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Michaelis-Menten Kinetics as a Model of Doctoral Supervisor-Supervisee Relationship

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Title: Michaelis-Menten Kinetics as a Model of Doctoral Supervisor-Supervisee Relationship
Language: English
Authors: Monteiro, L. H. A. (ORCID 0000-0002-2309-1254)
Source: International Journal of Mathematical Education in Science and Technology. 2023 54(1):145-150.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 6
Publication Date: 2023
Document Type: Journal Articles
Reports - Evaluative
Education Level: Higher Education
Postsecondary Education
Descriptors: Models, Kinetics, Equations (Mathematics), Chemistry, Scientific Concepts, Supervisor Supervisee Relationship, Doctoral Students, College Faculty, Simulation, Teaching Methods
DOI: 10.1080/0020739X.2022.2035002
ISSN: 0020-739X
1464-5211
Abstract: The process of turning a doctoral student into an independent researcher is usually guided by a professor. In this work, the supervisor-supervisee relationship is represented by a scheme inspired by Michaelis-Menten kinetics, which has been used to determine the rate of enzyme-catalysed reactions. Here, the time evolution of the number of supervisors with "k" supervisees is modelled by a system of linear ordinary differential equations. The long-term behaviour of these equations is analytically examined and illustrated by a numerical simulation. This model related to the PhD formation catalysed by a professor can be presented in a class on differential equations or chemical kinetics.
Abstractor: As Provided
Entry Date: 2023
Accession Number: EJ1377597
Database: ERIC
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  Value: <anid>AN0161624553;imt01jan.23;2023Feb03.04:10;v2.2.500</anid> <title id="AN0161624553-1">Michaelis–Menten kinetics as a model of doctoral supervisor–supervisee relationship </title> <p>The process of turning a doctoral student into an independent researcher is usually guided by a professor. In this work, the supervisor–supervisee relationship is represented by a scheme inspired by Michaelis–Menten kinetics, which has been used to determine the rate of enzyme-catalysed reactions. Here, the time evolution of the number of supervisors with k supervisees is modelled by a system of linear ordinary differential equations. The long-term behaviour of these equations is analytically examined and illustrated by a numerical simulation. This model related to the PhD formation catalysed by a professor can be presented in a class on differential equations or chemical kinetics.</p> <p>Keywords: Chemical kinetics; differential equation; doctoral supervision; dynamical system; Michaelis–Menten model; PhD student</p> <hd id="AN0161624553-2">1. Introduction</hd> <p>PhD supervision is the challenging process of transforming students into scientists (Carroll, [<reflink idref="bib3" id="ref1">3</reflink>]; Lloyd, [<reflink idref="bib8" id="ref2">8</reflink>]; Peelo, [<reflink idref="bib12" id="ref3">12</reflink>]). Since the supervisor can be seen as the catalyst of this academic journey, it seems natural to use the Michaelis–Menten enzyme kinetics to schematically represent this process.</p> <p>Enzymes are molecules that act as catalysts (Suzuki, [<reflink idref="bib13" id="ref4">13</reflink>]). In fact, they increase the rate of biochemical reactions by which reacting substances are converted into new substances. According to Michaelis and Menten ([<reflink idref="bib9" id="ref5">9</reflink>]), an intermediate complex <emph>ES</emph> is formed when the substrates <emph>S</emph> bind to a specific enzyme <emph>E</emph>. If the substrates react, the products <emph>P</emph> are released and the enzyme is free to catalyse another reaction. If the reaction does not occur, the complex can dissociate back into free enzyme and substrates. The scheme corresponding to this enzymatic transformation of substrates into products is (Johnson & Goody, [<reflink idref="bib6" id="ref6">6</reflink>]; Michaelis & Menten, [<reflink idref="bib9" id="ref7">9</reflink>]; Suzuki, [<reflink idref="bib13" id="ref8">13</reflink>]):</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>E</mi><mo>+</mo><mi>S</mi><mo>⇌</mo><mo>[</mo><mtext fontfamily="times">β</mtext><mo>]</mo><mrow><mi>α</mi></mrow><mi>E</mi><mi>S</mi><mover><mo>⟶</mo><mi>γ</mi></mover><mi>E</mi><mo>+</mo><mi>P</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib1" id="ref9">1</reflink>)</p> <p>in which <emph>α</emph>, <emph>β</emph>, and <emph>γ</emph> are the rate constants related to the complex formation, the complex dissociation, and the product formation, respectively. Notice that the enzyme-substrate binding is reversible, but the product generation is irreversible. The rates of change of the concentrations of <emph>E</emph>, <emph>S</emph>, <emph>ES</emph>, and <emph>P</emph> are governed by equations obtained from the law of mass action (Suzuki, [<reflink idref="bib13" id="ref10">13</reflink>]; Voit et al., [<reflink idref="bib15" id="ref11">15</reflink>]). This model, which is central in enzyme kinetics, has also been used to study, for instance, the adsorption of gaseous molecules by solid surfaces (Langmuir, [<reflink idref="bib7" id="ref12">7</reflink>]) and the metabolism of hydrogen producing bacteria (Chen et al., [<reflink idref="bib4" id="ref13">4</reflink>]).</p> <p>In this work, it is assumed that a scheme similar to Equation (<reflink idref="bib1" id="ref14">1</reflink>) can be used to express the supervisor–supervisee relationship. In the chemistry context, each enzyme can catalyse a single reaction. However, in the academic context, each professor can simultaneously supervise several students.</p> <p>Theoretical works based on the law of mass action on educational issues (Monteiro, [<reflink idref="bib10" id="ref15">10</reflink>]) or with didactic purposes (Abernethy, [<reflink idref="bib1" id="ref16">1</reflink>]) can be found in the literature. Here, a mathematical model is proposed to predict how the number of supervisors with <emph>k</emph> supervisees varies as time passes. This model, derived from the law of mass action, is formulated in terms of a set of linear ordinary differential equations and it is analysed from a dynamical systems theory perspective (Guckenheimer & Holmes, [<reflink idref="bib5" id="ref17">5</reflink>]; Tu, [<reflink idref="bib14" id="ref18">14</reflink>]). This model can be presented as a didactic example in courses in differential equations and also as an unusual example in courses in chemical kinetics.</p> <p>This manuscript is organized as follows. In Section 2, the proposed model is introduced; in Section 3, its asymptotical behaviour is analytically predicted; and in Section 4, the possible relevance of this work is stressed.</p> <hd id="AN0161624553-3">2. The model</hd> <p>Let</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> be the concentrations of doctoral students without supervisor, supervisors with <emph>k</emph> doctoral students, and students that successfully finished the doctorate, respectively. These variables are positive real numbers denoting the numbers of either students or professors per department of an educational institution at the instant <emph>t</emph>. Here,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> is taken as a constant; that is,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>S</mi></math> </ephtml> . This assumption means that there is a large excess of students looking for a supervisor.</p> <p>Assume that the following scheme holds:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>E</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>S</mi><mo>⇌</mo><mo>[</mo><mtext fontfamily="times">β</mtext><mo>]</mo><mrow><mi>α</mi></mrow><msub><mi>E</mi><mi>k</mi></msub><mover><mo>⟶</mo><mi>γ</mi></mover><msub><mi>E</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>P</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib2" id="ref19">2</reflink>)</p> <p>Equation (<reflink idref="bib2" id="ref20">2</reflink>) shows that the conversion of <emph>S</emph> in <emph>P</emph> is catalysed by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> . Observe that when a supervisor gets a new student, its label <emph>k</emph> increases by one unit; hence, during the supervision,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> changes to</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub></math> </ephtml> ; at the end of a successful or an unsuccessful supervision,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub></math> </ephtml> returns to</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> . The rate constants <emph>α</emph>, <emph>β</emph>, and <emph>γ</emph> are related to the beginning of a supervision, the end of an unsuccessful supervision, the end of a successful supervision, respectively. Assume also that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>K</mi></math> </ephtml> , in which <emph>K</emph> is the maximum number allowed of doctoral students per professor.