By Thomas Banitz & Volker Grimm

Ecology is the discipline of natural sciences that studies how organisms interact with each other and the environment. Together, the biotic organisms and their abiotic environment constitute ecosystems. The perspective of ecological systems analysis, also referred to as ecological modeling, aims at revealing causal relationships within these ecosystems. The approaches to perform such analysis can be differentiated into statistical modeling and mechanistic modeling (Radchuk et al. 2019). Yet, the boundary between these modeling categories is not always strictly defined.

## Statistical modeling

As in many other sciences, descriptive statistics is commonly used in ecology to summarize and visualize data, for example, calculating the mean and variance of a certain focal variable, or plotting a focal variable versus another, potentially explanatory variable (e.g. mortality vs. predator abundance, biomass production vs. nutrient concentration). A simple, and probably most widely used causal notion is embedded in such plots: a change in the dependent variable might be caused by a change in the independent variable.

More explicitly, statistical models are assumptions about a set of possible observations and their probabilities. These assumptions are expressed as probability distributions. As in other research fields, statistical models in ecology are used to analyze observational data and infer underlying probability distributions. This allows testing and accepting or rejecting hypotheses on the relationship between the state variables investigated (Hilborn and Mangel, 1997), given that the specified assumptions hold. It goes beyond the relationships between just two variables, for instance, using variation partitioning methods to assess the power of several candidate variables to explain the variation in a focal state variable.

However, detecting statistical relationships does not mean that these relationships are necessarily causal (“correlation does not imply causation”). It also neglects feedbacks on larger time or spatial scales. Therefore, causal interpretations of the hypotheses derived from statistical models should be supported by specific knowledge on their plausibility (e.g. on potential mechanisms for causal pathways, on the temporal order of events with causes preceding their effects) and, ideally, by additional targeted experiments (e.g. manipulating potential causes while holding other variables constant). Although causal reasoning based on statistical relationships is very common in ecology, it is not uncommon that the underlying assumptions are insufficiently discussed.

## Mechanistic modeling

Mechanistic models are used to describe the dynamics of ecological systems. This means that the processes that change ecological state variables are formalized into functional relationships between the changed variable and other variables affecting the change.

An important group are equation-based models, for instance, differential equations defining the rates of change of populations or of abiotic state variables (such as resources; Otto and Day 2011). Depending on how the specific equations for each state variable are defined, various forms of direct and indirect interactions between these variables can be modeled. Thus, the terms and parameter values in an equation-based model define the causal relationships between the modeled ecological variables. Often, the change in one variable is not just caused by others but also by the state of the variable itself, e.g. a population increase due to reproduction depends on the size of the population leading to negative feedbacks and nonlinearities.

For equation-based models, the resolution of biotic state variables described is often the population level, but lumping several populations into one state variable (“functional types”) or differentiating populations, e.g. into several age or stage classes, is possible (Caswell 2001). Also the level of further detail of such models may vary considerably. Equations may apply to the respective state variables for the whole ecosystem, or spatial differences may be taken into account (e.g. via partial differential equations). Another example is “negative density dependence”, denoting that certain fitness-enhancing processes get intensified at small (weakened at large) population sizes. This may be taken into account by defining process rates as a function of population size or by modeling the underlying mechanisms in more detail, e.g. defining process rates as a function of resources and modeling the dynamics of resource abundance as a state variable too.

Once an equation-based ecological model is defined, it can be analyzed. Fixed points can be identified for which certain state variables do not change and to which state variables may be attracted again upon perturbations. The initial conditions can be systematically varied, and in some cases analytical solutions of the equations can be obtained. More often, numerical simulations are used to obtain discretized solutions over time.

Simulations are also central for mechanistic models of ecosystems that, instead of a set of equations, are defined by a set of rules for how to propagate the state variables forward in time (an algorithm). This kind of models plays an increasingly important role in ecology (Hartig 2018). It allows more freedom in defining the modeled entities (biotic and abiotic), several state variables assigned to these entities, and the causal relationships between them. Importantly, the potentially highly complex ecosystem dynamics that result from the defined causal relationships need to be known *a priori* to a much lesser extent than in equation-based models. Rather, they emerge from simulating the dynamics of all state variables according to the defined rules. This aspect of simulation models is particularly important when not populations but single individuals are the modeled biotic entities (“individual-based models”, IBMs, often also referred to as “agent-based model”, ABMs; DeAngelis and Grimm 2014). IBMs define how individuals behave, how they change and respond to each other and their abiotic environment. The rules that need to be defined at this low level are often much better known and understood than the resulting dynamics of the whole ecosystem. IBMs facilitate the representation of local interactions and individual variation (Banitz 2019) as well as adaptive behavior, all of which are typical features of many ecological systems. Consequently, investigating the effects on the modeled ecosystem caused by these features gets possible.

