Session J42: Focus Session: Stochastic Population Dynamics I - Cyclic competition and population stability

Cyclic competition of four or more species: Results from mean field theory and stochastic simulations

Population dynamics is a venerable subject, dating back two centuries to Malthus, Verhulst, Lotka, Volterra, and many others. Nonetheless, new and interesting phenomena are continually being discovered. For example, the recent discovery of ``Survival of the Weakest'' in cyclic competition between 3 species with no spatial structure (Berr, Reichenbach, Schottenloher, and Frey, Phys. Rev. Lett. 102, 048102 (2009)) attracted considerable attention, e.g., http://www.sciencedaily.com/releases/2009/02/090213115127.htm. Considering a similar system with 4 or more species, we find a more intuitively understandable principle which appears to underpin all systems with cyclically competing species. We will present several interesting aspects of the 4 species system -- from non-linear dynamical phenomena in a deterministic mean-field approach to remarkable extinction probabilities in the stochastic evolution of a finite system. Some insights into the deterministic dynamics, gained from generalizing this system to one with any number of species with arbitrary pairwise interactions, will also be discussed.   

Varieties of extinction scenarios when four species compete cyclically

We study a stochastic system with $N$ individuals, consisting of four species competing cyclically: $A+B \longrightarrow A+A$, $\cdots$, $D+A \longrightarrow D+D$. Randomly choosing a pair and letting them react, $N$ is conserved but the fractions of each species evolve non-trivially. At late times, the system ends in a static, absorbing state $-$ typically, coexisting species $AC$ or $BD$. The master equation is shown and solved exactly for $N=4$, providing a little insight into the problem. For large $N$, we rely on simulations by Monte Carlo techniques (with a faster dynamics where a reaction occurs at every step). Generally, the results are in good agreement with predictions from mean field theory, after appropriate rescaling of Monte Carlo time. The theory fails, however, to describe extinction or predict their probabilities. Nevertheless, it can hint at many remarkable behavior associated with extinction, which we discover when studying systems with extremely disparate rates.   

The effects of mobility on the one-dimensional four-species cyclic predator-prey model

The dynamics of a one-dimensional lattice composed of four species cyclically dominating each other is very much dependent on the rates of mobility in the system. We realize mobility as the exchange of two particles located at two nearest neighbor sites with some species dependent rate s. Allowing for only one particle per site, the different species interact cyclically, with species dependent consumption rate k, such that $k + s \leq 1$. When varying the exchange rates, we see vastly different behavior when compared to the three-species model. The patterns of domain growth and decay still show an overall power law behavior, however the fundamental trend of domain growth does not follow the three-species case. We also look at the space-time diagrams to see precisely how the domains form, grow, and decay.   

Boundary conflicts and cluster coarsening: Waves of life and death in the cyclic competition of four species

In the cyclic competition among four species on a two-dimensional lattice, the partner particles, which swap positions on the lattice with some probability, produce clusters with a length that grows algebraically as $t^{1/z}$ where $z$ is the dynamical exponent. Further investigation of the dynamics at the boundary of the clusters is realized by placing one partner particle pair in the upper half of the system and the other pair in the lower half. Using this technique, results about the fluctuations of the interface are obtained. We also observe wave fronts in the case of non-symmetric reaction rates where extinction of a partner particle pair takes place.   

Discriminating the effects of spatial extent and population size in cyclic competition among species

Quantifying and understanding the stability and biodiversity of ecosystems is a major task in biological physics as well as in theoretical ecology. From the perspective of game theory, this is highly relevant for questions pertaining to the emergence of cooperation or the coexistence of cyclically competing species. The latter has been recently proposed as a paradigm for biodiversity and it has been shown that the mobility of individuals can support the stability of biodiversity by the formation of spirals. In this contribution, we present a population model for species under cyclic competition that extends earlier lattice models to allow the single cells to accommodate more than one individual by introducing a per cell carrying capacity. We confirm that the emergence of spirals induce a transition from an unstable to a stable regime. This transition however does not appear to be sharp and we find a broad intermediate regime that exhibits an ambiguous behavior. The separation of the two regimes by the usual scaling analysis is thus hampered. The newly introduced carrying capacity offers an alternative way of characterizing the transition. We thus overcome the original limitations by separately analyzing the effect of spatial extent and population size.   