</p> <p>First, consider that case <emph>K</emph> = 1; that is:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>S</mi><mo>⇌</mo><mo>[</mo><mtext fontfamily="times">β</mtext><mo>]</mo><mrow><mi>α</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mover><mo>⟶</mo><mi>γ</mi></mover><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>P</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib3" id="ref21">3</reflink>)</p> <p>which corresponds to the original scheme proposed by Michaelis and Menten (Johnson & Goody, [<reflink idref="bib6" id="ref22">6</reflink>]; Michaelis & Menten, [<reflink idref="bib9" id="ref23">9</reflink>]; Suzuki, [<reflink idref="bib13" id="ref24">13</reflink>]). The law of mass action states that the rate of a chemical reaction is proportional to the product of the concentrations of the reacting substances, with each concentration raised to a power equal to the coefficient that is used to balance such a chemical reaction (Voit et al., [<reflink idref="bib15" id="ref25">15</reflink>]). By applying this law, the rates of the reactions shown in Equation (<reflink idref="bib3" id="ref26">3</reflink>) are written as:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mo>−</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib4" id="ref27">4</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib5" id="ref28">5</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mi>γ</mi><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib6" id="ref29">6</reflink>)</p> <p>Since</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">d</mi></mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>></mo><mn>0</mn></math> </ephtml> , then</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> increases as time progresses. Notice that the variable</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> does not affect the time evolutions of</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> ; hence, Equation (<reflink idref="bib6" id="ref30">6</reflink>) can be neglected in the analysis of Equations (<reflink idref="bib4" id="ref31">4</reflink>)–(<reflink idref="bib5" id="ref32">5</reflink>). Notice also that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>+</mo><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mn>0</mn></math> </ephtml> . Consequently,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>N</mi></math> </ephtml> ; that is, the total number of professors with or without students remains constant and equal to <emph>N</emph>. By taking</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>N</mi><mo>−</mo><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> , Equation (<reflink idref="bib4" id="ref33">4</reflink>) can be rewritten as:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac><mo>=</mo><mo>(</mo><mo>−</mo><mi>α</mi><mi>S</mi><mo>−</mo><mtext fontfamily="times">β</mtext><mo>−</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><mi>N</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib7" id="ref34">7</reflink>)</p> <p>The case <emph>K</emph> = 2 corresponds to:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>S</mi></mtd><mtd><mi /><mo>⇌</mo><mo>[</mo><mtext fontfamily="times">β</mtext><mo>]</mo><mrow><mi>α</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mover><mo>⟶</mo><mi>γ</mi></mover><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>+</mo><mi>P</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib8" id="ref35">8</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>S</mi></mtd><mtd><mi /><mo>⇌</mo><mo>[</mo><mtext fontfamily="times">β</mtext><mo>]</mo><mrow><mi>α</mi></mrow><msub><mi>E</mi><mn>2</mn></msub><mover><mo>⟶</mo><mi>γ</mi></mover><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>P</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib9" id="ref36">9</reflink>)</p> <p>This case is described by the differential equations:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mo>−</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib10" id="ref37">10</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib11" id="ref38">11</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>−</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib12" id="ref39">12</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mi>γ</mi><mo>(</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib13" id="ref40">13</reflink>)</p> <p>Since</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>+</mo><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>+</mo><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mn>0</mn></math> </ephtml> , then</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>N</mi><mo>−</mo><mo>(</mo><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math> </ephtml> and