Abstract
Predicting the sign of press perturbation responses in ecological networks is challenging, due to the poor knowledge of the strength of the direct interactions among the species, and to the entangled coexistence of direct and indirect effects. We show in this paper that, for a class of networks that includes mutualistic and monotone networks, the sign of press perturbation responses can be qualitatively determined based only on the sign pattern of the community matrix, without any knowledge of parameter values. For other classes of networks, we show that a semi-qualitative approach yields sufficient conditions for community matrices with a given sign pattern to exhibit mutualistic responses to press perturbations; quantitative conditions can be provided as well for community matrices that are eventually nonnegative. We also present a computational test that can be applied to any class of networks so as to check whether the sign of the responses to press perturbations is constant in spite of parameter variations.
Highlights
One of the main goals in ecology is to understand the dynamics of communities of interacting species
We further show that, when a community matrix has only a limited number of negative entries, the responses to press perturbations can possibly be all mutualistic when the community matrix has the property of being eventually nonnegative[22, 23]: this property of Perron-Frobenius type implies that the negative direct interactions only have a transient effect on the dynamics, but leave no trace on the press perturbation response at steady state
Single and pairwise interactions that normally appear in this context are shown in Fig. 1(a), along with their graph representation. Assembling all of these elementary interactions among n species leads to a signed digraph (S) that represents the whole ecological network, where S is a n × n matrix whose entry [S]ij is +1 if a positive edge goes from j to i, −1 if a negative edge goes from j to i, and 0 otherwise
Summary
One of the main goals in ecology is to understand the dynamics of communities of interacting species. We show that for some classes of ecological networks, including mutualistic networks[17], and monotone networks[19,20,21], the sign of press perturbation responses can be determined in a purely qualitative manner from the sign pattern of the community matrix, without any information on the numerical value of its entries. We further show that, when a community matrix has only a limited number of negative entries, the responses to press perturbations can possibly be all mutualistic when the community matrix has the property of being eventually nonnegative[22, 23]: this (quantitative) property of Perron-Frobenius type implies that the negative direct interactions only have a transient effect on the dynamics, but leave no trace on the press perturbation response at steady state. When the influence sign is indeterminate, since it depends on the parameter values used in the model, semi-qualitative and quantitative approaches can be used
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