Diffusion-limited predator-prey dynamics in Euclidean environments: an allometric individual-based model

We claim that diffusion-limited rates of reaction can be an explanation for the altered population dynamics predicted by models incorporating local interactions and limited individual mobility. We show that the predictions of a spatially explicit, individual-based model result from reduced rates of predation and reproduction caused by limited individual mobility and patchiness. When these reduced rates are used in a mean-field model, there is better agreement with the predictions of the simulation model incorporating local interactions. We also explain previous findings regarding the effects of dimensionality on population dynamics in light of diffusion-limited reactions and Pólya random walks. In particular, we demonstrate that 3D systems are better "stirred" than 2D systems and consequently have a reduced tendency for diffusion-limited interaction rates.     

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.     

Adaptive superparasitism and host-parasitoid dynamics

We consider a host-solitary parasitoid system with three categories of individuals: parasitoids, healthy hosts and parasitized hosts. Parasitoids are assumed to discriminate perfectly between the two kinds of hosts and they can reject those which are already parasitized. If parasitoids systematically accept or reject superparasitism or behave randomly, the system is always unstable. Using an optimal foraging model, we determine the behavior of parasitoids which leads to maximization of the instantaneous reproductive rate. When following this adaptive decision rule, parasitoids accept or refuse superparasitism according to the densities of both healthy and parasitized hosts. We study the dynamics of the system when parasitoids follow the optimal rule and show that under certain conditions it possesses a locally stable equilibrium point. In addition, our model predicts that at equilibrium parasitoids show partial preferences for superparasitism.     

Scale-free extinction dynamics in spatially structured host-parasitoid systems

Much of the work on extinction events has focused on external perturbations of ecosystems, such as climatic change, or anthropogenic factors. Extinction, however, can also be driven by endogenous factors, such as the ecological interactions between species in an ecosystem. Here we show that endogenously driven extinction events can have a scale-free distribution in simple spatially structured host-parasitoid systems. Due to the properties of this distribution there may be many such simple ecosystems that, although not strictly permanent, persist for arbitrarily long periods of time. We identify a critical phase transition in the parameter space of the host-parasitoid systems, and explain how this is related to the scale-free nature of the extinction process. Based on these results, we conjecture that scale-free extinction processes and critical phase transitions of the type we have found may be a characteristic feature of many spatially structured, multi-species ecosystems in nature. The necessary ingredient appears to be competition between species where the locally inferior type disperses faster in space. If this condition is satisfied then the eventual outcome depends subtly on the strength of local superiority of one species versus the dispersal rate of the other.     

Periodicity, persistence, and collapse in host-parasitoid systems with egg limitation

There is an emerging consensus that parasitoids are limited by the number of eggs which they can lay as well as the amount of time they can search for their hosts. Since egg limitation tends to destabilize host-parasitoid dynamics, successful control of insect pests by parasitoids requires additional stabilizing mechanisms such as heterogeneity in the distribution of parasitoid attacks and host density-dependence. To better understand how egg limitation, search limitation, heterogeneity in parasitoid attacks, and host density-dependence influence host-parasitoid dynamics, discrete time models accounting for these factors are analyzed. When parasitoids are purely egg-limited, a complete anaylsis of the host-parasitoid dynamics are possible. The analysis implies that the parasitoid can invade the host system only if the parasitoid's intrinsic fitness exceeds the host's intrinsic fitness. When the parasitoid can invade, there is a critical threshold, CV*>1, of the coefficient of variation (CV) of the distribution of parasitoid attacks that determines that outcome of the invasion. If parasitoid attacks sufficiently aggregated (i.e., CV>CV*), then the host and parasitoid coexist. Typically (in a topological sense), this coexistence is shown to occur about a periodic attractor or a stable equilibrium. If the parasitoid attacks are sufficiently random (i.e. CV1. When CV<1, the parasitoid exhibits highly oscillatory dynamics. Alternatively, when parasitoid attacks are sufficiently aggregated but not overly aggregated (i.e. CV>1 but close to 1), the host and parasitoid coexist about a stable equilibrium with low host densities. The implications of these results for classical biological control are discussed.     

