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.     

Variation in plant quality and the population dynamics of herbivores: there is nothing average about aphids

In the attempt to use results from small-scale studies to make large-scale predictions, it is critical that we take into account the greater spatial heterogeneity encountered at larger spatial scales. An important component of this heterogeneity is variation in plant quality, which can have a profound influence on herbivore population dynamics. This influence is particularly relevant when we consider that the strength of density dependence can vary among host plants and that the strength of density dependence determines the difference between exponential and density- dependent growth. Here, we present some simple models and analyses designed to examine the impact of variable plant quality on the dynamics of insect herbivore populations, and specifically the consequences of variation in the strength of density dependence among host plants. We show that average values of herbivore population growth parameters, calculated from plants that vary in quality, do not predict overall population growth. Furthermore, we illustrate that the quality of a few individual plants within a larger plant population can dominate herbivore population growth. Our results demonstrate that ignoring spatial heterogeneity that exists in herbivore population growth on plants that differ in quality can lead to a misunderstanding of the mechanisms that underlie population dynamics.     

Refuge evolution and the population dynamics of coupled host—parasitoid associations

Summary We have investigated the theoretical consequences of character evolution for the population dynamics of a host—parasitoid interaction, assuming a monophagous parasitoid. In the purely ecological model it is assumed that hosts can escape parasitism by being in absolute refuges. A striking property of this model is a threshold effect in control of the host by the parasitoid, when host density dependence is weak. The approximate criteria for the parasitoid to regulate the host to low densities are (1) that the parasitoid's maximum population growth rate should exceed the host's and (2) that the maximum growth rate of the host in the refuge should be less than unity. We then use this ecological framework as a basis for a model which considers evolutionary changes in quantitative characters influencing the size of the absolute refuge. For each species, an increase in its refuge-determining character comes at a cost to maximum population growth rate. We show that refuge evolution can substantially alter the population dynamics of the purely ecological model, resulting in a number of emergent and sometimes counter-intuitive properties. In general, when the host has a high carrying capacity, systems are polarized either with low or minor refuge and top-down control of the host by the parasitoid or with a refuge and bottom-up control of the host by a combination of its own density dependence and the parasitoid. A particularly tantalizing result is that co-evolutionary dynamics can modify ecologically unstable systems into ones which are either stable or quasi-stable (with bouts of unstable dynamics, punctuating long-term periods of quasi-stable behaviour). We present five quantitative criteria which must all be met for the parasitoid to be the agent responsible for control of the host at a co-evolutionary equilibrium. The apparent stringency of this full set of requirements supports the empirically-based suggestion that monophagous parasitoid-driven systems should be less common in nature than those driven by multiple forms of density dependence. Further, we apply our theory to the question of whether exploiters may harvest their victims at maximum sustainable yields and to the evolutionary stability of biological control. Finally, we present a series of testable predictions of our theory and methods useful for testing them.     

Density-dependent vital rates and their population dynamic consequences

We explore a set of simple, nonlinear, two-stage models that allow us to compare the effects of density dependence on population dynamics among different kinds of life cycles. We characterize the behavior of these models in terms of their equilibria, bifurcations, and nonlinear dynamics, for a wide range of parameters. Our analyses lead to several generalizations about the effects of life history and density dependence on population dynamics. Among these are: (1) iteroparous life histories are more likely to be stable than semelparous life histories; (2) an increase in juvenile survivorship tends to be stabilizing; (3) density-dependent adult survival cannot control population growth when reproductive output is high; (4) density-dependent reproduction is more likely to cause chaotic dynamics than density dependence in other vital rates; and (5) changes in development rate have only small effects on bifurcation patterns. Received: 12 April 1999 / Published online: 3 August 2000     

Effects of Density Dependent Migrations on the Dynamics of a Predator Prey Model

We study the effects of density dependent migrations on the stability of a predator-prey model in a patchy environment which is composed with two sites connected by migration. The two patches are different. On the first patch, preys can find resource but can be captured by predators. The second patch is a refuge for the prey and thus predators do not have access to this patch. We assume a repulsive effect of predator on prey on the resource patch. Therefore, when the predator density is large on that patch, preys are more likely to leave it to return to the refuge. We consider two models. In the first model, preys leave the refuge to go to the resource patch at constant migration rates. In the second model, preys are assumed to be in competition for the resource and leave the refuge to the resource patch according to the prey density. We assume two different time scales, a fast time scale for migration and a slow time scale for population growth, mortality and predation. We take advantage of the two time scales to apply aggregation of variables methods and to obtain a reduced model governing the total prey and predator densities. In the case of the first model, we show that the repulsive effect of predator on prey has a stabilizing effect on the predator-prey community. In the case of the second model, we show that there exists a window for the prey proportion on the resource patch to ensure stability.     

