Modeling Language Shift

Exact solutions of a mathematical model describing competition and co-existence of different language speakers

“…In this section, we study in detail exact solution (11). First of all, we answer the question: When positive coefficients d 1 , d 3 , α 2 , α 4 and µ lead automatically to positive values of α 1 , α 3 , β 1 , β 3 , κ 1 and κ 2 in formulae (12)?…”

“…Thus, we can use the formulae derived above in order to construct examples of traveling fronts, to plot the relevant curves (using the package Maple) and to present their plausible interpretation. Figures 1-3 represent the exact solution (11) in Case (i) µ > 0 (Fig. 1-2) and Case (ii) µ < 0 (Fig.…”

“…Figure 1: Traveling fronts (11). Curves represent the functions u(t 0 , x) (blue represents the Russian speakers), v(t 0 , x) (red represents the bilingual speakers) and w(t 0 , x) (green represents the Ukrainian speakers) for the fixed time t 0 = 0.01 (left) and t 0 = 4 (right) and the parameters µ = 3 2 , d 1 = d 3 = 2, α 2 = α 4 = 5 (other parameters are calculated by formulae ( 12)).…”

“…In Fig. 3, the exact solution (11) is pictured in Case (ii) µ < 0, so that the traveling fronts are moving to the left. As a result, the relevant interpretation is different.…”

“…On the other hand, the rigorous mathematical models came to social sciences and humanities only recently. In particular, papers devoted to rigorous mathematical modeling interaction of communities (populations) of different language speakers were published only during the last two decades [5][6][7][8][9][10][11]. These models are based on nonlinear differential equations of reactiondiffusion type.…”