Stochastic demography and population dynamics in the red kangaroo Macropus rufus

1.  Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity is defined as the proportional change in population growth rate caused by a proportional change in a vital rate. Deterministic analyses may not necessarily be informative because large variation in a vital rate with a small deterministic elasticity may affect the population growth rate more than a small change in a less variable vital rate having high deterministic elasticity.
2.  We analyse a stochastic environment model of the red kangaroo ( Macropus rufus ), a species inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital rates.
3.  Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate.
4.  Even if changes in average rainfall have a larger impact on population growth rate, increased variability in rainfall may still be important also in long-lived species. The elasticity with respect to the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of vital rates.
5.  Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction of c . 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by more than c . 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how environmental stochasticity translates into population growth rate.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Population dynamics in variable environments. III. Evolutionary dynamics of r-selection

The boundary dynamics of a genetic model for an age-structured population in a temporally fluctuating environment are analyzed. The condition for invasion by a new allele identifies the logarithmic growth rate a of each life history phenotype as the fitness measure relevant to “r-selection.” An analytical formula is obtained for fitness a when temporal variance in life history characters is small. This formula reveals the major qualitative and quantitative effects of the average life history, fluctuations, and temporal autocorrelation on fitness. A similar approximation is obtained for the log-variance of population number so that the statistical distribution of population size can be estimated.     

Population dynamics in variable environments. VI. Cyclical environments

This paper studies the dynamics of an age-structured population which experiences cyclical variation in vital rates. The principal features of population behavior are found to be contained in an explicitly calculable response function. Three distinct regimes of qualitative behavior are described when cycle period is respectively much less than, of the order of, and much greater than the average generation length. These results make explicit the way in which transient properties corresponding to average vital rates determine population response to cycles.     

From stochastic environments to life histories and back

Environmental stochasticity is known to play an important role in life-history evolution, but most general theory assumes a constant environment. In this paper, we examine life-history evolution in a variable environment, by decomposing average individual fitness (measured by the long-run stochastic growth rate) into contributions from average vital rates and their temporal variation. We examine how generation time, demographic dispersion (measured by the dispersion of reproductive events across the lifespan), demographic resilience (measured by damping time), within-year variances in vital rates, within-year correlations between vital rates and between-year correlations in vital rates combine to determine average individual fitness of stylized life histories. In a fluctuating environment, we show that there is often a range of cohort generation times at which the fitness is at a maximum. Thus, we expect ‘optimal’ phenotypes in fluctuating environments to differ from optimal phenotypes in constant environments. We show that stochastic growth rates are strongly affected by demographic dispersion, even when deterministic growth rates are not, and that demographic dispersion also determines the response of life-history-specific average fitness to within- and between-year correlations. Serial correlations can have a strong effect on fitness, and, depending on the structure of the life history, may act to increase or decrease fitness. The approach we outline takes a useful first step in developing general life-history theory for non-constant environments.     

Understanding contributions of cohort effects to growth rates of fluctuating populations

1. Understanding contributions of cohort effects to variation in population growth of fluctuating populations is of great interest in evolutionary biology and may be critical in contributing towards wildlife and conservation management. Cohort-specific contributions to population growth can be evaluated using age-specific matrix models and associated elasticity analyses. 2. We developed age-specific matrix models for naturally fluctuating populations of stoats Mustela erminea in New Zealand beech forests. Dynamics and productivity of stoat populations in this environment are related to the 3-5 year masting cycle of beech trees and consequent effects on the abundance of rodents. 3. The finite rate of increase (lambda) of stoat populations in New Zealand beech forests varied substantially, from 1.98 during seedfall years to 0.58 during post-seedfall years. Predicted mean growth rates for stoat populations in continuous 3-, 4- or 5-year cycles are 0.85, 1.00 and 1.13. The variation in population growth was a consequence of high reproductive success of females during seedfall years combined with low survival and fertility of females of the post-seedfall cohort. 4. Variation in population growth was consistently more sensitive to changes in survival rates both when each matrix was evaluated in isolation and when matrices were linked into cycles. Relative contributions to variation in population growth from survival and fertility, especially in 0-1-year-old stoats, also depend on the year of the cycle and the number of transitional years before a new cycle is initiated. 5. Consequently, management strategies aimed at reducing stoat populations that may be best during one phase of the beech seedfall cycle may not be the most efficient during other phases of the cycle. We suggest that management strategies based on elasticities of vital rates need to consider how population growth rates vary so as to meet appropriate economic and conservation targets.     