Single‐ or multistage regulation in complex life cycles: does it make a difference?

Data on the different stages of complex life cycles are often rather unbalanced, especially those concerning the effects of density. How does this affect our understanding of a species’ population dynamics? Two discrete three‐stage models with overlapping generations and delayed maturation are constructed to address this question. They assume that survival or emigration in any life stage and/or reproduction can be density dependent. A typical pond‐breeding amphibian species with a well‐studied larval stage serves as an example. Numerical results show that the population dynamics resulting from density dependence at a single (e.g. the larval) stage can be decisively and unpredictably modified by density dependence in additional stages. Superposition of density‐dependent processes could thus be one reason for the difficulties in identifying density dependence in the field. Moreover, in a simulated source‐refuge system with habitat‐specific density‐dependent dispersal of juveniles density dependence in multiple stages can stabilize or destabilize the dynamics and produce misleading age structures. From an applied perspective this model shows that excluding multistage regulation prematurely clearly affects our ability to predict consequences of human impacts.     

Red environmental noise and the appearance of delayed density dependence in age-structured populations

Previous work suggests that red environmental noise can lead to the spurious appearance of delayed density dependence (DDD) in unstructured populations regulated only by direct density dependence. We analysed the effect of noise reddening on the pattern of spurious DDD in several variants of the density-dependent age-structured population model. We found patterns of spurious DDD in structured populations with either density-dependent fertility or density-dependent survival of the first age class, inconsistent with predictions from unstructured population models. Moreover, we found that nonspurious negative DDD always emerges in populations with deterministic chaotic dynamics, regardless of population structure or the type of environmental noise. The effect of noise reddening in generating spurious DDD is often negligible in the chaotic region of population deterministic dynamics. Our findings suggest that differences in species' life histories may exhibit different patterns of spurious DDD (owing to noise reddening) than predicted by unstructured models.     

Continuous-discrete models of the dynamics of an isolated population and of 2 competing species

The paper presents the analysis of various mathematical models for dynamics of isolated population and for competition between two species. It is assumed that mortality is continuous and birth of individuals of new generations takes place in certain fixed moments. Influence of winter upon the population dynamics and conditions of classic discrete model "deduction" of population dynamics (in particular, Moran-Ricker and Hassel's models) are investigated. Dynamic regimes of models under various assumptions about the birth and death rates upon the population states are also examined. Analysis of models of isolated population dynamics with nonoverlapping generations showed the density changes regularly if the birth rate is constant. Moreover, there exists a unique global stable level and population size stabilizes asymptotically at this equilibrium, i.e. cycle and chaotic regimes in various discrete models depend on correlation between individual productivity and population state in previous time. When the correlation is exponential upon mean population size the discrete Hassel model is realized. Modification of basis model, based on the assumption that during winter survival/death changes are constant, showed that population size at global level is stable. Generally, the dependence of population rate upon "winter parameters" has nonlinear character. Nonparametric models of competition between two species does not vary if the individual productivity is constant. In a phase space there are several stable stationary states and population stabilizes at one or other level asymptotically. So, in discrete models of competition between two species oscillation can be explained by dependence of population growth rate on the population size at previous times.     