Equations (<reflink idref="bib10" id="ref41">10</reflink>)–(<reflink idref="bib11" id="ref42">11</reflink>) can be rewritten as:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mo>−</mo><mi>α</mi><mi>S</mi><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib14" id="ref43">14</reflink>)</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac></mtd><mtd><mi /><mo>=</mo><mo>(</mo><mi>α</mi><mi>S</mi><mo>−</mo><mtext fontfamily="times">β</mtext><mo>−</mo><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mo>−</mo><mi>α</mi><mi>S</mi><mo>−</mo><mn>2</mn><mtext fontfamily="times">β</mtext><mo>−</mo><mn>2</mn><mi>γ</mi><mo>)</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><mi>N</mi></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib15" id="ref44">15</reflink>)</p> <p>In the next section, the long-term behaviours of the cases with <emph>K</emph> = 1 and <emph>K</emph> = 2 are analysed. Their dynamics are governed by Equation (<reflink idref="bib7" id="ref45">7</reflink>) and by Equations (<reflink idref="bib14" id="ref46">14</reflink>)–(<reflink idref="bib15" id="ref47">15</reflink>), respectively. The results can give hints for the cases with</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>≥</mo><mn>3</mn></math> </ephtml> .</p> <hd id="AN0161624553-4">3. Analytical results</hd> <p>Consider the system of first-order linear ordinary differential equations (Blanchard et al., [<reflink idref="bib2" id="ref48">2</reflink>]; Guckenheimer & Holmes, [<reflink idref="bib5" id="ref49">5</reflink>]; Tu, [<reflink idref="bib14" id="ref50">14</reflink>]):</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>x</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mtd></mtr><mtr><mtd><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>x</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>x</mi><mi>n</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mo>=</mo><mrow><mo>[</mo><mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>a</mi><mrow><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>a</mi><mrow><mn>12</mn></mrow></msub></mtd><mtd><mo>...</mo></mtd><mtd><msub><mi>a</mi><mrow><mn>1</mn><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>a</mi><mrow><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>a</mi><mrow><mn>22</mn></mrow></msub></mtd><mtd><mo>...</mo></mtd><mtd><msub><mi>a</mi><mrow><mn>2</mn><mi>n</mi></mrow></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd><mtd><mo>⋮</mo></mtd><mtd><mo>⋱</mo></mtd><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mi>a</mi><mrow><mi>n</mi><mn>1</mn></mrow></msub></mtd><mtd><msub><mi>a</mi><mrow><mi>n</mi><mn>2</mn></mrow></msub></mtd><mtd><mo>...</mo></mtd><mtd><msub><mi>a</mi><mrow><mi>n</mi><mi>n</mi></mrow></msub></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>x</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr><mtr><mtd><msub><mi>x</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mi>x</mi><mi>n</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></mtd></mtr></mtable><mo>]</mo></mrow><mo>+</mo><mrow><mo>[</mo><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>b</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>b</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><mo>⋮</mo></mtd></mtr><mtr><mtd><msub><mi>b</mi><mi>n</mi></msub></mtd></mtr></mtable><mo>]</mo></mrow></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib16" id="ref51">16</reflink>)</p> <p>This system can be compactly rewritten as:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mfrac><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mi mathvariant="bold">x</mi></mrow></mrow><mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mi mathvariant="bold">A</mi></mrow><mrow><mi mathvariant="bold">x</mi></mrow><mo>+</mo><mrow><mi mathvariant="bold">b</mi></mrow></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib17" id="ref52">17</reflink>)</p> <p>in which the state variables</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mi>i</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> (</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></math> </ephtml> ) form the state vector</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">x</mi></mrow><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> . The coefficients</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> </ephtml> (</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></math> </ephtml> ) and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mrow><mi>i</mi></mrow></msub></math> </ephtml> (</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></math> </ephtml> ) are constants that form the square matrix</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">A</mi></mrow></math> </ephtml> and the column vector</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">b</mi></mrow></math> </ephtml> , respectively.