Further Results on the Mathematical Theory of Motion of Researchers Between Research Organizations

Abstract

Recently, Vitanov and Dimitrova presented a mathematical theory of motion of researchers between research organizations. They obtained analytical results based on exact solutions for a specific case of a model system of ordinary differential equations for studying a system of research organizations. In this article, we investigate the system of model equations of Vitanov and Dimitrova analytically and numerically, obtaining several new results. We provide sufficient conditions for the concavity of the solutions of the system and make a comparison with an exact solution. We numerically examine the effect of several parameters on the concavity. We further inspect the influence of the parameters on whether the solution has overall increased or decreased compared with the initial condition.

1. Introduction

The movement of researchers between research organizations is a crucial aspect of the dynamics of science, influencing collaboration networks, knowledge transfer, and innovation [1,2,3]. Understanding the patterns and drivers of this mobility is essential for policymakers and research institutions aiming to foster a vibrant and productive scientific ecosystem [4,5,6]. This phenomenon is a manifestation of broader individual human mobility patterns, which have been extensively studied using large-scale datasets [7]. The structure of scientific communities and the patterns of collaboration are also critical in this context, often visualized and analyzed through co-authorship networks [8]. The intricate nature of researcher mobility, as with many complex systems, necessitates the use of models and objects to simplify analysis and design [9], drawing upon the foundational theories and mathematical frameworks for understanding complexity [10]. Furthermore, the application of adaptive network models enables the exploration of the dynamic interplay between the structure of research collaborations and the movement of researchers within this network [11]. The theoretical foundations for understanding such intricate systems often draw upon the principles of statistical mechanics applied to complex networks [12] and consider the collective dynamics that emerge within them [13]. Understanding connectivity and distance effects is also vital when analyzing transport phenomena in such networks [14].

Mathematical models have proven to be invaluable tools for studying complex phenomena across various domains, including the evolution of research organizations [6,15,16,17,18], network theory [19,20,21,22,23,24], and the motion of substances in networks [25,26]. These models range from simple representations to intricate systems of equations capable of capturing emergent behaviors [27,28]. Network flow theory provides a fundamental framework for analyzing these movements, with applications spanning diverse fields [29]. In the context of research, mathematical approaches have been used to analyze publication productivity [30,31] and the attraction of funding [32]. Furthermore, the dynamics of flows in networks have also been investigated through studies of movement and distribution processes [33,34,35]. The analysis of these networked activities can also reveal underlying patterns and structures, such as those described by rank-size laws [36,37]. Moreover, the study of evolving networks enables the understanding of how these systems change over time [38].

The modeling of human migration, which shares conceptual similarities with the movement of researchers, has also benefited from mathematical frameworks [39,40]. Various approaches, including box models [41], discrete-time models [42], and continuous-time models [43], have been employed to describe migration patterns and flows, drawing upon established theories of international migration that consider various socio-economic and network-based factors [44]. The social structure arising from these networks also plays a significant role in understanding mobility [45]. These studies often consider networks of interconnected locations, where individuals move based on a variety of factors. Recent work has further explored statistical distributions associated with such movements [46], the growth of networks influenced by migration [47], and the dynamics of flow in network channels with specific structures [48,49]. Nonlinear analysis is also crucial to understanding complex flows within these networks [50].

Building upon this rich history of mathematical modeling in complex systems and the specific context of researcher mobility, Vitanov and Dimitrova [51] recently introduced a mathematical theory to describe the motion of researchers between research organizations using a specific case of a system of ordinary differential equations. This foundational work provided analytical solutions for a specific scenario, laying the groundwork for further investigations into the intricate dynamics of researcher movement.

In this paper, the authors aim to expand on the work performed in [51] in the following ways: generalize the model presented in [51], in order for the model to be able to simulate more complex phenomena, and explore some properties of a particular case of this more general model, namely, monotonicity, concavity, and overall change in the system over a fixed interval of time. Understanding these properties will bring more clarity when using the model to study any applicable circumstance and will bring further clarity to the significance of the model.

This paper is structured in the following way: In Section 2, an overview of the model proposed in [51] is presented. A generalization of this model is examined, and general notation is established. In Section 3, an analytical analysis is conducted, focusing first on the case of constant coefficients and then on the concavity of a particular analytical solution. In Section 4, the numerical procedure used is described [52,53] and AN-stability [54] analysis is carried out. The study of concavity is continued numerically. Further, an investigation into the effect of the parameters on the outcome is performed, focusing on the change in researchers at the beginning of the investigated period and the end (i.e., researcher numbers have increased or decreased). This article concludes with finishing remarks and potential future developments in Section 5.

2. Mathematical Formulation of the Problem

In a recent study [51], a model describing the flow of a substance in a network of nodes (or cells) was introduced. The model’s scope encompasses the interactions between such nodes and is based on a system of ordinary differential equations. One of the first such proposed models can be found in [30], where the model is used to measure publication productivity in the scientific community. The model was generalized in [41] and was employed to describe migration. Further modifications to the model can be found in [43,48,55,56]. We will not delve deeper into the specific generalizations, due to the introduction of the model in [51], which is similar. Analogous discrete models can be found in [42,46,47,57].

Let us take a closer look at the model in [51] and the interactions of its elements. The nodes can form chains and exchange substance between each other via internal links. These chains can have connections with other chains, all of which are a part of a greater structure called a network. Further, this network is embedded in an environment. The following assumptions are made:

• Time is continuous.
• The substance in a given node moves only to adjacent nodes, that is, the previous or the next node.
• The movement of the substance can be both from the node to other chains in the network and the environment, and vice versa.

A single chain with $N+1$ elements is considered in detail. This is performed by the following functions:

• $i_k^e\left(t\right)$ and $o_k^e\left(t\right)$ represent the inflow and outflow of the substance between the environment and the k-th cell;
• $i_k^c\left(t\right)$ and $o_k^c\left(t\right)$ are the inflow and outflow from the k-th to the $(k+1)$-st cell;
• $i_k^n\left(t\right)$ and $o_k^n\left(t\right)$ are the amounts of the inflow and outflow of the substance exchanged between the k-th cell and the network.

These interactions are demonstrated in Figure 1. Let us denote the amount of the substance in cell k by $x_k$. Then this formulation gives rise to the system

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_0}{dt}}& =i_0^e\left(t\right)-o_0^e\left(t\right)+i_0^c\left(t\right)-o_0^c\left(t\right)+i_0^n\left(t\right)-o_0^n\left(t\right)\hfill \\ \hfill {\displaystyle \frac{d{x}_k}{dt}}& =i_k^e\left(t\right)-o_k^e\left(t\right)-i_{k-1}^c\left(t\right)+o_{k-1}^c\left(t\right)+i_k^c\left(t\right)-o_k^c\left(t\right)+i_k^n\left(t\right)-o_k^n\left(t\right),k=1,\dots ,N-1\hfill \\ \hfill {\displaystyle \frac{d{x}_N}{dt}}& =i_N^e\left(t\right)-o_N^e\left(t\right)-i_{N-1}^c\left(t\right)+o_{N-1}^c\left(t\right)+i_N^n\left(t\right)-o_N^n\left(t\right).\hfill \end{array}$$

(1)

A further restriction enforced on the model in [51] is that the inflows and outflows are proportional to their respective $x_k$ and that the substance does not travel backwards along the chain, that is, inflow from the k-th cell to the $(k-1)$-st one is not allowed, or $i_{k-1}^c\left(t\right)\equiv 0$, $k=1,\dots ,N$ (Figure 2). After some simplification, the model can be represented in the following way:

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_0}{dt}}& =-{\nu }_0\left(t\right)x_0\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{x}_k}{dt}}& =-{\nu }_k\left(t\right)x_k\left(t\right)+f_{k-1}\left(t\right)x_{k-1}\left(t\right),k=1,\dots N,\hfill \end{array}$$

(2)

with ${\nu }_k, f_k\left(t\right):{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, where ${\mathbb{R}}_{+}=\{x\in \mathbb{R}:x\ge 0\}$. In contrast with that, in this paper we will weaken some of these restrictions.

