Comparative dynamics of three models for host-parasitoid interactions in a patchy environment

Phenomenological models represent a simplified approach to the study of complex systems such as host-parasitoid interactions. In this paper we compare the dynamics of three phenomenological models for host-parasitoid interactions developed by May (1978), May and Hassell (1981) and May et al. (1981). The essence of the paper by May and Hassell (1981) was to define a minimum number of parameters that would describe the interactions, avoiding the technical difficulties encountered when using models that involve many parameters, yet yielding a system of equations that could capture the essence of real world interactions in patchy environments. Those studies dealt primarily with equilibrium and coexistence phenomena. Here we study the dynamics through bifurcation analysis and phase portraits in a much wider range of parameter values, carrying the models beyond equilibrium states. We show that the dynamics can be either stable or chaotic depending on the location of a damping term in the equations. In the case of the stable system, when host density dependence acts first, a stable point is reached, followed by a closed invariant curve in phase space that first increases then decreases, finally returning to an asymptotically stable point. Chaos is not seen. On the other hand, when parasitoid attack occurs before host density dependence, chaos is inevitably apparent. We show, as did May et al. (1981) and stated earlier byWang and Gutierrez (1980), that the sequence of events in host-parasitoid interactions is crucial in determining their stability. In a chaotic state the size of the host (e.g., insect pests) population becomes unpredictable, frequently becoming quite large, a biologically undesirable outcome. From a mathematical point of view the system is of interest because it reveals how a strategically placed damping term can dramatically alter the outcome. Our study, reaching beyond equilibrium states, suggests a strategy for biological control different from that of May et al. (1981).     

An implicit approach to model plant infestation by insect pests

Various spatial approaches were developed to study the effect of spatial heterogeneities on population dynamics. We present in this paper a flux-based model to describe an aphid-parasitoid system in a closed and spatially structured environment, i.e. a greenhouse. Derived from previous work and adapted to host-parasitoid interactions, our model represents the level of plant infestation as a continuous variable corresponding to the number of plants bearing a given density of pests at a given time. The variation of this variable is described by a partial differential equation. It is coupled to an ordinary differential equation and a delay-differential equation that describe the parasitized host population and the parasitoid population, respectively. We have applied our approach to the pest Aphis gossypii and to one of its parasitoids, Lysiphlebus testaceipes, in a melon greenhouse. Numerical simulations showed that, regardless of the number and distribution of hosts in the greenhouse, the aphid population is slightly larger if parasitoids display a type III rather than a type II functional response. However, the population dynamics depend on the initial distribution of hosts and the initial density of parasitoids released, which is interesting for biological control strategies. Sensitivity analysis showed that the delay in the parasitoid equation and the growth rate of the pest population are crucial parameters for predicting the dynamics. We demonstrate here that such a flux-based approach generates relevant predictions with a more synthetic formalism than a common plant-by-plant model. We also explain how this approach can be better adapted to test different management strategies and to manage crops of several greenhouses.     

Spatial scales in the study of reef fishes: A theoretical perspective

Abstract Theoretical models imply that spatial scale derives its greatest importance through interactions between density-dependent processes and spatial variation in population densities and environmental variables. Such interactions cause population dynamics on large spatial scales to differ in important ways from predictions based on measurements of population dynamics at smaller scales, a phenomenon called the scale transition. These differences can account for large-scale population stability and species coexistence. The interactions between density dependence and spatial variation that lead to the scale transition can be understood by the process of non-linear averaging, which shows how variance originating on various spatial scales contributes to large-scale population dynamics. Variance originating below the scale of density dependence contributes less to the scale transition as the spatial scale of the variation declines, while variation originating on or above the scale of density dependence contributes independently of the spatial scale of the variation.     

