Speed of expansion and extinction in experimental populations

Although the causes of population extinction are well understood, the speed at which populations decline to extinction is not. A testable, counter-intuitive prediction of stochastic population theory is that, on average, for any interior interval of the domain of biologically attainable population sizes, the expected duration of increase equals the expected duration of decline. Here we report the first empirical tests of this hypothesis. Using data from two experiments in which replicate populations of Daphnia magna were observed to go extinct under different experimental conditions, we failed to reject the null hypothesis of no difference between the growth and decline phases in populations under constant conditions and conditions with modest environmental variability, but find strong evidence to reject equal first passage time in highly variable environments. These results confirm the prediction of equal passage times entailed by diffusion models of population dynamics, supporting continued application in both population theory and conservation decision making under the restricted conditions where the approximation can be expected to hold.     

Density-dependent demographic variation determines extinction rate of experimental populations

Understanding population extinctions is a chief goal of ecological theory. While stochastic theories of population growth are commonly used to forecast extinction, models used for prediction have not been adequately tested with experimental data. In a previously published experiment, variation in available food was experimentally manipulated in 281 laboratory populations of Daphnia magna to test hypothesized effects of environmental variation on population persistence. Here, half of those data were used to select and fit a stochastic model of population growth to predict extinctions of populations in the other half. When density-dependent demographic stochasticity was detected and incorporated in simple stochastic models, rates of population extinction were accurately predicted or only slightly biased. However, when density-dependent demographic stochasticity was not accounted for, as is usual when forecasting extinction of threatened and endangered species, predicted extinction rates were severely biased. Thus, an experimental demonstration shows that reliable estimates of extinction risk may be obtained for populations in variable environments if high-quality data are available for model selection and if density-dependent demographic stochasticity is accounted for. These results suggest that further consideration of density-dependent demographic stochasticity is required if predicted extinction rates are to be relied upon for conservation planning.     

Effects of habitat quality and size on extinction in experimental populations

Stochastic population theory makes clear predictions about the effects of reproductive potential and carrying capacity on characteristic time-scales of extinction. At the same time, the effects of habitat size and quality on reproduction and regulation have been hotly debated. To trace the causal relationships among these factors, we looked at the effects of habitat size and quality on extinction time in experimental populations of Daphnia magna. Replicate model systems representative of a broad-spectrum consumer foraging on a continuously supplied resource were established under crossed treatments of habitat size (two levels) and habitat quality (three levels) and monitored until eventual extinction of all populations. Using statistically derived estimates of key parameters, we related experimental treatments to persistence time through their effect on carrying capacity and the population growth rate. We found that carrying capacity and the intrinsic rate of increase were each influenced similarly by habitat size and quality, and that carrying capacity and the intrinsic rate of increase were in turn both correlated with time to population extinction. We expected habitat quality to have a greater influence on extinction. However, owing to an unexpected effect of habitat size on reproductive potential, habitat size and quality were similarly important for population persistence. These results support the idea that improving the population growth rate or carrying capacity will reduce extinction risk and demonstrate that both are possible by improving habitat quality or increasing habitat size.     

Minimisation of extinction probabilities in reproducing populations

In an attempt to explain variability in clutch size among laying birds of the same species, models are considered in which birds have to choose a randomised strategy for clutch size in the face of random environments in order to minimise a probability of extinction. Three models emerge, depending upon whether this is the probability of extinction of the whole species or that of an individual line of descent, and whether change of strategy from one generation to the next is allowed. In two of these models an asymptotically optimal strategy, which typically is randomised, is predicted for large populations.     

Persistence and extinction in some Cepaea populations

Eight artificial populations of Cepaea nemoralis kept in enclosures excluding visual predators, built up to peak numbers and then crashed in periods of between 10 and 20 years. Fluctuations in numbers were caused by variation in recruitment, not in adult survival. One population survived through a very low population size (14 individuals or less), and shows no significant change in morph frequencies. The very slight fluctuation in morph frequency has a probabilty of chance occurrence of 7%. Studies of 250 years or more are probably needed to distinguish systematic effects on gene frequency from random variation. The lack of any major alteration in morph frequencies, however, suggests that there has been no genetic revolution in this population.     

