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Introduction

Many languages ​​are at risk of disappearing. It is therefore crucial to understand the mechanisms underlying changes in the number of speakers and to determine if there are measures that could help preserve some of them.

In their article,

Abrams, S. Strogatz, Modelling the dynamics of language death, Nature 424 (6951) (2003) 900

Abrams and Strogatz proposed a mathematical model for studying language competition.

This model fits well with several sets of empirical data: it satisfactorily accounts for historical data concerning the decline of Welsh, Scottish Gaelic, Quechua, and other endangered languages.

In the Abrams and Strogatz model, as well as in those derived from it, the population size is assumed to remain constant. Therefore, the variables are the proportions of the different speaker groups. In models that include bilingualism, the population consists of three groups: monolingual speakers of language A, monolingual speakers of language B, and bilingual speakers AB. The model is two-dimensional, with $\sigma_A$ representing the proportion of speakers of language A and $\sigma_B$ the proportion of speakers of language B ($\sigma_{AB} = 1 - \sigma_A - \sigma_B$).

Within any linguistic subpopulation, forces and influences lead members of one group to switch languages ​​in favor of another. In the Abrams and Strogatz model, the rate at which speakers of one language learn the second language depends on the attractiveness of the latter. In their most general definition of attractiveness, Abrams and Strogatz postulate that a language is more attractive the more speakers it has and the higher its prestige.

In their article,

C. Bernard, S. Martin, Building strategies to ensure language coexistence in presence of bilingualism, Appl. Math. Comput. (2012), doi:10.1016/j.amc.2012.02.041

The authors model only the transitions of the following four types: A → AB, AB → A, A → AB, and AB → B (with the transitions A → B and B → A being extremely rare in practice). They also assume an asymmetry between monolinguals and bilinguals: A → AB (or B → AB) occurs at a rate proportional to the attractiveness of monolingual speakers of A (or B); AB → A (or AB → B) occurs at a rate proportional to the attractiveness of the entire pool of speakers of A, including bilinguals (thus, some bilinguals can become monolingual speakers of A even if A has no monolingual speakers).
Consequently, the two-dimensional model is defined by:

$$
\left\{\begin{array}{ll}
\sigma_A' &= (1-\sigma_A-\sigma_B)(1-\sigma_B)^as_A-\sigma_A\sigma_B^as_B\\
\sigma_B' &= (1-\sigma_A-\sigma_B)(1-\sigma_A)^as_B-\sigma_B\sigma_A^as_A
\end{array}\right.
$$

where $s_A$ (resp. $s_B$) denotes the prestige of language A (resp. B), and $a$ is a parameter modeling how the attractiveness of a language varies with the proportion of its speakers.
For convenience, the authors assume that $s_A+s_B=1$, allowing the substitutions $s_A=s$ and $s_B=1-s$.

The problem of the coexistence of two languages

Bernard and Martin consider that prestige, $s$, can evolve (under the influence of public action, for example), but that its variation at each time step is limited. Thus,

$$
s' = u\in U=[\underline{u};\bar{u}]
$$

When $u>0$, the prestige of language A increases and that of B decreases; the reverse occurs when $u<0$.

The dynamics of the system are thus described by:

$$
\left\{\begin{array}{ll}
\sigma_A' &= (1-\sigma_A-\sigma_B)(1-\sigma_B)^as-\sigma_A\sigma_B^a(1-s)\\
\sigma_B' &= (1-\sigma_A-\sigma_B)(1-\sigma_A)^a(1-s)-\sigma_B\sigma_A^as\\
s' &= u\in U=[\underline{u};\bar{u}]
\end{array}\right.
$$

The authors propose seeking strategies based on prestige variations that ensure the coexistence of the two languages—that is, strategies that maintain a certain level of monolingual speakers for each language. The set of constraints can thus be written as follows:

$$
K:= [\underline{\sigma};1]\times[\underline{\sigma};1]\times [0;1]
$$

Solving this problem means finding the set of states $(\sigma_A, \sigma_B, s)$ from which it is possible to maintain a certain proportion of monolingual groups by manipulating the relative prestige of the two languages—that is, the viability kernel in the terminology of viability theory.