Host-Parasitoid Dynamics and the Success of Biological Control When Parasitoids Are Prone to Allee Effects

In sexual organisms, low population density can result in mating failures and subsequently yields a low population growth rate and high chance of extinction. For species that are in tight interaction, as in host-parasitoid systems, population dynamics are primarily constrained by demographic interdependences, so that mating failures may have much more intricate consequences. Our main objective is to study the demographic consequences of parasitoid mating failures at low density and its consequences on the success of biological control. For this, we developed a deterministic host-parasitoid model with a mate-finding Allee effect, allowing to tackle interactions between the Allee effect and key determinants of host-parasitoid demography such as the distribution of parasitoid attacks and host competition. Our study shows that parasitoid mating failures at low density result in an extinction threshold and increase the domain of parasitoid deterministic extinction. When proned to mate finding difficulties, parasitoids with cyclic dynamics or low searching efficiency go extinct; parasitoids with high searching efficiency may either persist or go extinct, depending on host intraspecific competition. We show that parasitoids suitable as biocontrol agents for their ability to reduce host populations are particularly likely to suffer from mate-finding Allee effects. This study highlights novel perspectives for understanding of the dynamics observed in natural host-parasitoid systems and improving the success of parasitoid introductions.     

Overcompensatory population dynamic responses to environmental stochasticity

1. To quantify the interactions between density-dependent, population regulation and density-independent limitation, we studied the time-series dynamics of an experimental laboratory insect microcosm system in which both environmental noise and resource limitation were manipulated. 2. A hierarchical Bayesian state-space approach is presented through which it is feasible to capture all sources of uncertainty, including observation error to accurately quantify the density dependence operating on the dynamics. 3. The regulatory processes underpinning the dynamics of two different bruchid beetles (Callosobruchus maculatus and Callosobruchus chinensis) are principally determined by environmental conditions, with fluctuations in abundance explained in terms of changes in overcompensatory dynamics and stochastic processes. 4. A general, stochastic population model is developed to explore the link between abundance fluctuations and the interaction between density dependence and noise. Taking account of time-lags in population regulation can substantially increase predicted population fluctuations resulting from underlying noise processes.     

Delayed density‐dependent onset of spring reproduction in a fluctuating population of field voles

Delayed density‐dependent demographic processes are thought to be the basis for multi‐annual cyclic fluctuations in small rodent populations, but evidence for delayed density dependence of a particular demographic trait is rare. Here, using capture–recapture data from 22 sites collected over nine years, we demonstrate a strong effect of population density with a one‐year lag on the timing of the onset of spring reproduction in a cyclically fluctuating population of field voles Microtus agrestis in northern England. The mean date for the onset of spring reproduction was delayed by about 24 days for every additional 100 voles ha^?1 in the previous spring. This delayed density dependence is sufficient to generate the type of cyclic population dynamics described in the study system.     

The dynamical consequences of developmental variability and demographic stochasticity for host-parasitoid interactions

Few age-structured models of species dynamics incorporate variability and uncertainty in population processes. Motivated by laboratory data for an insect and its parasitoid, we investigate whether such assumptions are appropriate when considering the population dynamics of a single species and its interaction with a natural enemy. Specifically, we examine the effects of developmental variability and demographic stochasticity on different types of cyclic dynamics predicted by traditional models. We show that predictions based on the deterministic fixed-development approach are differentially sensitive to variability and noise in key life stages. In particular, we find that the demonstration of half-generation cycles in the single-species model and the multigeneration cycles in the host-parasitoid model are sensitive to the introduction of developmental variability and noise, whereas generation cycles are robust to the intrinsic variability and uncertainty that may be found in nature.     

A dynamic refuge model and population regulation by insect parasitoids

1. The population dynamic effects of refuges, which hosts enter and leave by diffusive movement, in host–parasitoid interactions are explored using simple models in continuous time.
2. This type of refuge has a stabilizing effect on a host–parasitoid interaction, which is contrary to the implications of some previous models.
3. Stability can be explained by considering how depletion processes lead to a refuge proportion (proportion of hosts protected at a given instant) that increases as parasitoid density increases. This effect is synonymous with pseudointerference in the context of the model.
4. Very high rates of movement of host larvae largely destroy this stability process. Stability is greatest at intermediate levels of movement.
5. Density-dependent host movement can alter the effect of these refuges such that they are either more stabilizing, or tend to destabilize, the dynamics of host–parasitoid systems, depending on the type of density dependence assumed. The conclusion that intermediate movement rates are likely to generate stability with this general type of refuge is not altered in the presence of any type of density dependence, unless the density dependence is at levels which we consider unrealistically high and unlikely to be encountered in nature.
6. It is the assumption that larvae do not move into the refuge prior to becoming vulnerable to parasitism that ensures top-down population control in the model. Thus, parasitoids attacking very early instars make good candidates for biological control when faced with a structural refuge.     

