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.

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.

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”.

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

  1. Pingback: Guest Post about modeling plant-pollinator webs on theoreticalecology | Jochen Fründ

  2. Thank you for that well-written post! I have been interested in and only recently begun studying the population dynamics of mutualistic interactions. I’ve yet to work on networks specifically, but by association and necessity, I have found myself reading quite a bit on mutualistic networks because, like you mention, many empirical datasets have emerged against which we can test our theoretical models.

    The question underlying these studies on networks is: what is the structure of mutualist pairs in a community? Identifying mutualism is not easy. For food webs, it’s really easy–one species eats the other, and that is a sufficient interaction. But what about mutualists? The presence of a floral visitation, for instance, is not a effective mutualistic interaction–a recurrent, yet substantive aspect of mutualism that I don’t believe have been addressed in network studies. Why might that matter? Well, what we know about generalist and specialist interactions (i.e., the specialists would be the ends, the low-row-high-column values and the low-column-high-row elements; the generalists would be the core, or low-row-low-column elements), is that there is disparity in the effectiveness of the interactions, meaning that there would likely be a drastic shift in topology of an “effective” interaction matrix.

    A second, lesser concern, is the binary nature of these matrices. Adding the frequency of the interactions (or effective interactions, as I mentioned above) will also change the qualitative and quantitative aspects of the mutualism community.

    I also liked that you pointed out the usefulness (or lack thereof) of the r-K logistic solution of the differential logistic model (i.e., dN/dt = rN – aN^2). K is biologically and mathematically a phenomenological parameter that doesn’t have an ecological meaning aside from that dN/dt = 0, which means that adding anything (e.g., dimension, order) obscures its interpretation, like carrying capacity. As you point out and cite (e.g., Holland and DeAngelis’ work), explicit consideration for resources is vitally important to yield a biologically meaningful model or results.

    Thank you for the post. It was very thought-provoking, interesting, and had great figures.

    • Thanks for the kind comment.

      You’re right that a crude measure such as floral visitation doesn’t proof an effective mutualistic interaction (not sure whether there’s a fundamental difference to antagonistic interactions, though). Some studies have shown that visitation frequency is still a decent estimate, and I think it’s usefulness shouldn’t be discounted, because it will often not be feasible to get the most accurate measures for all species pairs. Still, we shouldn’t forget that it’s only a crude estimate.

      That being said, I think more and more mutualistic network studies are now trying to better differentiate between quantity and quality of interactions. You might be interested in this paper on this topic: http://onlinelibrary.wiley.com/doi/10.1111/ecog.00983/abstract

  3. Thanks for an interesting post – I was wondering: do you know of any model of plant-pollinator community dynamics fitted to data, in the sense of fitted to time series of abundance by least squares or such?

    These might not exist for large networks of course, but for small modules, such data could be interesting to know more about functional forms (at least I’d be very interested to see what is known on particular mutualistic systems). Most of your discussion requires implicit knowledge of the numbers of plant and pollinators and how they relate to their interaction rates or population dynamics. Ideally, one would want to infer functional forms from data as well as theorize them from ecological theory (as you rightly do above).

    I’d find time series data especially relevant to dynamic community models as any increase in mechanistic detail in the interaction between plants and pollinators – even a very relevant one – might be offset by the lack of other “realistic” factors elsewhere in the model changing plant and pollinator population growth rates (e.g. stochasticity in weather, crop rotation; making the dynamics much more intricate than logistic-like). Also, what’s the appropriate stability concept for the system at hand might become more evident from time series of abundance (e.g., persistence or variability rather than return rates to equilibrium, for widely fluctuating / non-equilibrium populations submitted to large environmental stochasticity).

    Or is there simply no such data for plant-pollinator systems of interest?

    • As far as we know, there are as yet no published studies of Lotka-Volterra-type models of plant-pollinator dynamics fitted to data. The available time series of whole plant-pollinator networks are usually really short (3-5 years). Longer time series probably exist for very specialized pollination systems (e.g. figs and fig wasps), but many of them involve seed predation as well as pollination and often the plant partners have much longer generation times and slower dynamics than the pollinators. I agree that we could potentially learn a lot from fitting models of plant-pollinator dynamics to data, but it will be a big challenge to separate the effects of the mutualism from all the other things that are going on in the real world (interactions with predators and competitors, environmental variation, observation errors etc.).

      • It’s quite striking to me that there are no published studies of L-V models of plant-pollinator dynamics fitted to data. Although I do agree that it’d be very difficult to isolate the direct effect of plants on pollinators on each other, I think that is not a stranger to other kinds of interactions. Most predator-prey and competition datasets are have all sorts of confounding factors or are idiosyncratic. I think that we could learn a great deal if we had those data or a model-type system, like with Paramecium.

        • Thanks for the links! The first (Yoshikawa & Isagi, 2013) looks like a nice long time series of multiple species, although the interactions are not purely mutualistic (some of the birds are flower predators, some are both pollinators and flower predators). Since these are citizen science data, one would have to account for variation in sampling effort when fitting a Lotka-Volterra type model.
          I have read the Burkle et al. paper (2013, Science), but I don’t think it would be possible to fit a population dynamics model to those data. As far as I understand, the data consist of only three snapshots that span a period of 120 years.

  4. Pingback: What’s wrong with null models? | theoretical ecology

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