Starting with pattern, focusing on robustness (The “Tinkerer”)
Alternatively, theory can be motivated by testing the robustness of a particular pattern. In this case, the role of the theorist is to explore the effects of tinkering with existing models. Questions with this framing could be, ‘Here's a pattern, how robust is it to changes in the underlying mechanism(s)?’ This role is typically associated with clarifying a mechanism (rather than proposing a new one), either by making a verbal model more concrete, or by clarifying the conditions under which a mechanism can generate a pattern. For example, Shaw et al.
[26] used one type of modelling approach (ordinary differential equations) to study the relative importance of insect life history and insect behaviour on the spread rate of vector-borne plant pathogens (pattern). Following this, Shaw et al.
[27] studied how robust this pattern was to the modelling approach used by developing an individual-based model to capture the same mechanisms of the same system. Another instance of a “Tinkerer” style model comes from Shoemaker and Melbourne
[28] who developed a framework with multiple ecological mechanisms thought to influence the pattern of spatial coexistence of species in competitive metacommunities. Using their framework, they calculated the relative contribution of these different ecological mechanisms in determining the strength of coexistence (here, a species’ low-density growth rate) for different metacommunity paradigms.
Starting with pattern, focusing on conflict (The “Mediator”)
Third, theory can be motivated by needing to reconcile conflicting patterns – within the empirical literature, between conceptual ideas, or between theoretical and empirical results. The goal of this type of theory is to mediate a conflict in patterns. Unlike the above two (“Explainer”, “Tinkerer”), this role is motivated by several patterns rather than a single one. Questions here could be framed accordingly, e.g., as ‘Here are multiple conflicting empirical data patterns, how do we reconcile them?’ For example, some empirical studies find the pattern that migratory individuals typically have greater parasite infection than resident individuals
[29,30], while other studies find the opposite pattern
[31,32]. Shaw et al.
[33] used a theoretical model to clarify the mechanism leading to these disparate patterns: migration can simultaneously lead to a higher richness of parasites and lower infection prevalence. Thus studies that use richness as a metric find one pattern, while studies that use prevalence find the opposite.
Questions could also be framed as ‘Here are multiple conflicting conceptual ideas, how do we reconcile them?’ For example, studies that aim to understand sex-specific patterns of how individuals move to find mates in sexually reproducing species have generated two contradicting predictions. Theory that focuses on mate finding shows that when individuals of one sex move more, individuals of the other sex should move less, creating sex bias in dispersal
[34]. In contrast, theory that focuses on dispersal distance shows that when dispersal affects the potential to find mates, selection favours both males and females to have similar dispersal kernels, creating no sex bias in dispersal
[35]. Shaw and Kokko
[36] built a single model to resolve this contradiction, showing that the mechanism details shape which results apply – e.g. if females moving more brings females closer to males (and thus males do not have to move) we get one outcome whereas if females moving more leads to males being left behind, we see the other outcome.
A final way that questions could be framed using this role is as ‘Here is a pattern in data that doesn’t match the existing theory, how do we reconcile this?’ For example, an observed empirical pattern is that seabirds have a diversity of foraging strategies
[37], which conflicts with theory suggesting that there should be a single optimal foraging strategy
[38]. Jeffries et al.
[38] reconciled this conflict by developing a model where seabirds have only partial knowledge of the patchy food distributions in their environment and showed that different foraging strategies are favoured by birds with different personalities along the bold-shy continuum.
Starting with mechanism, focusing on novelty (The “Advocate”)
In contrast to the three roles above (“Explainer”, “Tinkerer”, “Mediator”), theory can be framed as starting with a mechanism and determining the consequences in terms of the pattern (or patterns) generated. This kind of theory might result in advocating for an overlooked mechanism to be included. Theory that aims to describe a novel consequence of a mechanism could be framed as, 'Here is a mechanism that has been overlooked, what are the consequences of including it?' For example, Yamamichi and Letten
[39] developed a model with a new mechanism (rapid evolution in the face of temporal fluctuations with a tradeoff between competitive ability and speed of evolution) and determined how it affected the pattern of coexistence between competing species.
