Increasing temperature weakens the positive effect of genetic diversity on population growth

Genetic diversity and temperature increases associated with global climate change are known to independently influence population growth and extinction risk. Whether increasing temperature may influence the effect of genetic diversity on population growth, however, is not known. We address this issue in the model protist system Tetrahymena thermophila. We test the hypothesis that at temperatures closer to the species’ thermal optimum (i.e., the temperature at which population growth is maximal, or T _opt), genetic diversity should have a weaker effect on population growth compared to temperatures away from the thermal optimum. To do so, we grew populations of T. thermophila with varying levels of genetic diversity at increasingly warmer temperatures and quantified their intrinsic population growth rate, r. We found that genetic diversity increases population growth at cooler temperatures, but that as temperature increases, this effect weakens. We also show that a combination of changes in the amount of expressed genetic diversity (G) in r, plastic changes in population growth across temperatures (E), and strong G × E interactions underlie this temperature effect. Our results uncover important but largely overlooked temperature effects that have implications for the management of small populations with depauperate genetic stocks in an increasingly warming world.     

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.     

Environmental Uncertainty and Variable Diapause

We analyze a stage-structured model of a population that displays variable diapause in a randomly varying environment. The ruggedness of the environment is measured by the extent of random variation in per-capita reproductive success. We show how variable diapause and environmental characteristics affect the population′s stochastic growth rate. In rugged unpredictable environments, phenotypes that show some tendency to diapause are found to have a higher growth rate than nondiapausing phenotypes. In harsh rugged environments, some tendency to diapause may be all that permits population persistence. Positive serial autocorrelation causes the optimal diapause fraction to decrease, while negative autocorrelation causes that fraction to increase. The structured model behaves very differently from a scalar model for large diapause fractions even in uncorrelated environments, and in many cases predicts a broad optimum. The difference between models is due to the extreme variability of stage structure in populations subject to even small variability when diapause tendency is high.     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Anthropogenically-Mediated Density Dependence in a Declining Farmland Bird

Land management intrinsically influences the distribution of animals and can consequently alter the potential for density-dependent processes to act within populations. For declining species, high densities of breeding territories are typically considered to represent productive populations. However, as density-dependent effects of food limitation or predator pressure may occur (especially when species are dependent upon separate nesting and foraging habitats), high territory density may limit per-capita productivity. Here, we use a declining but widespread European farmland bird, the yellowhammer Emberiza citrinella L., as a model system to test whether higher territory densities result in lower fledging success, parental provisioning rates or nestling growth rates compared to lower densities. Organic landscapes held higher territory densities, but nests on organic farms fledged fewer nestlings, translating to a 5 times higher rate of population shrinkage on organic farms compared to conventional. In addition, when parental provisioning behaviour was not restricted by predation risk (i.e., at times of low corvid activity), nestling provisioning rates were higher at lower territory densities, resulting in a much greater increase in nestling mass in low density areas, suggesting that food limitation occurred at high densities. These findings in turn suggest an ecological trap, whereby preferred nesting habitat does not provide sufficient food for rearing nestlings at high population density, creating a population sink. Habitat management for farmland birds should focus not simply on creating a high nesting density, but also on ensuring heterogeneous habitats to provide food resources in close proximity to nesting birds, even if this occurs through potentially restricting overall nest density but increasing population-level breeding success.     

Estimation of growth regulation in natural populations by extended family of growth curve models with fractional order derivative: Case studies from the global population dynamics database

Estimating the trend in population time series data using growth curve models is a central idea in population ecology. Several models, mainly governed by differential or difference equations, have been applied to real data sets to identify general growth pattern and make predictions. In this article, we analyze ecological time series data by fitting mathematical models governed by fractional differential equations (FDE). The order of the FDE (α) is used to quantify the evidence of memory in the population processes. The application of FDE is exemplified by analyzing time series data on two bird species Phalacrocorax carbo (Great cormorant) and Parus bicolor (Tufted titmouse) and two mammal species Castor canadensis (Beaver) and Ursus americanus (American black bear) extracted from the global population dynamics database. Five different population growth models were fitted to these data; density-independent exponential, negative density-dependent logistic and θ-logistic model, positive density-dependent exponential Allee and strong Allee model. Both ordinary and fractional derivative representations of these models were fitted to the time series data. Markov chain Monte Carlo (MCMC) method was used to estimate the model parameters and Akaike information criterion was used to select the best model. By estimating the return rate for each of the time series, we have shown that populations governed by FDE with a small value of α (high level of memory) return to the stable equilibrium faster. This demonstrates a synergistic interplay between memory and stability in natural populations.     