Let us consider a general formulation for system (1), i.e., a system of nonlinear ordinary differential equations. In vector form, it takes the form

$${\displaystyle \frac{d\overrightarrow{x}}{dt}}=\overrightarrow{F}(t,\overrightarrow{x}\left(t\right),\overrightarrow{p}),$$

(3)

where $\overrightarrow{x}={(x_0,\dots ,x_k,\dots ,x_N)}^T,\overrightarrow{F}={(F_0,\dots ,F_k,\dots ,F_N)}^T,t\in [0,T]$ is a time variable, and $\overrightarrow{p}=(p_1, p_2,\dots ,p_M)$ is a vector of M different independent parameters describing the properties of the system under consideration.

If we consider a special case of this system, where the relationship is linear with respect to $\overrightarrow{x}\left(t\right)$, we would arrive at

$$\overrightarrow{F}(t,\overrightarrow{x}\left(t\right),\overrightarrow{p})=A(t,\overrightarrow{p})\overrightarrow{x}\left(t\right)+b(t,\overrightarrow{p}),$$

(4)

where $A(t,\overrightarrow{p}):\mathbb{R}\times {\mathbb{R}}^M\to {\mathbb{M}}_{(N+1)\times (N+1)}$ and $b(t,\overrightarrow{p}):\mathbb{R}\times {\mathbb{R}}^M\to {\mathbb{R}}^{N+1}$.

The matrix function $A(t,\overrightarrow{p})$ defines the interconnections between all the substances of the cells. Let us point out that this form generalizes (2) in two ways. The first is the assumption that the substance moves only via adjacent cells. In this formulation, movement of the substance between any two cells is allowed. If we enforce the original assumption, however, the matrix function A takes a specific form—a tridiagonal matrix:

$$A(t,\overrightarrow{p})=\left(\begin{array}{ccc}\begin{array}{ccc}{a}_{00}& a_{01}& 0\\ a_{10}& a_{11}& a_{12}\\ 0& a_{21}& a_{22}\\ ⋮& ⋮& ⋮\\ 0& 0& 0\\ 0& 0& 0\end{array}& \begin{array}{c}\dots \\ \dots \\ \dots \\ \ddots \\ \dots \\ \dots \end{array}& \begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ ⋮& ⋮& ⋮\\ a_{N-1,N-2}& a_{N-1,N-1}& a_{N-1,N}\\ 0& a_{N,N-1}& a_{N,N}\end{array}\end{array}\right)$$

(5)

where $a_{ij}:\mathbb{R}\times {\mathbb{R}}^M\to \mathbb{R}$, $i,j=0,\dots ,N$.

Reintroducing the restriction on the inflow from the next cell leads to $a_{k,k+1}=0$, $k=0,\dots ,N-1$. This condition could have a natural explanation and even be required depending on the context. For example, in [51], the authors examine this model in the context of the number of researchers in a given institution. The simplest possible separation of scientists is made—$x_0\left(t\right)$ for the young and $x_1\left(t\right)$ for the experienced. Here it is evident that $a_{k,k+1}=0$—for a researcher to have lost their experience seems contradictory in the general case. Even though this is feasible on rare and very specific occasions, such occurrences would be so rare so as not to be relevant to the model at hand. In this paper, we will also not take into consideration such events.

The second generalization has to do with the sign of the functions in front of $\overrightarrow{x}$. In order to explain its significance, we will continue with the tridiagonal matrix along with $a_{k,k+1}=0, k=0,\dots ,N-1$. An important restriction that is often used in [51] is that the substance’s rate of change in cell k is never self-reinforcing, i.e., $a_{kk}<0$. This leads to the introduction of a negative sign and the condition ${\nu }_k\left(t\right)>0$. In this paper, we will approach this system without such an assumption. In order to avoid confusion, we will express (2) as

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_0}{dt}}& ={\phi }_0\left(t\right)x_0\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{x}_k}{dt}}& ={\phi }_k\left(t\right)x_k\left(t\right)+f_{k-1}\left(t\right)x_{k-1}\left(t\right),k=1,\dots ,N,\hfill \end{array}$$

(6)

where we do not explicitly write the influence of the parameters $\overrightarrow{p}$, but they are still a part of the system. For simplicity, the explicit dependence of the parameter set will be omitted henceforth. The transition from (6) to (2) is clearly ${\phi }_k\left(t\right)=-{\nu }_k\left(t\right),k=0,\dots ,N$.

As in [51], we will mainly focus our attention on the case when $N=1$, i.e.,

$$A(t,\overrightarrow{p})=\left(\begin{array}{cc}{a}_{00}(t,\overrightarrow{p})& 0\\ a_{10}(t,\overrightarrow{p})& a_{11}(t,\overrightarrow{p})\end{array}\right),$$

or equivalently,

$$\begin{array}{cc}\hfill {\displaystyle \frac{d{x}_0}{dt}}& ={\phi }_0\left(t\right)x_0\left(t\right),\hfill \\ \hfill {\displaystyle \frac{d{x}_1}{dt}}& ={\phi }_1\left(t\right)x_1\left(t\right)+f_0\left(t\right)x_0\left(t\right).\hfill \end{array}$$

(7)

More specifically, we will consider the case when

$$\begin{array}{ccc}{\displaystyle {\phi }_0\left(t\right)=-{\nu }_0+\frac{1}{\alpha -t},}& {\displaystyle {\phi }_1\left(t\right)=-{\nu }_1-\frac{\kappa }{\alpha -t},}& {\displaystyle f_0\left(t\right)=f_0,}\end{array}$$

(8)

where $t\in [0;T]$, ${\nu }_0,{\nu }_1,\kappa \in \mathbb{R},$ $f_0\in [0;1]$, $\alpha >0,0<T<\alpha$. Thus, $\overrightarrow{p}=({\nu }_0,{\nu }_1,\alpha ,\kappa ,f_0)$. In [51], conclusions subject to all of the aforementioned restrictions for a particular analytical solution with this choice of functions are given.

In our current work, we aim to examine these relationships and formulate results that explain the behavior of the functions based exclusively on the parameters. As opposed to [51], in this paper we will at first shift our attention from investigating the increase and decrease of researchers based on the initial and final values to the monotonicity of the functions in $[0,T]$ (Section 3.2). We will conduct an analytical investigation of the system and specifically the convexity of the unknown functions (Section 3.3). After that we will proceed with a numerical investigation, seeking to find regions of monotonicity of $\overrightarrow{x}\left(t\right)$. This approach is beneficial in that it has a simple interpretation—the first and second derivatives representing the rate of change and the acceleration of the motion of the researchers, respectively. Since a point of interest to educational institutions would still be to find out what the outcome of such a process would be, that is, if the number of researchers of both types have increased or decreased, we will return to the interpretation in [51] and conclude with narrowing down the regions of increase and decrease in $x_k\left(t\right),k=0,1$.

3. Analytical Results

The form of the general solution of system (6), or at least the procedure of obtaining it, is well known and can be found in many textbooks on differential equations. Its form, as presented in [51], is

$$\begin{array}{cc}\hfill x_0\left(t\right)& =x_0\left(0\right)exp\left[\int {\phi }_0\left(t\right)dt\right],\hfill \\ \hfill x_k\left(t\right)& =exp\left[\int {\phi }_k\left(t\right)dt\right]\left\{{\Theta }_k+\int \left[f_{k-1}\left(t\right)x_{k-1}\left(t\right)exp\left(-\int {\phi }_k\left(t\right)dt\right)\right]dt\right\}\hfill \end{array}$$

(9)

for $k=1,\dots ,N$, where ${\Theta }_k$ are constants of integration.

