Simulating the Soft Power Effect in Two-Dimensional Models

of A. Lotka – V. Volterra

VLADIMIR BOBROV

Institute of Mathematics and Computer Sciences,

Moscow Pedagogical State University,

14 Krasnoprudnaya str., Moscow 107140,

RUSSIAN FEDERATION,

YURY BRODSKY

Simulation Systems and Operation Research Department,

Federal Research Center “Computer Science and Control” of the Russian Academy of Sciences,

40 Vavilova str., Moscow 119333,

RUSSIAN FEDERATION

Abstract: - The most famous models of A. Lotka and V. Volterra, “predator-prey” and “competition,” are consid-

ered in relation to social systems. The peculiarity of the social systems that distinguishes them from the biological

ones is a quick reaction to the current situation. It turns out that the straightforward struggle of preys against

predators is ineffective, whereas the methods of “soft power” are able to get rid of the predators completely. The

article considers the competition equations as mathematical model of cross-cultural interaction. The study of the

model reveals the possibility of a paradoxical situation when one of the cultures positively treats the other, though

this other one is actually harmful to it. Soft power, in this case, masks a negative attitude, presenting it as a friendly

one. Conversely, in some cases, a harmless culture may be mistakenly perceived as a very negative one because

of the “brutality” of some of its manifestations.

Key-Words: Soft Power, Double Standards, Simulation, Predator – Prey, Competition

1 Introduction

Soft power – they say that this term was introduced

in 1990 by Joseph Nye of Harvard University, [1], for

the first time, but something similar can be found even

in ancient times – for example, in Lao Tzu’s Tao Te

Ching [2]. It is possible to say that cultural values, ca-

pable of inducing others to want what is wanted by the

soft power operator, are the cornerstone of this con-

cept.

2 Predator – Prey Model

Take a look at the most famous model of A. Lotka and

V. Volterra, – “perdator-prey,” [3].

dt =αN 1−M

M,

dt =βM N

N−1.

(1)

Here Nis the population size of preys, Mis the

population size of predators, α, β are the Malthusian

factors, Nis the minimum number of preys necessary

to feed the predators, M– is the maximum number of

predators that the prey population can withstand.

Note that the essence of the model (1) was de-

scribed, including the oscillatory nature of its solu-

tions (although in natural language), by L.N. Tolstoy

in [4], almost 40 years before A. Lotka and V. Volterra.

2.1 Elements of Macro Analysis

We transform our equations (1):

dt =αN(M−M)

M,dM

dt =βM(N−N)

Next, – divide the first equation by the second:

dM =αNN(M−M)

βMM(N−N).(2)

Let us separate the variables:

M−M

βMM dM +N−N

αNN dN = 0,

or,

βM −dM

βM +dN

αN −dN

αN = 0.

Received: June 19, 2022. Revised: August 5, 2023. Accepted: September 11, 2023. Published: October 4, 2023.

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Figure 1: The Phase Plane in N, M coordinates.

Integrate both parts – is the next step.

βln M−M

M+1

αln N−N

N=C. (3)

Consider equality (3) as the function of Nand M.

C(N, M) = 1

αln N−N

N+1

βln M−M

M.

Let’s call the function C(N, M )the potential of

the predator-prey system, and the functions

C(M) = 1

βln M−M

M,

C(N) = 1

αln N−N

N,

potentials of predator and prey populations, respec-

tively. This definition of potential is good because

it has the same functional form for both populations.

Then C(N, M) = C(N) + C(M).

Finding a gradient C(N, M)at the point (N, M).

This is the vector

1

αN −1

αN ,1

βM −1

βM .

The appearance of the gradient suggests that the

function C(N, M)is strictly concave and reaches a

strict global maximum at the point N, M.

Indeed, the partial derivative of the function

C(N, M)by Nis

∂C (N, M)

∂N =1

α1

N−1

N.

We see that if N < N, the derivative is strictly

positive, at N=N, it is zero, and if N > N, it is

strictly negative.

Similarly, the partial derivative of the function

C(N, M)by Mis

∂C (N, M)

∂M =1

β1

M−1

M.

Again, we see that if M < M, the derivative

is strictly positive, at M=M, it is zero, and if

M > M, it is strictly negative. Since for any so-

lution N(t), M(t)of the predator-prey system, the

following is true: (N(t), M (t)) = const, the plane

C(N, M) = C, and the surface

C(N, M) = 1

αln N−N

N+1

βln M−M

M,

intersecting if const ≤C(N, M), this gives us the

trajectory of the system on the phase plane.