This role can be particularly effective at connecting scales in ecology by asking, 'Here is a mechanism at one scale, when does it affect patterns at another scale?' For example, linking the scales of individuals to populations, Shoemaker et al.
[40] first used experiments to measure how a pathogen changes the preferences of its aphid vector for feeding on different host plants (mechanism). Then they used a model to ask under what conditions these measured preferences affected the spread of pathogens across a host plant population (pattern).
Starting with mechanism, focusing on robustness (The “Semantician”)
Just like the “Tinkerer” role above, theory that starts with a mechanism can also have a focus on robustness by asking, 'Here is a mechanism, how does its effect on the pattern depend on how the mechanism is captured in a model?' This role is fundamentally focused on the semantics of a mechanism, that is, how the mechanism is expressed in the model(s). For example, previous work has shown that species with environmental sex determination are particularly sensitive to environmental change
[41,42]. Crowley and Labonne
[43] developed a model to ask, 'how robust are these conclusions to how evolutionary success (i.e., fitness) is encoded?' In this model, the authors measured how environmental quality and habitat loss (multiple mechanisms) could influence the fitness of populations (pattern) based on whether the model was formulated in terms of growth rate (
r) or lifetime reproductive success (
R0). Thus, such models can decipher how robust outcomes (or patterns) are to underlying differences in descriptions of the same mechanisms in different models.
Starting with mechanism, focusing on conflict (The “Instigator”)
Finally, theoretical models can be used to explore the consequences of several mechanisms in combination. In this kind of framing, the theorist instigates a potential conflict (or interaction) between multiple mechanisms and investigates the outcome. The question here could be framed in terms of relative importance, e.g., ‘Here are multiple mechanisms, what is their relative importance for generating a pattern?’ For example, abundance of grazers can dramatically impact reef cover dynamics
[44]. Dispersal of coral and macroalgae between reefs seems likely to be important for reef cover dynamics as well
[45], yet it has been less explicitly explored. Greiner et al.
[46] built a theoretical model to ask, 'what is the relative importance of these two mechanisms?'. Here the authors study the relative effects of grazer and dispersal behaviour (multiple mechanisms) which interact to influence spatial patterns of coral reefs (pattern).
Alternatively questions in this role could be framed as combining several mechanisms that have previously been considered separately to ask, ‘Here are multiple mechanisms, how might they interact to generate different patterns?’ For example, studies have examined how seed dormancy
[47] and facilitation
[48] each separately shape plant populations and communities. Leverett and Shaw
[49] developed a theoretical model with both mechanisms, in order to understand how facilitation between seedlings interacts with dormancy in seeds to shape patterns of plant population dynamics.
Other perspectives
Although the six roles as described in detail above form the core of our paper, we wanted to comment on a few other aspects of framing theory that readers might find useful. First, one could start by asking how many mechanisms and how many patterns they were interested in, and using that to choose a role (Figure 1). A single mechanism and single pattern would best fit one of the novelty roles of “Advocate” or “Explainer”. A single mechanism with two (or more) patterns would fit either robustness role (“Tinkerer”, “Semantician”) or the “Mediator”, while a single pattern with two (or more) mechanisms would fit the “Instigator” role of looking at the relative importance of the mechanisms. Multiple patterns and multiple mechanisms could also fit the “Instigator” role of aiming to understand the interaction among multiple mechanisms.