3.1. Constant Coefficients

Let us consider system (6) in an important subcase—the case of constant coefficients. In [51], this case, i.e., when ${\phi }_k\left(t\right)=c_k$, $f_j\left(t\right)=d_j$, $c_k,d_j\in \mathbb{R},k=0,\dots ,N$, $j=0,\dots ,N-1$, leads to the general solution

$$\begin{array}{cc}\hfill x_0\left(t\right)& =x_0\left(0\right)e^{c_{0}t}\hfill \\ \hfill x_k\left(t\right)& ={\Theta }_k{e}^{c_{k}t}+d_{k-1}{e}^{c_{k}t}\int x_{k-1}\left(t\right)e^{-c_{k}t}dt,\hfill \end{array}$$

with ${\Theta }_k\in \mathbb{R},k=1,\dots ,N$ being constants of integration. This solution is further analyzed under the restrictions $c_k\le 0$, $d_j\ge 0$, and $x_k\left(0\right)\ge 0$. The following cases are considered in [51]: $c_0<0$, in which $x_0\to 0$, and $c_0=0$, i.e., $x_0\left(t\right)\equiv x_0\left(0\right)$. In this second case it is concluded that every function tends to a stationary solution $x_k^{\ast }=-{\displaystyle \frac{d_{k-1}}{c_k}}{x}_{k-1}^{\ast }$ with $x_0^{\ast }=x_0\left(0\right)$, given that $c_k<0,k=1,\dots ,N$. Let us point out that if $c_0=c_1=0$, then $x_1\left(t\right)\to +\infty$. Therefore, the condition that $c_k<0,k=1,\dots ,N$ is essential to the appearance of stationary solutions.

Using induction, one could easily conclude that in this case, the stationary solutions take the form

$$x_k^{\ast }={(-1)}^k\left(\prod _{j=0}^{k-1}{\displaystyle \frac{d_j}{c_{j+1}}}\right)x_0^{\ast }.$$

Since $c_{j+1}$ are all negative and $d_j$ are nonnegative, for every stationary point, it holds that $x_k^{\ast }\ge 0$.

By relaxing the restrictions in the model, we can potentially get substantially different behavior. In order to demonstrate this, we will consider the simplest system of this kind—$k=0,1$. The solutions in this case are

$$\begin{array}{cc}\hfill x_0\left(t\right)& =x_0\left(0\right)e^{c_{0}t}\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill x_1\left(t\right)& =\left\{\begin{array}{cc}\left[x_1\left(0\right)-{\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}\right]e^{c_{1}t}+{\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}{e}^{c_{0}t},\hfill & c_0\ne c_1\hfill \\ \left[x_1\left(0\right)+d_0{x}_0\left(0\right)t\right]e^{c_{1}t},\hfill & c_0=c_1.\hfill \end{array}\right.\hfill \end{array}$$

(11)

Let us consider the special case of the coefficients $c_k$ being positive in contrast with the one in [51] ($c_k\le 0,k=0,1$). Other combinations of the coefficients are also applicable. The aim of the current paper is to present just one of the possible combinations and to demonstrate that the limits of the functions depend on the parameters of the problem. These parameters, in turn, depend on the policies of the institution in the considered case of the model of researchers’ motion.

To present a broader range of possible applications of the theorem, the mathematically feasible case of negative $d_0$ is considered. However, for the specific model examined in this article, the restriction $d_0\ge 0$ must be taken into account. This is because this coefficient describes the outflow of the cell, thus its sign has already been incorporated into the formulation of the mathematical model.

In the considered case of positive $c_k$, we always achieve $|x_k\left(t\right)|\to +\infty$ as $t\to \infty$ for $k=0,1$. This is formulated in the following.

Theorem 1. 

Let $c_0,c_1,x_0\left(0\right),x_1\left(0\right)>0$ in (10)–(11). Then the following apply:

(a) 

$x_0\left(t\right)\to +\infty$ as $t\to \infty$;

(b) 

$x_1\left(t\right)\to +\infty$ if any of the following holds:

• $d_0\ge 0$;
• $d_0<0$, $c_1>c_0$ and $x_1\left(0\right)\ge {\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}$.

(c) 

$x_1\left(t\right)\to -\infty$ if $d_0<0$ and either $c_0\ge c_1$ or $c_1>c_0$, and $x_1\left(0\right)<{\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}$.

Proof. 

In what follows, the expression for $x_1\left(t\right)$ when $c_0\ne c_1$ is utilized multiple times. For simplicity, we will use

$$x_1\left(t\right)=B_1{e}^{c_{1}t}+B_2{e}^{c_{0}t},$$

(12)

where $B_1=x_1\left(0\right)-{\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}$ and $B_2={\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}$. Taking into account that $c_0,c_1>0$, both terms diverge to infinity.

(a) Since $c_0>0$ and $x\left(0\right)>0$, it is clear from (10) that $x_0\left(t\right)\to +\infty$ as $t\to \infty$.

(b) Let $d_0\ge 0$. If $c_0=c_1$, then

$$\underset{t\to \infty }{lim}{x}_1\left(t\right)=\underset{t\to \infty }{lim}\left[x_1\left(0\right)+d_0{x}_0\left(0\right)t\right]e^{c_{1}t}=+\infty .$$

If $c_0>c_1$, then the dominant term in (12) is $B_2{e}^{c_{0}t}$. Due to $B_2>0$, it quickly follows that $x_1\left(t\right)\to \infty$. If $c_1>c_0$, then the determining factor becomes the sign of $B_1$. Since $B_1>0$, it follows that $x_1\left(t\right)\to \infty$.

Now let $d_0<0$, $c_1>c_0$ and $x_1\left(0\right)\ge {\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}$. Then both $B_i>0,i=1,2$. Thus, it is clear that $x_1\left(t\right)\to +\infty$ as $t\to \infty$.

(c) Let $d_0<0$. The case $c_0=c_1$ is easy to verify. Let $c_0>c_1$. Then $B_1>0$, and $B_2<0$, which leads to $x_1\left(t\right)\to -\infty$ as $t\to \infty$. When $c_1>c_0$ and $x_1\left(0\right)<{\displaystyle \frac{d_0{x}_0\left(0\right)}{c_0-c_1}}$, we can reach the same conclusion in a similar manner. □

This result shows that the constant coefficient subcase of (6) with two equations leads to unreasonable models in most cases. There could be some niche cases where $x_k$ would have no upper bound in practice, for example, if the system measured something abstract moving through the cells like knowledge, satisfaction, utility, or money. In some of these cases, even negative values could be considered, for instance, in the case of money, where all of the resources are flowing from $x_1$ to $x_0$. However, there are many cases when such solutions would be unacceptable. The subject of interest of this paper, that is, the motion of researchers, is one such case. The obvious upper bound on researchers (the population of the planet) and the incompatible-with-reality result of a negative number of researchers leads to the conclusion that such parameters would be best avoided, since they do not match the model situation.

3.2. Monotonicity

A point of interest to the scientific institution would be the efficiency of the applied policies. This efficiency could be measured by the change in researcher numbers in the institution or by the behavior of their flow through the institution. In [51], the first approach is examined, and a conclusion about the number of inexperienced and experienced researchers for a particular case of (7) is reached.