Moreover, from the above analysis of derivatives,

it follows that if N(t), M(t)and ˜

N(t),˜

M(t)are solu-

tions of our predator-prey system with different initial

conditions, (N(t), M(t)) = C,˜

N(t),˜

M(t)=˜

and C < ˜

C, then the phase trajectory ˜

N(t),˜

M(t)

lies strictly within the phase trajectory N(t), M(t).

The point N, Mis the global maximum point of the

function; thus, (N, M )lies within the phase path of

any solution. The example is shown in Fig. 2 below.

Figure 2: Level Planes C(N, M) = const.

Thus, we can say that our invariant

αln N−N

N+1

βln M−M

M=C,

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is the degree of proximity (topological, not metrical,

as the belonging to the phase trajectory C(N, M) =

const) of the phase trajectory to the equilibrium point

– the center N, M.On any solution N(t), M(t),

this degree of proximity is preserved. Phase trajecto-

ries with a larger value of C(N(t), M(t)) are inside

the phase trajectories with a smaller value, and the

stationary point N, Mis inside all the trajectories.

The above causes associations between the poten-

tial of the system and entropy. First, because it is

a measure of proximity to equilibrium – the station-

ary point of the system, where the maximum potential

N, Mis reached. Secondly, the essential role of the

logarithm in its definition.

2.2 Simulation Experiment with the Classic

Model

The Fig. 3 below shows examples of the model plots

with different initial conditions, built in the AnyLogic

system, [5].

Figure 3: Simulation of the Predator-Prey Model.

We see that the potential of the system is preserved

on the trajectory, although part of the potential of

preys can go to predators, and vice versa (just as when

a massive point moves around the center of gravity,

part of its potential energy can go into kinetics, and

vice versa but the total energy is preserved, and its

magnitude determines the type of trajectory). If the

phase trajectory of the system passes near the station-

ary point, the potentials of the populations can be con-

sidered approximately unchanged (their variances are

of the highest order of smallness compared to the tra-

jectory deviations from the stationary point).

Further, the higher the potential of the system, the

closer its trajectory to the stationary point (in Fig. 3,

N= 5000,M= 2000).

The graph also shows that the frequency of oscilla-

tions remains constant only in the small neighborhood

of the centerN, Mand with a decrease in the poten-

tial of the system, it also decreases. It might also be

noted that in the small neighborhood of the center, the

dynamics of populations are almost sines and cosines;

phase trajectories are circles, althoufg “in large” they

differ obviously.

2.3 Predator Control: Soft Power vs Fight

Suppose the preys resented their role as predator fod-

der in the model and decided to fight against oppres-

sion by the latter. The first thing that comes to mind,

and what the oppressed peoples did throughout the

20th century, is to kill predators, or at least signif-

icantly reduce their number. As the history of the

20th century shows, it is most likely not be possible

to eliminate everyone, but the reduction of the pop-

ulation size, as we will see later, does not solve the

problem.

Suppose that there were about 1,000 preys and

about 2,000 predators then the number of the lat-

ter was reduced to 300 at a time. The system sim-

ply jumped to another, more external trajectory, with

higher peaks for both predators and preys. In the sec-

ond example there were about 5,000 preys and more

than 6,000 predators, and reduced the number of the

latter was 1,500 that is more than four times less. The

trajectory jumped to the internal trajectory with small

oscillation amplitudes. Fundamentally, nothing has

changed in the model.

Figure 4: Instant Elimination of Part of the Predators.

What should preys do to improve their position in

the system if the elimination of some predators does

not have an effect?

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Let’s try to bring competition to the dynamics of

the prey population. Add the competition summand

with the environmental capacity N∗to the equation

of the dynamics of preys in (1):

dt =αN 1−N

N∗−M

M.

The center N, M(see Fig. 1) turns into a stable

focus and shifts left (see Fig. 5) to the point

N, M(1 −N

N∗).

As soon as the ratio N∗≤Nbegins to be fulfilled,

predators have no place in the system they die out (see

Fig. 5 and Fig. 6).

Figure 5: Competition among the Preys.

Note that the condition for the extinction of preda-

tors is that the entire resource is mastered in the pro-

cess of preys’ competition

N∗≥1.

At the same time, preys do not lose anything: their

number still fluctuates in the area N, and after the ex-

tinction of predators, it is possible to weaken the com-

petition – and increase N∗.

Note, that certain religions, as well as thinkers

such as L.N. Tolstoy, [4], M.K. Gandhi, [6], advocate

for non-resistance to evil trough violence, as an alter-

native to the retribution of “an eye for an eye”, and this

thesis often seems to us incomprehensible, impracti-

cal, idealistic, etc. Our simulations (Fig. 4) shows,

that direct fighting with the predators is not fruitful for

preys. Nevertheless, we see that preys can quite real-

istically get rid of predators without any violence, ex-

clusively by the self-organization – by increasing in-

ternal competition, i.e. through the self-government,

self-support, self-improvement.

