Abstract
In this paper, we establish the explicit connection between deterministic trait-based population-level models (in the form of partial differential equations) and species-level models (in the form of ordinary differential equations), in the context of eco-evolutionary systems. In particular, by starting from a population-level model of density distributions in trait space, we derive what amounts to an extension of the typical models at the species level known from adaptive dynamics literature, to account not only for abundance and mean trait values, but also explicitly for trait variances. Thus, we arrive at an explicitly polymorphic model at the species level. The derivations make precise the relationship between the parameters in the two classes of models and allow us to distinguish between notions of fitness on the population and species levels. Through a formal stability analysis, we see that exponential growth of an eigenvalue in the trait covariance matrix corresponds to a breakdown of the underlying assumptions of the species-level model. In biological terms, this may be interpreted as a speciation event: that is, we obtain an explicit notion of the blow-up of the variance of (possibly a linear combination of) traits as a precursor to speciation. Moreover, since evolutionary volatility of the mean trait value is proportional to trait variance, this provides a notion that species at the cusp of speciation are also the most adaptive. We illustrate these concepts and considerations using a numerical simulation.
Highlights
A range of models loosely termed ‘adaptive dynamics’ have been introduced to study the interaction between ecology and evolution, and link population dynamics to evolutionary stochastic deterministic royalsocietypublishing.org/journal/rsos R
Open Sci. 7: 200321 upscaling is insufficiently general to cover this case or the model becomes fundamentally 3 inapplicable. We argue that these interpretations are both correct: the species-level model breaks down when the assumption of clearly defined species is invalid, which is a fundamental assumption of the upscaling as well as the premise under which a species-level model is relevant. (When this assumption is not met, the prudent modeller will revert to a population-level model to analyse the system dynamics.)
The mismatch at intermediate times is due to the moment closure approximation: since the skewness of the intraspecific distribution of trait values cannot be captured by the mean and variance of traits, its influence on the evolution of the mean trait values is neglected, which is of modest importance during transitions as evidenced by the non-normal distribution seen in figure 4
Summary
A range of models loosely termed ‘adaptive dynamics’ have been introduced to study the interaction between ecology and evolution, and link population dynamics to evolutionary stochastic deterministic royalsocietypublishing.org/journal/rsos R. Since we are explicitly interested in polymorphic species-level models (a key component of the present contribution), it becomes natural to instead categorize models along the two axes of deterministic versus stochastic, and population versus species (figure 1) In this figure, and throughout the paper, we abbreviate ordinary and partial differential equations as ODEs and PDEs, respectively. We note that an objective similar to that of the current paper was pursued by Débarre et al [12] In their paper, they derive from a population-level model, the species-level equations for the change in the mean and variance of populations in the absence of population dynamics, which is sufficient when discussing conditions for evolutionary stationary equilibria.
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