Theorem 2 

([51]). Let us consider model (7) with functions and parameters of the form in (8). Let ${\phi }_0\left(t\right)<0$, ${\phi }_1\left(t\right)<0$ and the solution (9) be taken with $\kappa =-2$. Then

(a) 

$x_0\left(t\right)$ decreases with the increase in t as far as $t<\alpha$.

(b) 

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

By this result, we can conclude that for certain values of the parameters, we can expect a decrease in $x_0$, but the outcome for $x_1$ is not guaranteed to be an increase. We can also observe that the statement for $x_0$ is qualitative and shows that $x_0$ is monotone and decreasing. In the course of the proof of Theorem 2, the result about $x_1$ is reached not through an analysis of the derivative but through an analysis of the initial condition and the endpoint of the model.

In this paper, we would like to focus more on the qualitative aspects of the model, examining the monotonicity and concavity of the solutions instead of the outcome, much akin to Theorem 2(a). Of course, the outcome of the applied strategy by the educational institution is also important, and we will return to it in Section 4.3.

In order to produce the results on concavity, we will briefly make some comments on the monotonicity of the solutions. As far as $x_0$ is concerned, its monotonicity is determined purely by the sign of ${\phi }_0\left(t\right)$ (since $x_0>0$ by assumption). If ${\phi }_0\left(t\right)$ has a constant sign in the interval $t\in [0,T]$, then $x_0$ will be monotone—increasing for ${\phi }_0\left(t\right)>0$ and decreasing for ${\phi }_0\left(t\right)<0$ (as is the case in Theorem 2(a)). If ${\phi }_0\left(t\right)=0$ for $t\in [0,T]$, then we reach a constant solution.

When considering $x_1$, it is not as simple to determine its monotonicity. This is because $x_1^{\prime }\left(t\right)={\phi }_1\left(t\right)x_1\left(t\right)+f_0\left(t\right)x_0\left(t\right)$ is more complex than $x_0^{\prime }$. From $x_0$ and $x_1$ being positive by assumption, we can see that if ${\phi }_1\left(t\right)$ and $f_0\left(t\right)$ are positive for $t\in [0,T]$, then $x_1$ is increasing, whereas if ${\phi }_1\left(t\right)$ and $f_0\left(t\right)$ are both negative, then $x_1$ is decreasing. When the signs are different, we cannot make a claim about monotonicity. In the considered model (functions and parameters (8)), it is the case that $f_0\left(t\right)>0$. Thus, if ${\phi }_1\left(t\right)$ is positive, we can conclude that there will be an increase in $x_1$. However, as is the case in Theorem 2, when ${\phi }_1\left(t\right)<0$, we cannot reach a definitive conclusion. The case when $x_1^{\prime }=0$ leads to a constant solution but such a case would be very unlikely to be encountered if ${\phi }_1\left(t\right)\not\equiv 0$ and $f_0\left(t\right)\not\equiv 0$, since that would require a perfect balance of the inflow and outflow of experienced researchers in the academic institution.

We would like to point out that the same arguments could be applied to all t in a subinterval $[a,b]\subset [0,T]$ instead of $t\in [0,T]$ and analogous conclusions would be reached. The information gained from the first derivative will be of paramount importance for the following results. Since the functions in (8) change their signs in the general case, we will not be able to apply the above conclusions in a straightforward manner. Instead, we will opt for calculating the derivative either analytically when a solution is present (Section 3.3) or plugging a numerical approximation of $x_0$ and $x_1$ into (7) (Section 4).

3.3. Concavity Analysis

When considering the flow of the substance, one would be interested not only in the speed but also in its acceleration. If accumulation is the aim, then knowledge of the substance increasing would not be the optimal approach. The substance could be increasing and yet slowing down its rate of change. In this scenario, we would like to be able to manipulate the process in such a way so as to keep up the speed of amassing or even increase it. Conversely, if there were some upper limit that cannot be breached, e.g., the capacity of an academic institution for scientists, then a strategy that forces the speed to slow down or even to switch the flow to decreasing for a small period of time would be beneficial.

These considerations could very well be addressed by investigating the concavity of the model equations. By observing system (7), we can make conclusions for $x_0$ and $x_1$ separately and then unify the results. The equation for $x_0$ is not paired with $x_1$; hence, its analysis is simpler. Indeed, it is easy to verify that $x_0^{″}\left(t\right)=({\phi }_0^{\prime }\left(t\right)+{\phi }_0^2\left(t\right))x_0\left(t\right)$. Therefore (keeping in mind that $x_0\left(t\right)\ge 0$), the concavity of $x_0\left(t\right)$ is determined by ${\phi }_0\left(t\right)$. More concretely, $x_0\left(t\right)$ is strictly convex if and only if $({\phi }_0^{\prime }\left(t\right)+{\phi }_0^2\left(t\right))>0$, whereas strict concavity is observed when $({\phi }_0^{\prime }\left(t\right)+{\phi }_0^2\left(t\right))<0$. The same analysis for $x_1\left(t\right)$ is not so trivial and is presented in the next results. For brevity, we will leave out writing the explicit dependence of t when referring to ${\phi }_0\left(t\right),{\phi }_1\left(t\right)$, and $f_0\left(t\right)$, keeping in mind that such dependence remains.

Theorem 3. 

Let $x_1^{\prime }\left(t\right)>0$, ${\phi }_1\left(t\right)<0$, and $f_0\left(t\right)>0$ for a fixed $t\in [0,T]$:

(a) 

If ${\displaystyle \frac{f_0^{\prime }}{f_0}}\ge -({\phi }_0+{\phi }_1)$ and ${\phi }_1^{\prime }-{\phi }_0{\phi }_1-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1\ge 0$, then $x_1$ is convex.

(b) 

If ${\displaystyle \frac{f_0^{\prime }}{f_0}}\le -({\phi }_0+{\phi }_1)$ and ${\phi }_1^{\prime }-{\phi }_0{\phi }_1-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1\le 0$, then $x_1$ is concave.

Proof. 

Let us note that the sign of $x_1^{″}$ is the same as the sign of ${\displaystyle \frac{x_1^{″}}{x_1}}$, since $x_1>0$.

(a) From $x_1^{\prime }>0$, we get that ${\displaystyle \frac{x_0}{x_1}}>-{\displaystyle \frac{{\phi }_1}{f_0}}$. Then

$$\begin{array}{cc}\hfill x_1^{″}& ={\phi }_1^{\prime }{x}_1+{\phi }_1{x}_1^{\prime }+f_0^{\prime }{x}_0+f_0{x}_0^{\prime }\hfill \\ \hfill & ={\phi }_1^{\prime }{x}_1+{\phi }_1^2{x}_1+{\phi }_1{f}_0{x}_0+f_0^{\prime }{x}_0+f_0{\phi }_0{x}_0\hfill \\ \hfill & =({\phi }_1^2+{\phi }_1^{\prime })x_1+({\phi }_1{f}_0+f_0^{\prime }+{\phi }_0{f}_0)x_0.\hfill \end{array}$$

(13)

From ${\displaystyle \frac{f_0^{\prime }}{f_0}}\ge -({\phi }_0+{\phi }_1)$, it follows that ${\phi }_1{f}_0+f_0^{\prime }+{\phi }_0{f}_0\ge 0$ and

$$\begin{array}{cc}\hfill {\displaystyle \frac{x_1^{″}}{x_1}}& =({\phi }_1^2+{\phi }_1^{\prime })+({\phi }_1{f}_0+f_0^{\prime }+{\phi }_0{f}_0){\displaystyle \frac{x_0}{x_1}}\ge {\phi }_1^2+{\phi }_1^{\prime }-({\phi }_1{f}_0+f_0^{\prime }+{\phi }_0{f}_0){\displaystyle \frac{{\phi }_1}{f_0}}\hfill \\ \hfill & ={\phi }_1^2+{\phi }_1^{\prime }-{\phi }_1^2-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1-{\phi }_1{\phi }_0={\phi }_1^{\prime }-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1-{\phi }_1{\phi }_0\ge 0\hfill \end{array}$$

Therefore, $x_1$ is convex.

