# Lagrangian Formalism in Biology: I. Standard Lagrangians and their Role in
Population Dynamics

D.T. Pham1 and Z.E. Musielak2 1Department of Biology, University of Texas at
Arlington, Arlington, TX 76019, USA
2Department of Physics, University of Texas at Arlington, Arlington, TX 76019,
USA

###### Abstract

The Lagrangian formalism is developed for the population dynamics of
interacting species that are described by several well-known models. The
formalism is based on standard Lagrangians, which represent differences
between the physical kinetic and potential energy-like terms. A method to
derive these Lagrangians is presented and applied to selected theoretical
models of the population dynamics. The role of the derived Lagrangians and the
energy-like terms in the population dynamics is investigated and discussed. It
is suggested that the obtained standard Lagrangians can be used to identify
physical similarities between different population models.

###### keywords:

Lagrangian formalism , Calculus of variations , Hamilton’s principle ,
Standard lagrangian , Population dynamics

##  1 Introduction

The astounding progress of modern physics in discovering and understanding the
fundamental laws of Nature that govern the structure and evolution of
nonorganic matter has been caused by its powerful mathematical-empirical
approach. In this approach, physical theories are formulated using the
language of mathematics and their predictions are verified by experiments. At
the same time, life sciences, included biology, remain mainly empirical and
descriptive.

There have been attempts to formulate mathematical models of some biological
systems, and thereby to establish mathematically oriented theoretical biology
[1]. Different areas of mathematics have become increasingly more important in
biology in recent decades, specifically, statistics in experimental design,
pattern recognition in bioinformatics, and mathematical modeling in evolution,
ecology and epidemiology [2]; however, it is pointed out that some of these
attempts can be classified as ’uses’ but others must be considered as
’abuses’.

By combining theoretical and empirical biology, the mathematical-empirical
approach may be established with expectations that its description of living
organisms reaches the level of the description given by physics to nonorganic
matter. In modern theoretical physics, all fundamental equations describing
matter are derived by using the Lagrangian formalism [3-5], which requires a
prior knowledge of functions called Lagrangians [6-8]. A number of different
methods have been proposed [9-14] to obtain the Lagrangians for the most basic
equations of modern physics [5].

In theoretical biology, Kerner [15] was first who applied the Lagrangian
formalism to biology and obtained Lagrangians for several selected biological
systems described by first-order ordinary differential equations (ODEs).
Later, Paine [16] investigated the existence and construction of Lagrangians
for similar set of ODEs following the original work of Helmholtz [6]. More
recently, Nucci and Tamizhmani [17] derived Lagrangians for some models
representing the population dynamics by using the method based on Jacobi Last
Multiplier [10]. Moreover, Nucci and Sanchini [18] obtained Lagrangian for an
Easter Island population model.

The main goal of this paper is to develop the Lagrangian formalism for
biological systems. The formalism is based on standard Lagrangians, whose main
characteristic is the presence of the difference between the kinetic and
potential energy-like terms [5,9,10]. A method to derive these Lagrangians is
presented and applied to selected models of the population dynamics. The role
of the derived Lagrangians and the energy-like terms in these models is
investigated and discussed. It is suggested that the obtained standard
Lagrangians can be used to identify physical similarities between diverse
biological systems.

The paper is organized as follows. Section 2 presents a brief overview of the
Lagrangian formalism and standard Lagrangians. Section 3 describes and
discusses the models of the population dynamics, and Section 4 concludes the
paper.

##  2 Lagrangian formalism

###  2.1 General overview

Preliminary formulation of the Lagrangian formalism was originally done by
Euler in 1742, and then it was refined and applied to Newtonian dynamics by
Lagrange, who set up its currently used form in his Analytic Mechanics that
first appeared in 1788; however, see [3] for more recent edition. According to
Lagrange, the formalism deals with a functional $\mathcal{S}[x(t)]$, which
depends on a continuous and differentiable function $x(t)$ that describes
evolution of a property of any dynamical system (given by $x$) in time
(represented by $t$). This evolution is described by an equation of motion,
which can be derived from the Lagrangian formalism.

The functional $\mathcal{S}[x(t)]$ is called action and is defined by an
integral over a scalar function $L$ that depends on $\dot{x}(t)=dx/dt$, $x$
and on $t$, so $L(\dot{x},x,t)$ and it is called the Lagrangian function or
simply Lagrangian. According to the principle of least action, or Hamilton’s
principle [4,5], the functional $\mathcal{S}[x(t)]$ must obey the following
requirement $\delta\mathcal{S}=0$, which says that the variation represented
by $\delta$ must be zero to guarantee that the action is stationary (to have
either a minimum, maximum or saddle point). The necessary condition that
$\delta\mathcal{S}=0$, is known as the Euler–Lagrange (E–L) equation that can
be written as

${d\over{dt}}\left({{\partial L}\over{\partial{\dot{x}}}}\right)-{{\partial
L}\over{\partial x}}=0.$ (1)

The Euler–Lagrange equation leads to a second-order ordinary differential
equation (ODE) that can be further solved to obtain $x(t)$ that makes the
action stationary. The procedure forms the basis of calculus of variations,
and it works well when the Lagrangian $L(\dot{x},x,t)$ is already known.
Deriving the second-order ODE from the E–L equation is called the Lagrangian
formalism, and in this paper we deal exclusively with this formalism. It must
be pointed out that the formalism has been extensively used in modern physics
and that all its fundamental equations are derived by using it [5]. In this
paper, we develop the Lagrangian formalism in biology and use it to obtain
equations of motions for the selected models of population dynamics.

