A Mathematical Theory of Motion of Researchers between Research Organizations

Abstract

We discuss a mathematical model of motion of substance in a node of a network which has a structure consisting of a chain of cells. An exact solution for the model equations is obtained in the case when the structure of the node is a chain of two cells. The obtained general results are applied to the construction of a theory for the specific case of motion of young and experienced researchers between research organizations. For this case, we obtain analytical relationships for the time dependence of the number of two kinds of researchers in the studied research organization. These relationships are based on a specific choice of the time-dependent parameters of the model. The obtained analytical results show that the changes in the numbers of young and experienced researchers in a research organization may depend on the ratio between the initial numbers of the two kinds of researchers as well as on the parameters regulating the exchange of researchers between the research organization and the rest of the research environment.

1. Introduction

In this article, we discuss some aspects of the theory of motion of young and experienced researchers in a network of research organizations. In order to do this, we propose a model of motion of a substance in a network surrounded by an environment. We are interested in a specific node of the network and assume that this node has a structure. The structure contains a chain of cells which can exchange parts of the substance they contain. The cells can also exchange substance with the other nodes of the network as well as with the environment of the network.

The proposed model for the motion of substance includes a system of differential equations with time-dependent coefficients. The exact solution of this system is obtained for the structure of the node of interest containing an arbitrary number of cells. The solution includes integrals, and in general, one has to calculate this solution numerically. In order to illustrate the obtained solution, we consider a node of simple structure consisting of two cells. We apply this simple specific case of the model in order to better understand the processes connected to the motion of young and experienced researchers in a research organization.

Mathematical models are widely used in the study of complex systems [1,2,3,4,5]. Such models can be connected to the evolution of research organizations [6,7,8,9,10,11,12,13,14], and the theory of networks [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]. Our study will be based on a continuous-time model of motion of a substance on a channel of a network. The models of such a kind have different possible applications [30,31], such as network information flow [32], in the analysis of social structure [33], in evacuation planning [34], in traffic flow analysis [35,36,37,38], in petroleum and gas transportation [39,40], to model human and animal migration flows [41,42,43,44,45,46,47,48,49] or to model the publication activity in a research organization [50,51].

We will use a model of the above class in our study below. The new point with respect to the models [43,44,45,46,47,48,49] is that the solved model’s differential equations have time-dependent coefficients. This leads to the rich possible dynamics of the motion of researchers. In order to illustrate the selected interesting cases of this dynamics, we will assume a specific case of the time dependence of some of the coefficients of the model. This will allow us to obtain explicit forms of the solutions of the model. These solutions help us to illustrate the fact that the success of the efforts to increase the number of young researchers in a research organization may depend on the structure of the flows of researchers in the network and its environment as well as on the initial values of the ratio between the numbers of young and experienced researchers.

Another new point in the model discussed below in comparison to the models [43,44,45,46,47,48,49] is that one of the nodes of the network will have an internal structure. This will reflect the internal structure of the research organization, which is studied in the example of the application of the general model. The internal structure of the studied research organization is shaped by a specific characteristic of the researchers. In this article, we select the publication activity as such a characteristic. In addition, we consider the research organization as a part of a network of research organizations. This network can be considered to be the national network of research organizations. Furthermore, it is assumed that the network has an environment which consists of the other national networks of research organizations. Thus, we will study a complex system of networks, and we will account for the influence of the national and international environment on the processes in the studied research organization.

The text below is organized as follows. In Section 2, we formulate the model of motion of a substance in a channel consisting of finite number of nodes of a network. In Section 3, we apply the model to study a specific case of the time changes in the number of young and experienced researchers in an organization. The results show that the corresponding evolution can be quite complicated and, because of this, the research politics must be very carefully planned in order to lead to the desired positive effects. Several concluding remarks are summarized in Section 4.

2. Mathematical Formulation of the Problem

We consider the following problem. We have a network of nodes. An item is positioned at each node of the network (in our example below, we consider the specific case where the items are research organizations). This network is surrounded by the environment that consists of other items. We assume that each item has a structure. Below, we consider the specific case where we are interested in the structure of only one of the items. In general, the structure of items can be quite complicated. Below, we discuss the specific case in which the structure of the item of interest can be represented as a chain of cells from the point of view of the studied problem. The cells of this chain are numbered from 0 to N. The cells of the chain are connected by links. Each link connects two cells and each cell is connected to two links except for the 0-th cell and N-th cell. These two cells are connected to one link. These links are the internal links to the chain. External links are also possible. These links connect a cell of the chain to other items from the network or to other items from the environment of the network.

We assume that a certain substance is presented in the studied system and this substance can move across the network and its environment. The model is for the motion of a substance between the cells of the chain of the item of interest, among the items from the nodes of the network and among the items from the nodes of the network and the items from the environment. We will not specify the nature of the substance for the case of the general model. For the specific example below, we specify the substance to be young and experienced researchers who move among the research organizations from the network and its environment.

Returning to the general case, we assume that the motion of substance in the chain of cells (representing the structure of the item of interest) is controlled by the value of some quantity Q. There is a threshold value $Q_i$ for the quantity Q in the cell i. We assume that $Q_i>Q_{i-1}$. We assume that the substance can be separated into parts and each part is connected to some value of the quantity Q. If the value of the quantity is between $Q_{i-1}$ and $Q_i$, then the corresponding parts of the substance are in the i-th cell of the chain. We assume that $Q_{-1}=0$. If the amount of the quantity exceeds $Q_i$, then some amount of the substance moves to the cell $i+1$. The motion in the opposite direction is also possible. The two end cells of the chain (the nodes with numbers 0 and N) are special with respect to the motion of substance between the cells of the channel. There is no motion of the substance from the channel in a backward direction from cell 0 (as there is no cell with number $-1$). There is no motion of substance from the channel in the forward direction from cell N (as there is no cell of number $N+1$). Note that the motion of substance is possible (i) among the cells of the item of interest and (ii) among the other items from the network and other items from the environment.

