Discreteness of populations enervates biodiversity in evolution

Biodiversity widely observed in ecological systems is attributed to the dynamical balance among the competing species. The time-varying populations of the interacting species are often captured rather well by a set of deterministic replicator equations in the evolutionary game theory. However, intrinsic fluctuations arisen from the discreteness of populations lead to stochastic derivations from the smooth evolution trajectories. The role of these fluctuations is shown to be critical at causing extinction and deteriorating the biodiversity of ecosystem. We use children’s rock-paper-scissors game to demonstrate how the intrinsic fluctuations arise from the discrete populations and why the biodiversity of the ecosystem decays exponentially, disregarding the detail parameters for competing mechanism and initial distributions. The dissipative trend in biodiversity can be analogized to the gradual erosion of kinetic energy of a moving particle due to air drag or fluid viscosity. The dissipation-fluctuation theorem in statistical physics seals the fate of these originally conserved quantities. This concept in physics can be generalized to scrutinize the errors that might be incurred in the ecological, biological, and quantitative economic modeling for which the ingredients are all discrete in number. 1 ar X iv :1 00 5. 43 35 v1 [ qbi o. PE ] 2 4 M ay 2 01 0 Biodiversity is commonly used to indicate the stability of an ecosystem[1–3]. One of the central issues is to effectively promote the biodiversity while attracting more scientists attention from various fields[4–8]. The causes that threaten the biodiversity, for instance, climate change[4, 5], over-harvesting, habitat destruction[6], and population mobility[7], are well studied. Above those factors, Darwin’s theory of natural selection plays a crucial role in catalysis[9–12]. People are warned to reduce these effects in order to maintain and reserve the nature’s biodiversity. Nevertheless, a naive reversed statement can be checked, namely, without any hazardous factors, would ecosystem be perfectly stable? Here, we show the emergence of an intrinsic force originated from the fact that populations must be discredited. Furthermore, this force macroscopically jeopardizes the biodiversity of an ecosystem. However, populations in an ecosystem are discrete integers. Approximating these discrete populations by continuous variables inevitably introduces intrinsic fluctuations, which turn the evolutionary dynamics stochastic in nature. When external noises are introduced, the stochastic process has been shown to be capable of causing mass extinction[13] in analogous to the avalanche in the sand piles. How important is this tiny difference between discrete and continuous variables when the population size is large? Do the intrinsic fluctuations simply introduce small irregularities or will they ever accumulate and cause a drastic impact on the biodiversity of the ecosystem? To put the discussions on firm ground, we concentrate on the non-transitive rock-paperscissors game[14–19], known as a paradigm to illustrate the species diversity. When three subpopulations interact in this non-transitive way, we expect that each species can invade another when its population is rare but becomes vulnerable to the other species when over populated. The non-hierarchical competition[20–23] gives rise to the endlessly spinning wheel of species chasing species and the biodiversity of the ecosystem reaches a stable dynamical balance. This cyclic evolutionary dynamics has been found in plenty of ecosystems such as coral reef invertebrates[24], lizards in the inner Coast Range of California[25] and three strains of colicinogenic Escherichia coli [15, 26] in Petri dish. Although the oscillatory solutions for the replicator equations capture the main features, inclusion of mobility[7] or/and finite-population effects[22, 23] in the numerical simulations always jeopardizes the stable equilibrium and highlight the importance of stochasticity in the evolutionary dynamics. To measure the effects due to the discreteness of the populations, we introduce a bio-