For dynamical systems whose total energy is conserved, the existence of
Lagrangians is guaranteed by the Helmholtz conditions [6], which can also be
used to obtain Lagrangians. The procedure of finding Lagrangians is called the
inverse (or Helmholtz) problem of calculus of variations [7] and it shows that
there are three separate classes of Lagrangians, namely, standard [3,4],
nonstandard [8] and null [9] Lagrangians. Both standard and nonstandard
Lagrangians give the same equations of motion after they are substituted into
the E-L equation. However, null Lagrangians satisfy the E-L equation
identically and therefore they do not give any equation of motion.

Our main goal is to establish the Lagrangian formalism for known ODEs that
describe time evolution of different models of the population dynamics and
find their Lagrangians; in this paper, we concentrate exclusively on standard
Lagrangians.

###  2.2 Standrard Lagrangians

According to Lagrange [3], standard Lagrangians (SLs) are represented by
differences between kinetic and potential energies of dynamical systems. In
dynamics, if an oscillatory system of mass $m$ is displaced by $x$ in time
$t$, then its kinetic energy of oscillations is given by

$E_{\rm kin}={1\over 2}m\dot{x}(t),$ (2)

where $\dot{x}(t)$ is the time derivative of the dynamical variable $x(t)$, or
otherwise known as $\dot{x}=dx/dt$. The potential energy depends on a spring
constant $k$ of the oscillatory system and it can be written as

$E_{\rm pot}=kx^{2}(t).$ (3)

Thus, the standard Lagrangian for this system is

$L(\dot{x},x)=E_{\rm kin}-E_{\rm pot}={1\over
2}\left[m\dot{x}(t)-kx^{2}(t)\right].$ (4)

Substitution of this Lagrangian into the E-L equation (see Eq. 1) gives the
following equation of motion:

$m\ddot{x}(t)+kx(t)=0,$ (5)

where $\ddot{x}(t)$ is the second derivative of $x(t)$ with respect to time,
or $\ddot{x}=d^{2}x/dt^{2}$.

The presented above form of the SL is characteristic for oscillatory systems
that are not damped and not driven [2]. Obviously, the meaning of the variable
$x(t)$ may change from one physical system to another. Moreover, the meaning
of $x(t)$ in biological systems will be different than in physical systems, so
will the meaning of the constants $m$ and $k$. Nevertheless, a Lagrangian with
a kinetic-like energy term, which contains $\dot{x}^{2}(t)$, and a potential-
like energy term, which contains $x^{2}(t)$, will be identified as the SL. The
main objective of this paper is to derive SLs for different models of the
population dynamics and discuss the role of these SLs in mathematical
description of these models.

The Lagrangian formalism based on standard Lagrangians have been well-
established in most fields of modern physics (e.g., [5] and references
therein). Specifically, the formalism is commonly used to obtain equations of
motion for dynamical systems in Classical Mechanics [3,7,8]. More recently,
several methods were developed to solve the inverse Helmholtz problems for
physical systems described by ODEs (e.g., [10-14]). Some of these methods
developed for physical systems will be used in this paper to derive
Lagrangians for biological systems.

As already mentioned in the Introduction, there have been several attempts to
establish the Lagrangian formalism in biology [15-18], and different
Lagrangians were obtained for some selected biological systems. In this paper,
we concentrate on standard Lagrangians (SLs) because of their specific
physical meaning discussed above. Our goals are to derive SLs for some of the
model for population dynamics, and discuss their meaning and role of these
Lagrangians in biology.

###  2.3 Method to derive standard Lagrangians

The Lagrangian formalism requires prior knowledge of a Lagrangian. In general,
there are no first principle methods to obtain Lagrangians, which are
typically presented without explaining their origin. In physics, most
dynamical equations were established first and only then their Lagrangians
were found, often by guessing. Once the Lagrangians are known, the process of
finding the resulting dynamical equations is straightforward and it requires
substitution of these Lagrangians into the E-L equation. There has been some
progress in deriving Lagrangians for physical systems described by ODEs (e.g.,
[10-14]), and in this paper, we develop a new method to obtain standard
Lagrangians (SLs) for several models that describe interacting species of the
population dynamics. Let us point out that the same method can be used in
physics and in other natural sciences to obtain Lagrangians for systems
described by second-order ODEs.

The main objective is to solve the inverse (Helmholtz) problem of the calculus
of variations [6,7] and derive the standard Lagrangian for a given second-
order ODE. Let us consider the following ODE

$\ddot{x}+\alpha(x)\dot{x}^{2}+\beta(x)\dot{x}+\gamma(x)x=C_{0},$ (6)

where $\alpha(x)$, $\beta(x)$ and $\gamma(x)$ are at least twice
differentiable functions of the dependent variable only, and these functions
are to be specified by the selected models of the population dynamics. In
addition, $C_{0}$ is a constant that may appear in some of these models (see
Section 3). From a physical point of view, the above equation of motion
describes an oscillatory system that is affected by two damping terms with
$\alpha(x)$ and $\beta(x)$, and driven by the constant force $C_{0}$. In case,
$\alpha(x)=\beta(x)=C_{0}=0$, the equation represents a harmonic oscillator
[4,5].

