Stochastic population growth in spatially heterogeneous environments

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZ _t = μ Z _t dt + σ Z _t dW _t , t ≥ 0, where the conditional law of Z _t+Δt ? Z _t given Z _t = z has mean and variance approximately z μΔt and z ² σ ²Δt when the time increment Δt is small. The long-term stochastic growth rate ${\lim_{t \to \infty} t^{-1}\log Z_t}$ for such a population equals ${\mu -\frac{\sigma^2}{2}}$ . Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model ${{\bf X}_t = (X_t^1, \ldots, X_t^n)}$ , t ≥ 0, for the population abundances in n patches: the conditional law of X _t+Δt given X _t = x is such that the conditional mean of ${X_{t+\Delta t}^i - X_t^i}$ is approximately ${[x^i \mu_i + \sum_j (x^j D_{ji} - x^i D_{ij})] \Delta t}$ where μ _i is the per capita growth rate in the ith patch and D _ij is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of ${X_{t+\Delta t}^i - X_t^i}$ and ${X_{t + \Delta t}^j - X_t^j}$ is approximately x ⁱ x ^j σ _ij Δt for some covariance matrix Σ = (σ _ij ). We show for such a spatially extended population that if ${S_t = X_t^1 + \cdots + X_t^n}$ denotes the total population abundance, then Y _t = X _t /S _t , the vector of patch proportions, converges in law to a random vector Y _∞ as ${t \to \infty}$ , and the stochastic growth rate ${\lim_{t \to \infty} t^{-1}\log S_t}$ equals the space-time average per-capita growth rate ${\sum_i \mu_i \mathbb{E}[Y_\infty^i]}$ experienced by the population minus half of the space-time average temporal variation ${\mathbb{E}[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j]}$ experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multi-scale landscapes (e.g. insects on plants in meadows on islands). Our results provide fundamental insights into “ideal free” movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of density-dependent feedbacks, ideal-free dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.     

Mathematical analysis of the asymptotic behavior of the Leslie population matrix model

LetL be a Leslie population matrix. Leslie (1945) and others have shown that the matrixL has a leading positive eigenvalueλ ₀ and that in general: (1) $$\mathop {\lim }\limits_{t \to \infty } \frac{{L^t X}}{{\lambda _0^t }} = \gamma X_{\lambda _0 } $$ whereX _{λ 0} is an eigenvector corresponding toλ ₀,X is any initial population vector, and γ is a scalar quantity detormined byX. In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms ofλ ₀ and the entries of the matrixL the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is nonzero. To stabilize a population, the matrixL must be adjusted so thatλ ₀=1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stanbilization which yields as a byproduct an easily accessible scalar measure ofL's tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.     

Linking vital rates to invasiveness of a perennial herb

Invaders generally show better individual performance than non-invaders and, therefore, vital rates (survival, growth, fecundity) could potentially be used to predict species invasiveness outside their native range. Comparative studies have usually correlated vital rates with the invasiveness status of species, while few studies have investigated them in relation to population growth rate. Here, I examined the influence of five vital rates (plant establishment, survival, growth, flowering probability, seed production) and their variability (across geographic regions, habitat types, population sizes and population densities) on population growth rate (λ) using data from 37 populations of an invasive, iteroparous herb (Lupinus polyphyllus) in a part of its invaded range in Finland. Variation in vital rates was often related to habitat type and population density. The performance of the populations varied from declining to rapidly increasing independently of habitat type, population size or population density, but differed between regions. The population growth rate increased linearly with plant establishment, and with the survival and growth of vegetative individuals, while the survival of flowering individuals and annual seed production were not related to λ. The vital rates responsible for rapid population growth varied among populations. These findings highlight the importance of both regional and local conditions to plant population dynamics, demonstrating that individual vital rates do not necessarily correlate with λ. Therefore, to understand the role of individual vital rates in a species ability to invade, it is necessary to quantify their effect on population growth rate.     

