Self-optimization, community stability, and fluctuations in two individual-based models of biological coevolution

Abstract

We compare and contrast the long-time dynamical properties of two individual-based models of biological coevolution. Selection occurs via multispecies, stochastic population dynamics with reproduction probabilities that depend nonlinearly on the population densities of all species resident in the community. New species are introduced through mutation. Both models are amenable to exact linear stability analysis, and we compare the analytic results with large-scale kinetic Monte Carlo simulations, obtaining the population size as a function of an average interspecies interaction strength. Over time, the models self-optimize through mutation and selection to approximately maximize a community potential function, subject only to constraints internal to the particular model. If the interspecies interactions are randomly distributed on an interval including positive values, the system evolves toward self-sustaining, mutualistic communities. In contrast, for the predator-prey case the matrix of interactions is antisymmetric, and a nonzero population size must be sustained by an external resource. Time series of the diversity and population size for both models show approximate 1/f noise and power-law distributions for the lifetimes of communities and species. For the mutualistic model, these two lifetime distributions have the same exponent, while their exponents are different for the predator-prey model. The difference is probably due to greater resilience toward mass extinctions in the food-web like communities produced by the predator-prey model.

Modeling symbiosis by interactions through species carrying capacities

“…There are particular models describing the biological co-evolution of species by considering equations for some distributions, whether over size, or over age, or over phenotypes [23][24][25][26]. But our aim is to obtain the equations for the concentrations themselves, so that these equations could describe the variety of possible symbiotic relations.…”

Effects of demographic stochasticity on biological community assembly on evolutionary time scales

“…The limit σ F → 0 corresponds to the model considered in the body of this paper. We briefly show this solution for the sake of completeness and readers' convenience, although it is shown in detail in [18,19]. At the fixed point, the condition |P (R, {n * }) = 1/F is satisfied, where |P is the column vector of the reproduction probabilities.…”

“…The tanglednature model is an individual-based model, originally introduced by Hall and co-workers [17] and later simplified by Rikvold and Zia [20]. In the simplified models [18][19][20][21][22]30], the population evolves stochastically in discrete, non-overlapping generations. In these models, each individual of species I gives rise to F offspring with a reproduction probability P I before it dies.…”

“…With this value, the dynamics for a single species approaches its fixed point monotonically. These models have fixed points |n * , which can be calculated exactly [18]. Here |n * is a column vector of the equilibrium population sizes of all species present in the community.…”

Extreme events in population dynamics with functional carrying capacity

“…The feedback between the population density, associated with the Allee effect, can lead to the increase of the effective growth rate, in the case of sufficiently large populations, or to the rate decrease and species extinction for small-density populations [38][39][40][41]. In addition to the differential equations describing population dynamics, there exist as well difference equations [42,43] and integro-differential equations [44,45]. There are more complicated equations characterizing several factors, such as the population density, mass or weight dependence of individual members of different species, the dynamics of available food, and so on [44,46,47].…”