Apparent breeding success drives long-term population dynamics of a migratory swan

The ability of a species to adapt to environmental change is ultimately reflected in its vital rates – i.e. survival and reproductive success of individuals. Together, vital rates determine trends in numbers, commonly monitored using counts of species abundance. Rapid changes in abundance can give rise to concern, leading to calls for research into the biological mechanisms underlying variations in demography. For the northwest European population of Bewick's swan Cygnus columbianus bewickii, there have been major changes in the population trends recorded during nearly five decades of monitoring (1970–2016). The total number of birds increased to a maximum of ca 30 000 in 1995 and subsequently decreased to about 18 000 individuals in 2010. Such large fluctuation in population numbers is rare in long-lived species and understanding the drivers of this population change is crucial for species management and conservation. Using the integrated population model (IPM) framework, we analysed three demographic datasets in combination: population counts, capture–mark–resightings (CMR) and the proportion of juveniles in winter over a period of ~50 years. We found higher apparent breeding success in the years when the population had a positive growth rate compared to years with a negative growth rate. Moreover, no consistent trend in adult and yearling survival, and an increasing trend in juvenile survival was found. A transient life-table response experiment showed that apparent breeding success and adult survival contributed most to the variation in population trend. We explored possible explanatory variables for the different demographic rates and found a significant association between juvenile survival both with the water level in lakes during autumn migration, which affects food accessibility for the swans, and with summer temperatures. Such associations are important for understanding the dynamics of species with fluctuating population sizes, and thus for informing management and conservation decisions.     

Comparative statics and stochastic dynamics of age-structured populations

Arguments from the comparative statics of populations with fixed vital rates are of limited use in studying age-structured populations subject to stochastically varying vital rates. In an age-structured population that experiences a sequence of independently and identically distributed Leslie matrices, the expectation of the Malthusian parameters of the Leslie matrices has no exact interpretation either as the ensemble average of the long-run rate of growth of each sample path of the population (Eq. (3)) or as the long-run rate of growth of the ensemble average of total population size (Eq. (4)). On the other hand, the Malthusian parameter of the expectation of a sequence of Leslie matrices is exactly the logarithm of the finite growth rate of the ensemble average of total population size when Leslie matrices are independently and identically distributed (though not in general when Leslie matrices are sequentially dependent). These observations appear to contradict the claims of a recent study using computer simulation of age-structured populations with stochastically varying vital rates.     

Differential effects of herbivory and pathogen infestation on plant population dynamics

Plants often interact with antagonists such as herbivores or pathogens. Negative effects on individual plant performance are widely documented, but less is known about whether such effects translate into effects on population viability. In temperate forests, important herbivores include deer. During 2006–2009, I compared vital rates and population growth rates (calculated using integral projection models) between fenced exclosures and grazed control areas, using the perennial herb Phyteuma spicatum as a model species. Deer caused the largest damage to flowering individuals, removing about 24% of all inflorescences and 13% of the above-ground biomass. Only few vital rates seemed to be negatively affected by deer (mainly seed production) and this did not translate into effects on population growth rate. Contrary to expectations, population growth rates tended to be lower in the fenced exclosures in 1 year. This was likely caused by high-pathogen infestation rates, which negatively affected the probability of adult survival and growth. Population growth rate was more sensitive to changes in these vital rates than to changes in seed production. In summary, the results of this demographic study show that grazing effects may be small for long-lived herbs, and that negative effects on vital rates such as seed production may not always translate into effects on population growth rate. The findings also illustrate that other antagonists such as pathogens may be of greater relative importance for differences in population performance than herbivores.     

Demography of greater prairie-chickens: Regional variation in vital rates,sensitivity values,and population dynamics

