Indirect interactions in aphid–parasitoid communities

Indirect interactions between populations of different species can be important in structuring natural communities. Indirect effects are either mediated by changes in population densities (trophic or density-mediated effects) or by changes in the behavior of species that are not trophically connected (behavioral or trait-mediated effects). We reviewed the literature on aphids and their parasitoids to explore the various possible indirect interactions that can occur in such communities. The review was motivated by our study of a particular aphid–parasitoid community in a natural (i.e., nonagricultural) habitat, and by the wealth of information that exists about aphid–parasitoid systems in agricultural settings. We focused our review on aphid–parasitoid interactions, but considered how these were influenced by the other aphid natural enemies and also by aphid mutualists and host plants. We conclude that indirect effects are likely to have a major effect in structuring aphid–parasitoid communities, and that the latter are a valuable model system for testing ideas about community interactions. Received: December 20, 1998 / Accepted: January 12, 1999     

A discrete,size-structured model of phytoplankton growth in the chemostat: introduction of inhomogeneous cell division size

 We introduce inhomogeneous, substrate dependent cell division in a time discrete, nonlinear matrix model of size-structured population growth in the chemostat, first introduced by Gage et al. [8] and later analysed by Smith [13]. We show that mass conservation is verified, and conclude that our system admits one non zero globally stable equilibrium, which we express explicitly. Then we run numerical simulations of the system, and compare the predictions of the model to data related to phytoplankton growth, whose obtention we discuss. We end with the identification of several parameters of the system. Received: 9 February 2000 / Revised version: 10 October 2001 / Published online: 23 August 2002 RID="*" ID="*" Present address: Department of Mathematics and Statistics, University of Victoria, B.C., Canada. e-mail: jarino@math.uvic.ca Key words or phrases: Chemostat – Structured population models – Discrete model – Inhomogeneous division size     

Process and measurement errors of population size: their mutual effects on precision and bias of estimates for demographic parameters

Knowing the parameters of population growth and regulation is fundamental for answering many ecological questions and the successful implementation of conservation strategies. Moreover, detecting a population trend is often a legal obligation. Yet, inherent process and measurement errors aggravate the ability to estimate these parameters from population time-series. We use numerical simulations to explore how the lengths of the time-series, process and measurement error influence estimates of demographic parameters. We first generate time-series of population sizes with given demographic parameters for density-dependent stochastic population growth, but assume that these population sizes are estimated with measurement errors. We then fit parameters for population growth, habitat capacity, total error and long-term trends to the ‘measured’ time-series data using non-linear regression. The length of the time-series and measurement error introduce a substantial bias in the estimates for population growth rate and to a lesser degree on estimates for habitat capacity, while process error has little effect on parameter bias. The total error term of the statistical model is dominated by process error as long as the latter is larger than the measurement error. A decline in population size is difficult to document as soon as either error becomes moderate, trends are not very pronounced, and time-series are short (<10–15 seasons). Detecting an annual decline of 1% within 6-year reporting periods, as required for the European Union for the species of Community Interest, appears unachievable.     

Management of a German cockroach population with a juvenoid bait formulation: computer simulation

A computer model was applied to simulate responses of German cockroach populations to feeding on juvenoid baits during an equilibrium, exponential, and the earliest phase of population growth. The simulations show that the renewal potential of the cockroach populations will be exhausted when about 80% or more adults become sterile. The attainment of this activity threshold will be preceded by a lag time with no apparent control effect. The time delay appears substantially shorter for both growing populations (30–60 days) than for the equilibrious one (90–120 days), but, the former ones will continue growing throughout this lag. The simulations predict that even a worse baits' performance can give sufficient control effect in the field if the baits are laid in an early phase of population growth. Thus, optimum tactics should integrate administration of juvenoid baits with an introductory reduction of cockroach population.     

Searching for Spatial Patterns in a Pollinator–Plant–Herbivore Mathematical Model

This paper deals with the spatio-temporal dynamics of a pollinator–plant–herbivore mathematical model. The full model consists of three nonlinear reaction–diffusion–advection equations defined on a rectangular region. In view of analyzing the full model, we firstly consider the temporal dynamics of three homogeneous cases. The first one is a model for a mutualistic interaction (pollinator–plant), later on a sort of predator–prey (plant–herbivore) interaction model is studied. In both cases, the interaction term is described by a Holling response of type II. Finally, by considering that the plant population is the unique feeding source for the herbivores, a mathematical model for the three interacting populations is considered. By incorporating a constant diffusion term into the equations for the pollinators and herbivores, we numerically study the spatiotemporal dynamics of the first two mentioned models. For the full model, a constant diffusion and advection terms are included in the equation for the pollinators. For the resulting model, we sketch the proof of the existence, positiveness, and boundedness of solution for an initial and boundary values problem. In order to see the separated effect of the diffusion and advection terms on the final population distributions, a set of numerical simulations are included. We used homogeneous Dirichlet and Neumann boundary conditions.     