(b) The conclusions follow from the same arguments, only changing the direction of the inequalities. □

Even though we are assuming that $x_1^{\prime }\left(t\right)>0$ and we do not know the function a priori, in practice, data points could be collected, and the derivative could be approximated either by finite differences or by using the explicit model equation from (7) with the aid of a numerical approximation of $x_1\left(t\right)$, as mentioned in Section 3.2. If there are enough data points, numerical approximation might not be required. Thus, the hypothesis of Theorem 3 could be verified. If checking the condition ${\phi }_1^{\prime }-{\phi }_0{\phi }_1-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1\ge 0$ proves to be too difficult, the following corollary offers conditions that might be easier to work with.

Corollary 1. 

Let $x_1^{\prime }\left(t\right)>0$, ${\phi }_1\left(t\right)\le 0$, and $f_0\left(t\right)>0$ for a fixed $t\in [0,T]$:

(a) 

If ${\displaystyle \frac{f_0^{\prime }}{f_0}}\ge -({\phi }_0+{\phi }_1)$ and ${\phi }_1^2\ge -{\phi }_1^{\prime }$, then $x_1$ is convex.

(b) 

If ${\displaystyle \frac{f_0^{\prime }}{f_0}}\le -({\phi }_0+{\phi }_1)$ and ${\phi }_1^2\le -{\phi }_1^{\prime }$, then $x_1$ is concave.

Proof. 

(a) From ${\phi }_1\le 0$ and ${\displaystyle \frac{f_0^{\prime }}{f_0}}\ge -({\phi }_0+{\phi }_1)$, it follows that

$$\begin{array}{cc}\hfill {\phi }_1^{\prime }-{\phi }_0{\phi }_1-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1& \ge {\phi }_1^{\prime }-{\phi }_0{\phi }_1+({\phi }_0+{\phi }_1){\phi }_1\hfill \\ \hfill & \ge {\phi }_1^{\prime }+{\phi }^2\ge 0.\hfill \end{array}$$

Therefore, by Theorem 3, we can conclude that $x_1$ is convex.

(b) The proof is analogous to the one in (a). □

Let us point out that due to introducing another inequality in order to prove the result, Corollary 1 is a weaker claim and would give positive results in fewer cases. Even so, the ease of computation could be useful in certain circumstances.

Theorem 4. 

Let $x_1^{\prime }\left(t\right)<0$ and $f_0\left(t\right)>0$ for a fixed $t\in [0,T]$. Then ${\phi }_1<0$, and

(a) 

If ${\displaystyle \frac{f_0^{\prime }}{f_0}}\le -({\phi }_0+{\phi }_1)$ and ${\phi }_1^{\prime }-{\phi }_0{\phi }_1-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1\ge 0$, then $x_1$ is convex.

(b) 

If ${\displaystyle \frac{f_0^{\prime }}{f_0}}\ge -({\phi }_0+{\phi }_1)$ and ${\phi }_1^{\prime }-{\phi }_0{\phi }_1-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1\le 0$, then $x_1$ is concave.

Proof. 

From $x_1^{\prime }<0$, we get that ${\displaystyle \frac{x_0}{x_1}}<-{\displaystyle \frac{{\phi }_1}{f_0}}$. However, ${\displaystyle \frac{x_0}{x_1}}>0$. Then ${\phi }_1<0$.

(a) The following is very similar to the proof of Theorem 3:

$$\begin{array}{cc}\hfill {\displaystyle \frac{x_1^{″}}{x_1}}& =({\phi }_1^2+{\phi }_1^{\prime })+({\phi }_1{f}_0+f_0^{\prime }+{\phi }_0{f}_0){\displaystyle \frac{x_0}{x_1}}\ge {\phi }_1^2+{\phi }_1^{\prime }-({\phi }_1{f}_0+f_0^{\prime }+{\phi }_0{f}_0){\displaystyle \frac{{\phi }_1}{f_0}}\hfill \\ \hfill & ={\phi }_1^2+{\phi }_1^{\prime }-{\phi }_1^2-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1-{\phi }_1{\phi }_0={\phi }_1^{\prime }-{\displaystyle \frac{f_0^{\prime }}{f_0}}{\phi }_1-{\phi }_1{\phi }_0\ge 0\hfill \end{array}$$

Therefore, $x_1$ is convex.

(b) The conclusions follow from the same arguments, only changing the direction of the inequalities. □

An important caveat is that Theorems 3 and 4 hold point-wise. If we desire to check the concavity of $x_1\left(t\right)$ for an interval of time $[a,b]$, we would need to verify the inequalities hold for every $t\in [a,b]$. A few simple examples are shown in Figure 3 and Figure 4.

In order to demonstrate Theorems 3 and 4, we will use a particular solution of the form in (9), where $N=1$, with ${\phi }_0,{\phi }_1$, and $f_0$ of the form in (8), with $\kappa =1$. Since the theorems describe the concavity of $x_1$, we will not consider $x_0$. The general solution in this case, found in [51], after some simplification, is

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

(14)

where $Ei\left(x\right)=PV{\int }_{-\infty }^x{\displaystyle \frac{e^t}{t}}dt$ is the exponential integral understood in terms of the Cauchy principal value. In [51] the exponential integral is not used explicitly, instead being written down via the series [58]

$$Ei\left(x\right)=\gamma +ln\left|x\right|+\sum _{n=1}^{\infty }{\displaystyle \frac{x^n}{n·n!}},$$

where $x\ne 0$ and $\gamma$ is the Euler–Mascheroni constant.

In what follows, we will examine the change in the behavior of (14). By change in behavior we mean the change in the sign of either the first or the second derivative. For the purposes of demonstration, we have used the exact first derivative in order to check whether Theorem 3 or Theorem 4 is applicable. When the analytic solution is not known, one can approximate the first derivative by using (7) and an approximation of $x_1\left(t\right)$. When calculating either the first or the second derivative, we may find $t_0\in [0,T]$, where the derivative becomes 0 but maintains its sign around that value. Since we are interested in changing behavior, we have excluded these cases from the analysis. There are four cases left, which we will encode by an ordered pair. The first element is an arrow showing the sign of the first derivative (↗—increasing; ↘—decreasing), whereas the second represents the sign of the second derivative (∪—convex; ∩—concave).

We have considered the case $\alpha =2$ and $T=0.8\alpha =1.6$. By visualizing the effect of the other parameters in the plane, we can choose only two parameters for the axes. We have decided to examine the effect of ${\nu }_1$ on the behavior of the solution with time; i.e., we are observing the $\left(t{\nu }_1\right)$-plane. We would like to observe that any two parameters could be chosen for such an analysis. Given that, if none of the parameters is t, then the conclusions would be valid point-wise for the specific t chosen. The other parameters can also be fixed. In our case, we have taken ${\overrightarrow{x}}_0=(30,15)$ and $f_0=1$.

The results are visible in Figure 5 and Figure 6. We can clearly see that the sufficient conditions are not necessary. However, even though Theorems 3 and 4 do not give information for every possible value of the parameters, the information that they do give is accurate. By using these results, an educational institution could partially predict the characteristics of the solution without having to explicitly find it. This could lead to the early dismissal of plans that do not align with the goals of the institution.