Second, the process of model development can be seen as the act of ‘translating’ mechanisms from the complex language of biology to the more abstract language of mathematics, analysing the model, and then translating the results back again to understand the biological pattern produced
[50]. Just like translation of text across different languages, there can be multiple ‘correct’ ways to translate between math and biology. For example, a single biological system could be translated into e.g. a differential equation model or into an individual based model. Similarly, the same model result could be interpreted in the context of e.g. metapopulations or in the context of disease dynamics. Viewing theory development as translation, we can see that some of the roles described above effectively compare multiple translations. For example, one could start with a biological pattern and see if the pattern is disrupted if the same biological mechanism is translated in two different ways in the model (the “Semantician”). Similarly, one could start with a biological mechanism and vary how it is translated into a model to see whether the details of translation affect the pattern generated (the “Tinkerer”). In contrast, the remaining roles (the “Advocate”, “Explainer”, “Instigator”, “Mediator”) are more often about developing a single translation from biology to math and back again.
Flexibility in Framing
In the 'Framing theory' section above, we described each research project using a single role. However, the same project can often be framed in several different ways. Indeed, many theoretical papers use several roles when describing their study (e.g. in the abstract versus the discussion). For other projects, the role used at the start of the project may not be the same as the one used at the project’s end. In this section, we show how the same theoretical project can be framed using multiple roles.
Different ways of framing theoretical projects might be best suited to practitioners in different fields. For example, wildlife managers may be interested in starting with patterns to frame a theoretical project based on what they see in nature (e.g., observing the pattern that two plant species X and Y often coexist in a community that is affected by snowstorms). In contrast, developing a theoretical project starting with a mechanism may be more relevant to ecologists who are interested in understanding the effects of potential factors that influence ecological systems (e.g., asking how the mechanism of snowstorms affects the coexistence of plant species X and Y). By providing examples of different ways to frame the same theoretical project, we hope to inspire practitioners to continue asking their research questions using different framings.
Multiple ways to start with pattern (Sullivan et al. 2017, PNAS)
First, consider a theoretical project that starts with a pattern of interest and aims to understand the mechanisms underlying it. In Sullivan et al.
[25], the pattern of interest was species invasion dynamics and the large variation in speed of invasion (or range expansion) observed in many empirical systems. Understanding variation in speed of invasive species spread is crucial to devise strategies for their elimination or mitigation. In order to explain the pattern of varying invasive speeds for species, Sullivan and co-authors focused on two mechanisms: 1) a strong Allee effect (where populations cannot grow from low density) and 2) fluctuations in population size due to overcompensatory growth or density-dependence in dispersal. By building spatio-temporal mathematical models incorporating these mechanisms, the authors showed how demographic and dispersal dynamics jointly produced fluctuating invasion speeds as the population spread into new territory. Thus, framing the research question as an inquiry into mechanisms responsible for generating patterns of invasion speed variation led to novel scientific insights (i.e., pattern/novelty framing – the “Explainer”). However, one could approach the same project with a different framing: how robust is the pattern of variable invasion speeds found in nature to the specific forms of population dynamics and dispersal used in a model? Here, one could consider different existing population dynamics models (e.g. logistic growth, Allee effect) and study their interplay with different dispersal behaviours (e.g.,dispersal propensity, dispersal distance). In this framing, the model could be viewed as the “Tinkerer” role (i.e., pattern/robustness). Finally, one could frame a question in the role of “Mediator” (i.e., pattern/conflict) by noting that some invasive species spread at relatively constant rates. Here the question could be: how do we reconcile the fact that the spread rate of some species fluctuates while the spread rate of others does not? While the methods and results sections in both framings could remain identical, the motivation for the study would appear significantly different based on which above framing is used.
Multiple ways to start with mechanism (Miller et al. 2021, Ecology Letters)
Next, consider a theoretical project that starts with a mechanism of interest and aims to understand the pattern(s) it can generate. Miller et al.