Figure 6: Preys‘ Competition, as a Soft Power.

If you look at the competition in the predator-

prey model through the eyes of a reasonable predator,

then, on the contrary, in order to maintain the status

quo in the system, predators should avoid competi-

tion among preys in every possible way. For example,

isolate them from each other and feed them (such as

chickens or cows at a farm).

At the same time, in such a system, the average

number of preys may be much greater than it could be

in nature without reasonable predators. In this case,

the question arises: if we consider the average pop-

ulation to be the criterion for its success, then is the

presence of reasonable predators an evil, or, on the

contrary, a good for a prey population in such a sys-

tem?

3 Soft Power in a Competition Model

Now let us consider the next famous model of

A. Lotka and V. Volterra: the “competition,” [3],

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dt =αN 1−N

N∗−mM

M∗,

dt =βM 1−nN

N∗−M

M∗.

(4)

Here Nand M– populations sizes of competi-

tors, α, β are Malthusian factors, N∗and M∗are en-

vironmental capacities that is the maximal numbers

of each type of competitor, for which the system re-

source is steel enough, nand mare double standard

factors, they show how many times the competition

with aliens differs from the one with compatriots.

Double standards are characterized by different

applications of the principles, laws, rules, and esti-

mates to the same actions of various subjects, depend-

ing on the degree of loyalty of these subjects to the es-

timator or other reasons of benefit to him. We will see

that the double standards are effective control mech-

anisms in competitive systems [7].

It is interesting to consider system (4) not in the

biological domain, as in [3], but in the social one, [8],

where system (4) becomes the constraints of the dif-

ferential game – the capability of social systems to

change behavior in a short time in response to the cur-

rent situation – turns the dynamic system (4) into a po-

sitional differential game, where the double standard

factors nand mbecome the controls of players. That

is why double standards are so popular in interstate

relations.

We shall distinguish the following ranges of these

double standard factors:

• Supertolerance, if −∞ < n, m < 0,

nm < 1.

• Tolerance, if 0≤n, m < 1.

• Equal treatment (no double standards), if nor m

equals to 1.

• Intolerance, if 1< n, m < ∞.

It occurs, [7], that if the double standard factors

are less than one (tolerance), the cultures are friendly

– they can exist together. If the double standard co-

efficient of a culture is greater than one (intolerance),

this culture constitutes a real danger to another, and

may force it out of the system.

Now let us look at the situation, for example, from

the position of the culture Nrepresentative. First, the

value N

N∗is well-known to him because it is a way of

attitude to compatriots in the culture N– the manner

of correct behavior that has been taught since child-

hood. Secondly, the value mM

M∗is also known; it

is the competitive pressure of the culture M, which

the representatives of the culture Ndirectly observe

and feel because they are under this pressure. Most

likely, these values are not equal N

N∗6=mM

M∗, as the

cultures are really different.

Further, it is quite natural to assume that if N

N∗>

M∗, then the culture Mis pleasant to the represen-

tative of the culture N– usually it is pleasant to any-

body when the pressure upon him weakens. Perhaps

he assesses this situation approximately like: “Ah,

what darlings, these well-mannered people of M–

not as my rough compatriots!” On the contrary, if

N∗< m M

M∗, then the representative of Ndoes not

like the Mpopulation; very few people like the pres-

sure stronger than usual. Most likely, he will think,

“Well, and how savage these Mare! It is quite impos-

sible to live near them! They are not able to behave

at all!”

Actually, both first and second estimations can be

deeply wrong. In the system (4), nothing depends

upon the comparison between the values N

N∗and

M∗, as well as the comparison between M

M∗and

N∗. The behavior of the system (4) depends only

upon the combination of ranges of double standard

factors nand m, [7].

For example, if N

N∗>> m M

M∗, but at the same

time m > 1, the situation can be dangerous for the

culture N; it can disappear completely after a time be-

cause of the neighborhood with the “lovely and well-

mannered” people, especially if it puts n≤1, having

been under the illusion of Mculture friendship due to

the first inequality.

On the contrary, if N

N∗< m M

M∗and even

N∗<< m M

M∗, but m < 1, there is no danger for

the culture Nto disappear near the culture M. The

rival culture may be unpleasant since the competitive

pressure is greater due to it, but it is by no means fatal

since there is no danger of disappearing from such a

neighborhood. Moreover, if n > 1, then the culture

Nforces out the alien culture trough time.

