| Contents |
| Classic papers declare that the plant-animal interactions have played a very important role in the generation of Earth's biodiversity. The importance of mutualism, the beneficial interaction between two species, for biodiversity maintenance can be supported by the fat that more than 90\% of tropical plant species depend on animals for the dispersal of their seeds. These interactions are best represented by weighted mutualistic bipartite networks. Those networks have been repeatedly reported to show particular properties as truncated power law distribution of the degree or a nested structure. While several metrics for measuring nestedness in weighted mutualistic networks have been proposed, most dynamic models in the literature aim to reproduce just the binary nested structure ignoring the development of the weighted pattern.
In this study, we have proposed a new model modifying the classical logistic equation with additional terms to take into account interspecies interactions. We have developed a model of mutualism based on the aggregation of benefit for each species in its equivalent growth rate:
\begin{equation}
\begin{array}{lcl}
\displaystyle
\frac{1}{N^{a}_{i}}\frac{dN^{a}_{i}}{dt} = r_{i}+ \sum_{k=1}^{n_{p}} b_{ik}\, N^{p}_k - \left( \alpha_{i}+ c_{i} \sum_{k=1}^{n_{p}} b_{ik}\, N^{p}_k \right) N^{a}_{i} \nonumber\\
\displaystyle
\frac{1}{N^{p}_{j}}\frac{dN^{p}_{j}}{dt} = r_{j}+ \sum_{\ell=1}^{n_{a}} b_{j\ell}\, N^{a}_\ell - \left( \alpha_{j}+ c_{j} \sum_{\ell=1}^{n_{a}} b_{j\ell}\, N^{a}_\ell \right) N^{p}_{j}
\end{array}
\end{equation}
\noindent where the superscripts stand for each of the class of species: animals and plants. $N_i(N_j)$ is the population of the species $i(j)$; $r_i \, (r_j)$ is the intrinsic growth rate of population. The rate of mutualistic interactions between a species $i$ and another $j$ is given by $b_{ij}$, which can be seen as elements of a matrix encoding the mutualistic interaction network. Note that the matrix is not necessarily symmetric if the benefit of the interaction is different for the two species involved. Following Velhurst's idea for the logistic equation, this implies that the friction term, $\alpha_i$, must also depend on the mutualistic interactions. In order to keep the model simple, we assume that the effect of the mutualism on $\alpha$ is proportional to the benefit. Finally, $c_{i}$ is a proportionality constant.
With these equations, we have built a binomial stochastic simulator for the study of system dynamics. It allows the introduction of external perturbations such as step increases
in mortality by plagues, removal of links between species due to evolution, or overlapping of a predator foodweb. |
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