</p> <p>Let</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">x</mi></mrow></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup></math> </ephtml> be a stationary solution of Equation (<reflink idref="bib17" id="ref53">17</reflink>). Thus,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup></math> </ephtml> is a constant vector satisfying</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">A</mi></mrow><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup><mo>+</mo><mrow><mi mathvariant="bold">b</mi></mrow><mo>=</mo><mrow><mn mathvariant="bold">0</mn></mrow></math> </ephtml> ; that is,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">d</mi></mrow><mrow><mrow><mi mathvariant="bold">x</mi></mrow></mrow><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mrow><mn mathvariant="bold">0</mn></mrow></math> </ephtml> . The stability of this steady-state solution is determined by the eigenvalues <emph>λ</emph> of the matrix</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">A</mi></mrow></math> </ephtml> . The solution</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">x</mi></mrow></mrow><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msup><mrow><mi mathvariant="bold">x</mi></mrow><mo>∗</mo></msup></math> </ephtml> is globally asymptotically stable if all eigenvalues of</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">A</mi></mrow></math> </ephtml> have negative real part (Guckenheimer & Holmes, [<reflink idref="bib5" id="ref54">5</reflink>]; Tu, [<reflink idref="bib14" id="ref55">14</reflink>]). Recall that the eigenvalues <emph>λ</emph> of</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">A</mi></mrow></mrow></math> </ephtml> are obtained from</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo movablelimits="true" form="prefix">det</mo><mo>(</mo><mrow><mrow><mi mathvariant="bold">A</mi></mrow></mrow><mo>−</mo><mi>λ</mi><mrow><mi mathvariant="bold">I</mi></mrow><mo>)</mo><mo>=</mo><mn>0</mn></math> </ephtml> , in which</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">I</mi></mrow></mrow></math> </ephtml> is the identity matrix.</p> <p>By comparing Equations (<reflink idref="bib4" id="ref56">4</reflink>) to Equation (<reflink idref="bib17" id="ref57">17</reflink>); that is, by considering</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>−</mo><mo>(</mo><mi>α</mi><mi>S</mi><mo>+</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><mi>N</mi></math> </ephtml> , the stationary solution of the case <emph>K</emph> = 1 is:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mo>(</mo><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mfrac><mi>N</mi><mrow><mn>1</mn><mo>+</mo><mi>q</mi></mrow></mfrac><mo>,</mo><mi>q</mi><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib18" id="ref58">18</reflink>)</p> <p>with</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>≡</mo><mi>α</mi><mi>S</mi><mrow><mo>/</mo></mrow><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo></math> </ephtml> . Thus,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">d</mi></mrow><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mrow><mo>/</mo></mrow><mrow><mi mathvariant="normal">d</mi></mrow><mi>t</mi><mo>=</mo><mn>0</mn></math> </ephtml> for</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup></math> </ephtml> . This solution is globally asymptotically stable because</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>−</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>)</mo><mo><</mo><mn>0</mn></math> </ephtml> . Recall that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>=</mo><mi>N</mi><mo>−</mo><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup></math> </ephtml> .