Stochastic modeling of aphid population growth with nonlinear, power-law dynamics

This paper develops a deterministic and a stochastic population size model based on power-law kinetics for the black-margined pecan aphid. The deterministic model in current use incorporates cumulative-size dependency, but its solution is symmetric. The analogous stochastic model incorporates the prolific reproductive capacity of the aphid. These models are generalized in this paper to include a delayed feedback mechanism for aphid death. Whereas the per capita aphid death rate in the current model is proportional to cumulative size, delayed feedback is implemented by assuming that the per capita rate is proportional to some power of cumulative size, leading to so-called power-law dynamics. The solution to the resulting differential equations model is a left-skewed abundance curve. Such skewness is characteristic of observed aphid data, and the generalized model fits data well. The assumed stochastic model is solved using Kolmogrov equations, and differential equations are given for low order cumulants. Moment closure approximations, which are simple to apply, are shown to give accurate predictions of the two endpoints of practical interest, namely (1) a point estimate of peak aphid count and (2) an interval estimate of final cumulative aphid count. The new models should be widely applicable to other aphid species, as they are based on three fundamental properties of aphid population biology.     

On stochastic logistic population growth models with immigration and multiple births

This paper develops a stochastic logistic population growth model with immigration and multiple births. The differential equations for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model are reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the Normal approximation, and even more accurately using the saddlepoint approximation. It is shown that in comparison with the corresponding single birth model, the multiple birth mechanism increases the skewness and the variance of the equilibrium distribution, but slightly reduces its mean. Moreover, the type of density-dependent population control is shown to influence the sign of the skewness and the size of the variance.     

Temporal autocorrelation and stochastic population growth

How much does environmental autocorrelation matter to the growth of structured populations in real life contexts? Interannual variances in vital rates certainly do, but it has been suggested that between‐year correlations may not. We present an analytical approximation to stochastic growth rate for multistate Markovian environments and show that it is accurate by testing it in two empirically based examples. We find that temporal autocorrelation has sizeable effect on growth rates of structured populations, larger in many cases than the effect of interannual variability. Our approximation defines a sensitivity to autocorrelated variability, showing how demographic damping and environmental pattern interact to determine a population's stochastic growth rate.     

On the accuracy of a diffusion approximation to a discrete state–space Markovian model of a population

The traditional Kolmogorov equations treat the size of a population as a discrete random variable. A model is introduced that extends these equations to incorporate environmental variability. Difficulties with this discrete model motivate approximating the population size as a continuous random variable through the use of diffusion processes. The set of cumulants for both the population size and the environmental factors affecting the population size characterize the population–environmental system. The evolution of this set, as predicted by the diffusion approximation, closely matches the corresponding predictions for the discrete model. It is also noted that the simulation estimates of the cumulants against which the predictions of the diffusion model are checked can vary considerably between simulations — despite averaging over a large number of simulation runs. The precision of the simulation estimates–both over time and with differing cumulant order–is discussed.     

Fitting parameters of stochastic birth-death models to metapopulation data

Populations that are structured into small local patches are a common feature of ecological and epidemiological systems. Models describing this structure are often referred to as metapopulation models in ecology or household models in epidemiology. Small local populations are subject to demographic stochasticity. Theoretical studies of household disease models without resistant stages (SIS models) have shown that local stochasticity can be ignored for between patch disease transmission if the number of connected patches is large. In that case the distribution of the number of infected individuals per household reaches a stationary distribution described by a birth-death process with a constant immigration term. Here we show how this result, in conjunction with the balancing condition for birth-death processes, provides a framework to estimate demographic parameters from a frequency distribution of local population sizes. The parameter estimation framework is applicable to estimate parameters of disease transmission models as well as metapopulation models.     

Fly disturbance suppresses aphid population growth

1. The presence of predators is known to suppress prey populations not only by consumptive but also by non-consumptive effects, as it stresses the prey inducing costly changes to behaviour and physiology. However, there is recent evidence that disturbance from non-predacious, non-competing commensals can also negatively affect herbivore performance. 2. Populations of cherry-oat aphids (Rhopalosiphum padi L.) were initiated with adult aphids in mesocosms containing seedling wheat grass. Following aphid establishment, fruit flies (Drosophila melanogaster Meigen) were added at a relatively high density to half of the mesocosms and the aphids were left for another 5 days to reproduce. The experiment was performed over two blocks at 24 ± 4 °C. It was hypothesized that a relatively high density of commensals would stress the aphids and reduce their fitness, causing lower population growth and resulting in lower population sizes. 3. Aphid numbers were significantly lower in mesocosms with commensal flies after 5 days of fly presence across the two experimental blocks, documenting that fly disturbance suppresses aphid fitness and population growth. 4. The negative effect of fruit flies on aphid population growth must have come from the disturbance that flies imposed on the aphids in their search for food, indicating that the flies stressed the aphids. Thus, our study indicates that commensals may stress herbivores that do not distinguish between enemies and other active species in their environment, adding in the overall herbivore top-down control through fitness costs.     

