Journal of Robotics, Networking and Artificial Life
Vol. 8(4); March (2022), pp. 241–244
DOI: https://doi.org/10.2991/jrnal.k.211108.002; ISSN 2405-9021; eISSN 2352-6386
https://www.atlantis-press.com/journals/jrnal

Research Article

A Mathematical Model of Tritrophic Systems
Yasuhiro Suzuki*
Graduate School of Informatics, Nagoya University, Furocho Chikusa, Nagoya City, Aichi Prefecture 464/0814, Japan

ARTICLE INFOABSTRACT

Article History

Lotoka–Volterra, LV equations are to model predator–prey problem. In principle, the LV equations are belongs to a two-personsystem. Even if there are many-body, it is structurally in two-body, i.e., with three or more predators and a prey. On the otherhand, chemical ecology has shown that plants damaged by predation produce information chemicals (Hervibore Induced Plant
Volatile, HIPV) that attract natural enemies. Chemical ecology suggests that the ecosystem is a tri-trophic system consisting ofpredator–plant (HIPV)–prey. Therefore, chemical ecosystems are essentially different from LV equations. This paper proposes abasic equation for tri-trophic systems and investigates their stability.

Received 20 November 2020
Accepted 10 August 2021
Keywords
Chemical ecology
Lotoka–Volterra equation
tri-trophic systemmathematical biology© 2021 The Author. Published by Atlantis Press International B.V.
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION
Mathematical ecology has used the Lotoka–Volterra, i.e., LV equations which are the basic equation of the theory [1]. LV equations
are differential equations with the predator as y and the prey as x,
shown in Equations (1) and (2).				

dx= k1x - k2 xy , dt(1)dy= k2 xy - k3 y ,  (2)dt

The equilibrium point of the LV equation can be found as the point
where the derivative of x and y is zero; as shown in Equations (3)and (4);		 0 =dx= x(k1 - k2 y ), dt(3)dy		 0 = dt = y(k2 x - k3 ).  (4)We obtain the equilibrium points,
 k k  (x , y ) = (0, 0),  3 , 1   .
 k2 k2  The positive and negative differential values of x or y around equilibrium points give the behavior of the LV equations.

Email: ysuzuki@nagoya-u.jp.com; www.ysuzuki.info*

If y is more significant than k1/k2, x decreases, and if y is minor, itincreases. On the other hand, if x is more prominent than k3/k2,y increases, and if it is smaller, y decreases. Therefore, x and yoscillate around the equilibrium point; the solution to these equations are periodic (Figure 1) [1].

1.1. Chemical Ecology
The LV equations do not include plants. Therefore, the LV equations can also describe carnivores–herbivores relationships otherthan herbivores and carnivores [2].
Since the late of 1980s, chemicals are essential to ecosystems. Animals,insects, and plants produce non-volatile chemicals and volatile chemicals such as proteins. Non-volatile chemicals are used, for example,
for individual recognition between insects; personal credit is doneby touching proteins on the body surface of the other insect with theantennae. On the other hand, Volatile chemicals are Terpenes produced by plants; the Terpenes are called Herbivores Induced PlantVolatile, HIPV [3,4]. Terpene is a hydrocarbon composed of isoprene, a biological substance produced by plants, insects, and fungi.
When a plant is attacked, the plant chemically analyzes the salivaof the attacker and identify it. A plant can identify multiple attackers by its saliva. If the plant tells the attacker, it produces a volatile chemical that attracts the attacker’s natural enemies. About
three types of terpenes are used in the interaction between naturalenemies and plants. Different blends of these terpenes are used toattract suitable natural enemies.
In the case plants suffer mechanical damage, they synthesize salicylic acid to repair the damage. On the other hand, when attackers
damage a plant, the plant switching synthesizing salicylic acid toterpenes.

Y. Suzuki / Journal of Robotics, Networking and Artificial Life 8(4) 241–244

242

3. EQUATION OF THE TRI-TROPHICSYSTEM
In the tri-trophic system in chemical ecology, HIPV is crucial; inthe LV equations, carnivores can have herbivores as much as possible, while in the tri-tropihic chemical ecological system, carnivores
cannot have herbivores if there is no HIPV. Before herbivores takeplants, there is no HIPV, and it is processed in proportion to thenumber of plants eaten.

Figure 1 | Dynamics of LV equation.

