On February 6, 2013, the State Council executive meeting deployed and improved the vocational education investment mechanism. They decided to charge tuition fees to all newly admitted students enrolled in the National Admissions Program from the fall 2014 semester. The meeting determined three policies: improving the financial appropriation system, improving the policy system for vocational scholarships and grants, and establishing and improving the vocational education charging system. The news immediately aroused substantial repercussions in society [1]. As a part of higher education, vocational education shoulders the important task of cultivating talents. However, with the rapid expansion of the scale of vocational education in China, the contradiction between the quantity and quality of vocational education has become increasingly prominent. In particular, the quality of education has been questioned from many sources. Many domestic scholars have researched the current situation and development of vocational education. The research methods and conclusions mainly include qualitative and quantitative aspects.

Some scholars have proposed that the reform of the vocational training mechanism is a forward-looking, systematic, and overall project starting from the admission adjustment and the implementation of the tutor responsibility system. The recruitment work and the reform of the scholarship system are the entry point and critical points of the reform of the training mechanism. Some scholars consider the educational investment of China and the family and the proportion of high school graduates in the total population and establish a BP neural network model to determine the enrollment scale of colleges and universities. The results show that the growth rate of Chinese current educational investment is lower than the growth rate of the enrollment scale [2]. Some scholars have demonstrated the rationality of total occupational fees from public economics, cost-sharing, human capital, and reality. However, the complete charging of vocational education cannot fundamentally change the situation of single investment in education funds. This also raises the threshold for poor students to study. Some scholars have analyzed the current condition of vocational education in Chinese colleges and universities based on pedagogy, academic economics, management, and other theories. They applied the fuzzy comprehensive evaluation method to construct the evaluation model of the development level of vocational education in Chinese higher education institutions. At the same time, scholars put forward countermeasures for the practical problems faced by the development of vocational education in China.

Some scholars have discussed the strategies for the scale development of Chinese higher education through investigation and reference to many data, history, and domestic and international comparisons. The article pointed out that the annual growth rate of college students should be controlled at 2% to 7% before 2020. Some scholars have established a first-order linear differential equation of college enrollment scale and employment rate [3]. They analyzed the stability and conditions that should be met for enrollment expansion. Some scholars have quantitatively analyzed the impact of enrollment expansion on the employment rate of graduates at different levels and disciplines. They pointed out that enrollment development will bring about a decline in the employment rate. This is a manifestation of the expansion of investment in higher education, which leads to increased risks and lower returns of human capital investment. They proposed that the growing scale of investment should be controlled, and the hierarchical structure, professional structure, and regional structure of investment should be adjusted to optimize the allocation of higher education resources. Some scholars have established first-order, second-order, and third-order linear and nonlinear dynamic system models from different aspects based on theories such as “education cost-sharing.” They studied the existence and stability of the dynamical system solution in each case, which provided a theoretical basis for the government to carry out macro-control.

With the implementation of new policies, higher education research is also changing. We extract appropriate research subjects under the new policy, so our considerations are as follows:

Tuition is no longer an issue that needs to be considered too much. Before the new policy, some students do not need to pay tuition fees, while some need to pay tuition fees. And some professional tuition fees are as high as 12,000 yuan, and some majors only need 7,000 yuan. Tuition fees vary widely. However, the new policy requires all students to pay tuition, and the tuition fee is capped at 8,000 yuan [4]. The range that each university and each major can float is minimal.

Subsidy measures such as bursaries are reflected in the quality of teaching. The new policy stipulates that the bursary shall not be less than 6,000 yuan per person per year. This is the minimum bursary limit. The per capita bursary of different schools varies greatly, and the bursary reflects the school's comprehensive strength and teaching quality to a certain extent.

Enrollment scale and employment rate are still crucial issues to be considered. Vocational education aims to cultivate high-quality academic talents, and employment has always been an issue that has been widely concerned by all sectors of society.

Therefore, this paper will establish a three-dimensional dynamic model of enrollment scale, education quality, and employment rate [5]. We use the Routh-Hurwitz criterion and the stability criterion to determine the stability of the equilibrium point of the model. Further, we use the model results to estimate the optimal enrollment scale of colleges and universities. This provides a theoretical basis for the government and institutions of higher learning to carry out macro-control.

We use

We make the following assumptions through analysis:

_{1}: The larger the enrollment scale, to a certain extent, the more muscular the school's comprehensive strength and the higher the school's teaching quality. Therefore, the rate of change in enrollment is positively correlated with the quality of teaching;

_{2}: The rate of change in the enrollment scale has a negative quadratic correlation with the graduation cost of future occupational graduation [6]. The cost of employment is inversely proportional to the employment rate. According to the above assumptions, we obtain the kinetic equation of the enrollment scale
_{1} denotes the employment cost factor related to enrollment size. If the enrollment scale is too low, there will be a supply shortage between social demand and university supply. At this time _{1} < 0, and vice versa, _{1} > 0.