Intensification of rangeland management has coincided with population declines among obligate grassland species in the largest remaining tallgrass prairie in North America, although causes of declines remain unknown. We modeled population dynamics and conducted sensitivity analyses from demographic data collected for an obligate grassland bird that is an indicator species for tallgrass prairie, the greater prairie-chicken (Tympanuchus cupido), during a 4-year study in east-central Kansas, USA. We examined components of reproductive effort and success, juvenile survival, and annual adult female survival for 3 populations of prairie-chickens across an ecological gradient of human landscape alteration and land use. We observed regional differences in reproductive performance, survivorship, and population dynamics. All 3 populations of prairie-chickens were projected to decline steeply given observed vital rates, but rates of decline differed across a gradient of landscape alteration, with the greatest declines in fragmented landscapes. Elasticity values, variance-scaled sensitivities, and contribution values from a random-effects life-table response experiment all showed that the finite rate of population change was more sensitive to changes in adult survival than other demographic parameters in our declining populations. The rate of population change was also sensitive to nest survival at the most fragmented and least intensively grazed study site; suggesting that patterns of landscape fragmentation and land use may be affecting the relative influences of underlying vital rates on rates of population growth. Our model results indicate that 1) populations of prairie-chickens in eastern Kansas are unlikely to be viable without gains from immigration, 2) rates of population decline vary among areas under different land management practices, 3) human land-use patterns may affect the relative influences of vital rates on population trajectories, and 4) anthropogenic effects on population demography may influence the regional life-history strategies of a short-lived game bird. © 2012 The Wildlife Society.     

Demographic strategies of plant invaders in temporally varying environments

Plant populations may have evolved different demographic strategies to cope with temporal environmental variation. According to the demographic buffering hypothesis, vital rates that are most critical to population persistence are buffered against environmental variation and vary little over time, whereas the demographic lability hypothesis suggests that populations may track and benefit from environmental variation. While the hypotheses of demographic strategies have been widely tested in plant and animal species, they have not been explicitly examined for invasive plants, or in relation to different modelling methods (deterministic vs. stochastic). Here, we tested the demographic buffering and lability hypotheses for 23 populations of eight invasive plant species in relation to life form (woody vs. herbaceous species) and population growth rate using deterministic and stochastic modelling methods, and absolute and relative scales. We found that conclusions of demographic strategies depended on scale, with an absolute scale resulting in stronger negative correlations between the variability and importance of vital rates (i.e., buffering) than a relative scale. Conclusions of demographic strategies were also affected by life form that interacted with method. The populations of woody invaders exhibited buffering regardless of the method used, while for the populations of herbaceous species, deterministic calculations suggested buffering and stochastic calculations suggested lability. Overall, our findings emphasise the role of life form and methodological issues that need to be considered when exploring demographic strategies in fluctuating environments.     

Evolution and population dynamics in stochastic environments

Inter-generational temporal variability of the environment is important in the evolution and adaptation of phenotypic traits. We discuss a population-dynamic approach which plays a central role in the analysis of evolutionary processes. The basic principle is that the phenotypes with the greatest long-term average growth rate will dominate the entire population. The calculation of longterm average growth rates for populations under temporal stochasticity can be highly cumbersome. However, for a discrete non-overlapping population, it is identical to the geometric mean of the growth rates (geometric mean fitness), which is usually different from the simple arithmetic mean of growth rates. Evolutionary outcomes based on geometric mean fitness are often very different from the predictions based on the usual arithmetic mean fitness. In this paper we illustrate the concept of geometric mean fitness in a few simple models. We discuss its implications for the adaptive evolution of phenotypes, e.g. foraging under predation risks and clutch size. Next, we present an application: the risk-spreading egg-laying behaviour of the cabbage white butterfly, and develop a two-patch population dynamic model to show how the optimal solution diverges from the ssual arithmetic mean approach. The dynamics of these stochastic models cannot be predicted from the dynamics of simple deterministic models. Thus the inclusion of stochastic factors in the analyses of populations is essential to the understanding of not only population dynamics, but also their evolutionary dynamics.     

Mutator dynamics in fluctuating environments

Populations with high mutation rates (mutator clones) are being found in increasing numbers of species, and a clear link is being established between the presence of mutator clones and drug resistance. Mutator clones exist despite the fact that in a constant environment most mutations are deleterious, with the spontaneous mutation rate generally held at a low value. This implies that mutator clones have an important role in the adaptation of organisms to changing environments. Our study examines how mutator dynamics vary according to the frequency of environmental fluctuations. Although recent studies have considered a single environmental switch, here we investigate mutator dynamics in a regularly varying environment, seeking to mimic conditions present, for example, under certain drug or pesticide regimes. Our model provides four significant new insights. First, the results demonstrate that mutators are most prevalent under intermediate rates of environmental change. When the environment oscillates more rapidly, mutators are unable to provide sufficient adaptability to keep pace with the frequent changes in selection pressure and, instead, a population of 'environmental generalists' dominates. Second, our findings reveal that mutator dynamics may be complex, exhibiting limit cycles and chaos. Third, we demonstrate that when each beneficial mutation provides a greater gain in fitness, mutators achieve higher densities in more rapidly fluctuating environments. Fourth, we find that mutators of intermediate strength reach higher densities than very weak or strong mutators.     