Results

In the article cited above, the authors provided an analytical description of the viability kernel boundary, composed of integral curves originating from specific points on the boundary of the constraint set.
To illustrate their result, they reproduced the viability kernel for the parameter values ​​$a=1.41$, $\underline{u}=-0.06$, $\bar{u}=0.06$, and $\underline{\sigma}=0.1$.

Below, we present four 3D views of the viability kernel approximation corresponding to these parameter values, generated using the ViabLab software based on a $201 \times 201 \times 201$ grid of points (the black lines represent the edges of the constraint set):

As expected, the kernel is invariant under the transformation $\sigma_A\rightarrow \sigma_B$, $\sigma_B\rightarrow \sigma_A$, and $s\rightarrow 1-s$.

Non-viable situations correspond to states from which:

• even a continuous increase in the prestige of language A—at maximum intensity—up to its maximum value of 1 would not prevent the proportion of its speakers from crossing the threshold $\underline{\sigma}$,
• or even a continuous increase in the prestige of language B—at maximum intensity—up to its maximum value of 1 would not prevent the proportion of its speakers from crossing the threshold $\underline{\sigma}$.

Introduction

A major challenge in managing grazed pastures lies in reconciling conflicting objectives: increasing short-term profits while preserving the long-term viability of the rangeland. Increasing the stocking rate (number of animals per hectare) boosts short-term returns but can lead to virtually irreversible environmental degradation if the resulting grazing pressure is excessive. Conversely, maintaining an excessively low stocking rate can result in significant lost revenue.

In the article

Anderies, J., Janssen, M. & Walker, B. Grazing Management, Resilience, and the Dynamics of a Fire-driven Rangeland System. Ecosystems 5, 23–44 (2002). https://doi.org/10.1007/s10021-001-0053-9

The authors use the following model to account for the interactions between grass growth and grazing pressure:

$$
\left\{\begin{array}{ll}
c' &= r_cs-\delta_cc\\
s'&= c(a_c+r_ss)(1-\frac{s}{s^*}-\alpha_{ws}(\frac{w}{w^*})^\beta)-\gamma_gs\\
w'&=r_ww(1-\frac{w}{w^*})
\end{array}\right.
$$

Regarding the grass, $c(t)$ and $s(t)$ represent the biomass of the crown (roots plus growing points) and the shoot part, respectively, at time $t$. The parameter $s^*$ corresponds to the maximum shoot biomass per unit area, while $\gamma_g\leq 1$ represents the fraction of the shoot part removed by the grazing pressure exerted by the animals present. The crown grows at a rate $r_cs$ in the presence of shoot parts and declines at a rate $\delta_c$.
$a_c$ corresponds to the growth rate of shoot biomass per unit of crown biomass, and $r_s$ to the growth rate of shoot biomass per unit of crown biomass and per unit of shoot biomass.

Grass growth is also affected by the presence of shrubs: $w(t)$ represents shrub biomass, $r_w$ the intrinsic growth rate of shrubs, $w^*$ the maximum shrub biomass per unit area, and $w_s (=w/w^*)$ models the competitive effect of shrubs on the grass. The parameter $\beta$ measures the relative non-linearity of this effect.

The pasture manager's viability problem

The objective of rangeland management is to identify intervention strategies (adjustments to stocking rates) that reconcile two constraints: providing sufficient feed for the animals while preserving grass quality.

In the article

The authors study this viability problem using a simplified grass growth model, assuming that the maximum shoot biomass value, $s^*$, is 1 and that the impact of the shrub population is negligible—that is: 

$$
s^* = 1 \text{ et } \alpha_{ws} = 0.$$

The possible intervention on the system involves varying the grazing pressure—a variation assumed to be bounded—since purchasing or selling animals takes time :

$$
\gamma'(t) = u(t) \in [u_{min},u_{max}].
$$

The dynamics of the system are thus described by:

$$
\left\{\begin{array}{ll}
c' &= r_cs-\delta_cc\\
s'&= \frac{r_c}{\delta_c}s(a_c+r_ss)(1-s)-\gamma_gs\\
\gamma_g'&= u \in [u_{min},u_{max}]
\end{array}\right.
$$