Deterministic extinction effect of parasites on host populations

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

Coexistence of different serotypes of dengue virus

 We formulate a non–linear system of differential equations that models the dynamics of dengue fever. This disease is produced by any of the four serotypes of dengue arbovirus. Each serotype produces permanent immunity to it, but only a certain degree of cross–immunity to heterologous serotypes. In our model we consider the relation between two serotypes. Our interest is to analyze the factors that allow the invasion and persistence of different serotypes in the human population. Analysis of the model reveals the existence of four equilibrium points, which belong to the region of biological interest. One of the equilibrium points corresponds to the disease–free state, the other three equilibria correspond to the two states where just one serotype is present, and the state where both serotypes coexist, respectively. We discuss conditions for the asymptotic stability of equilibria, supported by analytical and numerical methods. We find that coexistence of both serotypes is possible for a large range of parameters. Received: 7 July 1998 / Revised version: 12 July 2002 / Published online: 26 September 2002 Keywords or phrases: Dengue fever – Primary and secondary infections – Serotype – Coexistence – Threshold – Basic reproduction number – Persistence     

Early development and quorum sensing in bacterial biofilms

 We develop mathematical models to examine the formation, growth and quorum sensing activity of bacterial biofilms. The growth aspects of the model are based on the assumption of a continuum of bacterial cells whose growth generates movement, within the developing biofilm, described by a velocity field. A model proposed in Ward et al. (2001) to describe quorum sensing, a process by which bacteria monitor their own population density by the use of quorum sensing molecules (QSMs), is coupled with the growth model. The resulting system of nonlinear partial differential equations is solved numerically, revealing results which are qualitatively consistent with experimental ones. Analytical solutions derived by assuming uniform initial conditions demonstrate that, for large time, a biofilm grows algebraically with time; criteria for linear growth of the biofilm biomass, consistent with experimental data, are established. The analysis reveals, for a biologically realistic limit, the existence of a bifurcation between non-active and active quorum sensing in the biofilm. The model also predicts that travelling waves of quorum sensing behaviour can occur within a certain time frame; while the travelling wave analysis reveals a range of possible travelling wave speeds, numerical solutions suggest that the minimum wave speed, determined by linearisation, is realised for a wide class of initial conditions. Received: 10 February 2002 / Revised version: 29 October 2002 / Published online: 19 March 2003 Key words or phrases: Bacterial biofilm – Quorum sensing – Mathematical modelling – Numerical solution – Asymptotic analysis – Travelling wave analysis     

Extinction risk assessments at the population and species level: implications for amphibian conservation

Amphibian populations are declining worldwide and this is causing growing concern. High levels of population declines followed by the expansion of red lists are creating demands for effective strategies to maximize conservation efforts for amphibians. Ideally, integrated and comprehensive strategies should be based on complementary information of population and species extinction risk. Here we evaluate the congruence between amphibian extinction risk assessments at the population level (Declining Amphibian Database––DAPTF) and at species level (GAA––IUCN Red List). We used the Declining Amphibian Database––DAPTF that covers 967 time-series records of amphibian population declines assigned into four levels of declines. We assigned each of its corresponding species into GAA––IUCN red list status, discriminated each species developmental mode, and obtained their geographic range size as well. Extinction risk assessments at the population and species level do not fully coincide across geographic realms or countries. In Paleartic, Neartic and Indo-Malayan realms less than 25% of species with reported population declines are formally classified as threatened. In contrast, more than 60% of all species with reported population declines that occur in Australasia and the Neotropics are indeed threatened according to the GAA––IUCN Red List. Species with aquatic development presented proportionally higher extinction risks at both population and species level than those with terrestrial development, being this pattern more prominent at Australasia, Paleartic, and Neartic realms. Central American countries, Venezuela, Mexico and Australia presented the highest congruence between both population and species risk. We address that amphibian conservation strategies could be improved by using complementary information on time-series population trends and species threat. Whenever feasible, conservation assessments should also include life-history traits in order to improve its effectiveness.     