We can also notice how many times the behavior changes for the current parameters. In Figure 5, there are places with one, two, and three changes, whereas in Figure 6, we get either one change or no change at all. Some of these cases are visualized in Figure 7 and Figure 8. These results could be used for determining regions of monotonicity if the institution desired to strictly increase or decrease the number of its experienced researchers. Not only that, but the institution would also gain information about the speed of the change and would have the ability to act accordingly. For example, in Figure 5, we see that by using Theorems 3 and 4, we get a large area where $x_1$ is increasing and concave. Thus, while the number of researchers is increasing, this increase is slowing down. This could be beneficial to a well-established institution that wants a more stable addition to its experienced researchers. However, for an institution that desires swift gain in such researchers, parameter values closer to the ones in Figure 6 would be better suited.

The behavior in this particular case has an intuitive explanation. When ${\nu }_0<0$, this leads to an increase in $x_0$ for a certain amount of time. Because $f_0=1>0$, $x_1^{\prime }$ increases more the bigger $x_0$ is. By influencing $x_0$’s growth via ${\phi }_0$, one could further cause $x_1$ to grow, even if ${\phi }_1$ is negative. If $x_0$ keeps increasing, this would lead to convexity of $x_1$ (Figure 6), whereas when ${\nu }_0>0$, we get a strictly decreasing $x_0$, slowing down the growth of $x_1$, i.e., making it more likely to be concave (Figure 5).

4. Numerical Results

In [51] it is pointed out that closed-form expressions of (9) with functions of the form in (8) exist only for $\kappa \in \mathbb{Z}$. Otherwise numerical approximations must be used. The approach used in this paper is to find an approximate solution to system (7) as opposed to approximating the integrals in (9). Such an approach could easily be generalized to (3).

In order to use a numerical method to solve system (7), we would require the solution to exist and to be unique. For this purpose we will use the well-known Picard–Lindelöf Theorem, which can be found in many textbooks, e.g., [59]. For the solution to exist in a neighborhood of $t_0\in [0,T]$, we would require that the function $\overrightarrow{f}(t,\overrightarrow{x})$ be continuous and bounded with respect to t and $\overrightarrow{x}$, and uniformly Lipschitz continuous with respect to $\overrightarrow{x}$ on an appropriate interval. We will take the norm used for $\overrightarrow{f}$ and $\overrightarrow{y}$ to be the usual Euclidean norm. It is clear that $\overrightarrow{f}(t,\overrightarrow{x})=({\phi }_0\left(t\right)x_0,{\phi }_1\left(t\right)x_1+f_0\left(t\right)x_0)$ is continuous with respect to $\overrightarrow{x}$ on any interval. Using (8), we can also see that if we restrict ourselves to $t\in [0,T]$ with $T=0.8\alpha$, $\overrightarrow{f}(t,\overrightarrow{x})$ is also continuous with respect to t.

We will take the norm used for $\overrightarrow{f}$ and $\overrightarrow{y}$ to be the usual Euclidean norm. It is clear that $\overrightarrow{f}(t,\overrightarrow{x})=({\phi }_0\left(t\right)x_0,{\phi }_1\left(t\right)x_1+f_0\left(t\right)x_0)$ is continuous with respect to $\overrightarrow{x}$ on any interval. Using (8), we can also see that if we restrict ourselves to $t\in [0,T]$ with $T=0.8\alpha$, $\overrightarrow{f}(t,\overrightarrow{x})$ is also continuous with respect to t.

Let us denote by $\mathcal{D}$ the region over which we will examine $\overrightarrow{p}$:

$$\begin{array}{cc}\hfill \mathcal{D}=\{\overrightarrow{p}\in {\mathbb{R}}^5:& {\nu }_0^{min}\le {\nu }_0\le {\nu }_0^{max},{\nu }_1^{min}\le {\nu }_1\le {\nu }_1^{max},\hfill \\ \hfill & {\kappa }^{min}\le \kappa \le {\kappa }^{max},f_0^{min}\le f_0\le f_0^{max},\alpha =2\}.\hfill \end{array}$$

Since $|{\phi }_0\left(t\right)|,|{\phi }_1\left(t\right)|$, and $|f_0\left(t\right)|$ are continuous on $\mathcal{D}$, then there exists a maximum of each of these functions. Let

$$\begin{array}{ccc}{\displaystyle {\phi }_{0\ast }=\underset{\overrightarrow{p}\in \mathcal{D}}{max}\left|{\phi }_0\left(t\right)\right|,}& {\displaystyle {\phi }_{1\ast }=\underset{\overrightarrow{p}\in \mathcal{D}}{max}\left|{\phi }_1\left(t\right)\right|,}& {\displaystyle f_{0\ast }=\underset{\overrightarrow{p}\in \mathcal{D}}{max}\left|f_0\left(t\right)\right|.}\end{array}$$

Analogously, assuming that $\overrightarrow{x}$ is continuous, let

$$x_{0\ast }=\underset{\overrightarrow{p}\in \mathcal{D},t\in [0,T]}{max}|x_0\left(t\right)|,x_{1\ast }=\underset{\overrightarrow{p}\in \mathcal{D},t\in [0,T]}{max}\left|x_1\left(t\right)\right|.$$

Then, by taking $M=\sqrt{({\phi }_{0\ast }^2+f_{0\ast }^2)x_{0\ast }^2+{\phi }_{1\ast }^2{x}_{1\ast }^2+2{\phi }_{1\ast }{f}_{0\ast }{x}_{0\ast }{x}_{1\ast }}$ and the Lipschitz constant $K=max\{\sqrt{{\phi }_{0\ast }^2+f_{0\ast }^2+2{\phi }_{1\ast }{f}_{0\ast }},\sqrt{{\phi }_{1\ast }^2+2{\phi }_{1\ast }{f}_{0\ast }}\}$, we can apply the Picard–Lindelöf Theorem to conclude that (7) has a unique solution in an appropriate neighborhood. Because these conclusions are true for every $t\in [0,T]$, if necessary, we could extend the solution to a maximal region of existence [59].

4.1. The Numerical Procedure

For the numerical investigation, we will use the well-known four-stage explicit Runge–Kutta method [52,53,54]. Due to its prominence and familiarity, we omit writing it out explicitly. We will use the same time interval as in Section 3.3, i.e., $\alpha =2$, $T=0.8\alpha =1.6$. We divide the interval $[0,T]$ into N equal parts with step $h={\displaystyle \frac{T}{N}}$ and denote $t_n=nh$, ${\overrightarrow{x}}_n=\overrightarrow{x}\left(t_n\right)$, $n=0,1,2\dots ,N$.

If the real solution is smooth enough, this method has a global error of $O\left(h^4\right)$. Assuming that $\overrightarrow{x}$ is continuous, it is not hard to verify by differentiating (7) that if ${\phi }_0,{\phi }_1,f_0$ are in $C^k[0,T]$, then $\overrightarrow{x}\in C^{k+1}[0,T]$. In our case the functions in (8) are infinitely differentiable; hence, $\overrightarrow{x}$ is infinitely differentiable.

Another question that needs to be addressed is the stability of the procedure. We will examine the AN-stability of the method for the given system of differential equations. For this purpose the model equation $y^{\prime }=q\left(t\right)y$ is used. In the case of systems, one takes $q_i\left(t\right)$ to be the i-th eigenvalue of the system and studies the equations $y^{\prime }=q_i\left(t\right)y$. In (7), it is clear that $i=1,2$, and the eigenvalues are $q_1\left(t\right)={\phi }_0\left(t\right)$ and $q_2\left(t\right)={\phi }_1\left(t\right)$. By substituting the model equation into the equations for the Runge–Kutta method, we arrive at the expression for the stability function

$$R\left(z_n,z_{n+{\displaystyle \frac{1}{2}}}\right)=1+{\displaystyle \frac{z_n^2{z}_{n+\frac{1}{2}}^2}{24}}+{\displaystyle \frac{z_{n+\frac{1}{2}}}{6}}\left(4+z_{n+{\displaystyle \frac{1}{2}}}\right)+{\displaystyle \frac{z_n}{6}}\left(2+2z_{n+{\displaystyle \frac{1}{2}}}+z_{n+{\displaystyle \frac{1}{2}}}^2\right),$$

(15)

where $z_n=h{q}_i\left(t_n\right)$ and $z_{n+{\displaystyle \frac{1}{2}}}=h{q}_i\left(t_n+{\displaystyle \frac{h}{2}}\right)$ for $n=0,1,2\dots ,N-1$ and $i=1,2$.