The model assumes continuous time. The substance in a cell of the chain of cells can participate in one of the four following processes:

(1)

The substance remains in the same cell;

(2)

The substance moves to the previous or to the next cell (i.e., the substance can move from the cell m to the cell $m+1$ or from the cell m to the cell $m-1$);

(3)

The substance ”leaks” from cell m for other items from the network or for other items from the environment of the network;

(4)

The substance is ”pumped” to cell m from other items of the network or from other items from the environment of the network.

Let us formalize the above considerations mathematically. The quantities which will be used in the model are as follows.

1.

$i_n^e\left(t\right)$ and $o_n^e\left(t\right)$ are the amounts of inflow and outflow of substance from the environment to the n-th cell of the chain (the upper index e means that the quantities are for the environment);

2.

$o_n^c\left(t\right)$ is the amount of outflow of substance from the n-th cell of the chain to the $(n+1)$-th cell of the chain (the upper index c means that the quantities are for the chain of nodes);

3.

$i_n^c\left(t\right)$ is the amount of the inflow of substance from the $(n+1)$ cell of the chain to the n-th cell of the chain;

4.

$o_n^n\left(t\right)$ and $i_n^n\left(t\right)$ are the amounts of outflow and inflow of substance between the n-th node of the chain and the network (the upper index n means that the quantities are for the network).

For the 0-th cell of the chain, we have the exchange between the substance and the environment (inflow and outflow); the exchange between the substance and the next cell of the chain (inflow and outflow); and the exchange (inflow and outflow) between the substance and the network. Thus, the change of the amount of substance in 0-th cell of the chain is described by the relationship

$$\frac{d{x}_0}{dt}=i_0^e\left(t\right)-o_0^e\left(t\right)-o_0^c\left(t\right)+i_0^c\left(t\right)-o_0^n\left(t\right)+i_0^n\left(t\right).$$

(1)

For the cells of the chain numbered by $i=1,\cdots ,N-1$, there is an exchange between the substance and the environment; an exchange between the substance and the network; and the exchange between the substance and $(i-1)$-th and $(i+1)$-th cells from the chain of cells. Thus, the change in the amount of substance in the i-th cell of the chain is described by the relationship

$$\begin{array}{c}\hfill \frac{d{x}_i}{dt}=i_i^e\left(t\right)-o_i^e\left(t\right)+o_{i-1}^c\left(t\right)-i_{i-1}^c\left(t\right)-o_i^c\left(t\right)+i_i^c\left(t\right)-o_i^n\left(t\right)+i_i^n\left(t\right),i=1,\dots ,N-1.\end{array}$$

(2)

For the N-th cell of the chain, there is an exchange between the substance and the environment; and an exchange between the substance and the network; and an exchange between the substance and the $(N-1)$-th cell from the chain of cells. Thus, the change in the amount of substance in the N-th cell of the chain of cells is described by the relationship

$$\begin{array}{c}\hfill \frac{d{x}_N}{dt}=i_N^e\left(t\right)-o_N^e\left(t\right)+o_{N-1}^c\left(t\right)-i_{N-1}^c\left(t\right)-o_N^n\left(t\right)+i_N^n\left(t\right).\end{array}$$

(3)

Equations (1)–(3) describe the general case of the motion of substance along a chain of cells connected to a network and to the environment of this network.

A specific case of the general model (1)–(3) will be discussed below. It is based on the following specific cases for the quantities of the model.

• Exchange between the chain of cells and the environment of the network

  $$\begin{array}{ccc}\hfill i_0^e\left(t\right)& =& {\sigma }_0\left(t\right)x_0\left(t\right);o_0^e\left(t\right)={\mu }_0\left(t\right)x_0\left(t\right),\hfill \\ \hfill i_i^e\left(t\right)& =& {\sigma }_i\left(t\right)x_i\left(t\right);o_i^e\left(t\right)={\mu }_i\left(t\right)x_i\left(t\right),i=1,\dots ,N-1,\hfill \\ \hfill i_N^e\left(t\right)& =& {\sigma }_N\left(t\right)x_N\left(t\right);o_N^e\left(t\right)={\mu }_N\left(t\right)x_i\left(t\right).\hfill \end{array}$$

  (4)
  Note that the input from the environment to the cell 0 of the chain of cells is proportional to the amount of the substance in the 0-th cell.
• Exchange between the chain of cells and the network

  $$\begin{array}{ccc}\hfill i_0^n\left(t\right)& =& {ϵ}_0\left(t\right)x_0\left(t\right);o_0^n\left(t\right)={\gamma }_0\left(t\right)x_0\left(t\right),\hfill \\ \hfill i_i^n\left(t\right)& =& {ϵ}_i\left(t\right)x_i\left(t\right);o_i^n\left(t\right)={\gamma }_i\left(t\right)x_i\left(t\right),i=1,\dots ,N-1,\hfill \\ \hfill i_N^n\left(t\right)& =& {ϵ}_N\left(t\right)x_N\left(t\right);o_N^n\left(t\right)={\gamma }_N\left(t\right)x_N\left(t\right).\hfill \end{array}$$

  (5)
• Exchange within the chain of cells

  $$\begin{array}{ccc}\hfill o_0^c\left(t\right)& =& f_0\left(t\right)x_0\left(t\right);i_0^c\left(t\right)={\delta }_1\left(t\right)x_1\left(t\right),\hfill \\ \hfill o_i^c\left(t\right)& =& f_i\left(t\right)x_i\left(t\right);i_i^c\left(t\right)={\delta }_{i+1}\left(t\right)x_{i+1}\left(t\right),i=1,\dots ,N-2,\hfill \\ \hfill o_{N-1}^c\left(t\right)& =& f_{N-1}\left(t\right)x_{N-1}\left(t\right);i_{N-1}^c\left(t\right)={\delta }_N\left(t\right)x_N\left(t\right).\hfill \end{array}$$

  (6)

The above assumptions are that the inflows and the outflows are proportional to the amount of substance in the corresponding cell. The coefficients of proportionality are time-dependent. Such a kind of assumption covers vast numbers of possible cases. What is not covered are the cases of very strong flows where the inflows and outflows are proportional to some power of the substance available in the cell.