Using genetic markers to estimate the pollen dispersal curve

Pollen dispersal is a critical process that shapes genetic diversity in natural populations of plants. Estimating the pollen dispersal curve can provide insight into the evolutionary dynamics of populations and is essential background for making predictions about changes induced by perturbations. Specifically, we would like to know whether the dispersal curve is exponential, thin-tailed (decreasing faster than exponential), or fat-tailed (decreasing slower than the exponential). In the latter case, rare events of long-distance dispersal will be much more likely. Here we generalize the previously developed TWOGENER method, assuming that the pollen dispersal curve belongs to particular one- or two-parameter families of dispersal curves and estimating simultaneously the parameters of the dispersal curve and the effective density of reproducing individuals in the population. We tested this method on simulated data, using an exponential power distribution, under thin-tailed, exponential and fat-tailed conditions. We find that even if our estimates show some bias and large mean squared error (MSE), we are able to estimate correctly the general trend of the curve - thin-tailed or fat-tailed - and the effective density. Moreover, the mean distance of dispersal can be correctly estimated with low bias and MSE, even if another family of dispersal curve is used for the estimation. Finally, we consider three case studies based on forest tree species. We find that dispersal is fat-tailed in all cases, and that the effective density estimated by our model is below the measured density in two of the cases. This latter result may reflect the difficulty of estimating two parameters, or it may be a biological consequence of variance in reproductive success of males in the population. Both the simulated and empirical findings demonstrate the strong potential of TWOGENER for evaluating the shape of the dispersal curve and the effective density of the population (d(e)).     

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.     

Population dynamics and the colour of environmental noise.

The effect of red, white and blue environmental noise on discrete-time population dynamics is analyzed. The coloured noise is superimposed on Moran-Ricker and Maynard Smith dynamics, the resulting power spectra are less than examined. Time series dominated by short- and long-term fluctuations are said to be blue and red, respectively. In the stable range of the Moran-Ricker dynamics, environmental noise of any colour will make population dynamics red or blue depending the intrinsic growth rate. Thus, telling apart the colour of the noise from the colour of the population dynamics may not be possible. Population dynamics subjected to red and blue environmental noises show, respectively, more red or blue power spectra than those subjected to white noise. The sensitivity to differences in the noise colours decreases with increasing complexity and ultimately disappears in the chaotic range of the population dynamics. These findings are duplicated with the Maynard Smith model for high growth rates when the strength of density dependence changes. However, for low growth rates the power spectra of the population dynamics with noise are red in stable, periodic and aperiodic ranges irrespective of the noise colour. Since chaotic population fluctuations may show blue spectra in the deterministic case, this implies that blue deterministic chaos may become red under any colour of the noise.     

Modelling density-dependent resistance in insect-pathogen interactions.

We consider a mathematical model for a host-pathogen interaction where the host population is split into two categories: those susceptible to disease and those resistant to disease. Since the model was motivated by studies on insect populations, we consider a discrete-time model to reflect the discrete generations which are common among insect species. Whether an individual is born susceptible or resistant to disease depends on the local population levels at the start of each generation. In particular, we are interested in the case where the fraction of resistant individuals in the population increases as the total population increases. This may be seen as a positive feedback mechanism since disease is the only population control imposed upon the system. Moreover, it reflects recent experimental observations from noctuid moth-baculovirus interactions that pathogen resistance may increase with larval density. We find that the inclusion of a resistant class can stabilise unstable host-pathogen interactions but there is greatest regulation when the fraction born resistant is density independent. Nonetheless, inclusion of density dependence can still allow intrinsically unstable host-pathogen dynamics to be stabilised provided that this effect is sufficiently small. Moreover, inclusion of density-dependent resistance to disease allows the system to give rise to bistable dynamics in which the final outcome is dictated by the initial conditions for the model system. This has implications for the management of agricultural pests using biocontrol agents-in particular, it is suggested that the propensity for density-dependent resistance be determined prior to such a biocontrol attempt in order to be sure that this will result in the prevention of pest outbreaks, rather than their facilitation. Finally we consider how the cost of resistance to disease affects model outcomes and discover that when there is no cost to resistance, the model predicts stable periodic outbreaks of the insect population. The results are interpreted ecologically and future avenues for research to address the shortfalls in the present model system are discussed.     