Preventing extinction and outbreaks in chaotic populations

Interactions in ecological communities are inherently nonlinear and can lead to complex population dynamics including irregular fluctuations induced by chaos. Chaotic population dynamics can exhibit violent oscillations with extremely small or large population abundances that might cause extinction and recurrent outbreaks, respectively. We present a simple method that can guide management efforts to prevent crashes, peaks, or any other undesirable state. At the same time, the irregularity of the dynamics can be preserved when chaos is desirable for the population. The control scheme is easy to implement because it relies on time series information only. The method is illustrated by two examples: control of crashes in the Ricker map and control of outbreaks in a stage-structured model of the flour beetle Tribolium. It turns out to be effective even with few available data and in the presence of noise, as is typical for ecological settings.     

Predicting how populations decline to extinction

Global species extinction typically represents the endpoint in a long sequence of population declines and local extinctions. In comparative studies of extinction risk of contemporary mammalian species, there appear to be some universal traits that may predispose taxa to an elevated risk of extinction. In local population-level studies, there are limited insights into the process of population decline and extinction. Moreover, there is still little appreciation of how local processes scale up to global patterns. Advancing the understanding of factors which predispose populations to rapid declines will benefit proactive conservation and may allow us to target at-risk populations as well as at-risk species. Here, we take mammalian population trend data from the largest repository of population abundance trends, and combine it with the PanTHERIA database on mammal traits to answer the question: what factors can be used to predict decline in mammalian abundance? We find in general that environmental variables are better determinants of cross-species population-level decline than intrinsic biological traits. For effective conservation, we must not only describe which species are at risk and why, but also prescribe ways to counteract this.     

Dynamics of populations on the verge of extinction

Theoretical considerations suggest that extinction in dispersal-limited populations is necessarily a threshold-like process that is analogous to a critical phase transition in physics. We use this analogy to find robust, common features in the dynamics of extinctions, and suggest early warning signals which may indicate that a population is endangered. As the critical threshold of extinction is approached, the population spontaneously fragments into discrete subpopulations and, consequently, density regulation fails. The population size declines and its spatial variance diverges according to scaling laws. Therefore, we can make robust predictions exactly in the range where prognosis is vital, on the verge of extinction.     

Pathogen invasion and host extinction in lattice structured populations

We examined the propagation of an infectious disease and the eventual extinction of the host population in a lattice-structured population. Both the host colonization and pathogen transmission processes are assumed to be restricted to act between the nearest neighbor sites. The model is analyzed by an improved version of pair approximation (IPA). Pair approximation is a technique to trace the dynamics of the number of nearest neighbor pairs having particular states, and IPA takes account of the clustering property of lattice models more precisely. The results are checked by computer simulations. The analysis shows: (i) in a one-dimensional lattice population, a pathogen cannot invade a host population no matter how large is the transmission rate; (ii) in a two-dimensional lattice population, pathogens will drive the host to extinction if the transmission rate is larger than a threshold. These results indicate that spatially structured population models may give qualitatively different results from conventional population models, such as Lotka-Volterra ones, without spatial structure.     