For the specific case, described by (4)–(6), the system of Equations (1)–(3) becomes

$$\begin{array}{c}\hfill \frac{d{x}_0}{dt}={\sigma }_0\left(t\right)x_0\left(t\right)-{\mu }_0\left(t\right)x_0\left(t\right)-f_0\left(t\right)x_0\left(t\right)+{\delta }_1\left(t\right)x_1\left(t\right)-{\gamma }_0\left(t\right)x_0\left(t\right)+{ϵ}_0\left(t\right)x_0\left(t\right),\end{array}$$

(7)

$$\begin{array}{c}\hfill \frac{d{x}_i}{dt}={\sigma }_i\left(t\right)x_i\left(t\right)-{\mu }_i\left(t\right)x_i\left(t\right)+f_{i-1}\left(t\right)x_{i-1}\left(t\right)-{\delta }_i\left(t\right)x_i\left(t\right)-f_i\left(t\right)x_i\left(t\right)+{\delta }_{i+1}\left(t\right)x_{i+1}\left(t\right)-\\ \hfill {\gamma }_i\left(t\right)x_i\left(t\right)+{ϵ}_i\left(t\right)x_i\left(t\right),\\ \hfill i=1,\dots ,N-1,\end{array}$$

(8)

$$\begin{array}{c}\hfill \frac{d{x}_N}{dt}={\sigma }_N\left(t\right)x_N\left(t\right)-{\mu }_N\left(t\right)x_N\left(t\right)+f_{N-1}\left(t\right)x_{N-1}\left(t\right)-{\delta }_N\left(t\right)x_N\left(t\right)-{\gamma }_N\left(t\right)x_N\left(t\right)+{ϵ}_N\left(t\right)x_N\left(t\right).\end{array}$$

(9)

We restrict the model further by the assumption of the absence of inflow from $i+1$-th cell to the i-th cell of the chain of cells. In this case, the system of Equations (7)–(9) becomes

$$\begin{array}{c}\hfill \frac{d{x}_0}{dt}=[{\sigma }_0\left(t\right)-{\mu }_0\left(t\right)-f_0\left(t\right)-{\gamma }_0\left(t\right)+{ϵ}_0\left(t\right)]x_0\left(t\right),\end{array}$$

(10)

$$\begin{array}{c}\hfill \frac{d{x}_i}{dt}=[{\sigma }_i\left(t\right)-{\mu }_i\left(t\right)-f_i\left(t\right)-{\gamma }_i\left(t\right)+{ϵ}_i\left(t\right)]x_i\left(t\right)+f_{i-1}\left(t\right)x_{i-1}\left(t\right),i=1,\dots ,N-1,\end{array}$$

(11)

$$\begin{array}{c}\hfill \frac{d{x}_N}{dt}=[{\sigma }_N\left(t\right)-{\mu }_N\left(t\right)-{\gamma }_N\left(t\right)+{ϵ}_N\left(t\right)]x_N\left(t\right)+f_{N-1}\left(t\right)x_{N-1}\left(t\right).\end{array}$$

(12)

Below, we set

$$\begin{array}{c}\hfill {\sigma }_i\left(t\right)-{\mu }_i\left(t\right)-f_i\left(t\right)-{\gamma }_i\left(t\right)+{ϵ}_i\left(t\right)=-{\nu }_i\left(t\right),\\ \hfill {\sigma }_N\left(t\right)-{\mu }_N\left(t\right)-{\gamma }_N\left(t\right)+{ϵ}_N\left(t\right)=-{\nu }_N\left(t\right).\end{array}$$

(13)

Then, we obtain from (10)–(12)

$$\begin{array}{c}\hfill \frac{d{x}_0}{dt}=-{\nu }_0\left(t\right)x_0\left(t\right),\end{array}$$

(14)

$$\begin{array}{c}\hfill \frac{d{x}_i}{dt}=-{\nu }_i\left(t\right)x_i\left(t\right)+f_{i-1}\left(t\right)x_{i-1}\left(t\right),i=1,\dots ,N-1,\end{array}$$

(15)

$$\begin{array}{c}\hfill \frac{d{x}_N}{dt}=-{\nu }_N\left(t\right)x_N\left(t\right)+f_{N-1}\left(t\right)x_{N-1}\left(t\right).\end{array}$$

(16)

The solution of (14) is

$$x_0\left(t\right)=x_0\left(0\right)exp\left[-\int dt{\nu }_0\left(t\right)\right].$$

(17)

On the basis of this solution, we can calculate step by step the solutions for $x_1$,…, $x_N$:

$$x_i\left(t\right)=exp\left[-\int dt{\nu }_i\left(t\right)\right]\left\{C_i+\int dt\left[f_{i-1}\left(t\right)x_{i-1}\left(t\right)exp\left(\int dt{\nu }_i\left(t\right)\right)\right]\right\},$$

(18)

for $i=1,\dots ,N$. $C_i$ are constants of integration.

For example, for a structure of the node of interest which consists of a chain of just two cells, we have $N=1$. For the node $i=0$, the solution is (17). For the node $i=N=1$, the solution is obtained from (18)

$$x_1\left(t\right)=exp\left[-\int dt{\nu }_1\left(t\right)\right]\left\{C_1+\int dt\left[f_0\left(t\right)x_0\left(t\right)exp\left(\int dt({\nu }_1\left(t\right)-{\nu }_0\left(t\right))\right)\right]\right\}.$$

(19)

An important specific case of the theory is the case where the parameters ${\sigma }_i$, ${\mu }_i$, $f_i$, ${\gamma }_i$, ${ϵ}_i$ do not depend on the time. In this case, the system of model equations becomes

$$\begin{array}{c}\hfill \frac{d{x}_0}{dt}=-{\nu }_0{x}_0\left(t\right),\end{array}$$

(20)

$$\begin{array}{c}\hfill \frac{d{x}_i}{dt}=-{\nu }_i{x}_i\left(t\right)+f_{i-1}{x}_{i-1}\left(t\right),i=1,\dots ,N-1,\end{array}$$

(21)

$$\begin{array}{c}\hfill \frac{d{x}_N}{dt}=-{\nu }_N{x}_N\left(t\right)+f_{N-1}{x}_{N-1}\left(t\right).\end{array}$$

(22)