A general framework for modeling population abundance data

Time-series data resulting from surveying wild animals are often described using state-space population dynamics models, in particular with Gompertz, Beverton-Holt, or Moran-Ricker latent processes. We show how hidden Markov model methodology provides a flexible framework for fitting a wide range of models to such data. This general approach makes it possible to model abundance on the natural or log scale, include multiple observations at each sampling occasion and compare alternative models using information criteria. It also easily accommodates unequal sampling time intervals, should that possibility occur, and allows testing for density dependence using the bootstrap. The paper is illustrated by replicated time series of red kangaroo abundances, and a univariate time series of ibex counts which are an order of magnitude larger. In the analyses carried out, we fit different latent process and observation models using the hidden Markov framework. Results are robust with regard to the necessary discretization of the state variable. We find no effective difference between the three latent models of the paper in terms of maximized likelihood value for the two applications presented, and also others analyzed. Simulations suggest that ecological time series are not sufficiently informative to distinguish between alternative latent processes for modeling population survey data when data do not indicate strong density dependence.     

Influence of parasitized adult reproduction on host-parasitoid dynamics: an age-structured model

In this work, we develop an age-structured model (based on delay-differential equations) to investigate the dynamics of host-parasitoid systems in which adults are the target of parasitism. The rare previous work dealing with such interactions assumes that hosts are functionally dead as soon as they are attacked. We relax this assumption and show that low reproduction rates of parasitized hosts can promote stability at the expense of cyclic behavior (either long term or generation cycles). Higher reproduction rates make the regulation of the host population by parasitoids impossible. As it is the case in models in which adults are subjected to attacks but do not reproduce, our model generates generation cycles for a larger set of parameter values than in models in which juveniles are attacked.     

Density-dependent parasitoid recruitment per parasitized host: effects on parasitoid-host dynamics

Models of parasitoid-host dynamics are analyzed that include direct density dependence in the host population and either parasitoid- or host-density-dependent variation in parasitoid recruitment per parasitized host (parasitoid "yield"). The principal question addressed is how these forms of density dependence in parasitoid dynamics combine with aggregated parasitism to affect the stability of the models, in relation to suppression of host abundance. When parasitoid yield is an overcompensating function of either parasitoid or host density, stability is enhanced for systems with host equilibria suppressed far below the host carrying capacity. Substantially less aggregation of parasitism is required for stability in this situation than in previous models assuming parasitoid yield is constant. However, this density dependence in parasitoid yield also reduces stability when the host equilibrium is suppressed only moderately below carrying capacity; this is especially true when parasitoid yield is more strongly decreased by high host density than is host per capita reproduction. At present there is little empirical evidence concerning the relationships of parasitoid recruitment to parasitoid and host densities. The substantial effects shown in these models suggest that these relationships should be considered in empirical studies.     

Population dynamics of three songbird species in a nestbox population in Central Europe show effects of density,climate and competitive interactions

Unravelling the contributions of density‐dependent and density‐independent factors in determining species population dynamics is a challenge, especially if the two factors interact. One approach is to apply stochastic population models to long‐term data, yet few studies have included interactions between density‐dependent and density‐independent factors, or explored more than one type of stochastic population model. However, both are important because model choice critically affects inference on population dynamics and stability. Here, we used a multiple models approach and applied log‐linear and non‐linear stochastic population models to time series (spanning 29 years) on the population growth rates of Blue Tits Cyanistes caeruleus, Great Tits Parus major and Pied Flycatchers Ficedula hypoleuca breeding in two nestbox populations in southern Germany. We focused on the roles of climate conditions and intra‐ and interspecific competition in determining population growth rates. Density dependence was evident in all populations. For Blue Tits in one population and for Great Tits in both populations, addition of a density‐independent factor improved model fit. At one location, Blue Tit population growth rate increased following warmer winters, whereas Great Tit population growth rates decreased following warmer springs. Importantly, Great Tit population growth rate also decreased following years of high Blue Tit abundance, but not vice versa. This finding is consistent with asymmetric interspecific competition and implies that competition could carry over to influence population dynamics. At the other location, Great Tit population growth rate decreased following years of high Pied Flycatcher abundance but only when Great Tit population numbers were low, illustrating that the roles of density‐dependent and density‐independent factors are not necessarily mutually exclusive. The dynamics of this Great Tit population, in contrast to the other populations, were unstable and chaotic, raising the question of whether interactions between density‐dependent and density‐independent factors play a role in determining the (in) stability of the dynamics of species populations.     