Recruitment limitation: A theoretical perspective

Abstract A theoretical analysis of the concept of recruitment limitation leads to the conclusion that most populations should he regarded as jointly limited by recruitment and interactions between individuals after recruitment. The open nature of local marine systems does not permit avoidance of density-dependent interactions; it simply may make them more difficult to detect. Local populations consisting of settled organisms may not experience density-dependent interactions under some circumstances, but the entire species population consisting of the collection of local populations and their planktonic larvae must have density-dependent dynamics. Any local population of settled individuals can escape density dependence if sufficient density dependence occurs among planktonic larvae or within other local populations. Common conceptions of density dependence are too narrow, leading too often to the conclusion that it is absent from a system. It is equally wrong to expect that density-dependent interactions after settlement determine local population densities independently of recruitment. Special circumstances allowing density dependence to act strongly and quickly are needed before density dependence can neutralize the effects of recruitment. Recruitment limitation and density-dependent interactions therefore should not be regarded as alternatives but as jointly acting to determine the densities of marine benthic populations. Moreover, the interaction between fluctuating recruitment and density dependence is potentially the most interesting feature of recruitment limitation. For example, this interaction may be an important diversity-maintaining mechanism for marine systems.     

Differential cannibalism and population dynamics in a host-parasitoid system

The effects of host cannibalism on a host-parasitoid system were explored through experiment and modelling. In individual encounters between parasitized and unparasitized Plodia interpunctella larvae, parasitized larvae were more likely to be cannibalized. Inclusion of this differential cannibalism into a simple Lotka-Volterra-type model of host-parasitoid population dynamics generates alternative stable states-including stable coexistence and extinction of the parasitoid — which depend on starting conditions. Possible mechanisms for differential cannibalism, and its implications for studies of host-parasitoid populations and biological control programmes are discussed.     

Evolutionary dynamics of predator-prey systems: an ecological perspective

 Evolution takes place in an ecological setting that typically involves interactions with other organisms. To describe such evolution, a structure is needed which incorporates the simultaneous evolution of interacting species. Here a formal framework for this purpose is suggested, extending from the microscopic interactions between individuals – the immediate cause of natural selection, through the mesoscopic population dynamics responsible for driving the replacement of one mutant phenotype by another, to the macroscopic process of phenotypic evolution arising from many such substitutions. The process of coevolution that results from this is illustrated in the context of predator–prey systems. With no more than qualitative information about the evolutionary dynamics, some basic properties of predator–prey coevolution become evident. More detailed understanding requires specification of an evolutionary dynamic; two models for this purpose are outlined, one from our own research on a stochastic process of mutation and selection and the other from quantitative genetics. Much of the interest in coevolution has been to characterize the properties of fixed points at which there is no further phenotypic evolution. Stability analysis of the fixed points of evolutionary dynamical systems is reviewed and leads to conclusions about the asymptotic states of evolution rather different from those of game-theoretic methods. These differences become especially important when evolution involves more than one species. Received 10 November 1993; received in revised form 25 July 1994     

Adaptive host preference and the dynamics of host-parasitoid interactions

Models of two independent host populations and a common parasitoid are investigated. The hosts have density-dependent population growth and only interact indirectly by their effects on parasitoid behavior and population dynamics. The parasitoid is assumed to experience a trade-off in its ability to exploit the two hosts. Three alternative types of parasitoid are investigated: (i) fixed generalists whose consumption rates are those that maximize fitness; (ii) "ideal free" parasitoids, which modify their behavior to maximize their rate of finding unparasitized hosts within a generation; and (iii) "evolving" parasitoids, whose capture rates change between generations based on quantitative genetic determination of the relative attack rates on the two hosts. The primary questions addressed are: (1) Do the different types of adaptive processes stabilize or destabilize the population dynamics? (2) Do the adaptive processes tend to equalize or to magnify differences in host densities? The models show that adaptive behavior and evolution frequently destabilize population dynamics and frequently increase the average difference between host densities.     

Temporal self-similarity created by spatial individual-based population dynamics

We report an individual-based single-species model producing temporal scale-free, self-similar dynamics in time. Individuals in the population renew in an explicit space with a large number of loci. We show that reproduction, subsequent dispersal of the offspring, and mortality will organise population fluctuations such that the emerging dynamics represent power law and scale-free structures. Further, we show that spatially structured population dynamics may show red frequency spectra, a property that the simple nonlinear population models are generally lacking.     