Let us consider the case $t\to \infty$. For this asymptotic case, the solution of the model (20)–(22) tends towards a stationary solution for ${\nu }_0>0$. We show this as follows. The solution of (20) is $x_0\left(t\right)=x_0\left(0\right)exp(-{\nu }_{0}t)$. For $t\to \infty$, $x_0\left(t\right)$ tends towards 0 except for the case ${\nu }_0=0$. In this case, $x_0$ tends to the stationary solution $x_0^{*}=x_0\left(0\right)$. For $x_i$, one obtains from (18)

$$x_i=C_{i}exp(-{\nu }_{i}t)+f_{i-1}exp(-{\nu }_{i}t)\int dt{x}_{i-1}exp\left({\nu }_{i}t\right).$$

(23)

The integration of parts in (23) leads to

$$x_i=C_{i}exp(-{\nu }_{i}t)+\frac{f_{i-1}}{{\nu }_i}{x}_{i-1}-\frac{f_{i-1}}{{\nu }_i}exp(-{\nu }_{i}t)\int \left[d{x}_{i-1}\right]exp\left({\nu }_{i}t\right).$$

(24)

Let us assume that $x_0$ does not depend on t. Then, at $t\to \infty$, we have $d{x}_0=0$ and $x_1\to \frac{f_0}{{\nu }_1}{x}_0$. Thus, $x_1$ tends to the stationary solution $x_1^{*}=\frac{f_0}{{\nu }_1}{x}_0.$ This leads to $d{x}_1\to 0$ at $t\to \infty$ and $x_2$ tends also to the stationary solution. Thus, it is important in the case of constant parameters of the model to study the stationary solution of the model, where $x_0$ is an arbitrary finite parameter and

$$x_i^{*}=\frac{f_{i-1}}{{\nu }_i}{x}_{i-1}^{*}.$$

(25)

3. Specific Case of Application of the Model: A Theory of Motion of Researchers in Research Organizations

We apply the model discussed above to model the motion of researchers between research organizations. In order to do this, we will use an idea discussed by Schubert and Glänzel [50], who discussed the publication activity of researchers from a research organization. The stationary solution of their model describes the distribution of publications among researchers from a research organization. This distribution is the heavy-tail Waring distribution.

The mathematical model of Schubert and Glänzel contains ordinary differential equations. Our model is also based on ordinary differential equations. There are several differences between the two models. One difference between the model of Schubert and Glänzel and the model used here is connected to the input of a substance to the 0-th node of the research structure. In the model of Schubert and Glänzel, this input is proportional to the substance available in the entire structure. In our model, the input is proportional to the substance available in the 0-th node. Another difference is that we will discuss time-dependent coefficients in the model equations.

We apply the model (10)–(12) to study the dynamics of young and experienced researchers in the research organization. We will use the idea of Schubert and Glänzel for the description of the distribution of the researchers with respect to their research production. This idea will be used in the description of the structure and the motion of researchers within a cell of interest for us (the research organization which is assumed to have a chain of cells structure with respect to the research production of the researchers). The structure of the research organization will be very simple: the research organization is represented by a chain having just two cells. Here, $N=1$. The researchers from this organization can write publications or can collect citations of their publications. There is a threshold value. If the number of publications or the number of citations is below the threshold value, the researcher is in the cell 0. If the number of publications or the number of citations is above the threshold value, the researcher is in the cell 1. We can set the threshold value appropriately, such that the researchers in the cell 0 are predominantly young researchers. Researchers in the cell 1 will be denoted as experienced researchers.

The researchers can move between the two cells by increasing the number of publications (respectively, by increasing the number of citations). The parameters of the model correspond to the inflow and outflow of researchers to the two cells of the structure of the research organization. ${ϵ}_i$ regulates the inflow from the network of the other research organizations. ${\gamma }_i$ regulates the outflow to the network of the other research organizations. ${\sigma }_i$ regulates the inflow from the environment of the network of research organizations. ${\mu }_i$ regulates the outflow to the environment of the network of research organizations. $f_i$ regulates the flow between the nodes of the structure of the studied research organizations.

In the case of a research organization represented by two cells, we have from (17) and (19)

$$\begin{array}{c}\hfill x=x_0+x_1=x_0\left(0\right)exp\left[-\int dt{\nu }_0\left(t\right)\right]+\\ exp\left[-\int dt{\nu }_1\left(t\right)\right]\left\{C_1+\int dt\left[f_0\left(t\right)x_0\left(0\right)exp\left(\int dt({\nu }_1\left(t\right)-{\nu }_0\left(t\right))\right)\right]\right\}.\end{array}$$

(26)

Thus, we can calculate the distribution of researchers in the cells of the organization

$$\begin{array}{c}\hfill y_0\left(t\right)=\frac{x_0}{x}=\left\{x_0\left(0\right)exp\left[-\int dt{\nu }_0\left(t\right)\right]\right\}/\{x_0\left(0\right)exp\left[-\int dt{\nu }_0\left(t\right)\right]+\\ \hfill exp\left[-\int dt{\nu }_1\left(t\right)\right]\left\{C_1+\int dt\left[f_0\left(t\right)x_0\left(0\right)exp\left(\int dt({\nu }_1\left(t\right)-{\nu }_0\left(t\right))\right)\right]\right\}\},\end{array}$$

(27)

$$\begin{array}{c}\hfill y_1\left(t\right)=\frac{x_1}{x}=\{exp\left[-\int dt{\nu }_1\left(t\right)\right]\{C_1+\\ \hfill \int dt\left[f_0\left(t\right)x_0\left(0\right)exp\left(\int dt({\nu }_1\left(t\right)-{\nu }_0\left(t\right))\right)\right]\left\}\right\}/\{x_0\left(0\right)exp\left[-\int dt{\nu }_0\left(t\right)\right]+\\ \hfill exp\left[-\int dt{\nu }_1\left(t\right)\right]\left\{C_1+\int dt\left[f_0\left(t\right)x_0\left(0\right)exp\left(\int dt({\nu }_1\left(t\right)-{\nu }_0\left(t\right))\right)\right]\right\}\}.\end{array}$$

(28)

Note that these distributions are time-dependent. The solutions (27) and (28) allow us to study the influence of the parameters of the model on the distribution of researchers in the research organization. We are able to analyze complicated situations. Usually, this analysis has to be performed numerically.