Deterministic extinction effect of parasites on host populations

 Experimental studies have shown that parasites can reduce host density and even drive host population to extinction. Conventional mathematical models for parasite-host interactions, while can address the host density reduction scenario, fail to explain such deterministic extinction phenomena. In order to understand the parasite induced host extinction, Ebert et al. (2000) formulated a plausible but ad hoc epidemiological microparasite model and its stochastic variation. The deterministic model, resembles a simple SI type model, predicts the existence of a globally attractive positive steady state. Their simulation of the stochastic model indicates that extinction of host is a likely outcome in some parameter regions. A careful examination of their ad hoc model suggests an alternative and plausible model assumption. With this modification, we show that the revised parasite-host model can exhibit the observed parasite induced host extinction. This finding strengthens and complements that of Ebert et al. (2000), since all continuous models are likely break down when all population densities are small. This extinction dynamics resembles that of ratio-dependent predator-prey models. We report here a complete global study of the revised parasite-host model. Biological implications and limitations of our findings are also presented. Received: 30 October 2001 / Revised version: 11 February 2002 / Published online: 17 October 2002 Work is partially supported by NSF grant DMS-0077790 Mathematics Subject Classification (2000): 34C25, 34C35, 92D25. Keywords or phrases: Microparasite model – Ratio-dependent predator-prey model – Host extinction – Global stability – Biological control     

Stochastic epidemics in dynamic populations: quasi-stationarity and extinction

Empirical evidence shows that childhood diseases persist in large communities whereas in smaller communities the epidemic goes extinct (and is later reintroduced by immigration). The present paper treats a stochastic model describing the spread of an infectious disease giving life-long immunity, in a community where individuals die and new individuals are born. The time to extinction of the disease starting in quasi-stationarity (conditional on non-extinction) is exponentially distributed. As the population size grows the epidemic process converges to a diffusion process. Properties of the limiting diffusion are used to obtain an approximate expression for τ, the mean-parameter in the exponential distribution of the time to extinction for the finite population. The expression is used to study how τ depends on the community size but also on certain properties of the disease/community: the basic reproduction number and the means and variances of the latency period, infectious period and life-length. Effects of introducing a vaccination program are also discussed as is the notion of the critical community size, defined as the size which distinguishes between the two qualitatively different behaviours. Received: 14 February 2000 / Revised version: 5 June 2000 / Published online: 24 November 2000     

Mobile DNA can drive lineage extinction in prokaryotic populations

Natural selection ultimately acts on genes and other DNA sequences. Adaptations that are good for the gene can have adverse effects at higher levels of organization, including the individual or the population. Mobile genetic elements illustrate this principle well, because they can self‐replicate within a genome at a cost to their host. As they are costly and can be transmitted horizontally, mobile elements can be seen as genomic parasites. It has been suggested that mobile elements may cause the extinction of their host populations. In organisms with very large populations, such as most bacteria, individual selection is highly effective in purging genomes of deleterious elements, suggesting that extinction is unlikely. Here we investigate the conditions under which mobile DNA can drive bacterial lineages to extinction. We use a range of epidemiological and ecological models to show that harmful mobile DNA can invade, and drive populations to extinction, provided their transmission rate is high and that mobile element‐induced mortality is not too high. Population extinction becomes more likely when there are more elements in the population. Even if elements are costly, extinction can still occur because of the combined effect of horizontal gene transfer, a mortality induced by mobile elements. Our study highlights the potential of mobile DNA to be selected at the population level, as well as at the individual level.     

Synchronous dynamics and rates of extinction in spatially structured populations

We explore extinction rates using a spatially arranged set of subpopulations obeying Ricker dynamics. The population system is subjected to dispersal of individuals among the subpopulations as well as to local and global disturbances. We observe a tight positive correlation between global extinction rate and the level of synchrony in dynamics among thesubpopulations. Global disturbances and to a lesser extent, migration, are capable of synchronizing the temporal dynamics of the subpopulations over a rather wide span of the population growth rate r. Local noise decreases synchrony, as does increasing distance among the subpopulations. Synchrony also levels off with increasing r: in the chaotic region, subpopulations almost invariably behave asynchronously. We conclude that it is asynchrony that reduces the probability of global extinctions, not chaos as such: chaos is a special case only. The relationship between global extinction rate, synchronous dynamics and population growth rate is robust to changes in dispersal rates and ranges.     

