All models are wrong, but which are useful for understanding the effect of nestedness on plant-pollinator dynamics?

A guest post by Gita Benadi and Jochen Fründ

Mutualistic networks have become an increasingly popular way of describing mutually beneficial interactions between species-rich communities, for example those between plants and their pollinators. Besides offering a nice method of visualizing these interactions, networks are also used to make inferences about the relationship between interaction structure and stability. If we try to summarize the literature on plant-pollinator networks, two points may appear to be the most established findings:

1. Real networks tend to have a nested structure, in which specialists interact mostly with generalists (Fig. 1);
2. Nestedness increases the stability of mutualistic networks, which likely is the reason for (1)

We are not yet convinced that we can be sure about (1), because such a pattern can too easily arise from sampling effects: In typical network datasets, rare species are represented by only few observations, which means they have few links and are unlikely to be observed interacting with each other (see Blüthgen et al. 2008). On the other hand, it’s easy to find biological examples for asymmetric specialization, so maybe many mutualistic networks are really nested – we won’t discuss this issue further in this post. So after (1) may or may not be true, what is the evidence for (2) based on, and can we accept the idea that nestedness increases stability as a fact?

Figure 1: Two plant-pollinator networks with different levels of nestedness. Black squares indicate interactions between plant species (rows) and pollinator species (columns). The interaction matrix on the right is more nested than the one on the left because most of its specialized species interact with subsets of the interaction partners of more generalized species.

Explanations for a nestedness-stability relationship

The notion of a positive relationship between nestedness and stability is mainly based on two types of models. First, there are simple “robustness” simulations that remove species from the network and check how many species lose all their partners. In this case a more connected network is clearly more stable (redundancy), while nestedness only increases stability when the least connected species are removed first; in fact, nested networks are very vulnerable to attacks targeted at the most connected species. When species are removed in a random sequence, nestedness has little influence on “robustness”. But “robustness” simulations are not very sophisticated and very crude representations of what’s going on with interacting species.

Secondly, there are dynamic models that consider the coupled population dynamics of species in the network. Such models have a long tradition in ecology; for some people they may even define what “theoretical ecology” is. Mutualistic interactions have a potential to destabilize such dynamic model systems. Wheras the simple Lotka-Volterra model of interspecific competition produces a stable equilibrium state of one or both species, the equivalent mutualism model can create a runaway process of ever-increasing mutualist populations (which is impossible in the real world). These runaway dynamics occur when the positive feedback produced by the mutualism is strong compared to the negative feedback of intraspecific competition. That is precisely why, in a Mercer Award – winning Science paper, Bascompte et al. (2006) found that asymmetric specialization makes mutualistic networks stable: If the product of mutualistic effects of two species on each other is small, the destabilising effect of mutualism is offset by intraspecific competition (see response by Holland et al., 2006).

Since then, there have been multiple papers finding that nestedness increases the stability of dynamic models of mutualistic networks (just a few examples from Nature and Science: Bastolla et al. 2009, Thebault & Fontaine 2010, Rohr et al. 2014). The models have incorporated more and more biological realism (saturating mutualistic effects, competition within a guild, trade-offs) and there are a lot of good things to say about many of these papers. An indefinite runaway “explosion” of mutualists is no longer possible in the refined models; eventually a limit is set on population sizes.

Looking closer at the models

Well, we know that “all models are wrong, but some are useful”. To us, the most important question about whether a model is useful is not whether it is completely realistic, but whether the processes leading to the conclusions would make sense in the real world. So let’s have a closer look at the processes that are represented in recently published models of mutualism:
In most dynamic models of mutualistic networks, the benefit of mutualists is an added term in the logistic equation for population growth:

dN/dt = r N (1 – N/K) + f(M) N

(Here, f(M) is the benefit of mutualism, which is a function of the mutualist’s density M. In the earliest models of mutualism, this term was a linear function of M, while in more recent papers it is usually saturating.)