The constraint regarding grass quality is modeled by a shoot biomass value exceeding a threshold, $s_{min}$, to ensure regrowth. The constraint regarding livestock feed is modeled by a grazing pressure that also exceeds a threshold, $\underline{g}$, over time. The set of constraints can thus be written as:

$$
K(s_{min},\underline{g}):=R^+\times [s_{min},+\infty[\times  [\underline{g},1]
$$

Solving this problem means finding the set of states $(s, \gamma_g)$ from which it is possible to feed the herd while maintaining high-quality grass by adjusting the number of animals—in other words, the viability kernel within the framework of viability theory.

Résultats

In the article cited above, the parameter values ​​are as follows: $r_c=1$, $\delta_c=1$, $a_c=0.1$,  $r_s=3$,  $s^*=1$, $u_{min} = -0.05$, $u_{max} = 0.05$, $s_{min} = 0.1$, $\underline{g}=0.65$.

We reproduce below the approximation of the viability kernel associated with these parameter values, generated by the ViabLab software using a 201×201×201 point grid:

As expected, starting from a higher grazing pressure results in a more restricted set of shoot and crown biomass combinations for which the system is viable.

This restriction is more pronounced when the capacity to reduce this pressure is low—that is, when $u_{min}$ approaches 0.

Introduction

A major challenge in managing grazed pastures lies in reconciling conflicting objectives: increasing short-term profits while preserving the long-term viability of the rangeland. Increasing the stocking rate (number of animals per hectare) boosts short-term returns but can lead to virtually irreversible environmental degradation if the resulting grazing pressure is excessive. Conversely, maintaining an excessively low stocking rate can result in significant lost revenue.

In the article

Anderies, J., Janssen, M. & Walker, B. Grazing Management, Resilience, and the Dynamics of a Fire-driven Rangeland System. Ecosystems 5, 23–44 (2002). https://doi.org/10.1007/s10021-001-0053-9

The authors use the following model to account for the interactions between grass growth and grazing pressure:

$$
\left\{\begin{array}{ll}
c' &= r_cs-\delta_cc\\
s'&= c(a_c+r_ss)(1-\frac{s}{s^*}-\alpha_{ws}(\frac{w}{w^*})^\beta)-\gamma_gs\\
w'&=r_ww(1-\frac{w}{w^*})
\end{array}\right.
$$

Regarding the grass, $c(t)$ and $s(t)$ represent the biomass of the crown (roots plus growing points) and the shoot part, respectively, at time $t$. The parameter $s^*$ corresponds to the maximum shoot biomass per unit area, while $\gamma_g\leq 1$ represents the fraction of the shoot part removed by the grazing pressure exerted by the animals present. The crown grows at a rate $r_cs$ in the presence of shoot parts and declines at a rate $\delta_c$.
$a_c$ corresponds to the growth rate of shoot biomass per unit of crown biomass, and $r_s$ to the growth rate of shoot biomass per unit of crown biomass and per unit of shoot biomass.

Grass growth is also affected by the presence of shrubs: $w(t)$ represents shrub biomass, $r_w$ the intrinsic growth rate of shrubs, $w^*$ the maximum shrub biomass per unit area, and $w_s (=w/w^*)$ models the competitive effect of shrubs on the grass. The parameter $\beta$ measures the relative non-linearity of this effect.

The pasture manager's viability problem

The objective of rangeland management is to identify intervention strategies (adjustments to stocking rates) that reconcile two constraints: providing sufficient feed for the animals while preserving grass quality.