[51] used a theoretical model to understand the effects of disturbance history (a mechanism) on plant community structure and invasion outcomes (patterns), given two resident plant species and an invader species population (i.e., mechanism/novelty – the “Advocate”). However, the most relevant way to define disturbance history in a model may depend on what aspects of disturbance ecologists are interested in understanding. For example, one may be interested in disturbance intensity (i.e., the strength of the disturbance force)
[52]. In contrast, one may be interested in disturbance frequency, which captures the number of disturbed events that have occurred per specified time period
[51]. Given different possible definitions of disturbance history, one could alternatively frame the project by asking how robust the model outcomes are to how the mechanism of disturbance history is defined in the model (i.e., mechanism/robustness framing – the “Semantician”). Alternatively, if disturbance history is viewed as the sum of several different elements, such as disturbance intensity and frequency, the same theory can be framed by understanding how two different mechanisms (disturbance intensity and frequency) combine to shape the dynamics of species community structure (i.e., mechanism/conflict framing – the “Instigator”). Thus, this same theoretical project could be framed in multiple ways that link these specific mechanisms to patterns in plant communities.
Changes in theoretical framing from start to finish
Finally, the framing used by researchers might change throughout the course of a research project. For example, a paper published in 2016 showed how migration from one environment to a second (migratory) environment could be used as a strategy to escape and recover from environmental-specific parasites in the first environment
[53]. When presenting this research in talks, a common audience question was 'what about parasites that are present in the migratory environment?'. To answer this, the authors built a model that included parasites in each environment in order to understand how the costs/risks of having parasites present in both environments shaped the evolution of migration. Walking through our roadmap (Figure 2), this would be a case of starting with mechanism (presence of parasites in the migratory environment), so taking the left branch of the decision tree, considering a single mechanism, and then advocating for the specific mechanism (since the mechanism had not previously been included in models). This leads to an “Advocate” (mechanism/novelty) role, i.e., what are the consequences of including parasites in the second environment (which has previously been overlooked)? The results were fairly intuitive: having parasites in the second environment narrows the range of conditions where migration is favoured, and whether migration is favoured or not comes down to the details of risk of infection (how fast the different parasites are transmitted in each environment) and the cost of infection (how costly the two parasites are compared to each other). However, the authors also discovered that the model produced an intriguing result: migrants typically had lower infection prevalence while simultaneously having higher parasite diversity compared to non-migrants
[33]. This result replicated an ongoing conflict in the empirical literature (described in the “Mediator” section above”), and they decided to revise the paper framing accordingly. Revising the roadmap, this would now be a case of starting with pattern(s) (pattern 1 is that some empirical studies find that migrants have higher infection, and pattern 2 is that other studies find the opposite), so taking the first right branch of the decision tree. At the next step, they were exploring known mechanisms underlying a set of patterns (since it is known empirically that parasites can be present in several environments), and finally at the next step the focus was on reconciling conflicting patterns. This leads to a “Mediator” role (pattern/conflict). The authors felt that a pattern/conflict framing would allow their work to reach a broader audience than a mechanism/novelty one, so they wrote the paper around the question: Here are conflicting empirical data patterns, how do we reconcile them?
Addressing misconceptions of theory
By providing a roadmap for how to formulate specific theoretical questions, we hope that our framework will pave the way for budding theorists to develop successful modelling projects. However, wading into theory can be daunting, particularly for empiricists, in no small part because there are many misconceptions about modelling that may discourage scientists from attempting their own theoretical work. Below, we use our framework to address these misconceptions by considering the example of R* theory – a seminal, mathematically-derived hypothesis in community ecology that suggests that coexistence between resource competitors is possible when competitors are limited by different resources
[54].