Competition and dispersal in predator-prey waves

Dispersing predators and prey can exhibit complex spatio-temporal wave-like patterns if the interactions between them cause oscillatory dynamics. We study the effect of these predator-prey density waves on the competition between prey populations and between predator populations with different dispersal strategies. We first describe 1- and 2-dimensional simulations of both discrete and continuous predator-prey models. The results suggest that any population that diffuses faster, disperses farther, or is more likely to disperse will exclude slower diffusing, shorter dispersing, or less likely dispersing populations, everything else being equal. It also appears that it does not matter whether time, space, or state are discrete or continuous, nor what the exact interactions between the predators and prey are. So long as waves exist the competition between populations occurs in a similar fashion. We derive a theory that qualitatively explains the observed behaviour and calculate approximate analytical solutions that describe, to a reasonable extent, these behaviours. Predictions about the cost of dispersal are tested. If strong enough, cost can reverse the populations' relative competitive strengths or lead to coexistence because of the effect of spiral wave cores. The theory is also able to explain previous results of simulations of coexistence in host-parasitoid models (Comins, H. N., and Massell, M. P., 1996, J. Theor. Biol. 183, 19-28).     

Environmental variability and population dynamics: do European and North American ducks play by the same rules?

Density dependence, population regulation, and variability in population size are fundamental population processes, the manifestation and interrelationships of which are affected by environmental variability. However, there are surprisingly few empirical studies that distinguish the effect of environmental variability from the effects of population processes. We took advantage of a unique system, in which populations of the same duck species or close ecological counterparts live in highly variable (north American prairies) and in stable (north European lakes) environments, to distinguish the relative contributions of environmental variability (measured as between‐year fluctuations in wetland numbers) and intraspecific interactions (density dependence) in driving population dynamics. We tested whether populations living in stable environments (in northern Europe) were more strongly governed by density dependence than populations living in variable environments (in North America). We also addressed whether relative population dynamical responses to environmental variability versus density corresponded to differences in life history strategies between dabbling (relatively “fast species” and governed by environmental variability) and diving (relatively “slow species” and governed by density) ducks. As expected, the variance component of population fluctuations caused by changes in breeding environments was greater in North America than in Europe. Contrary to expectations, however, populations in more stable environments were not less variable nor clearly more strongly density dependent than populations in highly variable environments. Also, contrary to expectations, populations of diving ducks were neither more stable nor stronger density dependent than populations of dabbling ducks, and the effect of environmental variability on population dynamics was greater in diving than in dabbling ducks. In general, irrespective of continent and species life history, environmental variability contributed more to variation in species abundances than did density. Our findings underscore the need for more studies on populations of the same species in different environments to verify the generality of current explanations about population dynamics and its association with species life history.     

Crossing regimes of temperature dependence in animal movement

A pressing challenge in ecology is to understand the effects of changing global temperatures on food web structure and dynamics. The stability of these complex ecological networks largely depends on how predator–prey interactions may respond to temperature changes. Because predators and prey rely on their velocities to catch food or avoid being eaten, understanding how temperatures may affect animal movement is central to this quest. Despite our efforts, we still lack a mechanistic understanding of how the effect of temperature on metabolic processes scales up to animal movement and beyond. Here, we merge a biomechanical approach, the Metabolic Theory of Ecology and empirical data to show that animal movement displays multiple regimes of temperature dependence. We also show that crossing these regimes has important consequences for population dynamics and stability, which depend on the parameters controlling predator–prey interactions. We argue that this dependence upon interaction parameters may help explain why experimental work on the temperature dependence of interaction strengths has so far yielded conflicting results. More importantly, these changes in the temperature dependence of animal movement can have consequences that go well beyond ecological interactions and affect, for example, animal communication, mating, sensory detection, and any behavioral modality dependent on the movement of limbs. Finally, by not taking into account the changes in temperature dependence reported here we might not be able to properly forecast the impact of global warming on ecological processes and propose appropriate mitigation action when needed.     