Below, we will consider a specific case of (27) and (28). We will select a specific form of time dependence of some of the coefficients in the model equations. The goal is to solve the integrals and to obtain explicit analytical results for $x_0$ and $x_1$. These results will illustrate the influence of the flows on the system and the influence of the initial ratio between the young and experienced researchers on the process of change in the numbers of young and experienced researchers in the course of time.

The formulation of the specific problem is as follows. We assume that the studied research organization does much to attract researchers to the cell 0. This will lead to an increased inflow of young researchers from abroad and from other research organizations of the network. No efforts are made to attract researchers to cell 1. Then, the researchers from the cell 1 may leave the research organization.

In order to discuss an analytically tractable example, we assume that there are efforts to attract young researchers to the cell 0 of the research organization and these efforts continue from $t=0$ to $t<\alpha$, where $\alpha$ is a parameter. We assume that, for the input from the environment to the node 0 and for the output to the environment from the node 1:

$${\sigma }_0\left(t\right)={\sigma }_0+\frac{1}{\alpha -t};{\mu }_1\left(t\right)={\mu }_1+\frac{\kappa }{\alpha -t}.$$

(29)

where ${\sigma }_0$ and $\kappa$ are parameters which do not depend on the time. We also assume that the other parameters of the model: ${\sigma }_1$, $f_0$, ${ϵ}_{1,2}$, ${\mu }_0$ and ${\gamma }_{1,2}$ do not depend on the time.

The example describes a situation in which the number of researchers coming to cell 0 from the environment (from abroad) increases in the course of time. At the same time, the experienced researchers tend to migrate from cell 1 to the environment of the network (migration abroad).

In this case, we can further assume that the following situation is connected to the motion of young and experienced researchers

$${\nu }_0\left(t\right)={\nu }_0-\frac{1}{\alpha -t}>0;{\nu }_1\left(t\right)={\nu }_1+\frac{\kappa }{\alpha -t}>0.$$

(30)

where ${\nu }_0=-({\sigma }_0-{\mu }_0-f_0-{\gamma }_0+{ϵ}_0)$ and ${\nu }_1=-({\sigma }_1-{\mu }_1-{\gamma }_1+{ϵ}_1)$. Thus, ${\nu }_0$ and ${\nu }_1$ are the parameters which do not depend on the time. (30) introduces a restriction about the time of running of the discussed scenario of motion of the researchers in the system.

For the numbers $x_0$ and $x_1$ of the researchers in the two nodes, we obtain from (17) and (19)

$$x_0\left(t\right)=x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t].$$

(31)

The results for the number of researchers in node 1 are

$$\begin{array}{c}\hfill x_1\left(t\right)={(\alpha -t)}^{\kappa }exp[-{\nu }_{1}t]\left[C_1+f_0{x}_0\left(0\right)\alpha \int dt{(\alpha -t)}^{-(1+\kappa )}exp\left[({\nu }_1-{\nu }_0)t\right]\right].\end{array}$$

(32)

For the integral in (32), we have

$$\begin{array}{c}\hfill \int dt{(\alpha -t)}^{-(1+\kappa )}exp\left[({\nu }_1-{\nu }_0)t\right]=-exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{\kappa }\int d\xi {\xi }^{-(1+\kappa )}exp\left(\xi \right),\end{array}$$

(33)

where $\xi =-({\nu }_1-{\nu }_0)(\alpha -t)$. This integral can be calculated for the integer values of $\kappa$. In order to obtain analytical results, we will assume that the values of $\kappa$ are integer ones. Then, for $\kappa =-1-n$, we obtain

$$\begin{array}{c}\hfill x_1\left(t\right)=\frac{exp[-{\nu }_{1}t]}{{(\alpha -t)}^{1+n}}\{C_1-f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)]({[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n-n{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)(\alpha -t)]+{(-1)}^{n}n!\left)\right]\}\end{array}$$

(34)

At $t=0$ $x_1(t=0)=x_1\left(0\right)$. This initial condition leads to the value of the integration constant $C_1$:

$$\begin{array}{c}\hfill C_1={\alpha }^{1+n}{x}_1\left(0\right)+f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]({[-({\nu }_1-{\nu }_0)\alpha ]}^n-n{[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)\alpha ]+{(-1)}^{n}n!\left)\right]\}.\end{array}$$

(35)

Thus,

$$\begin{array}{c}\hfill x_1\left(t\right)=\frac{exp[-{\nu }_{1}t]}{{(\alpha -t)}^{1+n}}\left\{\right\{{\alpha }^{1+n}{x}_1\left(0\right)+f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]({[-({\nu }_1-{\nu }_0)\alpha ]}^n-n{[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)\alpha ]+{(-1)}^{n}n!\left)\right]\}-\\ \hfill f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)]({[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n-n{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)(\alpha -t)]+{(-1)}^{n}n!\left)\right]\}.\end{array}$$

(36)

The second possibility is $1+\kappa =n$. In this case, $\kappa =n-1$. For $x_1\left(t\right)$, we obtain

$$\begin{array}{c}\hfill x_1\left(t\right)={(\alpha -t)}^{n-1}exp(-{\nu }_{1}t)\{C_1-f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}}-\dots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)(\alpha -t)]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)(\alpha -t)\mid +\frac{[-({\nu }_1-{\nu }_0)(\alpha -t)]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^3}{3·3!}+\dots +\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n}{n·n!}+\cdots \left)\right)\left]\right\}.\end{array}$$

(37)

$C_1$ is determined again from the initial condition $x_1(t=0)=x_1\left(0\right)$. We obtain

$$\begin{array}{c}\hfill C_1=x_1\left(0\right){\alpha }^{1-n}+f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)\alpha ]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)\alpha \mid +\frac{[-({\nu }_1-{\nu }_0)\alpha ]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^n}{n·n!}+\cdots \left)\right)]\end{array}$$

(38)

The substitution of (38) in (37) leads to

$$\begin{array}{c}\hfill x_1\left(t\right)={(\alpha -t)}^{n-1}exp(-{\nu }_{1}t)\{x_1\left(0\right){\alpha }^{1-n}+f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)\alpha ]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)\alpha \mid +\frac{[-({\nu }_1-{\nu }_0)\alpha ]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^n}{n·n!}+\cdots \left)\right)]-\\ \hfill f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)(\alpha -t)]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)(\alpha -t)\mid +\frac{[-({\nu }_1-{\nu }_0)(\alpha -t)]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n}{n·n!}+\cdots \left)\right)\left]\right\}.\end{array}$$