Misconception #1: Simplifying assumptions prevent models from being useful
A misconception that appears in almost any context in which theory is discussed is the idea that models are not useful because they rely on simplifying assumptions. No model can include every phenomenon at work in a natural system, but we emphasise that this misconception misunderstands the purpose of theory. Theoretical models are meant to draw connections between informed assumptions and their outcomes. Models cannot exist without assumptions, so this criticism is nonsensical. Our framework focuses on the central modelling question rather than belabouring what a model omits. This type of reframing can help theory-wary scientists better understand the consequences of the assumptions they make. For example, R* theory is concerned with coexistence between species engaged in resource competition, and therefore, in its most basic formulation, ignores the effects of other types of competition like apparent competition
[54]. While this type of interaction may be important in many natural systems, the predictions generated by R* remain useful for answering the central question regarding resource competition. Focusing on these assumptions as limitations of the model ignores the purpose of R*. If researchers, after considering the central theoretical question, find that their initial assumptions will prevent them from successfully addressing their question, we encourage them to amend or alter those assumptions – as some theorists have indeed done with R*, using it to consider the dynamics of apparent competition
[55]. This could result in theoretical work that advocates for the importance of a change in assumptions.
Misconception #2: Adding more model details always leads to predictive improvements
The freedom to amend modelling assumptions is key to successful theory, but that freedom often leads researchers to another misconception of theoretical work: the idea that a model can always be made more useful through the inclusion of additional details. It can be tempting to believe that the addition of another mechanism will make a model “truer to life” and improve results. In practice, additional details increase mathematical complexity and make results harder to interpret. Expanding a model also comes with the cost of time and energy. Empiricists face a similar trade-off: while additional experiments can always be performed (or more observations collected), more data is not worth the effort spent if the central question is already answered by the data in hand. We suggest to new theoreticians that models should only be expanded in pursuit of the specific modelling question. Many such expansions would fall under the role of “Advocate”, and thus should address why inclusion of novel mechanisms is important. When the aim of a model is prediction, it is more likely to be important to include additional phenomena
[9,56]. For example, if one is interested in the robustness of R* coexistence when resources follow seasonal fluctuations, the inclusion of these dynamics would be justified
[54]. Fluctuations in resource availability could also elucidate patterns of coexistence across temporal scales that R* is otherwise not situated to investigate
[54]. Adding a system detail is worthwhile if it has the potential to 1) change the predicted link between a model’s assumptions and its outcomes or 2) open up additional avenues of study through development of a new theoretical question. However, we emphasise that if the goal of a project is to increase understanding of a system, adding details beyond those sufficient to explain the pattern of interest will rarely improve theoretical insights
[9,56]. For example, adding spatial structure and consumers of competitors to R* for the purpose of increasing biological realism could make the model intractable without improving understanding of conditions of coexistence between resource competitors
[54]. We expect that our framework will help scientists new to theory make informed decisions about what to include and when to expand their models, though, in general, we recommend that models are kept as simple as possible.
Misconception #3: You can make a model show anything you want
Making decisions about what to include in a model can occasionally lead new practitioners of theory to believe that models can always be constructed to generate positive results supporting their preferred hypothesis. In some ways, this may be true: because theoretical models link assumptions and outcomes, if it is possible to think of a way that an assumption and an outcome can be connected, then a model can probably be constructed to reflect that. Generally, cases where this does not happen – i.e. when the process of model-building uncovers a flaw in our logic – are not framed this way when published, which can contribute to the sense that models always end up supporting our initial hypotheses. However, a model that produces an intuitive result is often just a starting point for addressing precise theoretical questions. For example, the R* result that competitors can coexist on shared resources may seem trivial until one frames a more precise question, which could be done using any of the six framings. For example, using a pattern/robustness framing (the “Tinkerer”), one could ask “under what conditions is coexistence between competitors possible” (rather than just ‘is coexistence possible’). Second, one could scale the understanding of this mechanism to an unlimited number of competing species (mechanism/novelty framing, the “Advocate”), and find that coexistence is theoretically possible as long as species are limited by different resources
[54]. We suggest to new theoreticians that both getting precise on the question framing and the thoughtful addition of system details can increase the range of possible questions a model can answer, reducing how trivial a model feels. Our framework supports scientists in extending their initial assumptions into these types of precise, insightful questions.