Stochastic extinction dynamics of HIV-1

We consider an HIV-1 within host model in which T cells are infected by the virus. Due to small numbers of molecules, stochastic effects play an important role in the dynamical outcomes in that two states are observed experimentally: a replication state in which the virus is active, or a dormant state leading to latency in which the virus becomes active after a delay. The two states are conjectured to be governed by the Tat gene protein transcription process, which does not possess two stable attractors. Rather, the active state is stable, while the dormant state is unstable. Therefore the dormant state can only be achieved through the dynamics of stochastic fluctuations in which noise organizes a path to dormancy. Here we use optimal path theory applied to a Tat gene stochastic model to show how random fluctuations generate the dormant state by deriving a path which optimizes the probability of achieving the dormant state. We explicitly show how the probability of achieving dormancy scales with the transition rate parameters.   

Experimental observation of critical slowing down before population collapse

Tipping points marking population collapse and other critical transitions in natural systems ($e.g.$ ecosystems, the climate) can be described by a fold bifurcation in the dynamics of the system. Theory predicts that the approach of bifurcations will result in an increasingly slow recovery from small perturbations, a phenomenon called critical slowing down. Here we demonstrate the direct observation of critical slowing down before population collapse using replicate laboratory populations of the budding yeast \textit{Saccharomyces cerevisiae}. We mapped the bifurcation diagram experimentally and found a significant increase in both the size and timescale of the fluctuations of population density near a fold bifurcation, in agreement with the theory. We further confirmed the utility of theoretically predicted warning signals by observing them in two different slowly deteriorating environments. To extend the application of warning signals to spatially extended populations, we proposed and identified several indicators based on the emergence of spatial patterns. Our results suggest that generic temporal and spatial indicators of critical slowing down can be useful in predicting tipping points in population dynamics.   

Extinction of Bacterial Populations: A Change of Paradigm?

It is now well-established that individual bacteria of many types switch stochastically between two phenotypes: fast-growing ``normals'' susceptible to antibiotics, and slowly-growing ``persisters'' hardly affected by the drug. In the competition of species during exponential growth, persisters are a burden, but they may become beneficial when introducing ``stress'' phases like drug treatment. We suggest to shift the focus to the \emph{persistence} of an established population. Due to fluctuations, the population will (after a long time) eventually go extinct; persisters act as a life insurance against this. We study a simple stochastic model of these processes. Using a WKB approximation, we find the most likely path to extinction and quantify the extinction risk under both favorable and adverse conditions. Analytical results are obtained both in the biologically relevant regime when the switching is rare compared with the birth and death processes, and in the opposite regime of frequent switching. We explain how persisters strongly reduce the extinction risk and show that rare switches are most beneficial to this end. [I. Lohmar and B. Meerson, \textit{Phys. Rev. E} \textbf{84} 051901 (2011)]   

Invasion, Coexistence, and Extinction Driven by Preemptive Competition and Sex Ratio

We investigate competitive invasion in a simple population dynamics model, where females can differ genetically in the sex ratio of their offspring, and males can differ in mortality. Analyzing of the mean-field dynamics, we obtain conditions for ecological stability of a given sex-ratio allele for any mortality rate parameters. We also found that stable coexistence of the two alleles is possible, but only males can differ; one female phenotype is present. Our results show that the success of invasion is determined by the female birth sex ratio. A lower female ratio never excludes a larger female sex ratio; in case of coexistence, the surviving female phenotype always has the greater female sex ratio. Finally, we identified an interesting invasion-to-extinction scenario: successful invasion followed by extinction occurs when the invader initially propagates with the resident allele, but after excluding the resident, cannot survive on its own.   

Vaccine enhanced extinction in stochastic epidemic models

We address the problem of developing new and improved stochastic control methods that enhance extinction in disease models. In finite populations, extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Using large deviation theory, we identify the location of the optimal path to extinction in epidemic models with stochastic vaccine controls. These models not only capture internal noise from random transitions, but also external fluctuations, such as stochastic vaccination scheduling. We quantify the effectiveness of the randomly applied vaccine over all possible distributions by using the location of the optimal path, and we identify the most efficient control algorithms. We also discuss how mean extinction times scale with epidemiological and social parameters.   

Fast stochastic algorithm for simulating evolutionary population dynamics

Evolution and co-evolution of ecological communities are stochastic processes often characterized by vastly different rates of reproduction and mutation and a coexistence of very large and very small sub-populations of co-evolving species. This creates serious difficulties for accurate statistical modeling of evolutionary dynamics. In this talk, we introduce a new exact algorithm for fast fully stochastic simulations of birth/death/mutation processes. It produces a significant speedup compared to the direct stochastic simulation algorithm in a typical case when the total population size is large and the mutation rates are much smaller than birth/death rates. We illustrate the performance of the algorithm on several representative examples: evolution on a smooth fitness landscape, NK model, and stochastic predator-prey system.