With this growth equation, the population grows faster and reaches a higher equilibrium density with the mutualism than without it. In other words, both the low-density growth rate and the carrying capacity of the population are increased by the interaction. In such models, mutualists are treated as a resource that is interchangeable with other resources, but free from competition. Hence, mutualism effectively decreases the strength of resource competition. But is that realistic?

For some types of mutualisms, the interaction may indeed provide more of an interchangeable resource, such as nitrogen fixed by symbiotic bacteria versus plant-accessible nitrogen in the soil. But then, the population’s carrying capacity should be determined by the availability of the resource that is most limiting to its growth, in accordance with Liebig’s Law of the Minimum. For example, if nitrogen-fixing bacteria provide a surplus of this nutrient, plant growth may become limited by phosphorus availability. In general, high mutualist availability cannot increase population sizes above the carrying capacity defined by other resources.

In the case of plant-pollinator interactions, the pollinators’ carrying capacity may or may not be determined by the available amount of floral resources. That depends on whether nectar and pollen or other resources such as nesting sites (e.g. for bees or birds) or larval food (e.g. for flies or butterflies) are in shorter supply. But the plants’ carrying capacity should definitely not depend on the availability of pollinators, because pollinators only affect plant seed production. Without sufficient abiotic resources (water, light, nutrients, space) a plant population cannot grow, no matter how many seeds it produces.

Hence, a more realistic model of plant-pollinator dynamics should at least incorporate an upper limit of plant population size independent of the pollinators’ density, and possibly also a similar limit to pollinator density (Fig. 2).

Figure 2: What happens if plant density is increased? Top row: what (can) happen in many mutualism models; bottom row: what we think has more meaningful dynamics. Both illustrated for the 2-species case.
Top row: step1: total service provided by the pollinator strongly (linearly) increases with number of plants; step2: pollinators can exceed K (carrying capacity enforced by other resources, illustrated by trap nests above); step3: plants can exceed K (illustrated by plant box below); total interaction strength can reach weird level (“mutualism party”).
Bottom row: step1: pollinator visits are distributed among more flowers, i.e. competition for pollinators increases among plants; step2: more floral resources per pollinator lead to an increase in pollinator population, within the bounds of K; step3: the increase of pollinator pollinations cannot increase plant populations above the limit set by K.

In addition, if mutualists are a limiting resource, there is competition for them – this should be true for pollinators (whose total pollination service is at least constrained by time) and plants (whose total rewards are at least constrained by energy). So far, competition for the services provided by mutualists has not been incorporated in most published models of mutualistic networks (but see Benadi et al., 2013). Since nestedness is an interaction pattern that maximizes the degree of overlap in the identity of interaction partners for a given number of interactions (links) per species, it should maximize the intensity of competition between species in the same group (e.g. competition between plants for pollinators). Intuitively, one could expect that a high intensity of competition reduces the number of species that can stably coexist, rather than increasing it (as suggested by Bastolla et al., 2009).

Conclusions and outlook

In conclusion, we think the question of whether nestedness increases community stability does still not have a definite answer. Of course, part of the difficulty of finding that answer is that stability in itself is a pretty vague concept (see Grimm & Wissel, 1997). For example, simply by applying a different stability criterion (“persistence”) to the exact same model used by Bastolla et al. (2009), James et al. (2012) reached the conclusion that nestedness is completely unrelated to a mutualistic network’s stability.

It remains a challenge to find the appropriate level of mechanistic detail in a dynamic model of mutualism. While it is of course desirable to keep the equations as simple as possible, the crucial thing is to make sure that the qualitative results remain the same when more mechanistic details are added. In this post, we have highlighted two details that we think could make a fundamental difference and deserve further investigation:. Viewing mutualists as providers of resources, (a) limits set by other resources will often not be changed by mutualists and (b) competition for mutualistic services might be important especially if competition for other resources is weak. We are looking forward to seeing which answers future analyses will provide about the influence of nestedness and other structural patterns on network stability.0

07.05.15: a typo in the title of this post was correct, it wrongly spelled before “all models all wrong”.