Host brood traits,independent of adult behaviours,reduce Varroa destructor mite reproduction in resistant honeybee populations

The ectoparasitic mite Varroa destructor is an invasive species of Western honey bees (Apis mellifera) and the largest pathogenic threat to their health world-wide. Its successful invasion and expansion is related to its ability to exploit the worker brood for reproduction, which results in an exponential population growth rate in the new host. With invasion of the mite, wild honeybee populations have been nearly eradicated from Europe and North America, and the survival of managed honeybee populations relies on mite population control treatments. However, there are a few documented honeybee populations surviving extended periods without control treatments due to adapted host traits that directly impact Varroa mite fitness. The aim of this study was to investigate if Varroa mite reproductive success was affected by traits of adult bee behaviours or by traits of the worker brood, in three mite-resistant honey bee populations from Sweden, France and Norway. The mite’s reproductive success was measured and compared in broods that were either exposed to, or excluded from, adult bee access. Mite-resistant bee populations were also compared with a local mite-susceptible population, as a control group. Our results show that mite reproductive success rates and mite fecundity in the three mite-resistant populations were significantly different from the control population, with the French and Swedish populations having significantly lower reproductive rates than the Norwegian population. When comparing mite reproduction in exposed or excluded brood treatments, no differences were observed, regardless of population. This result clearly demonstrates that Varroa mite reproductive success can be suppressed by traits of the brood, independent of adult worker bees.     

The finite element implementation of a K.P.P. equation for the simulation of tsetse control measures in the vicinity of a game reserve

An equation, strongly reminiscent of Fisher’s equation, is used to model the response of tsetse populations to proposed control measures in the vicinity of a game reserve. The model assumes movement is by diffusion and that growth is logistic. This logistic growth is dependent on an historical population, in contrast to Fisher’s equation which bases it on the present population. The model therefore takes into account the fact that new additions to the adult fly population are, in actual fact, the descendents of a population which existed one puparial duration ago, furthermore, that this puparial duration is temperature dependent. Artificially imposed mortality is modelled as a proportion at a constant rate. Fisher’s equation is also solved as a formality.The temporary imposition of a 2 % day⁻¹ mortality everywhere outside the reserve for a period of 2 years will have no lasting effect on the influence of the reserve on either the Glossina austeni or the G. brevipalpis populations, although it certainly will eradicate tsetse from poor habitat, outside the reserve. A 5 km-wide barrier with a minimum mortality of 4 % day⁻¹, throughout, will succeed in isolating a worst-case, G. austeni population and its associated trypanosomiasis from the surrounding areas. A more optimistic estimate of its mobility suggests a mortality of 2 % day⁻¹ will suffice. For a given target-related mortality, more mobile species are found to be more vulnerable to eradication than more sedentary species, while the opposite is true for containment.     

Modelling urban populations of the Australian White Ibis (Threskiornis molucca) to inform management

Since the 1970s, populations of the Australian White Ibis (Threskiornis molucca) have dramatically increased in many Australian urban centres. Managers of ibis are currently focusing on limiting this bird's reproductive success in order to reduce population sizes or at least halt further increases in urban areas. Here we use data on nesting success and survival for three populations of ibis around greater Sydney to develop an age-structured population model. The estimated growth rate for all populations combined was about 1.5 % per year and for individual sites were more variable at −1, −7, and 9 %. For all populations, growth rates were most sensitive (based on elasticity analyses) to the survival of adults and least sensitive to fecundity, especially of 3 year olds. Further exploration of the importance of fecundity rates, which are relatively poorly known for these populations, suggests that rates of <0.4 fledglings per nest per year is very likely to lead to a population decline (λ less than lower bound of 95 % CI). Conversely, positive population growth is nearly assured (λ greater than upper bound of 95 % CI) for fecundities of >0.7 fledgling per nest per year. The results suggest that ibis from other locations (probably their traditional breeding areas in inland Australia) have immigrated into urban environments as estimated growth rates cannot account for current population sizes. Management strategies must take these findings into account and also consider that ibis are declining in their traditional habitats to avoid exacerbating their decline at a regional scale.     