Let us consider a simplified form of this ODE by taking $\beta(x)=0$ and
$C_{0}=0$, and obtain

$\ddot{x}+\alpha(x)\dot{x}^{2}+\gamma(x)x=0.$ (7)

Despite the presence of the damping-like term $\alpha(x)\dot{x}^{2}$, the
equation is considered to be conservative, which means that its standard
Lagrangian can be obtained [11,20] and written as

$L(\dot{x},x)={1\over
2}\dot{x}^{2}e{{}^{2I(x)}}-\int\displaylimits^{x}\tilde{x}\gamma(\tilde{x})e^{2I(\tilde{x})}d\tilde{x},$
(8)

with

$I(x)=\int\displaylimits^{x}\tilde{\alpha}(\tilde{x})d\tilde{x}.$ (9)

The main reason for reducing Eq. (6) to Eq. (7) is that the term
$\beta(x)\dot{x}$ is by itself a null Lagrangian that identically satisfies
the E-L equation [21-23], therefore, its derivation from any SL is not
possible [11,12]. In the following, we propose to account for this term in a
novel way.

We write Eq. (7) in the following form

$\ddot{x}+\alpha(x)\dot{x}^{2}+\gamma(x)x=F(x,\dot{x}),$ (10)

where the force-like term becomes

$F(x,\dot{x})=C_{0}-\beta(x)\dot{x}.$ (11)

All population dynamics models considered in this paper (see Section 3) are
represented by second-order ODEs in the form of Eq. (10) with $F(x,\dot{x})$
given by Eq. (11).

Our results presented in the Appendix generalize the previous work [20] and
demonstrate that the Lagrangian given by Eq. (8) is also the SL for Eq. (10)
if, and only if, this SL is substituted into the E-L equation

${d\over{dt}}\left({{\partial L}\over{\partial{\dot{x}}}}\right)-{{\partial
L}\over{\partial x}}=F(\dot{x},x).$ (12)

This form of the E-L equation is commonly known in physics, and the presence
of the force-like term on the right-hand side of the equation is justified by
$F(\dot{x},x)$ being a force that does not arise from a potential (e.g., [4]);
for further discussion of this term, see Section 3.3.

It is straightforward to verify that substitution of Eq. (8) into Eq. (12)
gives the original equation given by Eq. (10). This makes our method easy to
find SLs for any ODEs of the form of Eq. (10) as it requires identifying the
functions $\alpha(x)$, $\gamma(x)$ and $F(\dot{x},x)$, and evaluating the
integral in Eq. (9) as well as the one in the SL given by Eq. (8).

Our method is simple and easy to use for the population dynamics models
considered in this paper. Let us point out that this method can be applied to
any dynamical system that is described by the second-order ODEs of the form of
Eq. (10) in any areas of natural sciences.

##  3 Applications to models of population dynamics

###  3.1 Selected models

The models of the population dynamics considered in this paper are listed in
Table 1. Our selection process was guided by the previous work, by Trubatch
and Franco[17], and Nucci and Tamizhmani [18]. Both papers considered the
well-known population models that involve two interacting species described by
coupled nonlinear ODEs, namely, the Lotka-Volterra, Gompertz, Verhulst and
Host-Parasite models as shown in Table 1. The authors of these papers
determined Lagrangians corresponding to the ODEs representing mathematically
the models, either ad hoc [19] or using the method of Jacobi Last Multiplier
[17]; the Lagrangians of the same form were obtained, and they were treated as
the generating functions for the ODEs representing the models [17,18].

As it is well-known, Lagrangians can be of different forms and yet they would
give the same equation of motion [5,7]. In most cases, the forms of these
Lagrangians do not resemble the SLs in which the kinetic and potential energy-
like terms can be identified [10-14]. However, the main objective of this
paper is to derive the SLs and for some selected models of the population
dynamics and compare the obtained SLs to those previously found [18,19].
Because of the specific physical meaning of the SLs derived here, we are able
to address the role and meaning of these SLs in the population dynamics.

In selecting models of the population dynamics, we used the four models used
in the previous studies [19,18]. In addition, we selected the SIR model (see
Table 1).

Population models | Equations of Motion
---|---
Lotka-Volterra Model | $\dot{w_{1}}=w_{1}(a\>+\>bw_{2}\>)$
| $\dot{w_{2}}=w_{2}(A\>+Bw_{1})$
Verhulst Model | $\dot{w}_{1}=w_{1}(A+\>Bw_{1}\>+f_{1}w_{2})$
| $\dot{w}_{2}=w_{2}(\>a+\>bw_{2}\>+f_{2}w_{1})$
Gompertz Model | $\dot{w}_{1}=w_{1}{(}A\log{\bigr{(}}{w_{1}\over m_{1}}{\bigr{)}}+B{w_{2}}{)}$
| $\dot{w}_{2}=w_{2}{(}\>a\log{\bigr{(}}{w_{2}\over
m_{2}}{\bigr{)}}+b{w_{1}})$
Host-Parasite Model | $\dot{w}_{1}=w_{1}(a\>-bw_{2})$
| $\dot{w}_{2}=w_{2}(A-B{w_{2}\over w_{1}})$
SIR Model | $\dot{w}_{1}=-bw_{1}w_{2}$
| $\dot{w}_{2}=\>\>\>bw_{1}w_{2}-aw_{2}$

The first four models of the population dynamics presented in Table 1 describe
two interacting species (preys and predators) of the respective populations
$w_{1}(t)$ and $w_{2}(t)$ that evolve in time $t$, which is denoted by the
time derivatives $\dot{w}_{1}(t)$ and $\dot{w}_{2}(t)$. The coefficients $a$,
$A$, $b$, $B$ $f_{1}$, $f_{2}$, $m_{1}$ and $m_{2}$ are real and constant
parameters that describe the interaction of the two species. The Lotka-
Volterra, Verhulst and Gompertz models are symmetric, which means that the
depedent variables can be replaced if, and only if, the constants are
replaced, $a\rightarrow A$, $b\rightarrow B$, $f_{1}\rightarrow f_{2}$ and
$m_{1}\rightarrow m_{2}$. However, the Host-Parasite model is asymmetric in
the dependent variables.

