DEBkiss or the quest for the simplest generic model of animal life history

https://doi.org/10.1016/j.jtbi.2013.03.011Get rights and content

Highlights

  • We present a simple energy-budget approach to model life-history traits of animals.

  • Model equations are derived from explicit assumptions that simplify biology.

  • The model covers the entire life cycle (embryo, juvenile, adult).

  • A case study for the pond snail is used to illustrate model performance.

Abstract

Understanding the life cycle of individual animals, and how it responds to stress, requires a model that causally links life-history traits (feeding, growth, development and reproduction). Dynamic Energy Budget (DEB) theory offers a powerful and formalised framework for building process-based models for organism life cycles. However, it takes some serious investment to understand the resulting equations and to implement them into software, and a substantial amount of data to parameterise. For many practical applications, there is therefore a need for further simplification. Here, we present a simple and transparent model that fully specifies the life cycle of an (invertebrate) animal, applies a strict mass balance, and has direct access to the primary parameters that determine the metabolic processes. We derive our ‘DEBkiss’ in a formalised manner, starting from an explicit formulation of the simplifying assumptions. The presented model can serve as a teaching tool and a smooth introduction into the much richer world of DEB theory. Furthermore, the model may prove useful as a building block for individual-based population modelling (where simplicity of the blocks is essential), and for the analysis of toxicity data (where ease of model verification and parameterisation is crucial). The model is illustrated using a fit on growth and reproduction data for the pond snail (Lymnaea stagnalis) at three food levels, and subsequent predictions for embryonic growth and respiration (oxygen use), and weight loss on starvation, for the same species.

Introduction

Simple is beautiful, but also practical, as embodied in the engineering principle of KISS (keep it simple, stupid). Complex things tend to break, and when they do, they are difficult to repair. But, as the quote often attributed to Albert Einstein warns us: “everything should be made as simple as possible, but not simpler.” Here, we are going to apply the KISS principle to modelling of life-history traits of an animal, while heeding Einstein's caution. How simple can we make a model for such traits of an individual, while still maintaining a degree of realism? This is one stage in a continuous quest for balancing simplicity and realism; a balancing act that will obviously depend on the purpose for which the model will be used. The specific purpose that we have in mind is to apply such a model for individuals to interpret the effects of stressors such as toxicants (Jager et al., 2006) or food limitation (Zimmer et al., 2012), and to translate the effects on the individual to the population level (Martin et al., 2012, Jager and Klok, 2010). The focus in our work is on small invertebrate animals.

At minimum, our model should provide a prediction of reproductive output over the life cycle of an animal, as a function of food availability (which might vary over time). Reproductive output is the most straightforward indicator of individual fitness, and clearly needed for the translation to the population level; in its simplest form we can think of population dynamics as the difference between births and deaths. However, the reproduction rate is not determined by the current food level alone; it also depends on the state of the individual. Body size is an obvious candidate for such a state, as it determines feeding rates (and thereby the available resources for reproduction), and is often an accurate indicator of whether the organism is capable of reproducing. Interpreting the effects of varying food levels and stressors on reproduction therefore requires (at least) following body size as a state variable. Furthermore, because the dynamics of populations often depend on feedbacks between a population and its prey, keeping track of body size (and the associated feeding rates) is an essential aspect in population models. Our model should thus provide us with a good description of at least body size and reproductive output over the entire life cycle (including the embryonic stage) as a function of food availability. It should be based on well-established principles (such as conservation of mass and energy, and consistency with thermodynamics) to ensure that the model behaviour is physically realistic. Furthermore, the model should include a (possibly crude) representation of biological processes such that we can model stressor effects on these processes. And finally, the core model should be generic and free from species- or stressor-specific argumentation as we do not want to build a new model for each species-stressor combination.

Dynamic Energy Budget (DEB) theory offers a powerful and formalised framework for building such models (Kooijman, 2001, Sousa et al., 2010, Nisbet et al., 2000). This power, however, comes at a price. Even though the concepts and underlying assumptions are simple, understanding how they lead to the equations of the ‘standard DEB model’ for animals (see Sousa et al., 2010) is not. Implementation of the model in software is certainly not straightforward, and the subsequent parameterisation requires an extensive data set. Although efficient procedures and software have been developed to aid the user and to accommodate limited data sets (Kooijman, 2009, Lika et al., 2011), it takes serious study to be able to apply them properly, and even more to verify the code. One would effectively have to rely on the derivations and programming of the developers, which can be an issue for potential users.

The standard model is the simplest complete DEB model, but it is often considered too complex (e.g., as a basis for population modelling, Nisbet et al., 2010). In many practical fields of application, the interest in dynamic models rapidly declines with the level of complexity. The standard animal model has been simplified, yielding the ‘scaled standard model’ (Kooijman et al., 2008, Jager et al., 2010) and ‘DEBtox’ (Jager and Zimmer, 2012). These simplifications, however, have their disadvantages. The use of scaling and compound parameters hampers interpretation of the equations and can lead to difficult-to-spot inconsistencies (e.g., transformation efficiency greater than one) for certain choices of parameter values. Furthermore, the use of compound parameters hampers the straightforward application of stress due to toxicants, which are assumed to affect metabolic processes and thus primary energy-budget parameters (Jager et al., 2010).

In short, we believe that there is room for a simple and transparent model that fully specifies the life cycle of an (invertebrate) animal, applies an explicit mass balance, and has direct access to the primary parameters that determine the metabolic processes. The model should be simple enough for users to check its consistency, implement into their own software of choice, and to parameterise it on easily obtained data sets without additional help. Such a model would be suitable for particular applications where simplicity is of key importance, but it may also provide a good teaching tool for theoretical biology in general, and DEB theory in particular. In this paper, we present such a simple model in a formalised manner (starting from an explicit formulation of the simplifying assumptions). We name the model ‘DEBkiss’ to emphasise that the work is highly inspired by DEB theory, with a strong focus on the KISS principle.

Section snippets

Model definition

The DEBkiss model we propose is schematically depicted in Fig. 1, showing the mass fluxes J(⁎) (in dry weight per unit of time). In the possible topologies for energy budget models of Lika and Kooijman (2011), it would classify as a κGR0R model. In the κ models, the fundamental split between investment in the soma and reproduction comes first (on the assimilates obtained from feeding). This contrasts ‘production models’, where maintenance costs are paid before the split (e.g., Lika and Nisbet,

Case study

To demonstrate how to link DEBkiss to real data, we provide an example for the pond snail Lymnaea stagnalis.

Main differences with existing DEB models

The DEBkiss model is very similar to the model that laid the foundation of DEB theory: the budget model of Kooijman and Metz (1984), which was also used in a few later studies (e.g., Klok and De Roos, 1996, Jager and Klok, 2010). If we include maturity maintenance (see Section 2.3), the resulting model is essentially equivalent to the simplified DEBtox model (Jager and Zimmer, 2012) in which the reserve density goes to zero (the ‘energy investment ratio’ goes to infinity). However, we here

Acknowledgements

This research has been financially supported by the European Union under the 7th Framework Programme (project acronym CREAM, contract number PITN-GA-2009-238148). We thank two anonymous reviewers for their constructive comments.

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