Definition 1 

([54]). The Runge–Kutta method is ‘AN-stable’ if function (15) is bounded in magnitude by 1 whenever $Re\left(z_n\right)<0,Re\left(z_{n+{\displaystyle \frac{1}{2}}}\right)<0$.

This definition has a simple interpretation. If $Re\left(q_i\left(t_n\right)\right)$ is negative, then the solution in the direction of the i-th eigenvalue of the system is decreasing in magnitude; thus, we desire the numerical method to exhibit the same property.

Because the Runge–Kutta method has a bounded region of existence, we need to establish conditions under which the method will be stable. For that purpose, we will consider

$$\mathcal{D}=\{\overrightarrow{p}\in {\mathbb{R}}^5:-1\le {\nu }_0\le 1,-1\le {\nu }_1\le 1,0\le \kappa \le 2,f_0=1,\alpha =2\}.$$

Proposition 1. 

If $h\le 0.464$ and $\overrightarrow{p}\in \mathcal{D}$, then the Runge–Kutta method applied to (7) is conditionally AN-stable.

Proof. 

Because the eigenvalues of (7) have only real values, we need to consider the cases when $q_i\left(t\right)<0$. By examining the partial derivatives of $q_i\left(t\right)$, $i=1,2$, we can conclude that

$$\begin{array}{cc}\hfill {q_1}_{min}& =\underset{\overrightarrow{p}\in \mathcal{D},t\in [0,T],h\in [0,0.464]}{min}h{\phi }_0\left(t\right)=-0.232,\hfill \\ \hfill {q_2}_{min}& =\underset{\overrightarrow{p}\in \mathcal{D},t\in [0,T],h\in [0,0.464]}{min}h{\phi }_1\left(t\right)=-2.784.\hfill \end{array}$$

Let us consider the function $u(x,y)=1+{\displaystyle \frac{x^2{y}^2}{24}}+{\displaystyle \frac{y}{6}}(4+y)+{\displaystyle \frac{x}{6}}(2+2y+y^2)$ in the region $(x,y)\in {[-2.784,0]}^2$. Since whenever $z_n$ and $z_{n+{\displaystyle \frac{1}{2}}}$ for $n=0,1,\dots ,N-1$ are negative, they are in the interval $[-2.784,0]$, then $u\left(z_n,z_{n+{\displaystyle \frac{1}{2}}}\right)$ is equal to (15). Thus, from $\left|u\right(x,y\left)\right|\le 1$, it would follow that (15) would be confined in magnitude by 1 for each $n\in 0,1,2\dots N-1$ and $i=1,2$. By routine calculations one can find that $0.059\le u(x,y)\le 1$, from which we conclude that $\left|u\right(x,y\left)\right|\le 1$. Thus, $\left|R\left(z_n,z_{n+{\displaystyle \frac{1}{2}}}\right)\right|\le 1$, and the Runge–Kutta method applied to (7) is conditionally AN-stable. □

To verify the order of the error and the stability of the scheme, we will use solution (14) along with the analytical solution of $x_0\left(t\right)=x_0\left(0\right){\displaystyle \frac{\alpha }{\alpha -t}}exp(-{\nu }_{0}t)$ and the errors

$$\begin{array}{cc}\hfill L_{\infty }& =\underset{n=1,2,\dots N}{max}|{\widehat{y}}_n-y\left(t_n\right)|\hfill \\ \hfill RMSE& =\sqrt{{\displaystyle \frac{1}{N-1}}\sum _{n=1}^N{[{\widehat{y}}_n-y\left(t_n\right)]}^2},\hfill \end{array}$$

where $\widehat{y}$ is the approximate solution and $y\left(t\right)$ is the analytical solution. We can see the results in

Table 1. Errors for (7) using the Runge–Kutta method with initial condition ${\overrightarrow{x}}_0=(30,15)$.

h	$\overrightarrow{\mathit{p}}$	${\mathit{L}}_{\mathit{\infty }}$		RMSE
		${\mathit{x}}_{\mathbf{0}}$	${\mathit{x}}_{\mathbf{1}}$	${\mathit{x}}_{\mathbf{0}}$	${\mathit{x}}_{\mathbf{1}}$
0.1	(0.55, 0.8, 2, 1, 1)	3.38 × 10−4	1.46 × 10−3	9.25 × 10−5	4.16 × 10−4
0.1	(0.55, −0.9, 2, 1, 1)	3.38 × 10−4	1.58 × 10−3	9.25 × 10−5	4.40 × 10−4
0.01	(0.55, 0.8, 2, 1, 1)	3.98 × 10−8	1.49 × 10−7	7.52 × 10−9	3.08 × 10−8
0.01	(0.55, −0.9, 2, 1, 1)	3.98 × 10−8	1.62 × 10−7	7.52 × 10−9	3.23 × 10−8
0.001	(0.55, 0.8, 2, 1, 1)	4.32 × 10−12	1.48 × 10−11	8.24 × 10−13	2.93 × 10−12
0.001	(0.55, −0.9, 2, 1, 1)	4.32 × 10−12	1.66 × 10−11	8.24 × 10−13	3.39 × 10−12

, where the errors are within the expected magnitude.

4.2. Numerical Study of Concavity

In this subsection, we will expand on the analysis performed in Section 3.3. Our focus will be on the effect of $\kappa$ on the behavior of the solution. The values of the first and second derivatives are required for such an analysis to occur. Since we do not have closed-form solutions to (7) for non-integer $\kappa$, we will approximate their values by finding $\overrightarrow{x}$ numerically and substituting it into (7) and (13). Due to potential higher values of the coefficients in (13), the results were generated with $h=0.001$ and a step in ${\nu }_1$ being $\Delta {\nu }_1=0.001$. One more thing to pay attention to is the size of the solutions. The model does not inherently distinguish between reasonable and unreasonable solutions. Thus, a solution that leads to a huge number of researchers, potentially more than the available workforce in the vicinity of the institution, or a negative amount would have to be manually disregarded. The lower limit is obviously 0, whereas the upper limit is more complex and could depend on a plethora of factors, such as the population, the financial capabilities, the location, the public opinion of the institution, etc. For the purposes of demonstration, we have decided to keep solutions for which $x_0\left(t\right)\le C{x}_0\left(0\right)$ and $x_1\left(t\right)\le C{x}_1\left(0\right)$, where $C=100$ for $t\in [0,T]$. This allows for an expansion that would be reasonable for most institutions.

To make the analysis a true extension of Section 3.3, we have created behavior plots in the $\left(t{\nu }_1\right)$-plane with ${\nu }_0=0.55,f_0=1$, and $\kappa \in [0,2]$. Numerical results for $\kappa \in \{0,0.4,0.8,1.2,1.6,2\}$ are presented in Figure 9.