Stochastic gene expression in fluctuating environments

Stochastic mechanisms can cause a group of isogenic bacteria, each subject to identical environmental conditions, to nevertheless exhibit diverse patterns of gene expression. The resulting phenotypic subpopulations will typically have distinct growth rates. This behavior has been observed in several contexts, including sugar metabolism and pili phase variation. Under fixed environmental conditions, the net growth rate of the population is maximized when all cells are of the fastest growing phenotype, so it is unclear what fitness advantage is conferred by population heterogeneity. However, unlike ideal laboratory conditions, natural environments tend to fluctuate, either periodically or randomly. Here we use a stochastic population model to show that, during growth in such fluctuating environments, a dynamically heterogenous bacterial population can sometimes achieve a higher net growth rate than a homogenous one. By using stochastic mechanisms to sample several distinct phenotypes, the bacteria are able to anticipate and take advantage of sudden changes in their environment. However, this heterogeneity is beneficial only if the bacterial response rate is sufficiently low. Our results could be useful in the design of artificial evolution experiments and in the optimization of fermentation processes.     

Stochastic LTRE analysis of the effects of herbivory on the population dynamics of a perennial grassland herb

Herbivores can have strong deleterious effects on vital rates (growth, reproduction, and survival) and thus negatively impact the population dynamics of plant species. In practice, however, these effects might be strongly correlated, for example as a result of tradeoffs between vital rates. To get better insights into the effects of herbivory on the population dynamics of the long‐lived grassland plant Primula veris population projection matrices were constructed from demographic data collected between 1999 and 2008 (nine annual transitions). Data were collected in two large grassland populations, each of which was subjected to two treatments (grazing by cattle versus a mowing treatment), yielding a total of 36 matrices. We applied a lower‐level vital rate life table response experiment (LTRE) using the small noise approximation (SNA) of the stochastic population growth rate to disentangle the contributions of changes in mean vital rates, variability in vital rates, correlations between vital rates and vital rate elasticities to the difference in the stochastic growth rate. Stochastic growth rates (a= log λ_S) were significantly lower in grazed than in mown plots (a= 0.0185 and 0.1019, respectively). SNA LTRE analysis showed that contributions of mean vital rates by far made the largest contribution to the observed difference in a between grazed and control plots. In particular, changes in sexual reproduction rates made the largest contributions to lower the stochastic growth rate in grazed plots: both adult flowering probabilities and flower and seed production were importantly lower in grazed populations, but these negative effects were largely buffered by increased establishment and seedling survival rates. Among the stochastic terms of the SNA decomposition, contributions of covariance and correlations between vital rates had the largest impact, whereas contributions of elasticities were smaller. The strongest correlation driver was the association between adult survival and seedling establishment, suggesting that environmental conditions favouring adult survival also are beneficial for seedling establishment. Overall, our results show that herbivory had a strong negative effect on the long‐term population growth rate of P. veris that was primarily mediated by differences in fecundity (flower and seed production) and germination.     

Response of population size to changing vital rates in random environments

Population size and population growth rate respond to changes in vital rates like survival and fertility. In deterministic environments change in population growth rate alone determines change in population size. In random environments, population size at any time t is a random variable so that change in population size obeys a probability distribution. We analytically show that, in a density-independent population, the proportional change in population size with respect to a small proportional change in a vital rate has an asymptotic normal distribution. Its mean grows linearly at a rate equal to the elasticity of the long-term stochastic growth rate λ _S while the standard deviation scales as $\sqrt t$ . Consequently, a vital rate with a larger elasticity of λ _S may produce a larger mean change in population size compared to one with a smaller elasticity of λ _S. But a given percentage change in population size may be more likely when the vital rate with smaller elasticity is perturbed. Hence, the response of population size to perturbation of a vital rate depends not only on the elasticity of the population growth rate but also on the variance in change in population size. Our results provide a formula to calculate the probability that population size changes by a given percentage that works well even for short time periods.     