Differential effects of abandonment on the demography of the grassland perennial Succisa pratensis

Abandonment of traditional land-use practices can have strong effects on the abundance of species occurring in agricultural landscapes. However, the precise mechanisms by which individual performance and population dynamics are affected are still poorly understood. To assess how abandonment affects population dynamics of Succisa pratensis we used data from a 4-year field study in both abandoned and traditionally grazed areas in moist and mesic habitats to parameterize integral projection models. Abandoned populations had a lower long-term stochastic population growth rate (λ _S = 0.90) than traditionally managed populations (λ _S = 1.08), while λ _S did not differ between habitat types. The effect of abandonment differed significantly between years and had opposed effects on different vital rates. Individuals in abandoned populations experienced higher mortality rates and lower seedling establishment, but had higher growth rates and produced more flower heads per plant. Population viability analyses, based on a population survey of the whole study area in combination with our demographic models, showed that 32 % of the populations face a high risk of extinction (>80 %) within 20 years. These results suggest that immediate changes in management are needed to avoid extinctions and further declines in population sizes. Stochastic elasticity analyses and stochastic life table response experiments indicated that management strategies would be most effective if they increase survival of small plants as well as seedling establishment, while maintaining a high seed production. This may be achieved by varying the grazing intensity between years or excluding grazers when plants are flowering.     

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.     

Testing single-sample estimators of effective population size in genetically structured populations

The effective population size (N _e ) is a key parameter in evolutionary and population genetics. Single-sample N _e estimation provides an alternative to traditional approaches requiring two or more samples. Single-sample methods assume that the study population has no genetic sub-structure, which is unlikely to be true in wild populations. Here we empirically investigated two single-sample estimators (onesamp and L d N e) in replicated and controlled genetically structured populations of Drosophila melanogaster. Using experimentally controlled population parameters, we calculated the Wright–Fisher expected N _e for the structured population ( ^Total N _e ) and demonstrated that the loss of heterozygosity did not significantly differ from Wright’s model. We found that disregarding the population substructure resulted in ^Total N _e estimates with a low coefficient of variation but these estimates were systematically lower than the expected values, whereas hierarchical estimates accounting for population structure were closer to the expected values but had a higher coefficient of variation. Analysis of simulated populations demonstrated that incomplete sampling, initial allelic diversity and balancing selection may have contributed to deviations from the Wright–Fisher model. Overall the approximate-Bayesian onesamp method performed better than L d N e (with appropriate priors). Both methods performed best when dispersal rates were high and the population structure was approaching panmixia.     

D-amino acids in the cell pool of bacteria

About 30 different bacterial species were tested for the possible presence of freed-amino acids in their cell pool. Gram-positive bacteria particularly the species of the genusBacillus have a fairly large pool of freely extractabled-amino acids. Varied quantities of freed-amino acids were detected inBacillus subtilis B₃,Bacillus subtilis Marburg,Bacillus licheniformis, Bacillus brevis, Bacillus stearothermophilus, Lactobacillus fermenti, Lactobacillus delbrueckii, Staphylococcus aureus andClostridium acetobutylicum. The individual components ofd-amino acids were identified in 5Bacillus species referred to above,d-alanine is the major component; the otherd-amino acids identified are aspartic acid, glutamic acid, histidine, leucines, proline, serine and tyrosine. Thed-amino acid pool size inBacillus subtilis B₃ varies with different culture conditions. The pool size is maximum when growth temperature is 30°C and it fluctuates with change in pH of the medium. The maximum quantity ofd-amino acids could be recovered when the culture was at mid log phase. O₂ supply to the medium has little effect ond-amino acid pool size. The starvation of cells leads to depletion of thed-amino acid pool which is exhausted almost completely within 4 hours by incubation in nutrient-free medium.     

On the basic reproduction number in a random environment

The concept of basic reproduction number $R_0$ in population dynamics is studied in the case of random environments. For simplicity the dependence between successive environments is supposed to follow a Markov chain. $R_0$ is the spectral radius of a next-generation operator. Its position with respect to 1 always determines population growth or decay in simulations, unlike another parameter suggested in a recent article (Hernandez-Suarez et al., Theor Popul Biol, doi:10.1016/j.tpb.2012.05.004, 2012). The position of the latter with respect to 1 determines growth or decay of the population’s expectation. $R_0$ is easily computed in the case of scalar population models without any structure. The main emphasis is on discrete-time models but continuous-time models are also considered.     