Irrespective of the particular type of mechanistic ecological model, a general strategy for disentangling causal relationships in the modeled ecosystem is to systematically modify the model and observe the consequences for model output. This entails modifying initial conditions, but also parameter values and even process definitions (rules or equations). For this “sensitivity analysis”, established methods exist that take into account that mechanisms interact (Thiele et al. 2014). Factor A may have strong influence on factor B, but only if factor C is within a certain range. So-called “global sensitivity analyses” allow ranking model parameters in terms of their overall importance for model output, but also their relative importance. For the latter, the above-mentioned variation partitioning methods are often used, but can be based on more comprehensive data than in empirical studies. The empirical counterpart to sensitivity analyses are experiments following factorial designs (Lorscheid et al. 2012).

Model output ideally contains different aspects characterizing the ecosystem (and its dynamics) at different organizational levels (e.g. from individuals to populations and communities, spatial or temporal, explicit or aggregated, biotic or abiotic). These different “patterns” and how they change in response to model modifications reveal the effects of the single causal relationships when embedded in the complex framework of all processes considered in the model (Grimm and Railsback 2012).

Understanding the causes of certain structure and behaviors that emerge in mechanistic models can be a challenge, because of the complexity of the model system. However, in contrast to real ecosystems, factors that are assumed to cause certain effects can simply be switched on and off to directly test this (“robustness analysis”; Grimm and Berger 2016).

In this manner, a mechanistic model can be used without empirical data (as a “generic” model) or confronted with empirical observations from a specific ecosystem. In the generic case, the model is developed to formalize an idea, a conceptual model, a certain set of assumptions into a complete and precise specification of all causal relationships that follow from these assumptions. As described, the multiple characteristic patterns that emerge from this set of interacting causal relationships are then used to observe and understand the wider consequences of the assumptions, assess their plausibility and eventually determine critical factors for the ecological dynamics of interest.

The same applies when the different patterns obtained from the model are also confronted with corresponding patterns obtained from empirical data. By iteratively modifying the model and comparing model output and data, one can at the same time approximate the realistic dynamics of the ecosystem of interest and understand the causal mechanisms that generate these dynamics (“pattern-oriented modeling”; Grimm and Railsback 2012). For this approach of confronting and fitting mechanistic ecological models to empirical data, again statistical models (statistical inference techniques) are highly useful, especially when the mechanistic models contain stochastic processes.

## References

Banitz, T. (2019). Spatially structured intraspecific trait variation can foster biodiversity in disturbed, heterogeneous environments. *Oikos*, *128*(10), 1478-1491.

Caswell, H. (2001) Matrix population models: construction, analysis, and interpretation. Sinauer, Sunderland.

DeAngelis, D. L., & Grimm, V. (2014). Individual-based models in ecology after four decades. *F1000prime reports*, *6*.

Grimm, V., & Berger, U. (2016). Robustness analysis: deconstructing computational models for ecological theory and applications. *Ecological Modelling*, *326*, 162-167.

Grimm, V., & Railsback, S. F. (2012). Pattern-oriented modelling: a ‘multi-scope’ for predictive systems ecology. *Philosophical Transactions of the Royal Society B: Biological Sciences*, *367*(1586), 298-310.

Hartig, F. (2018). Simulation Modeling. *arXiv preprint arXiv:1812.09137 *

Hilborn, R., & Mangel, M. (1997). The ecological detective: confronting models with data. Princeton University Press.

Lorscheid, I., Heine, B. O., & Meyer, M. (2012). Opening the ‘black box’ of simulations: increased transparency and effective communication through the systematic design of experiments. *Computational and Mathematical Organization Theory*, *18*(1), 22-62.

Otto, S. P., & Day, T. (2011). A biologist’s guide to mathematical modeling in ecology and evolution. Princeton University Press.

Radchuk, V., Kramer-Schadt, S., & Grimm, V. (2019). Transferability of mechanistic ecological models is about emergence. *Trends in Ecology and Evolution*, *34*(6), 487-488.

Thiele, J. C., Kurth, W., & Grimm, V. (2014). Facilitating parameter estimation and sensitivity analysis of agent-based models: a cookbook using NetLogo and R. *Journal of Artificial Societies and Social Simulation*, *17*(3), 11.