2. MODEL OF TRI-TROPHIC SYSTEM
2.1. Rule Dynamics
The group dynamics governing the elements can exist in a varietyof forms. Among them, there are mutually independent dynamicalrules that are fundamental. These basic dynamics rules are calledrules, and the motion in the rule space is called rule dynamics [2].
In this paper, a rule rewrites a multiset of the alphabet; we denotea set of alphabets as AL. A multiset, Ms, is defined by a set and apair of sup functions, <Ms, Sup>, where Sup is a function fromthe elements of the multiset to a natural number, including zero.
For example, SupMs(a) gives the number of occurrences of an in Ms
SupMs(a) = two means that Ms contains two a’s, while supMs(a) = 0means that Ms contains zero a’s. We denote a multiset explicitly as{a, a, b, b, b} = Msi; SupMsi(a) = 2, SupMSi(b) = 3 and SupMSi(c) = 0.
A multiset rewriting rule is defined by a pair of multisets, <{l}, {r}>.
If Ms, then the intersection of Ms and l is rewritten to r. If {l} ⊆ Ms
then {l} is deleted and into {r} is included. For example, the set ofrules of rule dynamical expression of LV equations are as follows:< {a} ,{a, a} >< {a, b} ,{b, b} >< {b} ,{Null} >r1,r 2,r 3,

where Null stands for and empty multiset; r1 transform {a, a, b} to{a, a, a, b}, r2 rewrites it into {a, a, b, b} then r3, {a, a, b}.
The tri-trophic system in chemical ecology is composed of plants,herbivores, carnivores and HIPV; in the system, plants grow, andif a herbivore eats a plant, the plant generates HIPV, and the herbivore grows; if there is HIPV and herbivores, a carnivore will be
attracted and get carnivores to increase population, and the plant
stopped generating HIPV. In this scenario, we ignore the increasing
population of herbivores and carnivores as the LV equations alsoignore the process.
We will denote herbivore as h, carnivores as c and HIPV as d. Basedon the scenario, a set of rules of a dynamical expression of thetri-trophic system are as follows.
< { p} , { p, p} >,< { p, h} , {h, h, v} >,< {c, h, v} , {c, c} >,< {c} , {Null} > .

r1,r 2,r 3,r4

The LV equations cannot describe the tri-trophic chemical ecosystem conventionally. LV equations with multi-bodies have been
proposed in Dimas Martins [5]. The n species is essentially in abiotrophic system [6–9]. And Sabelis et al. [10] proposed mathematical models of tritrophic system with modified LV equations
and game theory in two-species.
Since it is composed of multiple herbivores or carnivores, it can have
many bodies interactions between biotrophic layers and cannot
have interactions beyond these layers. Therefore, the generalized
LV equations cannot essentially describe the chemical ecosystem.

3.1. Transform Dynamical Rule Systems
to Differential Equations
We regard the dynamical rule system as a jump Markov process,where every rule is a transition rule; we obtain transition rules ofthe procedure as r1:p→p,pr2:h,p→h,h,vr3:c,h,v→c,cr4:c→Null,respectively.
We give transition probabilities as		

k1 p,
k1 p + k2 ph + k3chv + k4c(5)k2 ph		 k p + k ph + k chv + k c ,  (6)12

3

4k chv		 k p + k ph3+ k chv + k c ,  (7)1

3

2

4kc		 k p + k ph +4 k chv + k c .  (8)
12

3

4

Increments of each rule are (1, 0, 0, 0), (−1, 1, 0, 1), (0, −1, 1, –1)
and (0, 0, −1, 0), respectively. Hence, we obtain expectation valuesof each rule, (5)–(8) gives (9);
æ æ1ö
æ0ö
æ 0 öö
æ< p >ö
æ -1 ö
ç1÷
ç -1 ÷
ç 0 ÷÷
ç < h > ÷ 1 ç ç0÷
÷ = ç k1 p ç ÷ , k2 ph ç ÷ , k3chv ç ÷ , k4c ç ÷ ÷ ,    çç0÷ç1÷
ç -1 ÷ ÷ç < c > ÷ r ç ç0÷
ç1÷
ç -1 ÷
ç 0 ÷÷
ç<v >÷
ç ç0÷
øè
ø
è øøè
øèø
è
è

(9)

where ρ = k1p + k2ph + k3chv + k4c. If p, h, c are sufficiently large
and the change in probability value due to variation of each value
is negligible, the following deterministic equation (10)–(12) can beobtained;		d á pñ
= k1 p - k2 ph, dt(10)

Y. Suzuki / Journal of Robotics, Networking and Artificial Life 8(4) 241–244

243

v = 0 or y < a/bv, dy/dt < 0, while y > a/bv, dy/dt > 0 and in casev = 0 or x < c/bv, dx/dt > 0, while x > c/bv, dx/dt > 0. (Figure 2).