Nonlinear time-series modeling of vole population fluctuations

A central goal of population ecology is to understand and predict fluctuations in population numbers. Until recently, much of the debate focused on the issue of population regulation by density-dependent factors. In this paper, I describe an approach to nonlinear modeling of time-series data that is designed to go beyond this question by investigating the possibility of complex population dynamics, characterized by lags in regulation and periodic or chaotic oscillations. The questions motivating this approach are: what are relative contributions of endogenous vs. exogenous components of dynamics? Is the irregular component in fluctuations entirely due to exogenous noise, or do nonlinearities contribute to it, too? I describe the philosophy and the technical details of the nonlinear modeling approach, and then apply it to a collection of time-series data on vole population fluctuations in northern Europe. The results suggest that population dynamics of European voles undergo a latitudinal shift from stability to chaos. Dynamics in northern Fennoscandia are characterized by positive Lyapunov exponent estimates, and a high degree of short-term (one year ahead) predictability, suggesting a strong endogenous component. In more southerly populations estimated Lyapunov exponents are negative, and there is no one-step ahead predictability, suggesting that fluctuations are driven by exogenous factors.     

Soybean plant stage and population growth of soybean aphid

The soybean aphid, Aphis glycines Matsumura, is a newly invasive species of aphid in North America. Previous studies disagree as to whether soybean, Glycine max (L.) Merr., plant stage has an impact on aphid intrinsic rate of increase. Therefore, the growth rate of soybean aphids on soybean plants of different stages was examined at two different scales in the field. A planting date experiment was used to measure the population growth of soybean aphids on plants of different stages. Clip-cages were used to measure life history characteristics of individual aphids on plants of different stages. No differences were found in the population growth or dynamics of soybean aphids in the planting date experiment. The life history characteristics of individual aphids also showed no significant difference when feeding on different growth stages of soybean plants. The impact of these findings on soybean aphid management is discussed and the possible reasons why the results differ from previous estimates of the aphid growth-plant stage relationship are considered.     

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450-463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173-205].     

Sam Karlin and the stochastic theory of evolutionary population genetics

Sam Karlin’s role in the development of the stochastic theory of evolutionary population genetics is outlined, together with his work in developing BLAST theory.     

An age-dependent stochastic model of population growth

A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ<1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.     

Effect of PGPR on population growth parameters of cotton aphid

Plant-growth-promoting rhizobacteria (PGPR) help plants by increasing plant growth and defending them against pathogens and pests. There is considerable research supporting the use of these bacteria in plant pathogens management, but the number of research papers that have focused on their effect on insect pests control is quite few. In this research, we examined the ability of four native strains of Pseudomonas fluorescens to manage the cotton aphid, Aphis gossypii, on greenhouse-grown cucumbers in Iran. The strains UTPF (University of Tehran Pseudomonas fluorescens) 68, UTPF1, UTPF6 and PF169 of P. fluorescens were used as seed treatment to evaluate their impact on both cucumber growth and cotton aphid population growth parameters. The strains UTPF68 and PF169 of this bacterium influenced phenology of cucumber by decreasing the Growing Degree Days of flowering period. Also, the plants treated with these bacterial strains had a significant increase in yield weight of approximately 58%. In the other part of the study, various effects on developmental parameters of aphid were recorded among bacterial strains, but only PF169 caused significant decreasing effect on r_m value of aphid, which is the most important factor in aphid’s life table. We conclude that the PF169 treatment had significant negative effects on population growth rate of A. gossypii. The PF169 could be reliable in a control programme for A. gossypii on cucumber plants in greenhouses.     

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.     