Metapopulation extinction risk: Dispersal’s duplicity

Metapopulation extinction risk is the probability that all local populations are simultaneously extinct during a fixed time frame. Dispersal may reduce a metapopulation’s extinction risk by raising its average per-capita growth rate. By contrast, dispersal may raise a metapopulation’s extinction risk by reducing its average population density. Which effect prevails is controlled by habitat fragmentation. Dispersal in mildly fragmented habitat reduces a metapopulation’s extinction risk by raising its average per-capita growth rate without causing any appreciable drop in its average population density. By contrast, dispersal in severely fragmented habitat raises a metapopulation’s extinction risk because the rise in its average per-capita growth rate is more than offset by the decline in its average population density. The metapopulation model used here shows several other interesting phenomena. Dispersal in sufficiently fragmented habitat reduces a metapopulation’s extinction risk to that of a constant environment. Dispersal between habitat fragments reduces a metapopulation’s extinction risk insofar as local environments are asynchronous. Grouped dispersal raises the effective habitat fragmentation level. Dispersal search barriers raise metapopulation extinction risk. Nonuniform dispersal may reduce the effective fraction of suitable habitat fragments below the extinction threshold. Nonuniform dispersal may make demographic stochasticity a more potent metapopulation extinction force than environmental stochasticity.     

Population growth regulated by intraspecific competition for energy or time: Some simple representations

An examination is made of some of the ways populations can grow in response to changes in their own density. Under two different assumptions on birth and death rates, models for single-species population growth that incorporate intraspecific competition by interference but not exploitation are of logistic form. Where an individual's net energy input from feeding is inversely proportional to population size, population growth follows a convex curve, whether interference is included or not. Data of Smith (1963) on Daphnia populations are fit well by this kind of curve. Combination of the two kinds of growth can produce S-shaped curves whose inflection is displaced from that value—half the carrying capacity—given by the logistic; an upward displacement is favored by a high ratio of metabolic and replacement costs to feeding input. Inflection points from real curves are much more often higher than expected from the logistic. Nonmonotonic growth curves can arise when there is instantaneous feedback between consumers and resource availability; certain of these equations are of logistic or convex form at equilibrium. The possible effect of r- and K-selection on the biological parameters, such as feeding efficiency, used to construct the monotonie equations is discussed, and the equations are extended to 2-species competition. Table III characterizes some simple single-species growth curves.     

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.     

Changes in <Emphasis Type="Italic">Synechococcus</Emphasis> Population Size and Cellular Ribosomal RNA Content in Response to Predation and Nutrient Limitation

A mathematical model of predator-prey interactions was used to predict the relationship between population size and cellular growth rate in a two-tiered trophic system consisting of Synechococcus PCC 6301 and Tetrahymena pyriformis. As predicted, axenic chemostat cultures of Synechococcus responded to increased nutrient availability by expanding the equilibrium population size without a concurrent change in growth rate. Likewise, the addition of the predator Tetrahymena pyriformis decreased the Synechococcus population size by 85% and increased the Synechococcus growth rate. Synechococcus populations in the surface waters of the Gulf of Mexico were sampled to ascertain whether the relationship between population size and cellular 16S rRNA concentration conformed to that predicted by the model. Direct counts of autofluorescent cells in size-fractionated seawater samples provided an estimate of Synechococcus population size. The growth rate of in situ populations was estimated by measuring the extent of hybridization of an oligonucleotide probes complementary to Synechococcus 16S rRNA, based on evidence that ribosomal RNA content increases concurrently with growth rate. The comparison of in situ population sizes and specific growth rates revealed that relatively large Synechococcus populations were growing slowly, indicative of nutrient limitation, and that quickly growing populations were relatively small, as predicted for predator-limited populations.     