4. CONCLUSION
We proposed the equation of three species interactions. We
showed that the equilibrium points of this equation are x and yat infinite points including 0. Note that we are considering a positive number including 0, because it is a model of an ecosystem.
In this experiment, the initial values of Herbivores were set
to 10 and 100 in increments of 20, while the initial value ofcarnivores was set to 5. The behaviour of tri-trophic equation
was stable at all initial values and did not change significantly(Figure 3).
Figure 2 | Phase diagram of tri-trophic equations.

CONFLICTS OF INTEREST
The author declares no conflicts of interest.

REFERENCES

Figure 3 | (Left) time developments of 10 species tri-trophic equation,(right) LV equation.		

d áh ñ= k2 ph - k3chv ,  (11)dtd ác ñ		 dt = k3chv - k4c.  (12)Note that this equation is a three bodies LV equation if we ignore v.

3.2. The Behavior of Basic Equations ofChemical Ecosystems
To investigate the behaviors of the model, Equation (13) wasembedded in Equation (14) to make it an approximate equations;		

d á x ñ d áv ñ== ax - bxyv , dtdt		dá y ñ= bxyv - cy ,  (14)
dt

(13)

where initial value of x, y > 0, v = 0 and no x, y and v are positive
including zero. When v = 0, equilibrium point is (0, 0), v ≠ 0, (0, 0)
and (c/bv, a/bv). Hence, dynamics of x and y become stable; when[1] A.J. Lotka, Elements of Physical Biology, Williams and Wilkins
Company, Baltimore, 1925.
[2] Y. Aizawa, Y. Nagai, Bussei Kenkyu (Kyoto), Plant Signal. Behav.
48 (1987), 316–320 [in Japanese].
[3] A.R. War, H.C. Sharma, M.G. Paulraj, M.Y. War, S. Ignacimuthu,Herbivore induced plant volatiles: their role in plant defense forpest management, Plant Signal. Behav. 6 (2011), 1973–1978.
[4] J. Takabayashi, M. Dicke, Herbivore-induced plant volatiles
with multifunctional effects in ecosystems: a complex patternof biotic interactions, in: T. Abe, S.A. Levin, M. Higashi (Eds.),
Biodiversity, Springer, New York, NY, 1997.
[5] A. Dimas Martins, E. Gjini, Modeling competitive mixtures withthe Lotka-Volterra framework for more complex fitness assessment between strains, Front. Microbiol. 11 (2020), 572487.
[6] E.H. Kerner, Comment on Hamiltonian structures for the ndimensional Lotka–Volterra equations, J. Math. Phys. 38 (1997),
1218.
[7] E. Chauvet, J.E. Paullet, J.P. Previte, Z. Walls, A Lotka-Volterra
three-species food chain, Math. Mag. 75 (2002), 243–255.
[8] Y. Takeuchi, Global dynamical properties of Lotka-Volterra systems, World Scientific, Singapore, 1996.
[9] G. Bunin, Ecological communities with Lotka-Volterra dynamics, Phys. Rev. E 95 (2017), 042414.
[10] M.W. Sabelis, M. Van Baalen, B. Pels, M. Egas, A. Janssen,
Evolution of exploitation and defense in tritrophic interactions,in: U. Dieckmann, J.A.J. Metz, M.W. Sabelis, K. Sigmund (Eds.),
Adaptive Dynamics of Infectious Diseases: In Pursuit of VirulenceManagement, Cambridge University Press, Cambridge, 2002, pp.
297–322.

Y. Suzuki / Journal of Robotics, Networking and Artificial Life 8(4) 241–244

244

AUTHOR INTRODUCTION
Dr. Yasuhiro Suzuki
He is an Associate Professor of Graduate Schoolof Informatics, Nagoya University Japan.
Graduated from Japan Advanced Instituteof Science and Technology in 1995, received
D. Informatics from Kyoto University in 2001.