rest point on the boundary of the state space, crucially with the property p = 1 (so that L1 speakers have reverted to using null subjects all the time). This may help to explain the reported instability of the mixed NS status of Afro-Peruvian Spanish—the observation that the younger speakers in these communities are turning towards the standard Spanish grammar, with full null subjects (Sessarego 2014). Although Sessarego (2014) attributes this development mostly to sociolinguistic causes—to the prestige enjoyed by standard coastal Peruvian Spanish—the above considerations suggest an alternative, or at least complementary analysis: the loss of overt subjects results from the fact that there are fewer L2 speakers of the variety in the speech community, leading to fewer constructions totally lacking null subjects in the input data based on which L1 learners acquire their variety.