The uncertain role of diversity dependence in species diversification and the need to incorporate time-varying carrying capacities

Abstract

There is no agreement among palaeobiologists or biologists as to whether, or to what extent, there are limits on diversification and species numbers. Here, we posit that part of the disagreement stems from: (i) the lack of explicit criteria for defining the relevant species pools, which may be defined phylogenetically, ecologically or geographically; (ii) assumptions that must be made when extrapolating from population-level logistic growth to macro-evolutionary diversification; and (iii) too much emphasis being placed on fixed carrying capacities, rather than taking into account the opportunities for increased species richness on evolutionary timescales, for example, owing to increased biologically available energy, increased habitat complexity and the ability of many clades to better extract resources from the environment, or to broaden their resource base. Thus, we argue that a more effective way of assessing the evidence for and against the ideas of bound versus unbound diversification is through appropriate definition of the relevant species pools, and through explicit modelling of diversity-dependent diversification with time-varying carrying capacities. Here, we show that time-varying carrying capacities, either increases or decreases, can be accommodated through changing intrinsic diversification rates (diversity-independent effects), or changing the effects of crowding (diversity-dependent effects).

Keywords: ecological saturation; evolutionary innovation; extinction; logistic growth; speciation; species equilibrium.

Figure 1.

Depiction of the intrinsic per lineage rate of diversification (intercepts on the y-axis) and carrying capacities (intercepts on the x-axis) for the three ‘faunas' Sepkoski [4] used to model the diversity trajectory of marine Phanerozoic families (inset; used with permission from the Paleontological Society).

Figure 2.

Assumptions and their implications when extending the population-level notion of logistic growth to species-level logistic growth. Left: phylogenetic/single-clade approach. (a) Simple scaling up of population level logistic growth makes some assumptions that are unrealistic when viewing evolution in deep time. (b) Relaxing the assumption of a fixed carrying capacity renders the phylogenetic approach more reasonable, and allows for a role for diversity dependence even if non-equilibrial processes dominate. Right: whole biota approach. (c) This approach makes some simple assumptions that are valid on short timescales. (d) To use the approach on longer timescales, it is necessary to include speciation as a relevant process. (e) Palaeontologists have co-opted the whole biota island biogeography approach, where the whole Earth is viewed as an island. In addition, they have used it hierarchically. In both approaches, the use of a logistic formulation becomes more reasonable once one incorporates the idea that the carrying capacity might change through time.

Figure 3.

Simple logistic growth. Essential elements of logistic growth, including the relationship between the diversity-independent intrinsic diversification rate (r₀), the strength of the diversity dependence (γ) and the carrying capacity (K). (Online version in colour.)

Figure 4.

Diversity trajectories (right panels) when the carrying capacity changes linearly with time via an increase or decrease in the intrinsic diversification rate, r₀(t), the ability to generate species with no crowding (left panels). The diversity curves were generated solving equation (4.1) with Mathematica 10.1.0.0. In all cases, the initial diversification rate (r₀) was 0.4 lineages/lineage million years, and the strength of diversity dependence (γ) of 0.001 per lineage million years, yielding an initial carrying capacity (K_t=0) of 400. For the slower rates of change α′ = 0.004 (a 1% change/million years), while the faster rate was fivefold faster with α′ = 0.02 (5% change/million years). (Online version in colour.)

Figure 5.

Diversity trajectories (right panels) when the carrying capacity changes linearly with time via an increase or decrease in the strength of per-species diversity dependence, γ(t), the effect of crowding in inhibiting diversification (left panels). The diversity curves were generated solving equation (4.2) with Mathematica 10.1.0.0. The initial values of r₀, γ, and thus the initial carrying capacity (K_t=0) of 400, were the same as those used in figure 4. Similarly, we used the same α values as in figure 4, which, given that α = α′/γ in equation (4.2) (see appendix A), translated into a value of α = 4 for the slower rates (a 1% change/million years), and α = 20 species for the faster rates (5% change/million years). (Online version in colour.)