Studies on the demography and life history ofTaraxacum serotinum (Waldst. etKit.)Poir

The population dynamics ofTaraxacum serotinum (Waldst. etKit.)Poir. were studied in two stands of a loess grassland community (Salvio-Festucetum rupicolae). The two stands differed in their previous management: one of them had been grazed by sheep and the other had been mown before the start of the study. The fate of every rosette ofTaraxacum was followed in an area of 75 m² in each stand for nine years. The demographic characteristics of the populations differed in the two habitats. In the formerly mown site, the population size decreased considerably due to an increase in mortality after the cessation of mowing. In contrast, the population size in the formerly grazed site showed only a slight decrease with lower mortality rates. In both populations it was mostly the new offspring which died. Comparing the year-to-year demographic characterstics there were significant differences between the two sites in almost every year. However, if the same characteristics for the whole nine-year period were considered, there were no differences. The population growth rates calculated from the year-to-year transition matrices were almost always under the equilibrium value of 1, but the deviation was not significant. An elasticity analysis indicated that the greater contribution to the variability of population growth in both populations comes from those plants that remain in the same (adult) stage from year to year.     

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.     

Genetic parameters and provenance variation of Pinus radiata D. Don. ‘Eldridge collection’ in Australia 2: wood properties

Provenance variation and genetic parameters for wood properties of mature radiata pine (Pinus radiata D. Don) were studied by sampling three provenance/progeny trials in southeast Australia. Among the mainland provenances, Monterey and Año Nuevo had higher density and modulus of elasticity (at one site) than Cambria. Basic density and predicted modulus of elasticity (MoE) for the island provenances, Guadalupe and Cedros, were ～20% higher at Billapaloola compared to mainland provenances grown at Green Hills and Salicki, differences that may or may not be linked to site differences. Heritability estimates of density, predicted MoE and microfibril angle were significant and $ {\bar{h}^2} $ ?>?0.45, suggesting moderate to strong genetic control. The estimated genetic correlations between diameter at breast height and wood properties in the current study were weaker (less negative) than the mean estimated from the current breeding population generation in radiata pine. Of the wood properties, density showed the strongest adverse genetic correlations with growth (mean r _A ?=??0.23?±?0.09). Selection for MoE may produce greater gain than selection for density because MoE had almost twice the estimated additive genetic coefficient of variation ( $ {\overline {\text{CV}}_A} $ ) compared to density. Estimated type B genetic correlations (r _B) for all wood quality traits were typically high, conforming to the trend that wood properties have low genotype-by-environment interaction (G?×?E). Significant differences in wood properties among provenances, families and/or individual trees provide an opportunity for breeding programmes to select superior trees for solid wood production that will combine superior growth with desirable wood traits.     

Effect of nutrient enrichment on growth and photosynthesis of the zooxanthellate coral Stylophora pistillata

The effect of prolonged (9 week) nutrient enrichment on the growth and photosynthetic rates of the zooxanthellate coral Stylophora pistillata was investigated. The main questions were: (1) what is the exposure time needed to induce measurable change in growth rate? (2) which are the concentrations of nitrogen and phosphorus required to cause changes in these rates? (3) what is the recovery potential of the corals after the nutrient stress? For this purpose, three tanks (N, P, NP) were enriched with ammonium (N), phosphorus (P) or both nutrients (NP), respectively. A fourth tank (C) served as a control. The growth of 40 nubbins (10 in each tank) was monitored during four periods: period 1 (nutrient-poor conditions), period 2 (10?μm NH₄ and/or 2?μm PO₄ enrichment), period 3 (20?μm NH₄ and/or 2?μm PO₄) and period 4 (nutrient-poor conditions). Period 4 was performed to study the recovery potential of corals after a nutrient stress. During period 1, growth rates remained constant in all tanks. In the P tank, growth rates declined during the two enrichment periods, with a total decrease of 60% by the end of period 3. In the N tank, growth rates remained nearly constant during period 2 but decreased in period 3 (60% decrease). In the NP tank, 50% and 25% decreases were observed during periods 2 and 3. At the end of the recovery period, a regain in growth rate was observed in the N and NP tanks (35 and 30% increase, respectively, compared with the rates measured at the end of period 3) and growth rates returned to 60% of the initial rates. By contrast, in the P tank, there was no regain in growth and a further decrease of 5% was observed. Rates of photosynthesis were often higher during the enriched than the nutrient-poor period (up to 150% increase). Corals with the highest percent increases in maximal gross photosynthetic rate (P ^g _max ) had the smallest decreases in growth rate due to nutrient enrichment. In conclusion, high ammonium (20?μm) and relatively low phosphorus concentrations (2?μm) are required to induce a significant decrease in coral growth rate. The largest reduction was observed with both ammonium and phosphorus enrichment. The decrease in growth rate was rapid following nutrient enrichment, since a 10% decrease or more could be observed after the first week of treatment.     