(39)

On the basis of the obtained analytical relationships, we can calculate the time-dependent distributions of the researchers in the $y_0$ and $y_1$. In the case $\kappa =-1-n$

$$\begin{array}{c}\hfill y_0\left(t\right)=x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]/(x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]+\\ \hfill \frac{exp[-{\nu }_{1}t]}{{(\alpha -t)}^{1+n}}\left\{\right\{{\alpha }^{1+n}{x}_1\left(0\right)+f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]({[-({\nu }_1-{\nu }_0)\alpha ]}^n-n{[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}+n(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}-\cdots +\\ \hfill {(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)\alpha ]+{(-1)}^{n}n!\left)\right]\}-f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)]({[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n-n{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)(\alpha -t)]+{(-1)}^{n}n!\left)\right]\left\}\right),\end{array}$$

(40)

$$\begin{array}{c}\hfill y_1\left(t\right)=\left(\frac{exp[-{\nu }_{1}t]}{{(\alpha -t)}^{1+n}}\right\{\{{\alpha }^{1+n}{x}_1\left(0\right)+f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]({[-({\nu }_1-{\nu }_0)\alpha ]}^n-n{[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}+n(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}-\cdots +\\ \hfill {(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)\alpha ]+{(-1)}^{n}n!\left)\right]\}-f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)]({[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n-n{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)(\alpha -t)]+{(-1)}^{n}n!\left)\right]\left\}\right)/(\\ \hfill x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]+\frac{exp[-{\nu }_{1}t]}{{(\alpha -t)}^{1+n}}\left\{\right\{{\alpha }^{1+n}{x}_1\left(0\right)+f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]({[-({\nu }_1-{\nu }_0)\alpha ]}^n-n{[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}+n(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}-\cdots +\\ \hfill {(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)\alpha ]+{(-1)}^{n}n!\left)\right]\}-f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^{1+n}}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)]({[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n-n{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}+\\ \hfill n(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}-\cdots +{(-1)}^{n-1}n![-({\nu }_1-{\nu }_0)(\alpha -t)]+{(-1)}^{n}n!\left)\right]\left\}\right).\end{array}$$

(41)

In the case $\kappa =n-1$

$$\begin{array}{c}\hfill y_0\left(t\right)=x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]/(x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]+\\ \hfill {(\alpha -t)}^{n-1}exp(-{\nu }_{1}t)\{x_1\left(0\right){\alpha }^{1-n}+f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)\alpha ]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)\alpha \mid +\frac{[-({\nu }_1-{\nu }_0)\alpha ]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^n}{n·n!}+\cdots \left)\right)]-\\ \hfill f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)(\alpha -t)]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)(\alpha -t)\mid +\frac{[-({\nu }_1-{\nu }_0)(\alpha -t)]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n}{n·n!}+\cdots \left)\right)\left]\right\}),\end{array}$$

(42)

$$\begin{array}{c}\hfill y_1\left(t\right)=({(\alpha -t)}^{n-1}exp(-{\nu }_{1}t)\{x_1\left(0\right){\alpha }^{1-n}+f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)\alpha ]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)\alpha \mid +\frac{[-({\nu }_1-{\nu }_0)\alpha ]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^n}{n·n!}+\cdots \left)\right)]-f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)(\alpha -t)]}+\end{array}$$

(43)

$$\begin{array}{c}\hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)(\alpha -t)\mid +\frac{[-({\nu }_1-{\nu }_0)(\alpha -t)]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n}{n·n!}+\cdots \left)\right)\left]\right\})/\\ \hfill (x_0\left(0\right)\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]+{(\alpha -t)}^{n-1}exp(-{\nu }_{1}t)\{x_1\left(0\right){\alpha }^{1-n}+\\ \hfill f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[exp[-({\nu }_1-{\nu }_0)\alpha ](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)\alpha ]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)\alpha ]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)\alpha \mid +\frac{[-({\nu }_1-{\nu }_0)\alpha ]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)\alpha ]}^n}{n·n!}+\cdots \left)\right)]-\\ \hfill f_0{x}_0\left(0\right)\alpha exp\left[\alpha ({\nu }_1-{\nu }_0)\right]{[-({\nu }_1-{\nu }_0)]}^{n-1}[exp[-({\nu }_1-{\nu }_0)(\alpha -t)](-\frac{1}{(n-1){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-1}}-\\ \hfill \frac{1}{(n-1)(n-2){[-({\nu }_1-{\nu }_0)(\alpha -t)]}^{n-2}}-\cdots -\frac{1}{(n-1)![-({\nu }_1-{\nu }_0)(\alpha -t)]}+\\ \hfill \frac{1}{(n-1)!}(ln\mid -({\nu }_1-{\nu }_0)(\alpha -t)\mid +\frac{[-({\nu }_1-{\nu }_0)(\alpha -t)]}{1!}+\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^2}{2·2!}+\\ \hfill \frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^3}{3·3!}+\cdots +\frac{{[-({\nu }_1-{\nu }_0)(\alpha -t)]}^n}{n·n!}+\cdots \left)\right)\left]\right\}).\end{array}$$

The obtained results allow us to analyze numerous scenarios, even for the specific example considered in this text. Below, we analyze only one of the possible scenarios: the scenario for the case $\kappa =-1-n,n=1$. This case corresponds to the very favorable situation when the experienced researchers flow to cell 1 and the leadership applies measures to increase the number of young researchers in the cell 0.

Theorem 1.

Let us consider the model (14)–(16). Let the parameters of the model be ${\sigma }_0\left(t\right)={\sigma }_0+\frac{1}{\alpha -t}$; ${\mu }_1\left(t\right)={\mu }_1+\frac{\kappa }{\alpha -t}$, where ${\sigma }_0$ and κ are parameters which do not depend on the time and the other parameters of the model: ${\sigma }_1$, $f_0$, ${ϵ}_{1,2}$, ${\mu }_0$ and ${\gamma }_{1,2}$ do not depend on the time. Let, in the obtained solutions (31)–(36) of the model, one have $\kappa =-1-n,n=1$. Then:

1. 