Hydrogen isotopic variation in migratory bird tissues of known origin: implications for geographic assignment

Continent-wide variation in hydrogen isotopic composition of precipitation is incorporated into animal diets, providing an intrinsic marker of geographic location at the time of tissue growth. Feathers from migratory birds are now frequently analyzed for stable-hydrogen isotopes (δD) to estimate the location of individuals during a preceding molt. Using known-origin birds, we tested several assumptions associated with this emerging technique. We examined hydrogen isotopic variation as a function of age, sex, feather type and the timing of molt in a marked population of American redstarts (Setophaga ruticilla) breeding in southeastern Ontario. We measured δD in feathers and blood from individuals that bred or hatched at our study site during the year in which those tissues were grown. Juvenile tissues from 5- to 10-day-old birds had more negative δD values than those from adults, which most likely reflected age-related differences in diet. Within adults, primary feathers had more negative δD values than contour feathers. The mean δD value in adult primary feathers was relatively consistent among years and with the value expected for our study population. However, among-individual variation in δD corresponded to an estimated latitudinal range of 6–8° (650–900 km). We conclude that feathers sampled from recently hatched juveniles may not provide a reliable estimate of expected local isotopic signatures for comparison with adult feathers of unknown origin. Furthermore, we urge researchers to use caution when using δD values in feathers to infer geographic origin, and suggest that the best approach is to assign individuals to broad geographic zones within a species’ potential molting range.     

Degeneracy-Driven Self-Structuring Dynamics in Selective Repertoires

Numerous biological interactions, such as interactions between T cell receptors or antibodies with antigens, interactions between enzymes and substrates, or interactions between predators and prey are often not strictly specific. In such less specific, or “sloppy,” systems, referred to here as degenerate systems, a given unit of a diverse resource (antigens, enzymatic substrates, prey) is at risk of being recognized and consumed by multiple consumers (lymphocytes, enzymes, predators). In this study, we model generalized degenerate consumer-resource systems of Lotka–Volterra and Verhulst types. In the degenerate systems of Lotka–Volterra, there is a continuum of types of consumer and resource based on variation of a single trait (characteristic, or preference). The consumers experience competition for a continuum of resource types. This non-local interaction system is modeled with partial differential-integral equations and shows spontaneous self-structuring of the consumer population that depends on the degree of interaction degeneracy between resource and consumer, but does not mirror the distribution of resource. We also show that the classical Verhulst (i.e. logistic) single population model can be generalized to a degenerate model, which shows qualitative behavior similar to that in the degenerate Lotka–Volterra model. These results provide better insight into the dynamics of selective systems in biology, suggesting that adaptation of degenerate repertoires is not a simple “mirroring” of the environment by the “fittest” elements of population.     

Alternative stable states in host–phage dynamics

Bacteriophage are ubiquitous in nature, yet many central aspects of host–phage biology have not been integrated into mathematical models. We propose a novel model of host–phage population dynamics that accounts for the decreased ability of phages to lyse hosts as hosts approach their carrying capacity. In contrast to existing predator–prey-like models, we find a parameter regime in which phages cannot invade a host-only system but, nonetheless, can stably coexist with hosts at lower densities. The finding of alternative stable states suggests clear linkages with observed life history strategies of phages. In addition, we solve a limiting case of the proposed model and show that conservative predator-prey like systems do not inevitably exhibit population cycles. Finally, we discuss possible extensions of the present model and scenarios for experimental testing.     

Modelling and analysis of time-lags in some basic patterns of cell proliferation

 In this paper, we present a systematic approach for obtaining qualitatively and quantitatively correct mathematical models of some biological phenomena with time-lags. Features of our approach are the development of a hierarchy of related models and the estimation of parameter values, along with their non-linear biases and standard deviations, for sets of experimental data. We demonstrate our method of solving parameter estimation problems for neutral delay differential equations by analyzing some models of cell growth that incorporate a time-lag in the cell division phase. We show that these models are more consistent with certain reported data than the classic exponential growth model. Although the exponential growth model provides estimates of some of the growth characteristics, such as the population-doubling time, the time-lag growth models can additionally provide estimates of: (i) the fraction of cells that are dividing, (ii) the rate of commitment of cells to cell division, (iii) the initial distribution of cells in the cell cycle, and (iv) the degree of synchronization of cells in the (initial) cell population. Received: 15 September 1997/Revised version: 1 April 1998     

Fitting a Structured Juvenile–Adult Model for Green Tree Frogs to Population Estimates from Capture–Mark–Recapture Field Data

We derive point and interval estimates for an urban population of green tree frogs (Hyla cinerea) from capture–mark–recapture field data obtained during the years 2006–2009. We present an infinite-dimensional least-squares approach which compares a mathematical population model to the statistical population estimates obtained from the field data. The model is composed of nonlinear first-order hyperbolic equations describing the dynamics of the amphibian population where individuals are divided into juveniles (tadpoles) and adults (frogs). To solve the least-squares problem, an explicit finite difference approximation is developed. Convergence results for the computed parameters are presented. Parameter estimates for the vital rates of juveniles and adults are obtained, and standard deviations for these estimates are computed. Numerical results for the model sensitivity with respect to these parameters are given. Finally, the above-mentioned parameter estimates are used to illustrate the long-time behavior of the population under investigation.     