Quantifying the risk of mis-estimating correlation significance of climate–tree growth relationships

In dendroecology, sampling effort has a strong influence of both regional chronology properties and climate–tree growth relationships assessment. Recent studies evidenced that decreasing sample size leads to a weakening of the bootstrapped correlation coefficients ( ${\text{BCC}}$ BCC ). The present analysis focused on the risk of mis-estimating the significance of population ${\text{BCC}}\,\left( {{\text{BCC}}_{\text{POP}} } \right)$ BCC ( BCC POP ) from a sample of N trees, and then proposed an approach to detect and correct mis-estimations using the properties of the sample. The sample size effect and the limits of the correction were illustrated from 840 individual growth chronologies of Corsican pine (Pinus nigra Arnold ssp. laricio Poiret var. Corsicana) sampled in Western France. The 840 trees were used to assess the population characteristics, and the effect of sampling effort was investigated through a simulation approach based on a resampling procedure of N trees amongst 840 (N ? [5; 50]). Our results evidenced that the risk strongly varied amongst the climatic regressors. The highest risks were evidenced for significant ${\text{BCC}}_{\text{POP}}$ BCC POP , with a percentage of mis-estimation ranging from 25 to 80. On the contrary, small samples allowed providing an reliable estimation of the significance of non-significant ${\text{BCC}}_{\text{POP}}$ BCC POP . To a lesser extent, the risk slightly decreased with increasing N, according to a negative exponential trend. The detection and correction method was found relevant to detect mis-estimation only for significant ${\text{BCC}}_{\text{POP}}$ BCC POP ; otherwise, the ${\text{BCC}}_{\text{POP}}$ BCC POP significance was generally overestimated.     

l-Leucine 5-hydroxylase of Nostoc punctiforme is a novel type of Fe(II)/α-ketoglutarate-dependent dioxygenase that is useful as a biocatalyst

l-Leucine 5-hydroxylase (LdoA) previously found in Nostoc punctiforme PCC 73102 is a novel type of Fe(II)/α-ketoglutarate-dependent dioxygenase. LdoA catalyzed regio- and stereoselective hydroxylation of l-leucine and l-norleucine into (2S,4S)-5-hydroxyleucine and (2S)-5-hydroxynorleucine, respectively. Moreover, LdoA catalyzed sulfoxidation of l-methionine and l-ethionine in the same manner as previously described l-isoleucine 4-hydroxylase. Therefore LdoA should be a promising biocatalyst for effective production of industrially useful amino acids.     

Das asymptotische Wachstum der Fische — ein Nonsens?

Considering an incomplete growth curve of the codGadus morhua,Knight (1968) questioned the asymptotic curvature of growth in fishes, which easily may be shown in complete growth curves illustrating the attainment of maximum size. It is also possible to calculate maximum size employing theFord-Walford formula. Maximum size, calculated in this manner, may also be used in theBertalanffy function. For other mathematical growth formulations different maximum sizes would be obtained. The numerical value of the maximum size depends upon the mathematical interpretation of the growth process. Therefore it always represents a mathematical parameter without biological meaning. It is shown that data ofKetchen (Forrester 1966) on growth in ♂ ♂ of the flat fishEopsetta jordani may be much better represented by the new growth formula (Krüger 1965) than by theBertalanffy function.     

Evaluation of surface colonization kinetics in continuous culture

Two equations, describing surface colonization, were evaluated and compared using suspended glass slides in a continuous culture ofPseudomonas aeruginosa. These equations were used to determine surface growth rates from the number and distribution of cells present on the surface after incubation. One of these was the colonization equation which accounts for simultaneous attachment and growth of bacteria on surfaces: $$N = (A/\mu )e^{\mu t} - A/\mu $$ where N=number of cells on surface (cells field^?1); A=attachment rate (cells field^?1h^?1);μ=specific growth rate (h^?1); t=incubation period (h). The other was the surface growth rate equation which assumes that the number of colonies of a given size (C_i) will reach a constant value (C_max) which is equal to A divided byμ: $$\mu = \frac{{\ln \left( {\frac{N}{{C_i }} + 1} \right)}}{t}$$ Both equations gave similar results and the time required to approximate C_max may not be as long as was previously thought. In all cases both A andμ continuously decreased throughout the incubation period. These decreases may be due to various effects of microbial accumulation on the surface. Both equations accurately determined surface growth rates despite highly variable attachment rates. Growth rates were similar for both the liquid phase of the culture and the solid-liquid interface (0.4 h^?1). Use of the surface growth rate equation is favored over the use of the colonization equation since the former does not require a computer to solve forμ and the counting procedure is simplified.     