The SIR model presented in Table 1 describes the spread of a disease in a
population and the dependent variables $w_{1}(t)$ and $w_{2}(t)$ represent
susceptible and infectious populations, with $a$ and $b$ being the recovery
and infection rates, respectively. Similarly to the Host-Parasite model, the
SIR model is also asymmetric but the origin and nature of this asymmetry in
both models is significantly different.

###  3.2 Steps to derive standard Lagrangians

We solve the inverse (Helmholtz) calculus variational problem and derive
standard Lagrangians for all selected models of the population dynamics shown
in Table 1. The following steps must be undertaken to solve the problem:

  1. 1.

Convert the set of coupled nonlinear ODEs for each model into a second-order
nonlinear ODE for each dependent variable.

  2. 2.

Cast the derived second-order ODEs into the equation of the same form as Eq.
(10).

  3. 3.

Compare the obtained second-order ODEs to Eq. (10) and identify the functions
$\alpha(x)$, $\gamma(x)$ and $F(x,\dot{x})$ for each variable in each model.

  4. 4.

Evaluate the integral in Eq. (9) and then determine the exponential factor.

  5. 5.

Evaluate the integral in Eq. (8).

  6. 6.

Use the above integrals and Eq. (8) to find the standard Lagrangian for each
variable in each model.

  7. 7.

Verify the derived standard Lagrangian by substituting it into the Euler-
Lagrange equation given by Eq. (12).

As pointed out in [19], the choice between the first and second order
description of the population dynamics is typically motivated by the
information available. In case only one population is observed, then the
second-order ODE can be solved for this population. To make it more general,
we derive the second-order ODE for each variable in each model, so both
equations can be solved if both populations are observed. For this paper and
the presented results, the second-order ODEs are required because our method
to derive the SLs is valid only for such equations.

Our method to derive the SLs for the selected models is straightforward and
the above steps are easy to perform and they always give the standard
Lagrangian for any second-order ODEs of the same form as Eq. (10). In the
following, we present the derived second-order ODEs for each variable of the
model and the resulting SLs.

###  3.3 Standard Lagrangians for selected models

Our method to derive standard Lagrangians for the models presented in Table 1
requires that the systems of coupled nonlinear first-order ODEs are cast into
one second-order ODE for each variable. All derived second-order ODEs can be
expressed in the same form as Eq. (10), which can be written as

$\ddot{w}_{i}+\alpha_{i}(w_{i})\dot{w}_{i}^{2}+\gamma_{i}(w_{i})w_{i}=F_{i}(\dot{w}_{i},w_{i}),$
(13)

where $i=1$ and $2$. Since $w_{i}(t)$ represents the population of species,
its derivative with respect of time $\dot{w}_{i}(t)$ describes the rate with
which the population changes, and $\ddot{w}_{i}(t)$ its acceleration. Despite
the presence of the damping-like term $\alpha_{i}(w_{i})\dot{w}_{i}^{2}$, the
LHS of the above equation is conservative [11,20] and it describes
oscillations of the population of species with respect to its equilibrium.
These oscillations are modified by the force-like term on the RHS of the
equation. Let us now describe this term.

Typically, the presence of any term with $\dot{w}_{i}(t)$ corresponds to
friction forces in classical mechanics. In the approach presented in this
paper, all friction-like terms that explicitly depend on $\dot{w}_{i}(t)$ are
collected on the RHS of the equation as $F_{i}(\dot{w}_{i},w_{i})$, which
becomes the force-like term. Since $F_{i}(\dot{w}_{i},w_{i})$ arises directly
from the friction-like terms, its origin is not potential, and therefore this
force-like term may appear on the RHS of the E-L equation (see Eq. 12) as it
is shown in [4].

In our derivations of the standard Lagrangians for the models of the
population dynamics presented in Table 1, we follow the above steps. As an
example, we show the calculations required for each step for the Lotka-
Volterra model. However, for the remaining four models in Table 1, we only
present the final results.

##  4 Lotka-Volterra Model

The Lotka-Volterra model was developed by chemist Alfred Lotka in 1910 and
mathematician Vito Volterra in 1926 [24,25], and this model describes the
interaction of two populations (predator-prey) based on the assumptions that
the prey increases exponentially in time without the predator, and the
predator decreases exponentially without the prey [19]. The model is symmetric
and it is represented mathematically by a system of coupled nonlinear ODEs
given in Table 1.

Step 1: Using the coupled nonlinear ODEs for this model, we convert the
equations into the following second-order ODEs for the variables $w_{1}$ and
$w_{2}$

$\ddot{w}_{1}-{\dot{w}_{1}^{2}\over
w_{1}}-B\dot{w_{1}}w_{1}-A\dot{w_{1}}+aBw_{1}^{2}+aA{w_{1}}=0,$ (14a)

and

$\ddot{w}_{2}-{\dot{w}_{2}^{2}\over
w_{2}}-bw_{2}\dot{w}_{2}-a\dot{w_{2}}+Abw_{2}^{2}+Aa{w_{2}}=0.$ (14b)

Step 2: We cast these equations into the form of Eq. (13), and obtain

$\ddot{w}_{1}\underbrace{-{1\over
w_{1}}}_{\alpha_{1}(w_{1})}\dot{w}_{1}^{2}+\underbrace{(Bw_{1}+A)a}_{\gamma_{1}(w_{1})}w_{1}=\underbrace{(Bw_{1}+A)\dot{w}_{1}}_{F_{1}(\dot{w}_{1},w_{1})}$
(15a)

and

$\ddot{w}_{2}\underbrace{-{1\over
w_{2}}}_{\alpha_{2}(w_{2})}\dot{w}_{2}^{2}+\underbrace{(bw_{2}+a)A}_{\gamma_{2}(w_{2})}w_{2}=\underbrace{(bw_{2}+a)\dot{w}_{2}}_{F_{2}(\dot{w}_{2},w_{2})}$
(15b)