Pattern Formation,Long-Term Transients,and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.     

An adaptive management approach to an octocoral fishery based on the Beverton-Holt model

Coral reef species are frequently the focus of bio-prospecting, and when promising bioactive compounds are identified there is often a need for the development of responsible harvesting based on relatively limited data. The Caribbean gorgonian Pseudopterogorgia elisabethae has been harvested in the Bahamas for over a decade. Data on population age structure and growth rates in conjunction with harvest data provide an opportunity to compare fishery practices and outcomes to those suggested by a Beverton-Holt fishery model. The model suggests a minimum colony size limit of 7–9 years of age (21–28 cm height), which would allow each colony 2–4 years of reproduction prior to harvesting. The Beverton-Holt model assumes that colonies at or above the minimum size limit are completely removed. In the P. elisabethae fishery, colonies are partially clipped and can be repeatedly harvested. Linear growth of surviving colonies was up to 3 times that predicted for colonies that were not harvested and biomass increase was up to 9 times greater than that predicted for undisturbed colonies. The survival of harvested colonies and compensatory growth increases yield, and yields at sites that had previously been harvested were generally greater than predicted by the Beverton-Holt model. The model also assumes recruitment is independent of fishing intensity, but lower numbers of young colonies in the fished populations, compared to unfished populations, suggest possible negative effects of the harvest on reproduction. This suggests the need for longer intervals between harvests. Because it can be developed from data that can be collected at a single time, the Beverton-Holt model provides a rational starting point for regulating new fisheries where long-term characterizations of population dynamics are rarely available. However, an adaptive approach to the fishery requires the incorporation of reproductive data.     

Analysing the effect of movement on local survival: a new method with an application to a spatially structured population of the arboreal gecko Gehyra variegata

Mortality during movement between habitat patches is the most obvious cost of dispersal, but rarely it has been demonstrated empirically. An approach is presented, which uses capture–mark–recapture data of an arboreal gecko species to determine the effect of individual movement on local survival in a spatially structured population. Because capture–mark–recapture data are widely available for a range of animal species, it should be possible to extend their application to other species. The method is based on the assumption that the tendency to be a territorial animal or to be a floating animal is fixed during the study period. The advantage of our approach is that only one additional parameter has to be estimated for describing movement risks. We further tested the power of our approach to detect an association of movement and mortality with simulated capture histories. The study revealed a strong negative effect of movement on local survival. Hence, animals that moved more often between trees had a lower survival rate. Interestingly, the mean movement rate for males was significantly higher than for females, which should lead to a biased sex ratio towards females in the population. As there was an even sex ratio in the population, we discuss not mutually exclusive explanations for this finding like differences in emigration rates between sexes, differences in survival rates between sexes, or a skewed sex ratio in offspring.     

Monotone travelling fronts in an age-structured reaction-diffusion model of a single species

 We consider a partially coupled diffusive population model in which the state variables represent the densities of the immature and mature population of a single species. The equation for the mature population can be considered on its own, and is a delay differential equation with a delay-dependent coefficient. For the case when the immatures are immobile, we prove that travelling wavefront solutions exist connecting the zero solution of the equation for the matures with the delay-dependent positive equilibrium state. As a perturbation of this case we then consider the case of low immature diffusivity showing that the travelling front solutions continue to persist. Our findings are contrasted with recent studies of the delayed Fisher equation. Travelling fronts of the latter are known to lose monotonicity for sufficiently large delays. In contrast, travelling fronts of our equation appear to remain monotone for all values of the delay. Received: 1 November 2001 / Revised version: 10 May 2002 / Published online: 23 August 2002 Mathematics Subject Classification (2000): 35K57, 92D25 Key words or phrases: Age-structure – Time-delay – Travelling Fronts – Reaction-diffusion     

Coevolution of dispersal in a parasitoid–host system

Interspecific interactions and the evolution of dispersal are both of interest when considering the potential impact of habitat fragmentation on community ecology, but the interaction between these processes is not well studied. We address this by considering the coevolution of dispersal strategies in a host–parasitoid system. An individual-based host–parasitoid metapopulation model was constructed for a patchy environment, allowing for evolution in dispersal rates of both species. Highly rarefied environments with few suitable patches selected against dispersal in both species, as did relatively static environments. Provided that parasitoids persist, all the variables studied led to stable equilibria in dispersal rates for both species. There was a tendency toward higher dispersal rates in parasitoids because of the asymmetric relationships of the two species to the patches: vacant patches are most valuable for hosts, but unsuitable for parasitoids, which require an established host population to reproduce. High host dispersal rate was favoured by high host population growth rate, and in the parasitoid by high growth rates in both species. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.