</p> <p>The stationary solution of the case <emph>K</emph> = 2 is:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mo>(</mo><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>E</mi><mn>2</mn><mo>∗</mo></msubsup><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mfrac><mi>N</mi><mrow><mn>1</mn><mo>+</mo><mi>q</mi><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup></mrow></mfrac><mo>,</mo><mi>q</mi><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><mi>q</mi><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib19" id="ref59">19</reflink>)</p> <p>By taking</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> in Equation (<reflink idref="bib17" id="ref60">17</reflink>), the matrix</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">A</mi></mrow></mrow></math> </ephtml> for <emph>K</emph> = 2 is:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow><mrow><mi mathvariant="bold">A</mi></mrow></mrow><mo>=</mo><mrow><mo>[</mo><mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mo>−</mo><mi>α</mi><mi>S</mi></mtd><mtd><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi></mtd></mtr><mtr><mtd><mi>α</mi><mi>S</mi><mo>−</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo></mtd><mtd><mo>−</mo><mi>α</mi><mi>S</mi><mo>−</mo><mn>2</mn><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo></mtd></mtr></mtable><mo>]</mo></mrow></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib20" id="ref61">20</reflink>)</p> <p>In this case, the eigenvalues are the roots of the polynomial</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><msub><mi>θ</mi><mn>1</mn></msub><mi>λ</mi><mo>+</mo><msub><mi>θ</mi><mn>2</mn></msub><mo>=</mo><mn>0</mn></math> </ephtml> , in which</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>)</mo></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>2</mn></msub><mo>=</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><msup><mo>)</mo><mn>2</mn></msup><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> </ephtml> . The two eigenvalues have negative real part if</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></math> </ephtml> (Ogata, [<reflink idref="bib11" id="ref62">11</reflink>]). Since both these conditions are satisfied, the steady-state solution given by Equation (<reflink idref="bib19" id="ref63">19</reflink>) is globally asymptotically stable.</p> <p>In order to illustrate the dynamical behaviour of the case <emph>K</emph> = 2, Equations (<reflink idref="bib10" id="ref64">10</reflink>)–(<reflink idref="bib13" id="ref65">13</reflink>) were numerically solved with fictitious parameter values by using the Euler's method (Blanchard et al., [<reflink idref="bib2" id="ref66">2</reflink>]) with integration time step of 0.01. The parameter values were chosen as <emph>S</emph> = 1,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mn>0.3</mn></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext fontfamily="times">β</mtext><mo>=</mo><mn>0.1</mn></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mn>0.1</mn></math> </ephtml> , and <emph>N</emph> = 100 (thus,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>100</mn></math> </ephtml> for any instant <emph>t</emph>). The initial condition was taken as</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>100</mn></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math> </ephtml> . Figure 1 shows that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>0</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>→</mo><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>≃</mo><mn>21.1</mn></math> </ephtml> (solid green line),</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>1</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>→</mo><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>≃</mo><mn>31.6</mn></math> </ephtml> (dashed blue line), and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mn>2</mn></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>→</mo><msubsup><mi>E</mi><mn>2</mn><mo>∗</mo></msubsup><mo>≃</mo><mn>47.3</mn></math> </ephtml> (dotted red line) for</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>→</mo><mi mathvariant="normal">∞</mi></math> </ephtml> . Obviously, identical numbers are obtained from Equation (<reflink idref="bib19" id="ref67">19</reflink>). Notice that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> </ephtml> (dash-dot cyan line) increases as time passes. After the transient,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>≃</mo><mi>γ</mi><mo>(</mo><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>+</mo><msubsup><mi>E</mi><mn>2</mn><mo>∗</mo></msubsup><mo>)</mo><mi>t</mi></math> </ephtml> ; that is,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>≃</mo><mn>7.89</mn><mi>t</mi></math> </ephtml> in this simulation.