Step 3: Using the above equations, we identify the functions

$\alpha_{1}(w_{1})=-{1\over w_{1}}$
---
$\alpha_{2}(w_{2})=-{1\over w_{2}}$
$\gamma_{1}(w_{1})=a\>(Bw_{1}+A)$
$\gamma_{2}(w_{2})=A\>(bw_{2}+a)$
$F_{1}(\dot{w}_{1},w_{1})=(Bw_{1}+A)\dot{w}_{1}$
$F_{2}(\dot{w}_{2},w_{2})=(\>bw_{2}+a)\dot{w}_{2}$

Step 4: Having identified $\alpha(w_{1})$ and $\alpha(w_{2})$, the intergral
given by Eq. (9) can be evaluated, and the result is

$I(w_{1})=-{\int\displaylimits^{\tilde{w}_{1}}}{dw_{1}\over
w_{1}}=-\ln|w_{1}|,$ (16a)

and

$I(w_{2})=-{\int\displaylimits^{\tilde{w}_{2}}}{dw_{2}\over
w_{2}}=-\ln|w_{2}|.$ (16b)

Then, the factors $e^{2I(w_{1})}$ and $e^{2I(w_{2})}$ become

$e^{2I(w_{1})}=e^{-2\ln|w_{1}|}={1\over w_{1}^{2}},$ (17a)

and

$e^{2I(w_{2})}=e^{-2\ln|w_{2}|}={1\over w_{2}^{2}}.$ (17b)

Step 5: Since $\gamma(w_{1})$ and $\gamma(w_{2})$ are known, the intergral in
Eq. (8) can be calculated, and we find

$\int\displaylimits^{w_{1}}\tilde{w_{1}}\gamma(\tilde{w_{1}})e^{2I(\tilde{w_{1}})}d\tilde{w_{1}}=a(Bw_{1}+A\ln|w_{1}|),$
(18a)

and

$\int\displaylimits^{w_{2}}\tilde{w_{2}}\gamma(\tilde{w_{2}})e^{2I(\tilde{w_{2}})}d\tilde{w_{2}}=A(bw_{2}+a\ln|w_{2}|).$
(18b)

Step 6: Using the above results and the definition of Lagrangian given by Eq.
(8), the following standard Lagrangians for the two dependent variables of the
Lotka-Volterra model are obtained:

$L_{1}(\dot{w}_{1},w_{1})=\underbrace{{1\over 2}\biggl{(}{\dot{w}_{1}\over
w_{1}}\biggl{)}^{2}}_{kinetic\ \rm
term}-\underbrace{a\bigr{(}Bw_{1}+A\ln|w_{1}|\bigr{)}}_{potential\ \rm term}$
(19a)

and

$L_{2}(\dot{w}_{2},w_{2})=\underbrace{{1\over 2}\biggl{(}{\dot{w}_{2}\over
w_{2}}\biggl{)}^{2}}_{kinetic\ \rm
term}-\underbrace{A\bigr{(}bw_{2}+a\ln|w_{2}|\bigr{)}}_{potential\ \rm term}$
(19b)

Step 7: Substituting the derived standard Lagrangians and
$F(\dot{w}_{i},w_{i})$ into the following Euler-Lagrange equations

${d\over{dt}}\left({{\partial
L}\over{\partial{\dot{w}_{i}}}}\right)-{{\partial L}\over{\partial
w_{i}}}=F(\dot{w}_{i},w_{i})e^{2I(w_{i})},$ (20)

where $i=1$ and $2$, we obtain Eqs. (15a) and (15b). This verifies that the
presented method to derive the SLs is valid.

##  5 Verhulst Model

This logistical (or Verhulst-Pearl) equation was first introduced by the
Belgian statistician Pierre Francois Verhulst in 1838 [27]. This model
describes the organisms’ growth dynamics in a habitat of finite resources,
which means the population is limited by a carrying capacity. This model is
valueable for optimisation of culture media by developing strategies and
selection of cell lines.

In this paper, the Verhulst model describes the population of interacting
species by considering self-interacting terms that prevent the exponential
increase or decrease in the size in the populations observed in the Lotka-
Volterra model [19]. The system of coupled nonlinear ODEs given in Table 1
shows that this model is symmetric.

After performing steps 1 through 3, the following functions for this model are
obtained:

$\alpha_{1}(w_{1})=-(1+\>b\>){1\over w_{1}}$
---
$\alpha_{2}(w_{2})=-(1+B){1\over w_{2}}$
$\gamma_{1}(w_{1})=\bigr{(}f_{2}-b\>\bigr{)}Bw_{1}^{2}+\bigr{(}Af_{2}-2Ab-a)w_{1}+A\bigr{(}a-Ab\bigr{)}$
$\gamma_{2}(w_{2})=\bigr{(}f_{1}-B\bigr{)}Bw_{2}^{2}+\bigr{(}\>af_{1}-2ab-A)w_{1}+a\bigr{(}A-aB\bigr{)}$
$F_{1}(\dot{w}_{1},w_{1})=-\dot{w}_{1}\bigr{[}(2b-1)Bw_{1}-f_{2}w_{1}^{2}+(2Ab-a)+(f_{2}-b)B\bigr{]}$
$F_{2}(\dot{w}_{2},w_{2})=-\dot{w}_{2}\bigr{[}(2B-1)bw_{2}-f_{1}w_{2}^{2}+(2aB-A)+(f_{1}-B)b\bigr{]}$

Substitution of these functions into Eq. (13) gives the second-order ODEs for
the variables $w_{1}$ and $w_{2}$.

