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Population dynamics in educational institutions considering the student satisfaction

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Highlights

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    The dynamics of the number of students in an educational institution is modeled.

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    In this model, the institutional quality perceived by the students is considered.

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    The model is analyzed for two kinds of student influx.

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    Transcritical and period-doubling bifurcations can occur.

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    The analytical and numerical results are discussed from a management point of view.

Abstract

Here, a discrete-time mathematical model representing the temporal evolution of the number of students in an educational institution is proposed and analyzed. In this approach, the students are divided into two groups: the ones who are satisfied with their academic performances and learning environments and the ones who are not. The population dynamics of these two groups is ruled by the following assumptions: all newly enrolled students are supposed to be satisfied; students can spontaneously modify their state of satisfaction; (dis)satisfaction can be propagated via social contacts as contagious diseases; student dropout primarily reduces the dissatisfied population; and both kinds of students can earn their diplomas. Analytical and numerical results are obtained and discussed from an institutional management point of view.

Introduction

Forecasting the number of enrolled students in the coming years is a major concern of any educational establishment, because such a number affects its resources and expenses (e.g. [7]). In this context, it is relevant, for instance, to estimate the probability of a student dropping out of a course before obtaining the diploma (e.g. [18]) and to examine the influence of good and bad study habits on the academic performance (e.g. [5]) and, consequently, on the dropout rate.

Here, I propose and analyze a second-order system of difference equations that represent the population dynamics in educational institutions, by taking into account the institutional quality perceived by the students. In this model, at every time instant, every student is in one of two states: satisfied or dissatisfied. Satisfaction and dissatisfaction can be propagated through personal contacts as contagious diseases or social behaviors (e.g. [5], [8], [16], [17]). Students are also able of autonomously shifting from satisfied to dissatisfied, or vice versa. Satisfied students drop out only due to financial or health problems; dissatisfied students can abandon the educational institution due to financial or health problems, deficient academic achievement, or negative perception of the learning environment (e.g. [4], [6]). In addition, both groups of students can complete their degrees. The aim of this study based on the above assumptions is to evaluate how academic and administrative measures for quality enhancement can affect the composition of the student population. I did not find in the literature any study with this approach. Obviously, student satisfaction in the real world can be determined from interviews, questionnaires, surveys.

This paper is organized as follows. In Section 2, the discrete-time system is described, and the steady-state solutions and the corresponding stabilities are analytically derived for two particular choices of student influx. In Section 3, results of numerical simulations are presented to further explore the model dynamics. In Section 4, the possible relevance of this work is stressed and discussed from an institutional management point of view.

Section snippets

Model and analytical results

Consider an educational establishment, like a school, a college, a university, or even a fitness center (which receives and loses students/users on a daily basis). Let Φ(t) be the amount of students admitted at the time step t (which can be conveniently measured in days, weeks, months, semesters, or years). The numbers of satisfied and dissatisfied students at t are denoted by S(t) and D(t), respectively. Assume that both groups of students are homogeneously distributed over the space; that is,

Numerical results

In order to illustrate the model dynamics, numerical simulations were performed. In all figures, thin line represents S(t); thick line, D(t).

Figs. 1(a)–(i) exhibit the dynamic behaviors obtained from Eqs. (1) and (2) for Φ(t)=Φ1(t),b=0.05,d=0.10,e=0.05,g1=0.15,g2=0.05, and ϕ=200. For instance, g1=0.15 means that 15% of the S-students earn their diplomas at each time step. The values of a, c, r1, and r2 are varied in these figures to investigate the dissatisfaction spread. In Fig. 1(a), a=0.0001,

Discussion

Several managerial and governmental issues have been theoretically investigated and some of these works have been based on discrete-time approaches. For instance, the effects of positive and negative behavior features of employees on the workplace dynamics were examined with a cellular automaton model [13]; the impact of cash transfer programs implemented in several countries to improve the education and health of children was studied with multi-agent systems [15].

Here, an analytical model

Acknowledgments

The author is partially supported by CNPq. The author would like to thank Renê de Ávila Mendes for the inspiring conversations.

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