The role of transient dynamics in biological pest control: insights from a host-parasitoid community

1. Identifying natural enemies that can maintain pests at low abundances is a priority in biological control. Here, we show that experiments combined with models generate new insights into identifying effective control agents prior to their release in the field. Using a host-parasitoid community (the harlequin bug and its egg parasitoids) as a model system, we report three key findings. 2. The interplay between the host's self-limitation and the parasitoids' saturating functional response causes the long-term (steady-state) outcomes for pest suppression to differ from those of short-term (transient) dynamics. When the bug's self-limitation is moderately strong, the parasitoid with the higher attack rate and conversion efficiency (Ooencyrtus) achieves greater host suppression in the long term, but its longer handling time causes long periods of transient dynamics during which the bug can reach high abundances; when the bug's self-limitation is weak, host fluctuations amplify over time and Ooencyrtus fails at host suppression altogether. In contrast, the parasitoid with the lower attack rate and conversion efficiency but the shorter handling time (Trissolcus) induces only weak transient fluctuations of short duration and can maintain the host at low abundances regardless of the strength of the bug's self-limitation. 3. Release of multiple enemy species can compromise host suppression if an enemy that induces stronger transient fluctuations excludes one that induces weaker fluctuations. For instance, Ooencyrtus excludes Trissolcus despite having a longer handling time because of its higher conversion efficiency. The model correctly predicts the time to exclusion observed in experiments, suggesting that it captures the key biological features of the host-parasitoid interaction. 4. Intraspecific interference reduces long-term pest suppression but improves short-term pest control by reducing the magnitude and duration of transient fluctuations. 5. These results highlight the importance of transient dynamics in pest suppression. Pests are unlikely to be strongly self-limited because they attack crop monocultures. Hence, pest fluctuations are likely to dominate short-term dynamics even when the long-term outcome is a stable equilibrium. The tendency to induce strong transient fluctuations (e.g. through a long handling time) is therefore a crucial consideration when identifying effective pest control agents.     

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).     

Root dynamics and global change: seeking an ecosystem perspective

Changes in the production and turnover of roots in forests and grasslands in response to rising atmospheric CO₂ concentrations, elevated temperatures, altered precipitation, or nitrogen deposition could be a key link between plant responses and longer-term changes in soil organic matter and ecosystem carbon balance. Here we summarize the experimental observations, ideas, and new hypotheses developed in this area in the rest of this volume. Three central questions are posed. Do elevated atmospheric CO₂, nitrogen deposition, and climatic change alter the dynamics of root production and mortality? What are the consequences of root responses to plant physiological processes? What are the implications of root dynamics to soil microbial communities and the fate of carbon in soil? Ecosystem-level observations of root production and mortality in response to global change parameters are just starting to emerge. The challenge to root biologists is to overcome the profound methodological and analytical problems and assemble a more comprehensive data set with sufficient ancillary data that differences between ecosystems can be explained. The assemblage of information reported herein on global patterns of root turnover, basic root biology that controls responses to environmental variables, and new observations of root and associated microbial responses to atmospheric and climatic change helps to sharpen our questions and stimulate new research approaches. New hypotheses have been developed to explain why responses of root turnover might differ in contrasting systems, how carbon allocation to roots is controlled, and how species differences in root chemistry might explain the ultimate fate of carbon in soil. These hypotheses and the enthusiasm for pursuing them are based on the firm belief that a deeper understanding of root dynamics is critical to describing the integrated response of ecosystems to global change.     

Dynamical consequences of optimal host feeding on host-parasitoid population dynamics

This study examines the influence of various host-feeding patterns on host-parasitoid population dynamics. The following types of host-feeding patterns are considered: concurrent and non-destructive, non-concurrent and non-destructive, and non-concurrent and destructive. The host-parasitoid population dynamics is described by the Lotka-Volterra continuous-time model. This study shows that when parasitoids behave optimally, i.e. they maximize their fitness measured by the instantaneous per capita growth rate, the non-destructive type of host feeding stabilizes host-parasitoid dynamics. Other types of host feeding, i.e. destructive, concurrent, or non-concurrent, do not qualitatively change the neutral stability of the Lotka-Volterra model. Moreover, it is shown that the pattern of host feeding which maximizes parasitoid fitness is either non-concurrent and destructive, or concurrent and non-destructive host feeding, depending on the host abundance and parameters of the model. The effects of the adaptive choice of host-feeding patterns on host-parasitoid population dynamics are discussed.     

