1.1. Chemical Ecology

The LV equations do not include plants. Therefore, the LV equations can also describe carnivores–herbivores relationships other than 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 done by touching proteins on the body surface of the other insect with the antennae. On the other hand, Volatile chemicals are Terpenes produced by plants; the Terpenes are called Herbivores Induced Plant Volatile, 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 saliva of 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 natural enemies and plants. Different blends of these terpenes are used to attract 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 to terpenes.

2.1. Rule Dynamics

The group dynamics governing the elements can exist in a variety of forms. Among them, there are mutually independent dynamical rules that are fundamental. These basic dynamics rules are called rules, and the motion in the rule space is called rule dynamics [2]. In this paper, a rule rewrites a multiset of the alphabet; we denote a set of alphabets as AL. A multiset, Ms, is defined by a set and a pair of sup functions, <Ms, Sup>, where Sup is a function from the elements of the multiset to a natural number, including zero. For example, Sup_Ms(a) gives the number of occurrences of an in Ms Sup_Ms(a) = two means that Ms contains two a’s, while sup_Ms(a) = 0 means that Ms contains zero a’s. We denote a multiset explicitly as {a, a, b, b, b} = Ms_i; Sup_Msi(a) = 2, Sup_MSi(b) = 3 and Sup_MSi(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 of rules of rule dynamical expression of LV equations are as follows:

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, and if 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 also ignore the process.

We will denote herbivore as h, carnivores as c and HIPV as d. Based on the scenario, a set of rules of a dynamical expression of the tri-trophic system are as follows.

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 of the procedure as r1:p→p,pr2:h,p→h,h,vr3:c,h,v→c,cr4:c→Null, respectively.

We give transition probabilities as

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 values of each rule, (5)–(8) gives (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 be obtained;

Note that this equation is a three bodies LV equation if we ignore v.

3.2. The Behavior of Basic Equations of Chemical Ecosystems

To investigate the behaviors of the model, Equation (13) was embedded in Equation (14) to make it an approximate equations;

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

Figure 2

Phase diagram of tri-trophic equations.