Evolutionary capacitance may be favored by natural selection

Evolutionary capacitors phenotypically reveal a stock of cryptic genetic variation in a reversible fashion. The sudden and reversible revelation of a range of variation is fundamentally different from the gradual introduction of variation by mutation. Here I study the invasion dynamics of modifiers of revelation. A modifier with the optimal rate of revelation mopt has a higher probability of invading any other population than of being counterinvaded. mopt varies with the population size N and the rate theta at which environmental change makes revelation adaptive. For small populations less than a minimum cutoff Nmin, all revelation is selected against. Nmin is typically quite small and increases only weakly, with theta-1/2. For large populations with N>1/theta, mopt is approximately 1/N. Selection for the optimum is highly effective and increases in effectiveness with larger N>1/theta. For intermediate values of N, mopt is typically a little less than theta and is only weakly favored over less frequent revelation. The model is analogous to a two-locus model for the evolution of a mutator allele. It is a fully stochastic model and so is able to show that selection for revelation can be strong enough to overcome random drift.     

Against the trend: increasing numbers of breeding Northern Lapwings Vanellus vanellus and Black-tailed Godwits Limosa limosa limosa on a German Wadden Sea island

Capsule The increase in population sizes over the last 30 years cannot be explained by reproductive success.

Aims To establish whether the positive population trends are due to increasing and self-sustaining populations or to immigration.

Methods We studied the population development of breeding lapwings from 1971 until 2005 and of godwits from 1977 until 2005 on Wangerooge, a German Wadden Sea island. Both species increased significantly during the last three decades. For each species we used a logistic growth model to derive the reproductive output required to explain the past population development without assuming immigration. We compare the values derived by this model with empirical findings of reproductive output of the respective populations.

Results For neither lapwings nor godwits can empirical reproductive success explain the observed population development.

Conclusion Our results imply that the increase in breeding pairs of Northern Lapwing and Black-tailed Godwit on Wangerooge Island is not due to reproductive output. We propose that it is mainly caused by immigration onto the island.     

The evolution of growth rates on an expanding range edge

Individuals in the vanguard of a species invasion face altered selective conditions when compared with conspecifics behind the invasion front. Assortment by dispersal ability on the expanding front, for example, drives the evolution of increased dispersal, which, in turn, leads to accelerated rates of invasion. Here I propose an additional evolutionary mechanism to explain accelerating invasions: shifts in population growth rate (r). Because individuals in the vanguard face lower population density than those in established populations, they should (relative to individuals in established populations) experience greater r-selection. To test this possibility, I used the ongoing invasion of cane toads (Bufo marinus) across northern Australia. Life-history theory shows that the most efficient way to increase the rate of population growth is to reproduce earlier. Thus, I predict that toads on the invasion front will exhibit faster individual growth rates (and thus will reach breeding size earlier) than those from older populations. Using a common garden design, I show that this is indeed the case: both tadpoles and juvenile toads from frontal populations grow around 30 per cent faster than those from older, long established populations. These results support theoretical predictions that r increases during range advance and highlight the importance of understanding the evolution of life history during range advance.     

Little tern breeding success in artificial and natural habitats: modelling population growth under uncertain vital rates

As a consequence of habitat loss, breeding in man-made habitats has become increasingly common for many coastal breeding bird species. While artificial sites provide valuable substitutes, they may also be more attractive, and importantly, differ in quality from natural sites. Therefore, information on habitat specific breeding success and their potential for supporting stable populations are needed. We compared little tern (Sternula albifrons) breeding success (nest and hatching success) between natural habitat (sandy beaches) and artificial port habitat at Bothnian Bay, Finland from 2006 to 2011. We further reviewed published estimates on pre-fledging and adult survival for little terns and least terns (Sternula antillarum), and used these ranges to estimate plausible parameter spaces for population growth rates given our estimates of breeding success. Nest success was among the highest reported for little terns in the artificial habitat (82 %) while being lower in the natural habitat (58 %). This difference may have resulted from differences in colony sizes and levels of disturbance. Hatching success did not differ significantly, but the percentage of successful nests containing unhatched eggs was twice as high in the natural habitat. The parameter spaces for population growth rates indicated that the artificial habitat has good potential to sustain stable populations (66 % positive growth rate) while for the natural habitat this potential was lower (37 % positive growth rate). While our results suggest that artificial habitats can be very productive breeding sites for habitat deprived tern populations, management should concentrate on improving both habitats with emphasis on natural sites.     