In the article

Martin, Sophie & Deffuant, Guillaume & Calabrese, Justin. (2011). Defining Resilience Mathematically: From Attractors To Viability. 10.1007/978-3-642-20423-4_2. 

the authors study this viability problem using a simplified grass growth model, assuming that the shrub population remains constant at its equilibrium value and that the crown biomass instantly reaches the equilibrium value associated with the shoot biomass:

$$
w = w^* \text{ and  } c = \frac{r_c}{\delta_c}s.
$$

The possible intervention on the system involves varying the grazing pressure—a variation assumed to be bounded—since purchasing or selling animals takes time :

$$
\gamma'(t) = u(t) \in [u_{min},u_{max}].
$$

The dynamics of the system are thus described by:

$$
\left\{\begin{array}{ll}
\gamma_g'&= u \in [u_{min},u_{max}]\\
s'&= \frac{r_c}{\delta_c}s(a_c+r_ss)(1-\frac{s}{s^*}-\alpha_{ws})-\gamma_gs.
\end{array}\right.
$$

The constraint regarding grass quality is modeled by a shoot biomass value falling between two limits: a lower limit to ensure regeneration and an upper limit to preserve nutritional quality. The constraint regarding livestock feed is modeled by a grazing pressure that must also remain above a certain threshold over time. The set of constraints can thus be expressed as follows:

$$
K(\underline{g},s_{min},s_{max}):=[\underline{g},1]\times [s_{min},s_{max}].
$$

Solving this problem means finding the set of states $(s, \gamma_g)$ from which it is possible to feed the herd while maintaining high-quality grass by adjusting the number of animals—in other words, the viability kernel within the framework of viability theory.

Results

For the parameter values ​​from the article cited above ($r_c=1$, $\delta_c=1$, $a_c=0.3$, $r_s=3$, $s^*=1$, $\alpha_{ws}=0.5$, $u_{min} = -0.02$, $u_{max} = 0.02$, $\underline{g}=0$, $s_{min} = 0.1$, $s_{max} = 0.18$), the viability kernel can be obtained by calculating two integral curves (the authors of the article did not elaborate on this point).

Indeed, its boundary is defined by a closed curve formed by subsets of the boundary of the constraint set $K$ and two integral curves:

• the one originating from the point $(\gamma_g^1,s_{min}) $ with $s'(\gamma_g^1,s_{min})) = 0$ that follows the backward dynamics: 
  \begin{equation}
  \left\{\begin{array}{ll}
  \gamma_g'&= -u_{min}\\
  s'&= -(\frac{r_c}{\delta_c}s(a_c+r_ss)(1-\frac{s}{s^*}-\alpha_{ws})-\gamma_gs)\\ 
  \end{array}\right.
  \end{equation}
• the other originating from the point $(\gamma_g^2,s_{max}) $ with $s'(\gamma_g^2,s_{max})) = 0$ that follows that follows the backward dynamics: 
  \begin{equation}
  \left\{\begin{array}{ll}
  \gamma_g'&= -u_{max}\\
  s'&= -(\frac{r_c}{\delta_c}s(a_c+r_ss)(1-\frac{s}{s^*}-\alpha_{ws})-\gamma_gs)\\ 
  \end{array}\right.
  \end{equation}

The following figure shows the boundary of the viability kernel as a closed curve, along with the approximation obtained using the ViabLab software on a 2001 × 2001 grid of points. The result is consistent with the graph in the original article and is far more precise in the section of the kernel boundary that coincides with the boundary of the constraint set, as well as in the section defined by the integral curves.

Here is an example where we know the viability kernel.

Indeed, let's consider two-dimensional dynamics where the orbit of the points on the circle centered at the origin of the coordinate system and with radius $a$ is the same circle; where the orbit of the points with a norm strictly less than $a$ is a spiral tending towards the origin, and where the orbit of the points with a norm strictly greater than $a$ is a spiral which diverges.

This is the case for the solutions of the equation $x'=f(x)$ with

$$
f(x_1,x_2):=(\frac{||x||-a}{||x||}x_1+x_2 ; -x_1+\frac{||x||-a}{||x||}x_2)
$$   

Let us consider a square as a constraint set  : 

$$
K := \{(x_1,x_2)\in R^2\,|\, |x_1|\leq c \text{ et } |x_2|\leq c \}.
$$

Clearly, the set of points from which trajectories originate that remain within $K$ indefinitely is the disk centered at the origin with radius $a$.

This example makes it possible to test the precision of approximations obtained using generic software.

Below is a comparison, for $a = 0.7$ and $c=1.5$, between the theoretical result (the disk bounded by the red circle) and the approximate result obtained using the ViabLab software on a regular 2001×2001 grid (the green set bounded by the blue line):