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Mutualisms are cooperative interactions between members of different species. We focus on obligate mutualism, where each species cannot survive without the other. From a theoretical aspect, obligate mutualism is similar to the relationship between male and female. Empirical data indicate a sex-ratio selection: male and female have a specific ratio in their population sizes. In the present paper, we apply lattice model to obligate mutualism between two species, and present a theory of “ratio selection” which is a generalization of sex-ratio selection. Computer simulations are carried out by two methods: local and global interactions. In the former, interactions occur between neighbouring cells, while in the latter they occur between any pair of cells. Simulations in both interactions show the so-called Allee effect: both species can survive, when both densities are large in some extent. However, we find a large difference between local and global simulations. In the case of local interaction, restriction for survival is found to be extremely severe compared to global interaction. Both species require a proper ratio for their sustainability. This result leads to the theory of ratio selection: when interaction occurs locally, the ratio of both species is uniquely determined. We discuss that the ratio selection explains not only the evolution of endosymbionts from free-living ancestors but also the evolution from endosymbionts to organelles.

In recent years, the concern for mutualism is growing, since almost all species have mutualistic relationship with other species [

The relationship of obligate mutualism has the similar behaviour as the male-female relationship [

It is known that ESS is the most powerful tool to obtain the optimal strategy which beats the other strategies [

Lattice models have been applied to ecological problems, where simulations have been carried out by two methods: local and global interactions [

The most famous model of population dynamics in ecology is Lotka-Volterra equations (LVEs) [

In the next section, we explain our model and method of both local and global interactions. In Section 3, the prediction by mean-field theory is reported. We briefly describe the results obtained by Iwata et al. [

Consider a system consisting of two mutualistic species X and Y (

The reactions (1a) and (1c) respectively denote the birth and death processes of species X, and

Simulations of lattice model are usually carried out by either local or global interaction [

1) Initially, we distribute X, Y on a square lattice. Each cell is one of three sites: X, Y, and O (see

2) To update, we choose a target site randomly.

a) Reaction (1c) or (1d): If the target cell is X (or Y), then it becomes O by the rate

b) Reaction (1a) or (1b): If the target site is O, we choose another cell from the neighboring 8 sites (Moore neighborhood) around the target cell. When the second site is X (or Y), then the target cell become X (or Y) by the rate

3) We repeat step 2) by

4) We further continue the updates, until the system reaches stationary state.

The reproduction rates

, (2)

where we call

Next, we explain the method of global interaction (lattice gas model). Almost all procedures are the same as local interaction. However, in the global simulations, the birth process [reaction (1a) or (1b)] occurs between any pair of lattice sites. In mean-field limit, the reproduction rates are replaced as follows:

, (3)

where

If the reaction (1a) occurs between any pair of lattice sites, and if the lattice size

where the factor

Note that this equation is almost the same as in the male-female system [

In summary, two conditions are necessary for survival; one is the relation (6), and the other is that initial densities of two species are higher than the separatrix which is schematically shown in

We explicitly obtain equilibrium density in stable state, when

From Equations (5) and (7), we get

where

From Equation (8), we can find the following relation:

Inserting Equations (7) and (10) into (5), we have

where

Equation (11) denotes a typical Allee-effect equation [

It is therefore found in the mean-field limit that 1) the maximum value of total density (

This inequality is the same as equation (6) under the condition of

We report simulation results of individual-based (lattice) model. In the case of global interaction, the mean-field theory [Equation (5)] predicts that the dynamics has two phases. One is an extinction phase: both species always go extinct. The other is Allee-effect phase. A typical example of the latter phase is shown in

In the case of local interaction, the dynamics are similar to the predictions of lattice gas theory. For instance, the Allee effect can be observed: unless both densities of species are considerably high, the population goes extinct. However, in the case of local interaction, simulation results exhibit a distinct difference from those of global interaction. In

In

ratio of densities. However, it is necessary for the sustainability in local interaction that the value of

When

We have developed a lattice population of obligate mutualism. Local interaction strongly effects on the populations dynamics. The survival condition is extremely limited for local simulation. When both species have the same mortality rate (

The ratio selection is a generalization of sex-ratio selection. In fact, our model is also applicable to the male- female system. Equation (5) is almost the same as obtained in male-female system [

Many empirical data suggest the ratio selection; two-species systems of obligate mutualism usually have some mechanism to avoid an abrupt increase of one species [

yucca moth: too many yucca moths become harmful for yucca plant [

The authors sincerely thank to professors Jin Yoshimura for valuable comments. They also thank to Mr. Keiji Amemiya for the help of simulations.

Kei-Ichi Tainaka,Tsuyoshi Hashimoto, (2016) A Theory of Ratio Selection—Lattice Model for Obligate Mutualism. Open Journal of Ecology,06,303-311. doi: 10.4236/oje.2016.66030