Evolution of division of labour in mutualistic symbiosis

Abstract

Mutualistic symbiosis can be regarded as interspecific division of labour, which can improve the productivity of metabolites and services but deteriorate the ability to live without partners. Interestingly, even in environmentally acquired symbiosis, involved species often rely exclusively on the partners despite the lethal risk of missing partners. To examine this paradoxical evolution, we explored the coevolutionary dynamics in symbiotic species for the amount of investment in producing their essential metabolites, which symbiotic species can share. Our study has shown that, even if obtaining partners is difficult, 'perfect division of labour' (PDL) can be maintained evolutionarily, where each species perfectly specializes in producing one of the essential metabolites so that every member entirely depends on the others for survival, i.e. in exchange for losing the ability of living alone. Moreover, the coevolutionary dynamics shows multistability with other states including a state without any specialization. It can cause evolutionary hysteresis: once PDL has been achieved evolutionarily when obtaining partners was relatively easy, it is not reverted even if obtaining partners becomes difficult later. Our study suggests that obligate mutualism with a high degree of mutual specialization can evolve and be maintained easier than previously thought.

Keywords: division of labour; evolution of specialist and generalist; horizontal transmission; mutualism; symbiosis.

Figure 1.

The schematic diagram of our model when numbers of species and metabolite are two. Individuals of both species are engaged in symbiosis with probability p₁p₂. Species 1 invests the amount x₁₁ to producing metabolite 1 (blue triangles) and x₁₂ to producing metabolite 2 (red squares), while species 2 invests the amount x₂₁ to producing metabolite 1 and x₂₂ to producing metabolite 2. The total amount of investment is assumed to be subject to a constraint: x₁₁ + x₁₂ = 1 and x₂₁ + x₂₂ = 1. With the amount ξ of investment for producing a metabolite, its production is given by a production function f(ξ). The produced metabolites are shared evenly between species 1 and 2. The fitness of an individual of each species is assumed to be given by multiplying the total amount of essential metabolites produced in the symbiotic system: {(f(x₁₁) + f(x₂₁))/2}{(f(x₁₂) + f(x₂₂))/2}. (Online version in colour.)

Figure 2.

Typical phase portraits of the coevolutionary dynamics of the two-species two-metabolites model. (Left panel) When the chance of obtaining symbiotic partners is remote (p being less than 4/11; p = p₁ = p₂ = 0.35 in the panel), the coevolutionary dynamics shows multistability of perfect division of labour (PDL) (labelled ‘P’), hemi-perfect-division-of-labour (HPDL) (labelled ‘H’), and Jack-of-all-trades (JAT) (labelled ‘J’) states. (Middle panel) When the chance of obtaining partners is intermediate (4/11 < p < 2/3; p = 0.4 in the panel), HPDL becomes unstable, leaving only bi-stability between PDL and JAT. (Right panel) When the chance of obtaining partners is higher than two-thirds (p = 0.8 in the panel), JAT becomes unstable, leaving only PDL state as the equilibrium of coevolutionary dynamics. Closed and open circle, respectively, represent stable and unstable equilibrium, and solid curves are null isoclines of coevolutionary dynamics. The grey lines with arrows indicate the trajectories of coevolutionary dynamics. The other parameters: σ_i = 1 in all panels. The production function is quadratic f(ξ) = ξ². Schematic diagrams of the bottom represent the pattern of specialization of PDL, HPDL, and JAT states. Circles and squares represent species and metabolites, respectively. An arrow indicates which species produce which metabolites (strictly speaking, an arrow is drawn if the corresponding investment x_ij from species i to metabolite j is positive at the equilibrium).

Figure 3.

The boundaries for local stability of PDL (a) and JAT (b) in the n-species n-metabolites model (n = 2, 3, 4, 5). (a) The regions for the stability of PDL are plotted against the probability of encountering partners p (horizontal axis) and the coefficient ε for the linear component in production function f (vertical axis). A dashed line represents the boundary of the regions and becomes finer along n = 2, 3, 4. The solid line is the boundary with n = 5. (b) The same as (a) but showing the boundaries for the local stability of JAT. Note that p₁ = p₂ = p and σ₁ = σ₂ are assumed. The production function is f(ξ) = εξ + ξ².

Figure 4.

Fraction, for randomly varied initial conditions, eventually fell into PDL (black), JAT (white), and the others (grey) in the coevolutionary dynamics with n-species and n-metabolites (n = 2, 3, 4, 5). (a) When obtaining symbiotic partners is easy (p = 0.7), the frequency that PDL is attained through the coevolutionary process stays high even when the numbers of species and essential metabolites are increased up to 5. (b) When obtaining symbiotic partners is difficult (p = 0.3), the fraction of PDL becomes progressively lower as the number of species increases. Note that p₁ = p₂ = p and σ₁ = σ₂ are assumed. The production function is f(ξ) = ξ².

Figure 5.

Distributions of Euclidian distances between an attained equilibrium and JAT equilibrium in the n-species n-metabolites model (n = 2, 3, 4, 5). The distances quantify the degree of specialization in essential metabolite production in symbiotic species (see Simulation results). Upper and lower panels show the cases when obtaining symbiotic partners is easy (p = 0.7) and difficult (p = 0.3), respectively. In each panel, the histograms indicate the distribution of Euclidean distance from JAT equilibrium to initial points which are sampled randomly (grey) and attained equilibria (red, green, blue, and white) of the coevolutionary dynamics. The colours of attained states correspond to their abundance: red (enclosed by the solid line) indicates the most frequent state, green (with the dashed line) indicates the second most frequent, blue (with the dashed spaced line) indicates the third most frequent, and white indicates all the other states. Small schematic diagrams in the inset of panels show the pattern of specialization of the equilibria (the same as in figure 2). The colours of the diagrams correspond to that of histograms. The number of sampled initial points are 5 × 10⁶, the production function is f(ξ) = ξ² (that is, ε = 0 in f(ξ) = εξ + ξ²), any probabilities of encountering partners and any rates of evolution are the same as the others (i.e. p_i = p and σ_i = σ for all i). (Online version in colour.)