Then, we implement steps 4 through 6 and the resulting standard Lagrangians
for the variables $w_{1}$ and $w_{2}$ of the Verhulst model are

$L(\dot{w}_{1},w_{1})={1\over 2}\biggl{[}\biggl{(}{\dot{w}_{1}\over
w_{1}}\biggl{)}^{2}-{\>(f_{2}-b)B\over(1-b)}w_{1}^{2}-{2(Af_{2}-2Ab-a)\over(1-2b)}w_{1}+{A(a-Ab)\over
b}\biggl{]}w_{1}^{-2b}$

(21a)

and

$L(\dot{w}_{2},w_{2})={1\over 2}\biggl{[}\biggl{(}{\dot{w}_{2}\over
w_{2}}\biggl{)}^{2}-{(f_{1}-B)b\over(1-B)}w_{2}^{2}-{2(af_{1}-2aB-A)\over(1-2B)}w_{2}+{a(A-aB)\over
B}\biggl{]}w_{2}^{-2B}$

(21b)

The kinetic and potential energy-like terms are easy to recognize, and the
functions $F_{1}(\dot{w}_{1},w_{1})$ and $F_{2}(\dot{w}_{2},w_{2})$ are given
above. Substitution of these Lagrangians into the E-L equations (see Eq. 20)
validates the method.

##  6 Gompertz Model

English economist Benjamin Gompertz proposed a model to describe the
relationship between increasing death rate and age in 1825. This model is
useful in describing the rapid growth of a certain population of organisms
such as the growth of tumors [28]. As well as, modelling the amount of
medicine in the bloodstream.

Here, we follow [17,19] and consider the Gompertz model for the population
dynamics This model generalizes the Lotka-Volterra model by including self-
interaction terms that prevent an unbounded increase of any isolated
population [19]; the self-interacting terms in the Gompertz model are
different than those in the Verhulst model. The mathematical representation of
this model given by the coupled and nonlinear ODEs in Table 1 shows that the
model is symmetric.

The steps 1 through 3 allow us to identify the following functions in Eq. (13)

$\alpha_{1}(w_{1})=-{1\over w_{1}}$
---
$\alpha_{2}(w_{2})=-{1\over w_{2}}$
$\gamma_{1}(w_{1})=\bigr{[}A\log\bigr{(}{w_{1}\over
m_{1}}\bigr{)}\bigr{]}w_{1}$
$\gamma_{2}(w_{2})=\bigr{[}a\>\log\bigr{(}{w_{2}\over
m_{2}}\bigr{)}\bigr{]}w_{2}$
$F_{1}(\dot{w}_{1},w_{1})=[Am_{1}+\>bw_{1}+g_{1}(\dot{w}_{1},w_{1})]\dot{w}_{1}-g_{1}(\dot{w}_{1},w_{1})Aw_{1}$
$F_{2}(\dot{w}_{2},w_{2})=[\>am_{2}+Bw_{2}+g_{2}(\dot{w}_{2},w_{2})]\dot{w}_{2}-g_{2}(\dot{w}_{2},w_{2})aw_{2}$

where

$g_{1}(\dot{w}_{1},w_{1})\>=\>a\log\>\biggr{[}{1\over
m_{2}B}\>\biggr{(}{\dot{w}_{1}\over w_{1}}-A\log\biggr{(}{w_{1}\over
m_{1}}\biggr{)}\biggr{)}\biggr{]},$ (22a) and

$g_{2}(\dot{w}_{2},w_{2})\>=A\log\>\biggr{[}{1\over
m_{1}b}\>\>\biggr{(}{\dot{w}_{2}\over w_{2}}-a\log\biggr{(}{w_{2}\over
m_{2}}\biggr{)}\biggr{)}\biggr{]}.$ (22b)

Then, the remaining steps result in the following standard Lagrangians for the
two dependent variables of the Gompertz model

$L_{1}(\dot{w}_{1},w_{1})={1\over 2}\biggr{(}{\dot{w}_{1}\over
w_{1}}\biggr{)}^{2}-A\biggr{[}\log\biggr{(}{w_{1}\over
m_{1}}\biggr{)}-1\biggr{]}w_{1}$ (23a)

and

$L_{2}(\dot{w}_{2},w_{2})={1\over 2}\biggr{(}{\dot{w}_{2}\over
w_{2}}\biggr{)}^{2}-a\biggr{[}\log\biggr{(}{w_{2}\over
m_{2}}\biggr{)}\>-1\biggr{]}w_{2}.$ (23b)

In both Lagrangians the kinetic and potential energy-like terms are seen, and
the forcing functions $F_{1}(\dot{w}_{1},w_{1})$ and
$F_{2}(\dot{w}_{2},w_{2})$ are given above. If we substitute these Lagrangians
into Eq. (20), the second-order ODEs for the variables $w_{1}$ and $w_{2}$ are
obtained.

##  7 Host-Parasite Model

This model describes the interaction between a host and its parasite. The
model takes into account nonlinear effects of the host population size on the
growth rate of the parasite population [19]. The system of coupled nonlinear
ODEs (see Table 1) is asymmetric in the dependent variables $w_{1}$ and
$w_{2}$.