As can be seen from the plot, the behavior changes in a non-obvious way. At the beginning ($\kappa =0$), most choices of ${\nu }_1$ lead to an increasing convex solution, whereas those that do not are concave only for a short time and then return to being convex. Gradually, the concavity of the solution becomes more predominant ($\kappa =0.4$), reducing the convex area and eventually leading to a decreasing part of the plot ($\kappa =0.8$). The most complex behavior can be seen at $\kappa =1.2$, where all four types of behavior are exhibited. By the end, the increasing convex part of the plot disappears completely ($\kappa =2$), whereas the solution becomes decreasing for a greater portion of ${\nu }_1$ and t. However, in that transition, it appears that $x_1$ begins to shift to a convex solution yet again in the upper-right corner of the plot ($\kappa =2$). Some of the solutions for ${\nu }_1=0.95$ are visualized in Figure 10.

From Figure 9 an academic institution could make some conclusions. If a rapid increase in academic staff is required, a strategy that is modeled by (7) with parameters (8) in $\mathcal{D}$ should aim at lower $\kappa$. If $\kappa$ can be brought close to zero, a negative ${\nu }_1$ keeps $x_1$ convex, thus accelerating the inflow of new experienced researchers. If $\kappa$ cannot be made appropriately small but can still be kept below $0.8$, then a reasonable choice for ${\nu }_1$ would be a positive number so as to avoid the decreasing zone of the behavior plot. If there is a surplus of experienced researchers, the institution could adopt a strategy with a bigger $\kappa$, where bigger values of ${\nu }_1$ lead to longer periods of decrease.

4.3. Numerical Study of the Outcome

Until now we have studied the behavior of the solution and its change with respect to ${\nu }_0,{\nu }_1$, and $\kappa$. This gives some nuanced information about how the number of researchers will change, but these predictions are purely qualitative. If a given institution would prefer a quantitative prediction, these plots might not be helpful in some cases. If the plots are like the ones in Figure 6 and Figure 9 with $\kappa \in \{0,0.4\}$, then since the whole area of the plot shows that $x_1$ is increasing, the outcome would lead to an inflow of experienced researchers. If the plot showed only decreasing regions, then that would analogously mean that there would be an outflow of experienced researchers if such a strategy were to be adopted by the institution. The problem becomes more complex when we have both periods of increase and decrease. Then it would not be clear whether the increasing part would outweigh the decreasing one, or vice versa.

In such a scenario, it would be informative if we observed the outcome that the model predicts—whether there would be an inflow or an outflow of researchers. Then we would be interested not only in the change in $x_1$, but also in the change in $x_0$. There are four possible outcomes:

• $x_0$ and $x_1$ have both decreased, i.e., $x_0\left(0\right)>x_0\left(T\right)$ and $x_1\left(0\right)>x_1\left(T\right)$;
• $x_0$ and $x_1$ have increased, i.e., $x_0\left(0\right)<x_0\left(T\right)$ and $x_1\left(0\right)<x_1\left(T\right)$;
• $x_0$ has decreased, whereas $x_1$ has increased, i.e., $x_0\left(0\right)<x_0\left(T\right)$ and $x_1\left(0\right)>x_1\left(T\right)$;
• $x_0$ has increased, whereas $x_1$ has decreased, i.e., $x_0\left(0\right)<x_0\left(T\right)$ and $x_1\left(0\right)>x_1\left(T\right)$.

Of course, the number could potentially remain the same, but for the purposes of this study, we can ignore such cases. Indeed, since the model is continuous and $\overrightarrow{x}$ is not restricted to the integers, landing on the exact same value as the initial condition is highly unlikely. We will encode these cases by a pair of arrows, where the first one will show the change in $x_0$ and the second one the change in $x_1$. For example, $(\nearrow ,\searrow )$ means that $x_0\left(0\right)<x_0\left(T\right)$ and $x_1\left(0\right)>x_1\left(T\right)$. For the purposes of demonstration, we have used the same parameter space $\mathcal{D}$ as in the previous subsection. The experiment consisted of switching the values of ${\nu }_0$ and ${\nu }_1$ (both with a step $\Delta {\nu }_0=\Delta {\nu }_1=0.01$), solving system (7) numerically with $h=0.01$ and comparing the endpoint with the initial conditions. The results can be seen in Figure 11.

The figures show that for this choice of parameters, we observe a strict increase in $x_0$. Therefore, using this strategy would lead to an influx of young researchers. On the other hand, an increase in $x_1$ is not guaranteed. One thing worth noting is that the conclusions of the previous subsection can mostly be applied. The lower the value of $\kappa$, the more area that leads to an increase in $x_1$ is observed. The same tendency is observed with ${\nu }_0$ and ${\nu }_1$. These interactions can be explained by $\kappa$ and ${\nu }_1$ contributing to the derivative of $x_1$ with a negative sign, thus pushing $x_1^{\prime }$ more and more into the negatives. The effect of ${\nu }_0$ is similar—it influences the growth of $x_0$. The smaller it is, the quicker $x_0$ grows. Since $f_0>0$, this contributes to $x_1^{\prime }$ with positive values and allows $x_1$ to remain increasing for a longer period of time. One last thing worth pointing out is that even though there is a sliver of an area of decrease in $x_1$ when $\kappa =0.8,{\nu }_0=0.55$, we can see from Figure 11 that this has not led to a decrease in $x_1$; that is, the increase before this change in behavior has not been overcome by the outflow of experienced researchers. Thus, the convexity plots can only offer general recommendations. When reaching a conclusion, it would be best if one would incorporate both plots in order to take into account both the outcome and the trajectory with which the institution would be left after the expiry of the chosen strategy.

5. Concluding Remarks

In this work, we have continued the analysis of the model of a substance moving through a network, first proposed in [51]. We have outlined a generalization that could lead to a richer interaction between the cells of a particular chain and have analyzed a slightly less restricted version of the original model. We have expanded upon the analysis of the constant coefficients case and have analyzed the convexity of $x_1$, providing sufficient conditions for the behavior of $x_1$ based only on the coefficients in the system of differential equations in (7) and verifying these results graphically. We have further investigated the influence of the parameters numerically using the fourth-order global error explicit Runge–Kutta method. This part consisted of convexity and outcome analyses.

The obtained results show that in the case of constant coefficients, allowing for a positive feedback loop of the cell ($c_0>0,c_1>0$) leads the solutions to tend to infinity, thus leaving this model with limited applicability to real-world problems. As for the convexity analysis, the results show that lower values of $\kappa$, ${\nu }_0$, and ${\nu }_1$ generally lead to a bigger influx of experienced researchers. Further, for a reasonable interpretation of the results, it would be advantageous to incorporate both the behavior and outcome plots, including the qualitative and quantitative analyses necessary for an optimal strategy for reaching the researcher goals of the given institution.

In the course of this analysis, the following questions remain open:

• A broader examination of the constant coefficient case could be considered. For example, the behavior when there are more than two cells in the chain and the evolution of the cells when $c_0<0$.
• A system with more unknowns could be investigated. Instead of system (7), an investigation of the more general system in (6) would lead to a more complex chain structure that requires a deeper analysis of the interactions between the cells.
• The tackling of a more generalized version of the model could provide a richer chain framework with more versatile connections between the substance amounts in the cells. Such versions could be the full matrix linear version in (4) or a nonlinear version of the form in (3). A possible first step into a general nonlinear modeling of the network could involve a quadratic polynomial of $x_k$ for each derivative in the system.
• The choice of $T=0.8\alpha$ in this paper is for demonstration purposes only. Other than $T<\alpha$, a reasonable endpoint for this model is unclear.
• The merits of a model can be measured truly by its application and forecasting abilities. A study focusing on fitting the model to real data would be a guide to what formulation of the general model could be useful in real situations. This way, theoretical analysis of such a model would be immensely useful for modeling any situation that might be describable by the model.