The following assumptions are made through the analysis:

_{3}: Because teaching resources are limited, the larger the enrollment scale, the smaller the per capita teaching resources. This causes the quality of teaching to decrease as the size of the enrolment increases.

_{4}: The higher the teaching quality, the higher the overall quality of the students. The higher the employment rate, the change in teaching quality is proportional to the employment rate [7]. According to the above assumptions, we get the dynamic equation of teaching quality
_{2} > 0 denotes the employment rate factor related to teaching quality.

Changes in employment rates are directly proportional to the quality of teaching. It is inversely proportional to the enrollment scale [8]. So the dynamic equation of employment rate

_{3} > 0 denotes the enrollment scale factor related to the employment rate. Our analysis of _{1} shows that _{3} can be positive or negative and always has the same sign as _{1}.

To sum up, we get the three-dimensional dynamic system of vocational enrollment scale

First, we find the equilibrium point of the system (4). Assuming that ^{*}, ^{*},

Since all variables are positive, _{3}^{*} = _{1}_{2}. Because _{2} > 0, _{1} and _{3} have the same sign, the existence of the equilibrium point has the following theorem: Theorem 1: System (4) has a unique positive equilibrium point when _{1} > 0

The theorem shows that the system has a positive equilibrium point when the enrollment scale is larger than the social demand. This is of practical significance. Because when the enrollment scale is lower than the social demand, the short supply situation will inevitably prompt colleges and universities to expand enrollment so that the supply and demand equal or even exceed the collection [9]. We determine whether the status quo corresponding to this equilibrium point is healthy and sustainable. Below, we discuss the conditions that the equilibrium point stability should be satisfied.

The Jacobian matrix of system (4)

The Jacobian matrix corresponding to the equilibrium point E

The characteristic equation of matrix (8)

According to the Routh-Hurwitz criterion and the stability criterion, we have the following theorem:

The equilibrium point (6) is stable when _{3} > 1.

remember

Where

We let

Equation (10) is stable when
_{3} > 1. The theorem is proven.

With the development of the economy, people's demand for knowledge is increasing, and the coverage of vocational education will be broader and broader. However, career expansion should not be blind [10]. To determine the optimal enrollment scale and rationalize teaching resources, we need to use the above model to determine the optimal enrollment scale.

When the growth rate of education quality is greater than the growth rate of the enrollment scale, there is a “surplus” in teaching grades. When the growth rate of education quality is lower than the growth rate of enrollment scale, leading quality will be in short supply. Therefore, the optimal enrollment scale must correspond to the growth rate of teaching quality equal to the growth rate of enrollment scale, namely

Solving Equation (11) to get

Whether positive or negative, _{1} is always positive and _{2} is always negative. Therefore, the following suggestions are made for the estimation of the enrollment scale of colleges and universities:

If the current actual enrollment is _{t}_{1}, then the school should expand the enrollment to _{1}. This can maximize the utilization of teaching resources.

If the current actual enrollment number of the school is _{t}_{2}, then the school belongs to the large enrollment scale. We should gradually reduce the enrollment size until _{t}_{2}.

This can maximize the utilization of teaching resources.

Equation (12) shows that to increase the upper limit of the optimal enrollment scale under the condition of certain teaching quality

We assume that _{3} is corrected to the following relationship: “When the enrollment scale is too small, the teaching resources cannot be fully utilized, and the rate of change in teaching quality is positively correlated with the enrollment scale [12]. When the enrollment scale is too large, the teaching resources cannot meet the teaching requirements, and the rate of change in teaching quality negatively correlated with enrollment size.” At the same time, we introduce the enrollment base _{m}

At this time, system (4) is transformed into

Fixed point

System Stability Conditions

Are established.

Equation (15) is more theoretically general than Equation (2). At present, the enrollment scale is too large, and the teaching resources cannot meet the teaching requirements, so formula (2) is feasible in practice. Conditions (15) and (16) are too complicated to apply practically. It will not be discussed in detail here.

On analyzing the new policy, we extract three aspects that vocational training should be paid attention to enrollment scale, education quality, and employment rate. Here we establish a three-dimensional nonlinear dynamical system and give the conditions that the equilibrium point stability of the system should satisfy. Further, we use the model results to estimate the value of the optimal occupational enrollment scale. The results can provide decision-making references for education management departments.

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