The shape of density dependence and the relationship between population growth, intraspecific competition and equilibrium population density

Abstract

Many ecologists and evolutionary biologists use the logistic growth model to capture density dependence.However, assumptions and limitations of this popular model are not well appreciated. Here, we derive population growth models from underlying consumer-resource dynamics and show that the logistic is likely not applicable to many biological systems.We first validate that filter feeders (type I functional response) using abiotic resources generally follow a convex density-regulation function, fully described by the continuous-time Beverton-Holt model. Furthermore, we show that saturating consumers (type II functional response) exhibit density-regulation functions that can switch from concave to convex. We derive a density-regulation function for saturating feeders on abiotic resources and show that more complex consumer dynamics can be well approximated with a continuous-time Maynard Smith-Slatkin model. Importantly, we show how population level parameters, such as intrinsic rates of increase and equilibrium population densities are not independent, but are functions of the same underlying parameters.Our work highlights that the commonly assumed positive relationship between equilibrium population density and competitive ability is typically invalid. Alternatively, we propose simple and general relationships between intrinsic rates of increase and equilibrium population densities that capture the essence of different consumer-resource systems. Finally, we expand our considerations to include multiple consumer and resource species.Relating population level models to underlying mechanisms allows us to discuss applications to evolutionary outcomes and how these models depend on environmental conditions, like temperature via metabolic scaling. Finally, we use time-series from microbial food chains to fit population growth models and validate theoretical predictions.Density-regulation functions need to be chosen carefully as their shapes will depend on the study system's biology. Importantly, we provide a mechanistic understanding of correlations between model parameters, which has implications for theory and for formulating biologically sound and empirically testable predictions.Other authors have acknowledged the dynamic relationship between populations and their resources that causes density dependence by resorting to using more mechanistic consumer-resource models. For instance Matessi and Gatto (1984) show how, in order to understand density-dependent selection, resource dynamics and especially consumer traits have to be taken into account (see also . Such consumer-resource models provide a framework that can be used in an eco-evolutionary context (for a detailed discussion see McPeek, 2017) because model parameters linked to resource use (search efficiency, handling time) are related to real, individual-level traits that can be subject to evolutionary change (Rueffler et al., 2006;Govaert et al., 2019). Importantly, bottom up population regulation due to renewing, depletable resources, as assumed in such consume...

Evolution under pH stress and high population densities leads to increased density‐dependent fitness in the protistTetrahymena thermophila

“…As recently discussed by Fronhofer et al. (, see also chapter 5 in Thieme ), using this model provides a better fit to microcosm data compared to less mechanistic models (for example, an r ‐ K population growth model, which captures the density‐regulation of microcosms less well) and readily allows for a biological interpretation of its parameters. The Beverton–Holt model is given by the equation with the intraspecific competitive ability (α) being Here, N corresponds to population size, r 0 corresponds to the intrinsic rate of increase, α to the intraspecific competitive ability (hereafter referred to as competitive ability), and d to the death rate of individuals in the population.…”

“…3), and by selection for maximizing fitness under pH stress (abiotic conditions) and high population density (biotic factor). Firstly, evolution is constrained in the sense that the intrinsic rate of increase (r 0 ) is positively correlated with competitive ability (α; see also Mueller and Ayala 1981;Reznick et al 2002;Fronhofer et al 2018, for a different view see Joshi et al 2001). This implies that fast growing genotypes will compete more strongly within the population than slow growing genotypes for available resources when densities increase, which is expected to slow down population growth rate at higher densities.…”

“…In order to analyze population growth dynamics of ancestral and evolved populations, we fit a continuous-time version of the Beverton-Holt population growth model (Beverton and Holt 1993). As recently discussed by Fronhofer et al (2018, see also chapter 5 in Thieme 2003, using this model provides a better fit to microcosm data compared to less mechanistic models (for example, an r -K population growth model, which captures the density-regulation of microcosms less well) and readily allows for a biological interpretation of its parameters. The Beverton-Holt model is given by the equation…”

Evolution under pH stress and high population densities leads to increased density-dependent fitness in the protistTetrahymena thermophila

“…3), and by an interaction between demography (high population density; biotic factor) and pH stress (abiotic conditions). Firstly, evolution is constrained in the sense that the intrinsic rate of increase (r 0 ) is positively correlated with competitive ability (α) (see also Mueller and Ayala, 1981;Reznick et al, 2002;Fronhofer et al, 2018, for a different view see Joshi et al 2001). This implies that fast growing genotypes will compete more strongly within the population than slow growing populations for available resources when densities increase, which is expected to slow down population growth rate at higher densities.…”

“…In order to analyze population growth dynamics of ancestral and evolved populations, we fit a continuous-time version of the Beverton-Holt population growth model (Beverton and Holt, 1993). As recently discussed by Fronhofer et al (2018) (see also Thieme, 2003), using this model provides a better fit to microcosm data and readily allows for a biological interpretation of its parameters. The Beverton-Holt model is given by the equation…”