Influence of activity in a heterogeneous environment on the dynamics of fish growth: an individual-based model of roacl

A spatio-temporal individual-based model (IBM), including bioenergetic principles, is used to investigate how energy costs coupled to activity represent a variable fraction in a heterogeneous environment and how this is related to another energetic criterion, the potential individual growth, under the conditions of a structured environment. With this approach, it is possible to connect simulated spatio-temporal activity patterns with the energetic needs required for these activities. By using simple foraging rules and a modified random walk model it is possible to reproduce spatial distributions and length frequency distributions. The simulated spatial distribution of roach Rutilus rutilus in Lake Belau, Germany, results in a mean weekly activity multiplier [(standard metabolic rate + activity costs) (standard metabolic rate)⁻¹] of 1–9 with deviations >100% during a simulated year. These deviations are of key importance to differences in the growth rate of individual simulated roach.     

INDISIM-YEAST: an individual-based simulator on a website for experimenting and investigating diverse dynamics of yeast populations in liquid media

Simulation modelling can be used to capture and mimic real-world microbial systems that, unlike the real-world, can then be experimented upon as a new kind of experimental milieu. Individual-based models, in which individuals interact dynamically with each other as structural elements in the model world, exemplify this view of simulation modelling. These models are more difficult to analyze, understand and communicate than traditional analytical models. It is good practice to provide executable versions that perform simulation results. INDISIM-YEAST, developed to deal with yeast populations in liquid media, models the evolution of a set of yeasts by setting up rules of behavior for each individual cell according to its own biological regulations and characteristics. The aim of this work is to develop and present a website from which INDISIM-YEAST is accessible, and how to carry out yeast simulations to further the skills associated with the use of this individual-based simulator. A good and useful way to analyze this yeast simulator is to experiment and explore the manner in which it reacts to changes in parameter values, initial conditions or assumptions. The application results in a very versatile program that could be used in controlled simulation experiments via the Internet.     

Predicting shifts in dynamics of cannibalistic field populations using individual-based models

The occurrence of qualitative shifts in population dynamical regimes has long been the focus of population biologists. Nonlinear ecological models predict that these shifts in dynamical regimes may occur as a result of parameter shifts, but unambiguous empirical evidence is largely restricted to laboratory populations. We used an individual-based modelling approach to predict dynamical shifts in field fish populations where the capacity to cannibalize differed between species. Model-generated individual growth trajectories that reflect different population dynamics were confronted with empirically observed growth trajectories, showing that our ordering and quantitative estimates of the different cannibalistic species in terms of life-history characteristics led to correct qualitative predictions of their dynamics.     

Individual variability and population regulation: an individual-based model

To study the influence of individual variability on population dynamics an individual-based model of the dynamics of a single population consisting of different individuals is constructed. The model is based on differences in individual assimilation rates due to intraspecific competition and variability of initial weights. The model exhibits imperfect regulation, i.e., the number of individuals in the population oscillates and sooner or later the population becomes extinct. When individual variability is included, the model produces longer population extinction times than without individual variability. The average extinction time is not however a monotonic function of the degree of individual variability.     

Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications

A mathematical model of the spatio-temporal dynamics of a two host, two parasitoid system is presented. There is a coupling of the four species through parasitism of both hosts by one of the parasitoids. The model comprises a system of four reaction-diffusion equations. The underlying system of ordinary differential equations, modelling the host-parasitoid population dynamics, has a unique positive steady state and is shown to be capable of undergoing Hopf bifurcations, leading to limit cycle kinetics which give rise to oscillatory temporal dynamics. The stability of the positive steady state has a fundamental impact on the spatio-temporal dynamics: stable travelling waves of parasitoid invasion exhibit increasingly irregular periodic travelling wave behaviour when key parameter values are increased beyond their Hopf bifurcation point. These irregular periodic travelling waves give rise to heterogeneous spatio-temporal patterns of host and parasitoid abundance. The generation of heterogeneous patterns has ecological implications and the concepts of temporary host refuge and niche formation are considered.