Numerical simulations of multilingual competition dynamics with nonlocal derivative

Highlights

• We consider the dynamics of language competition model involving the Riesz fractional derivative.

• Novel numerical approximation methods based on the Fourier spectral algorithm are formulated.

• The choice of fractional parameter could seriously affect the evolution of population speakers.

Abstract

The dynamics of the language competition model is considered in this paper. The classical system is converted to non-integer order case by replacing the second-order partial derivative with the Riesz fractional derivative. A well-known numerical approximation methods based on the Fourier spectral algorithm in space and the third-order exponential time-differencing scheme are formulated to numerically simulate the three component fractional-in-space reaction-diffusion system in one and high dimensions for different values of α. Numerical results indicate α ∈ (1, 1.5] as the key control parameter that can influence the coexistence of various speakers over a period of time.

Introduction

Language competition, in earlier days, refers to the group of processes which are generally modelled by the interpersonal relationship of heterogeneous speakers as an example of collective scenarios in the consensus dynamics [25], [29], [35]. Obviously, language competition is a global affair. Two languages can compete in such a way that at any point in time, the speakers may decided to switch from one language to another, as a result of the interaction (or other existing factors) between speakers of language A and B. Different languages exist and coexist within different societies and social groups of a particular region or geographical location. We consider Nigeria as a case study, where the inhabitants speak over 350 different local languages which are re-grouped under the three major ones, namely the Hausa, Igbo and Yoruba languages, which are denoted with parameters U, V and W, respectively.

Numerous studies based on language competitions have been reported for the purpose of revealing the degree of endangerment of particular languages, looking for a common ground, where the choice of language can be related to ethnicity, popularity, infrastructural development and social networks in the language competition dynamics [34]. In the late 2003, the research paper by Abrams and Strogatz [1] on the model for endangered languages has stirred up a coherent effort to examine and understand the mechanisms that are involved in the study of language dynamics within and outside the local linguistic research. The language shift and extinction problems from various perspective can be addressed by characterizing and modelling the phenomena of language competition mathematically. A two-component model similar to that of population systems of two biological species is considered in their study, in which one represents the speakers of either a language A or another language by letter B. In addition, other reports adopt discrete dynamic based models in conjunction with speakers of more than one language, see for instance [31], [33]. A recent review of bilingual and multilingual competition models can be found in [16].

This paper gives an extension to the scenario discussed above, we consider the three major populations of speakers of different languages in Nigeria which interact in a nonlinear fashion to form a reaction-diffusion system

$$\begin{array}{ccc}\hfill \frac{\partial u}{\partial t}-d_1\frac{{\partial }^{2}u}{\partial x^2}& =& f_1={\tau }_{1}u(1-\frac{u}{\kappa -v-w})-m_{1}uv+m_{2}uw,\hfill \\ \hfill \frac{\partial v}{\partial t}-d_2\frac{{\partial }^{2}v}{\partial x^2}& =& f_2={\tau }_{2}v(1-\frac{v}{\kappa -u-w})-m_{3}uv+m_{4}vw,\hfill \\ \hfill \frac{\partial u}{\partial t}-d_3\frac{{\partial }^{2}w}{\partial x^2}& =& f_3={\tau }_{3}w(1-\frac{w}{\kappa -u-v})\hfill \\ & & -(m_{2}u+m_{4}v)w+(m_1+m_3)uv,\hfill \end{array}$$

subject to the boundary conditions $\partial (·)/\partial \nu =0$ and x ∈ ∂Ω. ∂/∂ν denotes the outer normal derivation. The unknown functions given as u(x, t), v(x, t) and w(x, t) represent the frequencies of speakers of languages U, V and W on the spatial and temporal variables x and t, respectively. The term ∂( · )/∂t is the temporal change of the frequencies. The diffusion terms $d_i(i=1,2,3)$ describe language dispersal, and the reaction terms $f_i(i=1,2,3)$ account for the local kinetics or geographical inhomogeneities within the system. The first factor to the right-hand side in Eq. (1.1) denotes an intrinsic growth part that models cultural and biological reproduction within each subgroup of individuals, as bounded logistic growth with natural intrinsic rates τ_i expressed as the difference between the birth and the death rates. Population growth is restricted with the common carrying capacity κ. Other parameters such as ${\tau }_i,i=1,2,3,$ κ and $m_i,i=1,2,3,4$ are assumed positive constants.

The interest of this paper is not really in dynamics with the local derivatives. Since we are not aware of any work done on language competition dynamics with nonlocal representation. Hence, we are motivated by replacing the integer second-order partial derivative with respect to space variable in Eq. (1.1) with the fractional derivative operator. So without further ado, we present the space-fractional form of the language competition model as

$$\begin{array}{ccc}\hfill \frac{\partial u}{\partial t}& =& d_1\frac{{\partial }^{\alpha }u}{\partial \left|x\right|{}^2}+{\tau }_{1}u(1-\frac{u}{\kappa -v-w})-m_{1}uv+m_{2}uw,\hfill \\ \hfill \frac{\partial v}{\partial t}& =& d_2\frac{{\partial }^{\alpha }v}{\partial \left|x\right|{}^2}+{\tau }_{2}v(1-\frac{v}{\kappa -u-w})-m_{3}uv+m_{4}vw,\hfill \\ \hfill \frac{\partial u}{\partial t}& =& d_3\frac{{\partial }^{\alpha }w}{\partial \left|x\right|{}^2}+{\tau }_{3}w(1-\frac{w}{\kappa -u-v})\hfill \\ & & -(m_{2}u+m_{4}v)w+(m_1+m_3)uv,\hfill \end{array}$$

where $\frac{{\partial }^{\alpha }(·)}{\partial \left|x\right|{}^2}$ is defined as the Riesz derivative of fractional order α, for 1 < α ≤ 2.

Fractional calculus is the branch of applied mathematics where the derivatives and integrals of fractional order are considered as an excellent modelling tool to provide a better explanation of memory effect and hereditary properties of complex dynamics. Over the years, it has gained a lot of interest and become a powerful tool for modelling real-world phenomena [3], [27], [30]. The application of non-integer-order partial differential equations are increasingly popular in various fields of applied engineering and science. Many scholars have demonstrated that the fractional partial differential equations are vital for understanding different aspects of non-locality and spatial heterogeneity when compared to the classical order cases. The fractional derivative has been applied in groundwater and fractals [2], [3], control theory [8], [32] finance [17], [28], [37], hydrology and geo-hydrology [3], [14], image processing [7] and many other applications for problems in biology, chemistry and physics [4], [5], [23].

There have been a lot of extensive reports, and applications of space-fractional reaction-diffusion equations. Pindza and Owolabi [26] proposed the Fourier spectral algorithm for solving a range of two-component systems of higher order space-fractional reaction-diffusion problems. In [20], a robust and adaptive numerical technique was suggested for simulating nonlinear partial differential equations of fractional order in high dimensions. A class of high-order numerical algorithms for Riesz derivatives were established by Ding and Li [10], through constructing of new generating functions. Some of the recent applications of space-time fractional derivatives can be found in [6], [10], [11], [12], [20], [21], [22], [23], [27], [30] and references therein.

The remainder part of this paper is structured as follows. Preliminary definitions and lemma based on the Riesz fractional derivative is given in Section 2. Numerical methods for the spatial and temporal discretization are presented in Section 3. Numerical results are given in Section 4. Finally, conclusions are given in Section 5.