Although IGP is common in nature, models often predict that it is unlikely. This study shows that this discrepancy can be partly due to how models are analyzed. Invasion analysis, though commonly used, may predict that coexistence is impossible when it may be readily possible through locally stable attractors (e.g. stable equilibrium). In fact, locally stable attractors exist in wide parameter regions when individual variation in adaptive behavior is considered. These results suggest that both individual variation and analysis methods are important for examining the dynamics of the IGP module.
The result about the analysis methods (i.e. mutual invasibility is not necessary for coexistence) is valid no matter which model is considered. In other words, neither adaptive behavior nor individual variation is necessary. For example, the species can coexist even at very high productivity levels without an additional ecological factor (i.e., fixed behavior without individual variation). However, the coexistence possibility further expands with inclusions of adaptive behavior and individual variation. This difference between invasibility and coexistence exists due to nonlinear interactions (i.e., saturating functional responses). The nonlinear functions allow the IGP module to have alternative equilibria (e.g. all three species present and only two species present) [1
]. When all per capita interactions are linear functions, internal attractors would not exist when the mutual invasibility fails.
One may argue that invasion analysis is a more conservative way to study coexistence. For example, even if an internal attractor exists (i.e., coexistence is possible), a community may not persist if it experiences a strong perturbation. However, the domain of attraction is not negligible (Figure ). When considering why ecological communities in the field can persist even though models predict otherwise, the existence of internal attractors cannot be ignored. In addition to being conservative, another reason one might focus on invasibility is that communities are assembled by invasions, and initially colonizing species are likely rare. In invasion analysis, this is modeled by assuming the density of an invading species is 0 (i.e., completely negligible). In ecological invasions with finite population sizes, however, only few invading individuals can substantially violate the assumption. Results of invasion analysis can change if this assumption is slightly relaxed (i.e., the density of an invading species is assumed to be very small but not nil). Under these scenarios, internal attractors become more relevant to understanding real ecological dynamics.
The interpretation of the effect of adaptive foraging in the IGP module is not straightforward. Here we make two comparisons. One comparison is between the fixed behavior model and the adaptive behavior model without individual variation. This comparison shows that the inclusion of optimal foraging enhances the possibility of coexistence (Figures and ). The other comparison is between the adaptive behavior models with and without individual variation, which shows that individual variation enhances coexistence and also stabilizes non-equilibrium dynamics. Because individual variation reduces the average per-capita fitness of the IGpredator, these two comparisons show qualitatively different results. The former implies that optimal foraging has a positive effect, and the latter comparison result implies that optimal foraging has a negative effect on coexistence. When both fixed behavior and perfect adaptive behavior do not allow the community to persist, individual variation may be able to give the balance needed for the community to persist. Similar results relating to suboptimal behavior enhancing coexistence exist in other food web modules [27
]. Optimal foraging behavior of individuals leading to extinction of the population has also been reported in other studies e.g. [32
Abrams and Fung [12
] studied the IGP module in which the IGpredator exhibits partial preference while this study focused on individual variation (difference between partial preference and individual variation is discussed above), and both studies show that sub-optimal adaptive behavior can increase coexistence possibilities. Because of some differences between these studies Abrams and Fung (e.g. [12
] mainly focused on invasion criteria while this study focused on internal attractors, and the two studies consider different parameter values), it is difficult to directly compare differences between partial preference vs. individual variation. However, it has been reported in a study on a different food web module that the difference between partial preference and individual variation can cause ecologically significant differences [27
]. Although (empirical) studies have commonly considered individual variation as a factor to account for the type I error in statistical hypothesis tests, characterizing the patterns of variation will likely to provide rich information to studying ecological dynamics.