However, if the system (4) becomes a differential

game, the double standard factors nand mare not

observed directly. For the representative of the culture

Nto define m, it is necessary to compare given him

in feelings mM

M∗with M

M∗, but the last value, as a

rule, is unknown to him: studying foreign cultures is

a destiny of the rather narrow circle of specialists.

The only true measure of the culture is this culture

itself, not any other one. The competitive pressure of

a foreign culture is to be compared with its own in-

ternal competition, but by no means with the internal

competition of the native culture.

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4 Conclusion

The soft power (competition among preys) is able

to solve the problem of preys in the “predator-prey”

model, but the armed struggle of preys against the

predators cannot. This brings to mind the works by

L.N. Tolstoy and M.K. Gandhi about non-resistance

to evil trough violence.

It is also interesting to look at the model through

the eyes of a predator. It turns out that it is necessary

to put the preys in cages like chickens or in stalls like

cows at a farm and feed them to their fullest to exclude

the competition. Does this remind you of something

(like the chickencoop of cities with universal basic in-

come)?

The elementary “competition” model teaches us

that it is incorrect to measure one culture by the gauge

of another; such a measurement is not valid because

it is not informative. The only true yardstick for the

culture is this culture itself, but by no means any other

cultures.

In the authors’ subjective view, this paradox il-

lustrates why Russian cutting through a “window to

Europe” was not too successful during the last 300

years. Because here m > 1, though, N

N∗>> m M

M∗,

was fulfilled, which was attractive but not healthy

for the Npopulation in terms of our competition

model. Slavs once lived in Europe, but little of them

remained. At the same time, Russians survived un-

der the Horde Yoke, and under the Ottoman Empire,

Southern Slavs did (m < 1), though very unpleasant

memories about these historical periods remained in

the folklore of the survivors ( N

N∗<< m M

M∗).

These simplest models of A. Lotka and V. Volterra,

which laid the foundation of mathematical biology at

the beginning of the twentieth century, being trans-

ferred to the domain of social systems, cannot claim

the accuracy of numerical prediction, of course, be-

cause of their primitiveness.

However, they draw the researcher’s attention to

the following qualitative questions about the social

systems’ dynamics:

1. Are the appeals of L.N. Tolstoy and M.K. Gandhi

to the non-resistance as naive as they seem at first

glance? The twentieth century was one of social

“preys” armed struggle against the social “preda-

tors”. Was the result good enough? Maybe im-

proving sustainable development would be bet-

ter?

2. Do our politicians, who face problems of cross-

cultural interaction, understand well with whom

to be friends and on whom to keep an eye with

great wariness? The model of the soft power ef-

fect in a competition system shows that it is very

easy to make a wrong choice even in a simple sys-

tem of two differential equations with two vari-

ables!

It can be argued that social systems are much more

complicated than simple two-dimensional differential

equations. For instance, they are comprised of agents

who have their own behaviors and goals. However, it

turns out that the effects of the soft power and double

standards described here also occur in the agent ana-

logues of systems (1) and (4), implemented by cellular

automata, [9].

References:

[1] Nye J.S. Soft Power: The Means to Success in

World Politics, Public Affairs, NY, 2004.

[2] Lao Tzu Tao Te Ching, SHAMBALA, 2011.

[3] Volterra V. Leçons sur la Théorie Mathématique

de la Lutte pour la Vie, GauthierVillars, Paris,

1931.

[4] Tolstoï L.N. My Religion, Thomas Crowell, 1885.

https://ia903401.us.archive.org/15/items/myre-

ligion0000tols/myreligion0000tols.pdf, last

accessed 2023/05/29.

[5] Grigoryev I. AnyLogic 7 in Three Days, 2 ed,

AnyLogic, 2015.

[6] Gandhi M.K., An Autobiography or My Experi-

ments With Truth, Beacon Press, 1993.

[7] Brodsky Y.I. Tolerance, Intolerance, Identity:

simple math models of cultures’ interaction, LAP

LAMBERT Academic Publishing, 2011.

[8] Brodsky Y.I. Cross-Cultural Interaction as a

Positional Differential Game, Facing an Un-

equalWorld: Challenges for Russian Sociol-

ogy, Editor-in-Chief V. Mansurov, Moscow-

Yokohama, 2014, pp. 313-316.

[9] Bobrov V.A., Brodsky Y.I., Modeling of

double standards and soft power in cellu-

lar automata competition systems, E3S Web

of Conf., vol. 405, 02016, 2023, pp. 1-9.

DOI:10.1051/e3sconf/202340502016.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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