When invasive exotic populations are threatened with extinction

An important question that arises is what to do when an invasive exotic is a species threatened with extinction within its original distribution and there are few cases in the world illustrating this situation. These species potentially compete with local species for resources and may displace native species or, may in some cases, weaken the gene pool of the native species. The simple eradication of the invasive population could reduce the species′ gene pool, and the eradication process might affect local sympatric species. We recommend a program including identification of areas within the natural range where the species is extinct, removal of the causes of extinction in those areas, then gradual removal of the species from its introduced range and release in the relocation areas following proper guidelines for reintroduction of species.     

Biodiversity and reduced extinction risks in spatially isolated rodent populations

Ecologists have rarely explored the potential influence of local (alpha) biodiversity on the stability and local extinction of spatially isolated populations. Twenty years of annual counts of a small, grazing rodent (Utah prairie dogs, Cynomys parvidens ) from 20 different isolated local populations (colonies) in southern Utah, U.S.A. were analysed. These prairie dogs exhibited large fluctuations and repeated extinctions at individual colonies during the census period. Frequency of extinction at a colony declined dramatically as the number of locally occurring plant species increased. This pattern was not explained by differences among colonies in plant productivity, plant species composition, colony size, or variability in annual counts. Thus, lower extinction risk of consumer populations may be associated with greater resource diversity, and maintaining high local plant diversity may help sustain spatially isolated herbivore populations in fragmented habitats.     

Tail autotomy and extinction in Mediterranean lizards. A preliminary study of continental and insular populations

Tail autotomy is one of the main anti-predator mechanisms of lacertid lizards, but it has been predicted that it is only retained in its full capacity when its benefits exceed its costs (Arnold, 1988). To test this hypothesis, ease of tail shedding was examined in a number of continental and insular lacertid lizard populations, each of which showed a different shedding capacity. Tails are shed more easily in those continental and insular populations where there is a greater probability of predation. In insular populations not subjected to strong predation, the tail tends to be retained. The relationship of these findings to insular Mediterranean lizard populations and to the extinction of the Balearic lizard, Podarcis lilfordi are discussed.     

Localization and extinction of bacterial populations under inhomogeneous growth conditions

The transition from localized to systemic spreading of bacteria, viruses, and other agents is a fundamental problem that spans medicine, ecology, biology, and agriculture science. We have conducted experiments and simulations in a simple one-dimensional system to determine the spreading of bacterial populations that occurs for an inhomogeneous environment under the influence of external convection. Our system consists of a long channel with growth inhibited by uniform ultraviolet (UV) illumination except in a small "oasis", which is shielded from the UV light. To mimic blood flow or other flow past a localized infection, the oasis is moved with a constant velocity through the UV-illuminated "desert". The experiments are modeled with a convective reaction-diffusion equation. In both the experiment and model, localized or extinct populations are found to develop, depending on conditions, from an initially localized population. The model also yields states where the population grows everywhere. Further, the model reveals that the transitions between localized, extended, and extinct states are continuous and nonhysteretic. However, it does not capture the oscillations of the localized population that are observed in the experiment.     

Habitat selection reduces extinction of populations subject to Allee effects

Theoretical studies indicate that a single population under an Allee effect will decline to extinction if reduced below a particular threshold, but the existence of multiple local populations connected by random dispersal improves persistence of the global population. An additional process that can facilitate persistence is the existence of habitat selection by dispersers. Using analytic and simulation models of population change, I found that when habitat patches exhibiting Allee effects are connected by dispersing individuals, habitat selection by these dispersers increases the likelihood that patches persist at high densities, relative to results expected by random settlement. Populations exhibiting habitat selection also attain equilibrium more quickly than randomly dispersing populations. These effects are particularly important when Allee effects are large and more than two patches exist. Integrating habitat selection into population dynamics may help address why some studies have failed to find extinction thresholds in populations, despite well-known Allee effects in many species.