</p> <p>PHOTO (COLOR): Figure 1. Time evolutions of the concentrations E0(t) (solid green line), E1(t) (dashed blue line), E2(t) (dotted red line), and P(t) (dash-dot cyan line) for S = 1, α=0.3, β=0.1, γ=0.1, and N = 100 from the initial condition E0(0)=100, E1(0)=0, E2(0)=0, and P(0)=0. This plot was made by numerically integrating Equations (<reflink idref="bib10" id="ref68">10</reflink>)–(<reflink idref="bib13" id="ref69">13</reflink>) with Euler's method. In this simulation, E0(t)→E0∗≃21.1, E1(t)→E1∗≃31.6, E2(t)→E2∗≃47.3, and P(t)→7.89t as t→∞.</p> <p>The reader is invited to analyse the cases with</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>≥</mo><mn>3</mn></math> </ephtml> . For instance, the stationary solution for <emph>K</emph> = 3 is:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mo>(</mo><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>E</mi><mn>2</mn><mo>∗</mo></msubsup><mo>,</mo><msubsup><mi>E</mi><mn>3</mn><mo>∗</mo></msubsup><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mfrac><mi>N</mi><mrow><mn>1</mn><mo>+</mo><mi>q</mi><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup></mrow></mfrac><mo>,</mo><mi>q</mi><msubsup><mi>E</mi><mn>0</mn><mo>∗</mo></msubsup><mo>,</mo><mi>q</mi><msubsup><mi>E</mi><mn>1</mn><mo>∗</mo></msubsup><mo>,</mo><mi>q</mi><msubsup><mi>E</mi><mn>2</mn><mo>∗</mo></msubsup><mo>)</mo></mrow></mtd></mtr></mtable></math> </ephtml> (<reflink idref="bib21" id="ref70">21</reflink>)</p> <p>and the eigenvalues of the corresponding matrix</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">A</mi></mrow></mrow></math> </ephtml> are the roots of the polynomial</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>3</mn></msup><mo>+</mo><msub><mi>θ</mi><mn>1</mn></msub><msup><mi>λ</mi><mn>2</mn></msup><mo>+</mo><msub><mi>θ</mi><mn>2</mn></msub><mi>λ</mi><mo>+</mo><msub><mi>θ</mi><mn>3</mn></msub><mo>=</mo><mn>0</mn></math> </ephtml> , in which</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>)</mo></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>2</mn></msub><mo>=</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><msup><mo>)</mo><mn>2</mn></msup><mo>(</mo><mn>3</mn><mo>+</mo><mn>4</mn><mi>q</mi><mo>+</mo><mn>3</mn><msup><mi>q</mi><mn>2</mn></msup><mo>)</mo></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>3</mn></msub><mo>=</mo><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><msup><mo>)</mo><mn>3</mn></msup><mo>(</mo><mn>1</mn><mo>+</mo><mi>q</mi><mo>+</mo><msup><mi>q</mi><mn>2</mn></msup><mo>+</mo><msup><mi>q</mi><mn>3</mn></msup><mo>)</mo></math> </ephtml> . According to the Routh–Hurwitz criterion, asymptotical stability requires</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>1</mn></msub><mo>></mo><mn>0</mn></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>2</mn></msub><mo>></mo><mn>0</mn></math> </ephtml> ,</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>3</mn></msub><mo>></mo><mn>0</mn></math> </ephtml> , and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>1</mn></msub><msub><mi>θ</mi><mn>2</mn></msub><mo>−</mo><msub><mi>θ</mi><mn>3</mn></msub><mo>></mo><mn>0</mn></math> </ephtml> (Ogata, [<reflink idref="bib11" id="ref71">11</reflink>]). Because these four inequalities are satisfied, the stationary solution given by Equation (<reflink idref="bib21" id="ref72">21</reflink>) is globally asymptotically stable.</p> <hd id="AN0161624553-5">4. Conclusion</hd> <p>Here, the PhD supervision was modelled as an enzymatic reaction, in which the supervisor catalyses the candidate's transition to an academic role. This model predicts how</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>E</mi><mi>k</mi><mo>∗</mo></msubsup></math> </ephtml> (the steady-state number of supervisors with <emph>k</emph> doctoral students for</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>K</mi></math> </ephtml> ) depends on the parameters</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>S</mi><mrow><mo>/</mo></mrow><mo>(</mo><mtext fontfamily="times">β</mtext><mo>+</mo><mi>γ</mi><mo>)</mo></math> </ephtml> and <emph>N</emph>. Perhaps, the ideal scenario is</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mi>γ</mi></math> </ephtml> (inflow of students and outflow of PhDs at the same rates) and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext fontfamily="times">β</mtext><mo>=</mo><mn>0</mn></math> </ephtml> (no student dropout). The reader should easily prove that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>E</mi><mi>k</mi><mo>∗</mo></msubsup></math> </ephtml> for</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>K</mi><mo>≥</mo><mn>3</mn></math> </ephtml> depends only on the parameters <emph>q</emph> and <emph>N</emph>.