Semiclassical approximations of stochastic epidemiological processes towards parameter estimation using as prime example the SIS system with import

In this paper we investigate several schemes to approximate the stationary distribution of the stochastic SIS system with import. We begin by presenting the model and analytically computing its stationary distribution. We then approximate this distribution using Kramers–Moyal approximation, van Kampen's system size expansion, and a semiclassical scheme, also called WKB or eikonal approximation depending on its different applications in physics. For the semiclassical scheme, done in the context of the Hamilton–Jacobi formalism, two approaches are taken. In the first approach we assume a semiclassical ansatz for the generating function, while in the second the solution of the master equation is approximated directly. The different schemes are compared and the semiclassical approximation, which performs better, is then used to analyse the time dependent solution of stochastic systems for which no analytical expression is known. Stochastic epidemiological models are studied in order to investigate how far such semiclassical approximations can be used for parameter estimation.     

Generalized aphid population growth models with immigration and cumulative-size dependent dynamics

Mechanistic models in which the per-capita death rate of a population is proportional to cumulative past size have been shown to describe adequately the population size curves for a number of aphid species. Such previous cumulative-sized based models have not included immigration. The inclusion of immigration is suggested biologically as local aphid populations are initiated by migration of winged aphids and as reproduction is temperature-dependent. This paper investigates two models with constant immigration, one with continuous immigration and the other with restricted immigration. Cases of the latter are relatively simple to fit to data. The results from these two immigration models are compared for data sets on the mustard aphid in India.     

Defence variation within a guild of aphid-tending ants explains aphid population growth

• Mutualism studies often focus on the service provided by single species, while variation in benefits provided by multiple partners is less understood. Ant-aphid food-for-protection mutualisms are suitable to study the implications of intra-guild service variation because they often involve several ant species with varying levels of aggressiveness.
• We studied an aphid species and its associated ant guild to address whether intra-guild defence variation against aphid natural enemies explains aphid performance on plants (thistles). We surveyed plants with natural abundances of aphids associated with different ant species and estimated aphid population growth. We conducted confrontation experiments between ant species and aphid natural enemies (ladybugs and hoverfly larvae). In plants patrolled by the most aggressive ant species, we determined the ant's probability of expelling aphid natural enemies and tested whether ant exclusion affects the abundance of aphids and their natural enemies.
• The ant Dorymyrmex tener was the most abundant and frequent species on plants and the most aggressive against aphid natural enemies. Aphid populations grew faster on plants patrolled by D. tener compared to plants patrolled by Camponotus distinguendus or D. richteri. Field experiments confirmed that D. tener effectively expels aphid natural enemies from plants. When D. tener was excluded, the density of aphids decreased, while the abundance of aphid natural enemies increased.
• The disruption of aphid predation by aggressive and numerically dominant ant species is a determinant of aphid population dynamics. This study illustrates the importance of considering intra-guild service variation to better understand multi-partner mutualisms.

     

Nonlinear age-dependent models for prediction of population growth

In this paper, some new algorithms are proposed to estimate parameter functions in nonlinear age-dependent population models by practical data. These algorithms together with a numerical method are applied to compute the human population using the data provided by the United Nations Demographic Yearbook.     

Asymmetric effects of a leaf‐chewing herbivore on aphid population growth

1. Plant responses to herbivory are often specific to the feeding guild of the attacking herbivore. These phytochemical responses to herbivore damage can affect herbivore performance and activity. Comprehensive studies on the ecological consequences of multi‐herbivore plant interactions are key to understanding plant–herbivore community dynamics. 2. This study examined how feeding damage by co‐occurring herbivores from separate feeding guilds, Myzus persicae (Sulzer), a sucking herbivore, and Leptinotarsa decemlineata (Say), a chewing herbivore, alter plant chemistry and indirectly affect herbivore performance. Performance was measured when each insect fed on plants individually, sequentially, or simultaneously in laboratory and field experiments. Phytohormone and glycoalkaloid content were measured for each feeding sequence to evaluate plant responses to herbivory by each guild. Mid‐season and end‐of‐season tuber yield were evaluated in the field study. 3. Damage by L. decemlineata negatively impacted M. persicae performance in both laboratory and field settings. Damage by M. persicae did not affect L. decemlineata performance in laboratory assays. However, L. decemlineata performance was positively affected by M. persicae herbivory in the field, but this effect was temporary. Although phytohormones and plant defences varied across treatments, they provide little resolution on interaction outcomes. 4. These results confirm that the presence of multiple feeding guilds on a single plant can affect these chewing and sucking herbivores differentially, but given the variability in our phytochemical analyses compared with other studies, the mechanism remains unclear. The study's findings show that aphids are negatively affected by chewing herbivores across systems, while aphids temporarily affected beetles positively.