Equivalence relationships between stage-structured population models

Predicting how defaunation-induced changes in seed predation and dispersal will affect tropical tree populations

The loss of large animals due to overhunting and habitat loss potentially affects tropical tree populations and carbon cycling. Trees reliant on large-bodied seed dispersers are thought to be particularly negatively affected by defaunation. But besides seed dispersal, defaunation can also increase or decrease seed predation. It remains unclear how these different defaunation effects on early life stages ultimately affect tree population dynamics. We reviewed the literature on how tropical animal loss affects different plant life stages, and we conducted a meta-analysis of how defaunation affects seed predation. We used this information to parameterize models that altered matrix projection models from a suite of tree species to simulate defaunation-caused changes in seed dispersal and predation. We assessed how applying these defaunation effects affected population growth rates. On average, population-level effects of defaunation were negligible, suggesting that defaunation may not cause the massive reductions in forest carbon storage that have been predicted. In contrast to previous hypotheses, we did not detect an effect of seed size on changes in seed predation rates. The change in seed predation did not differ significantly between exclosure experiments and observational studies, although the results of observational studies were far more variable. Although defaunation surely affects certain tree taxa, species that benefit or are harmed by it and net changes in forest carbon storage cannot currently be predicted based on available data. Further research on how factors such as seed predation vary across tree species and defaunation scenarios is necessary for understanding cascading changes in species composition and diversity.

The genealogical decomposition of a matrix population model with applications to the aggregation of stages

Matrix projection models are a central tool in many areas of population biology. In most applications, one starts from the projection matrix to quantify the asymptotic growth rate of the population (the dominant eigenvalue), the stable stage distribution, and the reproductive values (the dominant right and left eigenvectors, respectively). Any primitive projection matrix also has an associated ergodic Markov chain that contains information about the genealogy of the population. In this paper, we show that these facts can be used to specify any matrix population model as a triple consisting of the ergodic Markov matrix, the dominant eigenvalue and one of the corresponding eigenvectors. This decomposition of the projection matrix separates properties associated with lineages from those associated with individuals. It also clarifies the relationships between many quantities commonly used to describe such models, including the relationship between eigenvalue sensitivities and elasticities. We illustrate the utility of such a decomposition by introducing a new method for aggregating classes in a matrix population model to produce a simpler model with a smaller number of classes. Unlike the standard method, our method has the advantage of preserving reproductive values and elasticities. It also has conceptually satisfying properties such as commuting with changes of units.

Matrix models for size-structured populations: unrealistic fast growth or simply diffusion?

Matrix population models are widely used to study population dynamics but have been criticized because their outputs are sensitive to the dimension of the matrix (or, equivalently, to the class width). This sensitivity is concerning for the population growth rate (λ) because this is an intrinsic characteristic of the population that should not depend on the model specification. It has been suggested that the sensitivity of λ to matrix dimension was linked to the existence of fast pathways (i.e. the fraction of individuals that systematically move up a class), whose proportion increases when class width increases. We showed that for matrix population models with growth transition only from class i to class i + 1, λ was independent of the class width when the mortality and the recruitment rates were constant, irrespective of the growth rate. We also showed that if there were indeed fast pathways, there were also in about the same proportion slow pathways (i.e. the fraction of individuals that systematically remained in the same class), and that they jointly act as a diffusion process (where diffusion here is the movement in size of an individual whose size increments are random according to a normal distribution with mean zero). For 53 tree species from a tropical rain forest in the Central African Republic, the diffusion resulting from common matrix dimensions was much stronger than would be realistic. Yet, the sensitivity of λ to matrix dimension for a class width in the range 1-10 cm was small, much smaller than the sampling uncertainty on the value of λ. Moreover, λ could either increase or decrease when class width increased depending on the species. Overall, even if the class width should be kept small enough to limit diffusion, it had little impact on the estimate of λ for tree species.

Complex population dynamics and complex causation: devils, details and demography

Population dynamics result from the interplay of density-independent and density-dependent processes. Understanding this interplay is important, especially for being able to predict near-term population trajectories for management. In recent years, the study of model systems-experimental, observational and theoretical-has shed considerable light on the way that the both density-dependent and -independent aspects of the environment affect population dynamics via impacting on the organism's life history and therefore demography. These model-based approaches suggest that (i) individuals in different states differ in their demographic performance, (ii) these differences generate structure that can fluctuate independently of current total population size and so can influence the dynamics in important ways, (iii) individuals are strongly affected by both current and past environments, even when the past environments may be in previous generations and (iv) dynamics are typically complex and transient due to environmental noise perturbing complex population structures. For understanding population dynamics of any given system, we suggest that 'the devil is in the detail'. Experimental dissection of empirical systems is providing important insights into the details of the drivers of demographic responses and therefore dynamics and should also stimulate theory that incorporates relevant biological mechanism.