</p> <p>The proposed model comprises a system of linear ordinary differential equations with constant coefficients. Hence, it can be presented in an introductory course in differential equations. The predictions of this model can be compared to real-world data, in order to test its validity. The model can be improved by considering, for instance, a limited inflow of students and/or the institutional policy of allotment of students. In addition, the values of the parameters <emph>β</emph> and <emph>γ</emph>, related to PhD failure/completion, are influenced by the socioeconomic conditions. These parameters can also be taken as functions of</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub></math> </ephtml> , in order to reflect the past experience and the acquired research skills of the supervisor. Thus, the differential equations can become nonlinear, which may lead to richer dynamical behaviours.</p> <hd id="AN0161624553-6">Disclosure statement</hd> <p>No potential conflict of interest was reported by the author.</p> <ref id="AN0161624553-7"> <title> References </title> <blist> <bibl id="bib1" idref="ref9" type="bt">1</bibl> <bibtext> Abernethy, G. M. (2018). Zombies: A simple discrete model of the apocalypse. International Journal of Mathematical Education in Science and Technology, 49 (8), 1260 – 1277. https://doi.org/10.1080/0020739X.2018.1455229</bibtext> </blist> <blist> <bibl id="bib2" idref="ref19" type="bt">2</bibl> <bibtext> Blanchard, P., Devaney, R. L., & Hall, G. R. (1998). Differential equations. Brooks-Cole.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref1" type="bt">3</bibl> <bibtext> Carroll, M. (2010). Supervision: Critical reflection for transformational learning (part 2). 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  Label: Title
  Group: Ti
  Data: Michaelis-Menten Kinetics as a Model of Doctoral Supervisor-Supervisee Relationship
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  Data: English
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  Data: <searchLink fieldCode="AR" term="%22Monteiro%2C+L%2E+H%2E+A%2E%22">Monteiro, L. H. A.</searchLink> (ORCID <externalLink term="http://orcid.org/0000-0002-2309-1254">0000-0002-2309-1254</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2023 54(1):145-150.
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  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Label: Page Count
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  Data: 6
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  Label: Publication Date
  Group: Date
  Data: 2023
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Evaluative
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Kinetics%22">Kinetics</searchLink><br /><searchLink fieldCode="DE" term="%22Equations+%28Mathematics%29%22">Equations (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Chemistry%22">Chemistry</searchLink><br /><searchLink fieldCode="DE" term="%22Scientific+Concepts%22">Scientific Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22Supervisor+Supervisee+Relationship%22">Supervisor Supervisee Relationship</searchLink><br /><searchLink fieldCode="DE" term="%22Doctoral+Students%22">Doctoral Students</searchLink><br /><searchLink fieldCode="DE" term="%22College+Faculty%22">College Faculty</searchLink><br /><searchLink fieldCode="DE" term="%22Simulation%22">Simulation</searchLink><br /><searchLink fieldCode="DE" term="%22Teaching+Methods%22">Teaching Methods</searchLink>
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  Data: 10.1080/0020739X.2022.2035002
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  Data: 0020-739X<br />1464-5211
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The process of turning a doctoral student into an independent researcher is usually guided by a professor. In this work, the supervisor-supervisee relationship is represented by a scheme inspired by Michaelis-Menten kinetics, which has been used to determine the rate of enzyme-catalysed reactions. Here, the time evolution of the number of supervisors with "k" supervisees is modelled by a system of linear ordinary differential equations. The long-term behaviour of these equations is analytically examined and illustrated by a numerical simulation. This model related to the PhD formation catalysed by a professor can be presented in a class on differential equations or chemical kinetics.
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