Effects of delay,truncations and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models

Understanding effects of hypotheses about reproductive influences, reproductive schedules and the model mechanisms that lead to a loss of stability in a structured model population might provide information about the dynamics of natural population. To demonstrate characteristics of a discrete time, nonlinear, age structured population model, the transition from stability to instability is investigated. Questions about the stability, oscillations and delay processes within the model framework are posed. The relevant processes include delay of reproduction and truncation of lifetime, reproductive classes, and density dependent effects. We find that the effects of delaying reproduction is not stabilizing, but that the reproductive delay is a mechanism that acts to simplify the system dynamics. Density dependence in the reproduction schedule tends to lead to oscillations of large period and towards more unstable dynamics. The methods allow us to establish a conjecture of Levin and Goodyear about the form of the stability in discrete Leslie matrix models.This research was supported in part by the US Environmental Protection Agency under cooperation agreement CR-816081     

A single climate driver has direct and indirect effects on insect population dynamics

Weather drives population dynamics directly, through effects on vital rates, or indirectly, through effects on the population's competitors, predators or prey and thence on vital rates. Indirect effects may include non-additive interactions with density dependence. Detection of climate drivers is critical to predicting climate change effects, but identification of potential drivers may depend on knowing the underlying mechanisms. For the butterfly Speyeria mormonia, one climate driver, snow melt date, has multiple effects on population growth. Snow melt date in year t has density-dependent indirect effects. Through frost effects, early snow melt decreases floral resources, thence per-capita nectar availability, which determines fecundity in the lab. Snow melt date in year t?+?1 has density-independent direct effects. These effects explain 84% of the variation in population growth rate. One climate parameter thus has multiple effects on the dynamics of a species with non-overlapping generations, with one effect not detectable without understanding the underlying mechanism.     

Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations

In this paper we develop a general mathematical model describing the spatio-temporal dynamics of host-parasitoid systems with forced generational synchronisation, for example seasonally induced diapause. The model itself may be described as an individual-based stochastic model with the individual movement rules derived from an underlying continuum PDE model. This approach permits direct comparison between the discrete model and the continuum model. The model includes both within-generation and between-generation mechanisms for population regulation and focuses on the interactions between immobile juvenile hosts, adult hosts and adult parasitoids in a two-dimensional domain. These interactions are mediated, as they are in many such host-parasitoid systems, by the presence of a volatile semio-chemical (kairomone) emitted by the hosts or the hosts food plant. The model investigates the effects on population dynamics for different host versus parasitoid movement strategies as well as the transient dynamics leading to steady states. Despite some agreement between the individual and continuum models for certain motility parameter ranges, the model dynamics diverge when host and parasitoid motilities are unequal. The individual-based model maintains spatially heterogeneous oscillatory dynamics when the continuum model predicts a homogeneous steady state. We discuss the implications of these results for mechanistic models of phenotype evolution.P. Schofield gratefully acknowledges the financial support of the BBSRC and The Wellcome Trust.     

The effects of asymmetric competition on the life history of Trinidadian guppies

The effects of asymmetric interactions on population dynamics has been widely investigated, but there has been little work aimed at understanding how life history parameters like generation time, life expectancy and the variance in lifetime reproductive success are impacted by different types of competition. We develop a new framework for incorporating trait‐mediated density‐dependence into size‐structured models and use Trinidadian guppies to show how different types of competitive interactions impact life history parameters. Our results show the degree of symmetry in competitive interactions can have dramatic effects on the speed of the life history. For some vital rates, shifting the competitive superiority from small to large individuals resulted in a doubling of the generation time. Such large influences of competitive symmetry on the timescale of demographic processes, and hence evolution, highlights the interwoven nature of ecological and evolutionary processes and the importance of density‐dependence in understanding eco‐evolutionary dynamics.