Life History and Production of the Western Gray Whale’s Prey,Ampelisca eschrichtii Kr?yer, 1842 (Amphipoda,Ampeliscidae)

Ampelisca eschrichtii are among the most important prey of the Western North Pacific gray whales, Eschrichtius robustus. The largest and densest known populations of this amphipod occur in the gray whale’s Offshore feeding area on the Northeastern Sakhalin Island Shelf. The remote location, ice cover and stormy weather at the Offshore area have prevented winter sampling. The incomplete annual sampling has confounded efforts to resolve life history and production of A. eschrichtii. Expanded comparisons of population size structure and individual reproductive development between late spring and early fall over six sampling years between 2002 and 2013 however, reveal that A. eschrichtii are gonochoristic, iteroparous, mature at body lengths greater than 15 mm and have a two-year life span. The low frequencies of brooding females, the lack of early stage juveniles, the lack of individual or population growth or biomass increases over late spring and summer, all indicate that growth and reproduction occur primarily in winter, when sampling does not occur. Distinct juvenile and adult size cohorts additionally indicate growth and juvenile production occurs in winter through spring under ice cover. Winter growth thus requires that winter detritus or primary production are critical food sources for these ampeliscid populations and yet, the Offshore area and the Eastern Sakhalin Shelf ampeliscid communities may be the most abundant and productive amphipod population in the world. These A. eschrichtii populations are unlikely to be limited by western gray whale predation. Whether benthic community structure can limit access and foraging success of western gray whales is unclear.     

Life history of an invasive and unexploited population of Nile tilapia (Oreochromis niloticus) and geographical variation across its native and non-native ranges

Although the life history traits of Nile tilapia, Oreochromis niloticus have been studied since the early 20^th century, the potential range of life history parameters in unexploited populations and geographical variability in life history traits are still poorly understood. We explored life history traits (age composition, growth rate, mortality, size, and age at maturity) of an invasive and unexploited population in the Tabaru River, Yonaguni-jima Island, southwestern Japan, through comparisons with exploited populations across the species’ global distribution. Analysis of sectioned otoliths from 307 fish revealed that growth and maximum age were sexually dimorphic (females growing less but having greater longevity). Large-scale comparisons with exploited populations revealed that the unexploited Tabaru River population had a greater life span than exploited populations in other regions, but the growth rate was in the middle of the range of observed values. Although a high variation in life history parameters was observed among populations (L _∞, K, maximum age), we found no significant variation in life history traits by latitude or between African and non-African populations. Such a combination of long life span and high variability in life history traits in response to environmental and fishing pressures may aid the success of non-native Nile tilapia in various environments.     

Non-autonomous logistic equations as models of populations in a deteriorating environment

The non-autonomous logistic equation

$\text{d}\text{x(t)}\text{d}\text{t}\text{= r(t)x(t)[1 ?}\text{x(t)}\text{K(t)}\text{]}$

is studied under conditions that include an environment which is completely deteriorating. In this setting, when the population's growth rate, r, is large on the average, solutions track the environment with a consequent extinction of the population. However, when both r and rK^?1 are small in the sense that they are in L¹[0,∞) then an asymptotic equivalence, where all solutions tend to positive limits as t approaches infinity, results and the population is persistent, independent of initial density. The asymptotic equivalence produces an unreasonable overshoot of carrying capacity which leads to concern about employing the logistic equation in the above form as a population model when growth rates are close to zero.A re-interpretation of the parameters of the logistic equation leads to the alternative logistic formulation

$\text{d}\text{x(t)}\text{d}\text{t}\text{= x(t)[r(t) ?}\text{c}\text{B(t)}\text{x(t)], (c > 0)}$

. A biological interpretation of the parameters is presented and this equation is compared with the classical logistic model in the case where the parameters are constant. If the alternative logistic model is applied in a situation with time-varying parameters, then a deteriorating environment always leads to extinction of the population regardless of the behavior of r. Similarly, a growth rate which is small on the average results in extinction regardless of the behavior of B. Furthermore, r and B have limiting values as t approaches infinity then so does x and the terminal value of x is equal to the terminal value of the carrying capacity of the population. In general, the alternative formulation seems to be the more reasonable model in situations where perturbations lead to severe decreases in environmental quality and growth rates.