Using steps 1 through 3, we find

$\alpha_{1}(w_{1})=-{1\over w_{1}}\Bigr{(}1+{B\over bw_{1}}\Bigr{)}$
---
$\alpha_{2}(w_{2})=-{2\over w_{2}}$
$\gamma_{1}(w_{1})=aA$
$\gamma_{2}(w_{2})=A(bw_{2}-a)$
$F_{1}(\dot{w}_{1},w_{1})=B{\>\>a^{2}\over b}\>+\Bigr{(}A-{2aB\over
bw_{1}}\Bigr{)}\dot{w}_{1}$
$F_{2}(\dot{w}_{2},w_{2})=(bw_{2}-a-A)\dot{w}_{2}$

After substituting these functions into Eq. (13), the second-order ODEs for
the variables $w_{1}$ and $w_{2}$ are obtained, and the equations are
asymmetric, which means that the remaining steps 4 through 6 must be applied
to each dependent variable ($w_{1}$ or $w_{2}$) separately.

The standard Lagrangian for the variable $w_{1}$ is

$L_{1}(\dot{w}_{1},w_{1})={1\over 2}\Biggr{(}{\dot{w}_{1}\over
w_{1}}\Biggr{)}^{2}e^{2B/bw_{1}}+aAEi\>\Biggr{(}{2B\over bw_{1}}\Biggr{)},$
(24a)

where the exponential integral $Ei(2B/bw_{1})$ is a special function defined
as

$Ei(z)=\int_{\infty}^{z}{{e^{\tilde{z}}}\over{\tilde{z}}}d\tilde{z},$ (25)

with $z=2B/bw_{1}$. It must be noted that $Ei(z)$ is not an elementary
function. Now, the standard Lagrangian for the variable $w_{2}$ is given by

$L_{2}(\dot{w}_{2},w_{2})={1\over 2}\Biggr{(}{\dot{w}_{2}\over
w_{2}}\Biggr{)}^{2}{1\over w_{2}^{2}}-A\Biggr{[}{1\over 2}{a\over
w_{2}}-b\Biggr{]}{1\over w_{2}}.$ (26a)

It is seen that there are significant differences between the Lagrangian for
$w_{2}$ and that for $w_{1}$ in both the kinetic and potential energy-like
terms. The differences are especially prominent in the potential energy-like
terms, whose explicit dependence on the exponential integral $Ei(2B/bw_{1})$
is a new phenomenon. The differences are caused by the asymmetry between the
dependent variables in the original equations (see Table 1), which makes this
model different than the fully symmetric Lotka-Volterra, Verhulst and Gompertz
models, whose standard Lagrangians are also fully symmetric.

$g_{1}(\dot{w}_{1},w_{1})\>=\>a\log\>\biggr{[}{1\over
m_{2}B}\>\biggr{(}{\dot{w}_{1}\over w_{1}}-A\log\biggr{(}{w_{1}\over
m_{1}}\biggr{)}\biggr{)}\biggr{]},$ (27a)

and

$g_{2}(\dot{w}_{2},w_{2})\>=A\log\>\biggr{[}{1\over
m_{1}b}\>\>\biggr{(}{\dot{w}_{2}\over w_{2}}-a\log\biggr{(}{w_{2}\over
m_{2}}\biggr{)}\biggr{)}\biggr{]}.$ (27b)

##  8 SIR Model

Kermack and McKendrick in 1927 derived the system of the first ODEs (see Table
1) describing the spread of a disease in a population [29]. It is the one of
the simplest model, dividing the population into three distinct sub-
populations: a susceptible population denoted by $w_{1}(t)$, the infectious
population represented by $w_{2}(t)$, and a recovered population, we denote as
$w_{3}(t)$.

It is seen that the dependent variable $w_{3}(t)$ does not appear explicitly
in the set of ODEs given in Table 1 because it is related to $w_{1}(t)$ and
$w_{2}(t)$ through the following population conservation law:
$d/dt(w_{1}+w_{2}+w_{3})=0$, which means that the sum of the three populations
must remain constant in time. Moreover, $a>0$ is the recovery rate and $b>0$
is the rate of infection, which means that the terms $-bw_{1}w_{2}$ and
$-aw_{2}$ represent newly infected and recovered individuals, respectively.

After performing steps 1 through 3, we obtain

$\alpha_{1}(w_{1})=-{1\over w_{1}}$
---
$\alpha_{2}(w_{2})=-{1\over w_{2}}$
$\gamma_{1}(w_{1})=0$
$\gamma_{2}(w_{2})=abw_{2}$
$F_{1}(\dot{w}_{1},w_{1})=(bw_{1}-a)\dot{w}_{1}$
$F_{2}(\dot{w}_{2},w_{2})=-bw_{2}\dot{w}_{2}$

Then, steps 4 through 6 give the following standard Lagrangians

$L_{1}(\dot{w}_{1},w_{1})={1\over 2}\biggr{(}{\dot{w}_{1}\over
w_{1}}\biggr{)}^{2},$ (28a)

and

$L_{2}(\dot{w}_{2},w_{2})={1\over 2}\biggr{(}{\dot{w}_{2}\over
w_{2}}\biggr{)}^{2}-abw_{2}.$ (28b)

The fact that the SIR model is asymmetric is shown the lack of the potential
energy-like term in $L_{1}(\dot{w}_{1},w_{1})$ and its presence in
$L_{2}(\dot{w}_{2},w_{2})$. However, the kinetic energy-like terms are the
same for the SLs for both variables, and they are also similar to such terms
in the SLs obtained for the other population dynamics models.