Stochastic gene expression in switching environments

Organisms are known to adapt to regularly varying environments. However, in most cases, the fluctuations of the environment are irregular and stochastic, alternating between favorable and unfavorable regimes, so that cells must cope with an uncertain future. A possible response is population diversification. We assume here that the cell population is divided into two groups, corresponding to two phenotypes, having distinct growth rates, and that cells can switch randomly their phenotypes. In static environments, the net growth rate is maximized when the population is homogeneously composed of cells having the largest growth rate. In random environments, growth rates fluctuate and observations reveal that sometimes heterogeneous populations have a larger net growth rate than homogeneous ones, a fact illustrated recently through Monte-Carlo simulations based on a birth and migration process in a random environment. We study this process mathematically by focusing on the proportion f(t) of cells having the largest growth rate at time t, and give explicitly the related steady state distribution π. We also prove the convergence of empirical averages along trajectories to the first moment , and provide efficient numerical methods for computing .      

Unusual allometry between in situ growth of freshwater phytoplankton under static and fluctuating light environments: possible implications for dominance

The effects of fluctuating light fields on the growth of phytoplanktonare not well understood and conclusions in the literature havebeen equivocal. Most studies have examined responses such asproductivity and chlorophyll a content (laboratory culture andfield tests) or growth rates (laboratory culture tests). Inthis study we examined the in situ growth rates of differenttypes of phytoplankton within two natural populations. Comparisonswere made between populations grown in a static environment(suspended in a fixed position in the water column) and an equivalentpopulation moving through the water column simulating the mixingof entrained phytoplankton. Growth under fluctuating light fieldsin this experiment only significantly (P < 0.05) increasedthe growth of the diatom Skeletonema and decreased the growthof Anabaena circinalis, Microcystis aeruginosa and Scenedesmussp. All other phytoplankton, including the genera Nitzschia,Fragilaria and Dactylococcopsis, did not have growth rates thatwere significantly different between static and fluctuatinglight treatments. A general pattern where diatoms grew best,followed by chlorophytes with the toxicogenic cyanophytes M.aeruginosa and A. circinalis growing least well, was distinguishedunder fluctuating irradiance. This seems consistent with thecommon occurrence of these groups of phytoplankton in the naturalenvironment. The cyanophytes Dactylococcopsis and Aphanothecedid not follow this pattern, with the former growing betterunder fluctuating light and the latter exhibiting an unusualgrowth pattern where growth was higher under lower light intensities.     

Patchy populations in stochastic environments: critical number of patches for persistence

We introduce a model for the dynamics of a patchy population in a stochastic environment and derive a criterion for its persistence. This criterion is based on the geometric mean (GM) through time of the spatial-arithmetic mean of growth rates. For the population to persist, the GM has to be >/=1. The GM increases with the number of patches (because the sampling error is reduced) and decreases with both the variance and the spatial covariance of growth rates. We derive analytical expressions for the minimum number of patches (and the maximum harvesting rate) required for the persistence of the population. As the magnitude of environmental fluctuations increases, the number of patches required for persistence increases, and the fraction of individuals that can be harvested decreases. The novelty of our approach is that we focus on Malthusian local population dynamics with high dispersal and strong environmental variability from year to year. Unlike previous models of patchy populations that assume an infinite number of patches, we focus specifically on the effect that the number of patches has on population persistence. Our work is therefore directly relevant to patchily distributed organisms that are restricted to a small number of habitat patches.     

Harvesting in seasonal environments

Most harvest theory is based on an assumption of a constant or stochastic environment, yet most populations experience some form of environmental seasonality. Assuming that a population follows logistic growth we investigate harvesting in seasonal environments, focusing on maximum annual yield (M.A.Y.) and population persistence under five commonly used harvest strategies. We show that the optimal strategy depends dramatically on the intrinsic growth rate of population and the magnitude of seasonality. The ordered effectiveness of these alternative harvest strategies is given for different combinations of intrinsic growth rate and seasonality. Also, for piecewise continuous-time harvest strategies (i.e., open / closed harvest, and pulse harvest) harvest timing is of crucial importance to annual yield. Optimal timing for harvests coincides with maximal rate of decline in the seasonally fluctuating carrying capacity. For large intrinsic growth rate and small environmental variability several strategies (i.e., constant exploitation rate, linear exploitation rate, and time-dependent harvest) are so effective that M.A.Y. is very close to maximum sustainable yield (M.S.Y.). M.A.Y. of pulse harvest can be even larger than M.S.Y. because in seasonal environments population size varies substantially during the course of the year and how it varies relative to the carrying capacity is what determines the value relative to optimal harvest rate. However, for populations with small intrinsic growth rate but subject to large seasonality none of these strategies is particularly effective with M.A.Y. much lower than M.S.Y. Finding an optimal harvest strategy for this case and to explore harvesting in populations that follow other growth models (e.g., involving predation or age structure) will be an interesting but challenging problem.