Datasets on prey–pollinator networks

We examined datasets on plant–pollinator networks and suggested a model for growing them. We used plant–pollinator data from the Interaction Web Database ( http://www.ecologia.ib.usp.br/iwdb). After analyzing the structure of the network, we proposed a model that reproduces the structure of the observed mutualistic networks. We summarized the 14 plant–pollinator networks analyzed in this study. The number of nodes and links in the ecological network was small and the network was sparse.

Nonlinear evolutionary model of a mutualistic network

The principle driving the generation of the degree of distribution function in a mutualistic network is not well understood. Here, we developed a growth network model to reproduce the degree of distribution function observed in plant–pollinator networks. The degree of distribution function follows a power law or a stretched exponential distribution. Therefore, we considered the linear or nonlinear preferential attachment in bipartite-growing mutualistic networks. Fig. 1 shows a model of a growing bipartite mutualistic network. The network growth begins with the initial core network. During each update, we selected a plant with the probability q_p and a pollinator with the probability q_A=1–q_p. The incoming node has links l_P for the plant and links l_A for the pollinator.

We controlled the connecting probability of the incoming links when a node such as a plant or a pollinator becomes attached to the existing network node. We then examined the linear and nonlinear preferential attachments for new incoming links. The linear preferential attachment introduced by Barabasi and Albert (1999), known as the Barbasi–Albert model, exhibits a scale-free degree distribution. The nonlinear preference attachment model can explain the stretched exponential function of the degree of distribution (Krapivsky & Redner, 2001; Maeng et al., 2019). We applied the general nonlinear preferential attachment of incoming links for plants and pollinators. The incoming links for the plant or pollinator were chosen as the target nodes according to the nonlinear preferential attachment with an attaching probability of $A_{k_i}=k_i^{\lambda }A/{{\displaystyle {\sum }_i^{}k}}_i^{\lambda }A$ for the pollinators and $P_{k_i}=k_i^{\lambda }p/{{\displaystyle {\sum }_i^{}k}}_i^{\lambda }p$ for the plants. We then repeated the attachment process to reach the target network size.

Solutions for a nonlinear evolutionary model of a mutualistic network

The growing mutualistic network follows the power law of the degree of distribution as λ_{A or P}=1. When the parameter is less than one (0<λ<1), the degree of distribution of the plant or pollinator shows a stretched exponential function expressed as P(k)~exp(–k^1–λj), where j=A or P. We set a master equation for the mean number of plants and pollinators. Maeng et al. (2012) obtained the degree of distribution for the growing networks. For λ_X=1, the degree of distribution follows the power law:

where the power-law exponent is obtained as ${\gamma }_X=2+\frac{P_X{l}_X}{P_Y{l}_Y}$ and X=(A or P). For λ_X<1, the degree distribution shows a stretched exponential function such that

where ${\mu }_X=1+\frac{P_X{l}_X}{P_Y{l}_Y}$.

Structure of growing bipartite mutualistic networks

We generated the growing bipartite mutualistic networks from the initial core networks for a given set of parameter sets (P_A,P_P,l_A,l_P), as shown in Fig. 1. We simulated a mutualistic network using linear or nonlinear growth exponents for preferential attachment. Fig. 2 shows the cumulative degree of distribution of the simulated growing bipartite mutualistic network with nonlinear exponents for λ_A=1.0 for the plant and λ_P=0.5 for the animal and for the given set of parameter sets ( $P_A=\frac{2}{3},P_P=\frac{1}{3},l_A=1,l_P=2$). The cumulative degree of distribution shows the power law for the animal with λ_A=1.0 and the stretched exponential distribution for the plant with λ_P=0.5. The simulated results were consistent with the analytical prediction.

Applications of the model for growing bipartite networks to real mutualistic networks

We applied the model of the growing bipartite mutualistic network to 14 empirical plant–pollinator networks. The incoming number of links l_P for plants is much larger than that of the animal l_A. There were found to be more plants than animals in the mutualistic networks, as shown in Table 2; the nonlinear growth exponents show significant asymmetry. In most of the networks in Table 2, we recorded the relation λ_A>λ_P except in the case of the Hocking network. This implies a significantly strong competition between a new animal and existing pollinators, in contrast with the relatively weak competition between plants. The restriction on the number of available plant species is a more important factor in shaping the mutualistic community than the restriction on the animal species available, which is likely related to the difference in their survival and reproduction rates. Plants with a large degree of distribution have the advantage of high abundance, and are screened by the competition between animals characterized by λ_P<1, which leads the degree of distribution to take the stretched exponential form. This is not the case for “Hocking” in Fig. 3, where it has been reported in Hocking (1968) that competition between plants is more significant than that between the pollinators, implying λ_A>λ_P .

In summary, we determined the degree of distribution for the plant–pollinator network and introduced a growing model for bipartite mutualistic networks. We can reproduce the power law or stretched exponential degree of distribution for a mutualistic network for plants and animals. Thus, we report that competition among animals is stronger than that among plants.