Misconception #4: Successful modelling requires extensive mathematical background
Finally, a misconception that represents a major barrier to the greater adoption of theoretical work in biology is the idea that extensive mathematical background is essential to developing and evaluating theoretical (especially mathematical) models. We believe that framing is much more important to successful theoretical work than mathematical acumen. This is because even practised theoreticians rarely have the need to create new models from scratch since the same mathematical representation can be translated into myriad biological meanings. Model development instead tends to consist of combining and altering pieces of existing models in interesting ways to answer new questions
[17]. Since “most models are variants of previous models”
[17], there are many resources available to aid the mathematical evaluation of any biological model, such as Otto and Day’s “A Biologist’s Guide to Mathematical Models in Ecology and Evolution”
[57] and similar primers. Armed with these guides and the computational power of platforms like MATLAB or R, new practitioners of theory will be able to derive mathematically sound insights from models. However, because model development relies on altering existing models, we suggest that framing is essential to contextualise these models so that they can be appropriately amended for a researcher’s specific purpose. For example, amending R* theory to consider different functional responses in competing species would reflect the work of a “Tinkerer”, asking how robust the pattern of coexistence is to biologically relevant changes in the way the underlying mechanism is translated into mathematical terms. By understanding the framing of an existing model, researchers can appropriately adapt models to answer their own questions without the need for extensive mathematical training.
Theoretical modelling is an incredibly powerful research tool, one that is made richer and more useful by the inclusion of scientists who generally consider themselves empiricists. We hope that our framework, by addressing common misconceptions of theory, can assuage many of the anxieties that scientists new to theory may face and encourage them, particularly those early in their research careers, to take informed theoretical risks. As a final note, we emphasise that the utility of models does not hinge on their validation by empirical experiments or observation
[12]. Models do not require data to be proven, in the same way that it is unnecessary to build a theoretical model to verify the results of an experiment done in the lab. Modelling is one way of knowing in science, most useful when paired with other types of investigation to improve our understanding of a given phenomenon. It is our hope that our framework helps scientists build generative questions to test with theory. Clarifying theoretical questions as we do here will make it easier to put modelling results in their proper context with findings from both lab and field work.
Conclusion
Here, we have argued that developing a clear verbal framing is critical when undertaking a new theoretical research project. Furthermore, we posit that the verbal framing is even more important than the mathematical framing when it comes to communicating the goal and results of a theoretical model with non-theorists, echoing a point from Ou et al.
[14] on the importance of emphasising the narrative reasoning behind models. Here, we present six types of verbal framings of theoretical questions that start with either a pattern or a mechanism, and that focus on novelty, robustness, or conflict. Together, these lead to six roles that one might play as a theorist: the “Advocate”, “Explainer”, “Instigator”, “Mediator”, “Semantician”, and “Tinkerer”. We recognize that there are other ways to frame theory and hope this paper inspires discussion among those with other perspectives on how to frame theory in biology. We have shown that these framings also serve as tools to help avoid misconceptions of theoretical research. Finally, although our focus has been on the field of biology and the approach of mathematical theory, we expect that some of these ways of thinking in terms of being explicit about pattern and mechanism apply broadly to other fields as well (e.g.
[58]). In sum, we hope the six roles we present here can serve as blueprints that researchers who are relatively new to theoretical research can use when developing their theoretical research questions.
Acknowledgements
Thanks to Miao He, Kate Meyer, Matt Michalska-Smith, Maria Napolitani, Aarcha Thadi, Jess Valiarovski, and Steph Varghese for helpful feedback. And to many others for asking questions that inspired writing this paper.
Funding
We were supported in part by funding from the National Science Foundation under Grant No. DEB-2109965 (for NN, PL, AKS), Grant No. IOS-2019304 (for ATB), an INTERN supplement on Grant No. DEB-1654609 (for DK), and an REU on Grant No. DEB-1947406 (for CS); a Doctoral Dissertation Fellowship (for DK) and a Torkse Klubben Fellowship (for MT) from the University of Minnesota; and Startup Funds from the University of Minnesota to AKS (for RA, AKS).
References