$x_0\left(t\right)$ decreases by an increase to t as far as $t<\alpha$.

2. 

The increase in $x_1\left(t\right)$ is not guaranteed.

Proof. 

In the case $\kappa =-1-n,n=1$, $x_0\left(t\right)$ is given by (31) and, for $x_1\left(t\right)$, we obtain from (36)

$$\begin{array}{c}\hfill x_1\left(t\right)=\frac{exp[-{\nu }_{1}t]}{{(\alpha -t)}^2}\{{\alpha }^2{x}_1\left(0\right)+f_0{x}_0\left(0\right)\alpha \frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^2}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]\left([-({\nu }_1-{\nu }_0)\alpha ]-1\right)-\\ \hfill exp[-({\nu }_1-{\nu }_0)(\alpha -t)]\left([-({\nu }_1-{\nu }_0)(\alpha -t)]-1\right)\left]\right\}\end{array}$$

(44)

From (31), we obtain

$$\frac{x_0\left(t\right)}{x_0\left(0\right)}=\frac{\alpha }{\alpha -t}exp[-{\nu }_{0}t]=\frac{\alpha }{(\alpha -t)exp\left({\nu }_{0}t\right)}=\alpha /F$$

(45)

where $F=(\alpha -t)exp\left({\nu }_{0}t\right)$. We note that we must have $t<\alpha$. When F increases, then $\alpha /F$ decreases and $\frac{x_0\left(t\right)}{x_0\left(0\right)}$ decreases too. When F decreases, then $\alpha /F$ increases and $\frac{x_0\left(t\right)}{x_0\left(0\right)}$ also increases. F increases when $dF/dt>0$. F decreases when $dF/dt<0$. We have the relationship

$$\frac{dF}{dt}=[{\nu }_0(\alpha -t)-1]exp\left({\nu }_{0}t\right)$$

(46)

Above, we assume that $-{\nu }_0\left(t\right)=-({\nu }_0-\frac{1}{\alpha -t})<0$. Then, ${\nu }_0>\frac{1}{\alpha -t}>0$ (as $t<\alpha$). In this case, $exp\left({\nu }_{0}t\right)>0$. As ${\nu }_0(\alpha -t)>1$, $dF/dt>0$ and F will increase. Then, $\alpha /F$ decreases, which means that $\frac{x_0\left(t\right)}{x_0\left(0\right)}$ decreases. Then, $x_0\left(t\right)$ decreases by an increase to t as far as $t<\alpha$.

Let us now consider $x_1\left(t\right)$. We have that ${\nu }_0>0$ and ${\nu }_1>0$ (because of (13)). Let us also assume that the structure of the flows is such one that ${\nu }_1>{\nu }_0$. Let the discussed scenario end at $t=\pi \alpha$, where $\pi <1$. At the end of the scenario, we have

$$\begin{array}{c}\hfill \frac{x_1(t=\alpha \pi )}{x_1\left(0\right)}=\frac{exp[-{\nu }_1\alpha \pi ]}{{(1-\pi )}^2}\{1+f_0\frac{x_0\left(0\right)}{x_1\left(0\right)}\frac{1}{\alpha }\frac{exp\left[\alpha ({\nu }_1-{\nu }_0)\right]}{{[-({\nu }_1-{\nu }_0)]}^2}[\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha ]\left([-({\nu }_1-{\nu }_0)\alpha ]-1\right)-\\ \hfill exp[-({\nu }_1-{\nu }_0)\alpha (1-\pi )]\left([-({\nu }_1-{\nu }_0)\alpha (1-\pi )]-1\right)\left]\right\}\end{array}$$

(47)

We want to study the possibility of the scenario $\frac{x_1(t=\alpha \pi )}{x_1\left(0\right)}<1$. This leads to a requirement for the ratio $\frac{x_0\left(0\right)}{x_1\left(0\right)}$. This requirement is

$$\begin{array}{c}\hfill \frac{x_0\left(0\right)}{x_1\left(0\right)}\left\{-\alpha ({\nu }_1-{\nu }_0)-1+exp\left[\alpha \pi ({\nu }_1-{\nu }_0)\right][\alpha ({\nu }_1-{\nu }_0)-\alpha \pi ({\nu }_1-{\nu }_0)+1]\right\}<\\ \hfill \frac{\alpha {({\nu }_1-{\nu }_0)}^2}{f_0}[{(1-\pi )}^{2}exp\left({\nu }_1\alpha \pi \right)-1].\end{array}$$

(48)

Let now ${\nu }_1\ge {\nu }_0$, ${\nu }_1>\frac{1}{\alpha \pi }ln\left[\frac{1}{{(1-\pi )}^2}\right]$, and $exp\left[\alpha \pi ({\nu }_1-{\nu }_0)\right][\alpha ({\nu }_1-{\nu }_0)-\alpha \pi ({\nu }_1-{\nu }_0)+1]>\alpha ({\nu }_1-{\nu }_0)+1$. Then, the requirement $\frac{x_1(t=\alpha \pi )}{x_1\left(0\right)}<1$ leads to

$$\begin{array}{c}\hfill \frac{x_0\left(0\right)}{x_1\left(0\right)}<\frac{\alpha {({\nu }_1-{\nu }_0)}^2\left[{(1-\pi )}^{2}exp\left({\nu }_1\alpha \pi \right)-1\right]}{f_0\left\{-\alpha ({\nu }_1-{\nu }_0)-1+exp\left[\alpha \pi ({\nu }_1-{\nu }_0)\right][\alpha ({\nu }_1-{\nu }_0)-\alpha \pi ({\nu }_1-{\nu }_0)+1]\right\}}.\end{array}$$

In other words, in the presence of (49), the increase in $x_1$ with an increasing t is not guaranteed. This completes the proof of the theorem. □

Several remarks connected to the above theorem follow:

• Let us now take into account that ${\nu }_0=-({\sigma }_0+{\mu }_0+f_0+{\gamma }_0-{ϵ}_0)$. We remember that $\alpha$ is the parameter which regulates the increase in ${\sigma }_0\left(t\right)$ with the increase in t (see (31)) and the assumption is that $\alpha >0$. We observe that the situation depends very much on the value of ${\nu }_0$ which depends on the flows between the 0-th node and the rest of the studied complex system. For ${\nu }_0>0$, the number of young researchers will continue to decrease for the duration of the measures (even if they return from abroad to the research organization!). In order to reverse this trend, one has to change the structure of the structure of the inflows to node 0 and outflows from node 0 in such a way that ${\nu }_0\left(t\right)$ becomes negative.
• The above analysis shows that the success of the measures for the increase in the number of young researchers may depend not only on the local parameter $\alpha$, but also on the more global parameter ${\nu }_0$, which regulates a number of flows between 0-th node and the rest of the system.
• The question about the number $x_1\left(t\right)$ of experienced researchers is: Can $x_1\left(t\right)$ decrease with an increase in t despite the inflow of experienced researchers from the environment of the network of organizations? The answer is that, in the presence of (49), the increase in the number of experienced researchers is not guaranteed.

The summary of our example is the following. We have a research organization with a simple structure with respect to the experience of its researchers: young researchers (cell 0) and experienced researchers (cell 1). The leadership of the organization decides to improve the conditions for attracting young researchers from abroad (the environment of the network of research organizations of the country). The success of these attempts, however, is not guaranteed. This success may depend on: (i) the structure of the the flows between the research organization and the rest of the complex network of organizations; and (ii) on the initial condition—the ratio between young researchers $x_0\left(0\right)$ and experienced researchers $x_1\left(0\right)$.

Figure 1 shows the influence of the model parameters on the right-hand side of the relationship (49), which is denoted by R. The relationship $R\left({\nu }_0\right)$ is plotted for fixed values of the remaining parameters except for one parameter which has several values corresponding to the different lines of the corresponding figure. The area below the corresponding line is the area where (49) is satisfied. We can study the influence of the movements connected to the young researchers in the studied research organization on the possibility for a decrease in the number of experienced researchers in the organization. Figure 1a shows the influence of the parameter $\alpha$ on $R\left({\nu }_0\right)$. The parameter $\alpha$ regulates the inflow of young researchers and the outflow of experienced researchers from the research organization. The increase in the value of $\alpha$ leads to an increase in the area where (49) is satisfied and this effect is larger for smaller values of ${\nu }_0$. Figure 1b shows the influence of the parameter $f_0$ on the relationship $R\left({\nu }_0\right)$. $f_0$ is the parameter which regulates the movement between the cells containing the young and the experienced researchers in the research organization. Large values of $f_0$ mean that more young researchers move to the category of experienced researchers. We observe that the larger values of the parameter $f_0$ lead to a decrease in the area of validity of (49) and to a decrease the area of validity of the scenario $\frac{x_1(t=\alpha \pi )}{x_1\left(0\right)}<1$. Thus, to some extent, the larger inflow from the group of young researchers to the group of experienced researchers can prevent a decrease in the number of experienced researchers in the research organization.

Figure 1c shows the influence of the parameter ${\nu }_1$ of the relationship $R\left({\nu }_0\right)$. The parameter ${\nu }_1$ regulates the motion of experienced researchers to and out of the studied research organization. The increase in the value of ${\nu }_1$ leads to an increase in the area of validity of (49). This means that the possibility for larger movements of experienced researchers in the network can lead to a decrease in their number in the studied research organization. Finally, Figure 1d shows the influence of the parameter $\pi$ on the relationship $R\left({\nu }_0\right)$. $\pi$ is a parameter which regulates the amount of time for running the scenario. We observe that the influence of this parameter is relatively weaker with respect to the influence of the other parameters. The increase in value of $\pi$ leads to an increase in the validity of the relationship (49).

Figure 2 shows the influence of the parameter $\alpha$ on the relationship $R\left(f_0\right)$ for fixed values of the other parameters in (49). The parameter $\alpha$ regulates the inflow of young researchers and the outflow of experienced researchers from the research organization. We observe that larger values of $\alpha$ increase the validity of (49), especially for the case of smaller values of the parameter $f_0$, which regulates the process of converting young researchers to experienced ones. Then, the small number of young researchers who move to the class of experienced researchers can lead to a decrease in the number of experienced researchers, despite the possibility of a larger inflow of young researchers in the research organization (which can be compensated by a larger outflow of experienced researchers from the research organization).

4. Concluding Remarks

In this article, we present a model of motion of substance in a channel of a network in the presence of exchange of substance between the channel, the rest of the network, and the environment of the network. We obtained a solution of the model for the case of model parameters which depend on time.

The model can have numerous applications, e.g, to the situations discussed in [43,44,45,46,47,48,49]. Above in the text, we use the model as the basis of a theory for the motion of researchers between research organizations. For simplicity, we consider an example where the calculations can be made analytically. The obtained analytical relationships help us better understand the situation connected to the motion of young and experienced researcher in a research organization. The results show that the number of young and experienced researchers in an organization has an evolution which depends on the flows of researchers among the organization and the rest of the considered network of organizations. In addition, the number of young and experienced researchers in the organization depends on the initial number of young and experienced researchers. A negative scenario of evolution is possible: the number of young researchers could decrease despite the inflow of such researchers from abroad. An even more negative scenario is possible: the number of young researchers and the number of experienced researchers could decrease.

The obtained analytical relationships allow us to study the influence of the values of the parameters of the model on the area of validity of the aforementioned scenarios in the parameter spaces. The example shows that the increase or decrease in the number of young and experienced researchers may depend on the ratio between the initial numbers of young and experienced researchers in the organization as well as on the exchange between young and experienced researchers between the research organization and the rest of the research environment. The possibility for the movement of experienced researchers around the network of research organizations can lead to large negative consequences for the number of these researchers in a research organization, even in the case of a relatively large inflow of young researchers into research organization.

We note that the model presented above in the text allows the consideration of various empirical cases of movements of researchers. This can be achieved by calibrating the model parameters. In the general case, the obtained solution will be complicated, and it must be studied numerically. In this way, the entire richness of the possible scenarios can be explored. Such scenarios will be discussed in our future research.