Age structure, growth and mortality estimates of an endemic Ptychobarbus dipogon (Regan, 1905) (Cyprinidae: Schizothoracinae) in the Lhasa River, Tibet

Ptychobarbus dipogon is an endemic fish in the Yarlung Tsangpo River, but its biology is poorly known. We sampled 582 specimens (total length, TL, between 70.6 and 593.0 mm) from April 2004 to August 2006 in the Lhasa River, Tibet. We estimated ages based on the counts of alternating opaque and translucent zones (annuli) in thin transverse sections of lapilli otoliths. Ages ranged from 1⁺ to 23⁺ years for males and 1⁺ to 44⁺ for females. The observed 44⁺ years was the oldest reported for schizothoracine fishes. Females attained a larger size than males. The TL–weight relationship was W?=?7.12?×?10^?6 TL ^3.006 for combined sexes. The growth parameters fitted von Bertalanffy growth functions were $L_\infty = 598.66\,{\text{mm}}$ , k?=?0.0898 year^?1, t ₀?=??0.7261 year and $W_\infty = 1585.38\;{\text{g}}$ for females and $L_\infty = 494.23{\text{ mm}}$ , k?=?0.1197 year^?1, t ₀?=??0.7296 year and $W_\infty = 904.88{\text{ g}}$ for males. The longevities of 32.7 year for females and 24.3 year for males were similar to the observed ages. Using an empirical model we estimated the instantaneous rate of total mortality (Z) at 0.28 per year in the lower reaches. Z in the upper and middle stocks was close to the M because of unexploited or lightly exploited stock. Protracted longevity, slow growth, low natural mortality and large body size were typical characteristics of P. dipogon. The current declining trend of P. dipogon could be prevented by altering fishing regulations.     

Transportable and Non-transportable Inhibitors of L-glutamate Uptake Produce Astrocytic Stellation and Increase EAAT2 Cell Surface Expression

Astrocytic excitatory amino acid transporters (EAATs) regulate excitatory transmission and limit excitotoxicity. Evidence for a functional interface between EAATs and glial fibrillary acidic protein (GFAP) relevant to astrocytic morphology led to investigations of actions of transportable (d-Aspartate (d-Asp) and (2S,3S,4R)-2-(carboxycyclopropyl)glycine (l-CCG-III)) and non-transportable (dl-threo-β-benzyloxyaspartate (dl-TBOA)) inhibitors of Glu uptake in murine astrocytes. d-Asp (1 mM), l-CCG-III (0.5 mM) and dl-TBOA (0.5 mM) produced time-dependent (24–72 h) reductions in ³[H]d-Asp uptake (approximately 30–70%) with little or no gliotoxicity. All drugs induced a profound change in phenotype from cobblestone to stellate morphology and image analysis revealed increases in the intensity of GFAP immunolabelling for l-CCG-III and dl-TBOA. Cytochemistry indicated localized changes in F-actin distribution. Cell surface expression of EAAT2, but not EAAT1, was elevated at 72 h. Blockade of Glu uptake by both types of EAAT inhibitor exerts longer-term effects on astrocytic morphology and a compensatory homeostatic rise in EAAT2 abundance.     

Persistence in fluctuating environments for interacting structured populations

Individuals within any species exhibit differences in size, developmental state, or spatial location. These differences coupled with environmental fluctuations in demographic rates can have subtle effects on population persistence and species coexistence. To understand these effects, we provide a general theory for coexistence of structured, interacting species living in a stochastic environment. The theory is applicable to nonlinear, multi species matrix models with stochastically varying parameters. The theory relies on long-term growth rates of species corresponding to the dominant Lyapunov exponents of random matrix products. Our coexistence criterion requires that a convex combination of these long-term growth rates is positive with probability one whenever one or more species are at low density. When this condition holds, the community is stochastically persistent: the fraction of time that a species density goes below \(\delta >0\) approaches zero as \(\delta \) approaches zero. Applications to predator-prey interactions in an autocorrelated environment, a stochastic LPA model, and spatial lottery models are provided. These applications demonstrate that positive autocorrelations in temporal fluctuations can disrupt predator-prey coexistence, fluctuations in log-fecundity can facilitate persistence in structured populations, and long-lived, relatively sedentary competing populations are likely to coexist in spatially and temporally heterogenous environments.