A model for the relationship between the interaction pattern of ecosystems and their fate

The growth of isolated species can be described by simple laws but the fate of ecosystems, where several species interact, is difficult to predict. A crucial point seems to be the identification of regularities in the species interaction patterns that determine the fate of ecosystems. We approach this problem by using an ecosystem of three species whose populations are governed by generalized Lotka-Volterra differential equations. In our model, the species undergo both inter-specific and intra-specific interactions. The inter-specific interactions are positive, null or negative and are parameterized by interaction coefficients ɛij which take values: +1, 0 and −1, respectively. In these conditions, 138 different patterns of interactions (up to species re-labeling) are possible. Two extreme cases for the three intra-specific interactions (self-interactions) are considered: ɛii = −1 and ɛii = 0. We also define derived parameters, calculated from the interaction coefficients, which are relevant to determine the survival of species. Comparison of particular patterns shows that the relationship between interaction structure and survival can be subtle and, a priori, unexpected. For instance, we found that adding a negative interaction to a given pattern may be detrimental, indifferent or even beneficial. The same may happen when adding a positive interaction. To make a systematic study of the relationship between structure and fate, first, we perform a “microscopic” study based on the analysis of individual patterns. As a result, we obtained certain general rules or “theorems” concerning the extreme cases of coexistence and extinction of the three species. Second, in order to cover intermediate cases, where precise rules could not be found, we performed “statistical” studies. These provide the probability of survival and coexistence of species as a function of the values of the interaction derived parameters. Among other results, we found that reciprocal cyclic structures, of both positive and negative interactions, and evenly distributed sign balances of input interactions to the species appear to favor survival and coexistence.