###  8.1 Comparisons of Lagrangians and models

The derived standard Lagrangians for the considered models of the population
dynamics are characterized by the terms corresponding to the kinetic and
potential energy as well as to the forcing function. These terms are easy to
identify (see Eqs. 19a and 19b) and they can be used to make comparisons
between the Lagrangians and models they represent. The models considered in
this paper can be divided into two families, namely, symmetric (Lotka-
Volterra, Verhulst and Gompertz) and asymmetric (Host-Parasite and SIR)
models. The SLs derived for these models are different than the Lagrangians
previously obtained [19,17]; the main difference is the explicit time-
dependence of those Lagrangians as compared to the SLs derived in this paper.
In the following, we describe the general characteristics of the derived SLs
and comment on their explicit time-independence.

The kinetic energy-like terms in all four models have the same factor
$(\dot{w}_{i}/w_{i})^{2}/2$, where $i$ = 1 and 2, which represents the ratio
at which the population changes to its value at a given time. For the Verhulst
and Host-Parasite models, this ratio is modified by the other factors that
depend on the concentration of species at a given time. It is interesting that
the kine tic energy-like terms in the Lotka-Volterra, Gompertz and SIR models
are independent from any constant parameters but for the other two models they
are; in case of the Host-Parasite models only the variable $w_{1}$ shows such
a dependence.

The potential energy-like terms of the Lotka-Volterra model depends linearly
on the concentration of species; however, the Verhulst, Gompertz and Host-
Parasite models also have nonlinear (second-order) terms in the concentration
of species. The SIR model is exceptional as its SL for the variable $w_{1}$
does not depend on any potential energy-like term. On the other hand, the SL
for the variable $w_{2}$ does depend on the potential energy-like term that is
linear in this variable. In all models, the potential energy-like terms depend
on the constant parameters that appear in the derived second-order ODEs for
these models. An interesting result is the presence of logarithmic terms in
the Lotka-Volterra and Gompertz models, and the exponential integral $Ei$ for
the variable $w_{1}$ for the Host-Parasite model. It must be also noted that
the form of the potential energy-like term for the SIR model is the simplest
among all the models considered here.

Since the kinetic and potential energy-like terms depend on the square of the
rate with which the populations change and the square of their concentration,
respectively, one may conclude that the derived SLs describe oscillatory
systems. The fact that none of the obtained SLs depends explicitly on time
indicates that the considered models oscillate in time with certain
frequencies around the equilibrium, and that these oscillations are not
directly affected by any ’physical damping’ associated with the presence of
terms that depend on $\dot{w}_{i}$, which are absent in derived the SLs; this
is the main difference between the SLs of this paper and those previously
obtained [19,17].

However, we must keep in mind that all damping terms represented by
$\dot{w}_{i}(t)$ have been moved to the force-like functions denoted by
$F_{i}(\dot{w}_{i},w_{i})$, which significantly vary for different models.
First, let us point out that the force-like functions become null Lagrangians
[21-23], which means that they identically satisfy the E-L equation, and
therefore they cannot contribute to any equation of motion. As a result, no
standard Lagrangian can properly account for them [11,12] because the
presented Lagrangian formalism is valid only for conservative systems.

Second, the force-like functions may depend only on $\dot{w}_{i}(t)$ and on
$\dot{w}_{i}(t)w_{i}(t)$, and the constant parameters, or may depend on higher
powers of thess variables that are arguments of the logarithmic functions. As
the presented results demonstrate, the forms of the force-like functions
significantly differ for different models, with the simplest being for the SIR
and Lotka-Volterra models, and then the increasing complexity for the Host-
Parasite and Verhulst models. The most complex form of
$F_{i}(\dot{w}_{i},w_{i})$ is found for the Gompertz model.

Third, the observed increase in complexity of the force-like function is
caused by the role played by the terms that depend on $\dot{w}_{i}$ as well as
on the combination of terms with $\dot{w}_{i}w_{i}$. In other words, the forms
of $F_{i}(\dot{w}_{i},w_{i})$ significantly affect the oscillatory behavior of
the systems represented by the SLs (see discussion above). The force-like
function may modify this behavior by causing the systems to reach the
equilibrium faster or diverge from it; detailed analysis requires solutions to
the ODEs representing the considered systems, which is out of the scope of
this paper.

Finally, let us point out that since the energy-like terms in the SLs have
specific meanings for the models, we suggest that they may be used to identify
physical similarities between different models as well as they may be utilized
to classify the models into categories that have similar biological
characteristics. Despite the fact that the SLs were derived only for the
population dynamics models, the developed method and the above discussion can
be easily applied to a broad range of biological systems, which will be
explored in succeeding papers.

##  9 Conclusions

We developed Lagrangian formalism for the following population dynamics
models: Lotka-Volterra, Verhulst, Gompertz, Host-Parasite and SIR models. For
ODEs that represent these models, we solved the inverse (Helmholtz)
variational calculus problem and derived standard Lagrangians for the models.
The main characteristic of these Lagrangians is that their kinetic and
potential energy-like terms are identified and that they can be used to make
comparisons between the obtained Lagrangians as well as the models.

The comparison demonstrates the role of these terms in the population dynamics
models and gives new insights into the models by showing their similarities
and differences. Moreover, the analogy between the derived standard
Lagrangians and that known for a harmonic oscillator in physics is used to
discuss the oscillatory behavior of the models with respect to their
equilibrium. In our approach, we collected the terms with the first-order
derivatives in time and identified them as the force-like functions, which
happened to be significantly different for each model and strongly depend on
the parameters of each model. By separating the force-like functions, we were
able to see the effects of these functions on the model’s oscillatory
behavior.

Our method of solving the inverse calculus of variation problem and deriving
standard Lagrangians is applied to the models of population dynamics. However,
the presented results show that the method can be easily extended to other
biological systems whose equations of motion are known.

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