Population dynamics, life stage and ecological modeling in Chrysomya albiceps (Wiedemann) (Diptera: Calliphoridae)

In this study we investigated the population dynamics of Chrysomya albiceps (Wiedemann) with laboratory experiments, employing survival analysis and stage structure mathematical models, emphasizing survival among life stages. The study also assessed the theoretical influence of density dependence and cannibalism during immature stages, on the population dynamics of the species. The survival curves were similar, indicating that populations of C. albiceps exhibit the same pattern of survival among life stages. A strong nonlinear trend was observed, suggesting density dependence, acting during the first life stages of C. albiceps. The time-series simulations produced chaotic oscillations for all life stages, and the cannibalism did not produce qualitative changes in the dynamic behavior. The bifurcation analysis shows that for low values for survival, the population reaches a stable equilibrium, but the cannibalism results in chaotic oscillations practically over all the parametric space. The implications of the patterns of dynamic behavior observed are discussed.     

Density dependence in group dynamics of a highly social mongoose, Suricata suricatta

1.?For social species, the link between individual behaviour and population dynamics is mediated by group-level demography. 2.?Populations of obligate cooperative breeders are structured into social groups, which may be subject to inverse density dependence (Allee effects) that result from a dependence on conspecific helpers, but evidence for population-wide Allee effects is rare. 3.?We use field data from a long-term study of cooperative meerkats (Suricata suricatta; Schreber, 1776) - a species for which local Allee effects are not reflected in population-level dynamics - to empirically model interannual group dynamics. 4.?Using phenomenological population models, modified to incorporate environmental conditions and potential Allee effects, we first investigate overall patterns of group dynamics and find support only for conventional density dependence that increases after years of low rainfall. 5.?To explain the observed patterns, we examine specific demographic rates and assess their contributions to overall group dynamics. Although per-capita meerkat mortality is subject to a component Allee effect, it contributes relatively little to observed variation in group dynamics, and other (conventionally density dependent) demographic rates - especially emigration - govern group dynamics. 6.?Our findings highlight the need to consider demographic processes and density dependence in subpopulations before drawing conclusions about how behaviour affects population processes in socially complex systems.     

The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting

We analyze the effects of a strategy of constant effort harvesting in the global dynamics of a one-dimensional discrete population model that includes density-independent survivorship of adults and overcompensating density dependence. We discuss the phenomenon of bubbling (which indicates that harvesting can magnify fluctuations in population abundance) and the hydra effect, which means that the stock size gets larger as harvesting rate increases. Moreover, we show that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates.     

A model for resolving the plankton paradox: coexistence in open flows

1. Recent developments in the field of chaotic advection in hydrodynamical/environmental flows encourage us to revisit the population dynamics of competing species in open aquatic systems.
2. We assume that these species are in competition for a common limiting resource in open flows with chaotic advection dynamics. As an illustrative example, we consider a time periodic two-dimensional flow of viscous fluid (water) around a cylindrical obstacle.
3. Individuals accumulate along a fractal set in the wake of the cylinder, which acts as a catalyst for the biological reproduction process. While in homogeneous, well mixed environments only one species could survive this competition, coexistence of competitors is typical in our hydrodynamical system.
4. It is shown that a steady state sets in after sufficiently long times. In this state, the relative density of competitors is determined rather by the fractal nature of the spatial distribution of the advected species, and by their initial conditions, than by their competitive abilities. We argue that two factors, the strong chaotic mixing along a fractal set and the boundary layer around the obstacle, are responsible for the coexistence.     

Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect

One of the main challenges in ecology is to determine the cause of population fluctuations. Both theoretical and empirical studies suggest that delayed density dependence instigates cyclic behavior in many populations; however, underlying mechanisms through which this occurs are often difficult to determine and may vary within species. In this paper, we consider single species population dynamics affected by the Allee effect coupled with discrete time delay. We use two different mathematical formulations of the Allee effect and analyze (both analytically and numerically) the role of time delay in different feedback mechanisms such as competition and cooperation. The bifurcation value of the delay (that results in the Hopf bifurcation) as a function of the strength of the Allee effect is obtained analytically. Interestingly, depending on the chosen delayed mechanism, even a large time delay may not necessarily lead to instability. We also show that, in case the time delay affects positive feedback (such as cooperation), the population dynamics can lead to self-organized formation of intermediate quasi-stationary states. Finally, we discuss ecological implications of our findings.     

Neural coding for the retrieval of multiple memory patterns

We investigate the retrieval dynamics in a feature-based semantic memory model, in which the features are coded by neurons of the Hindmarsh-Rose type in the chaotic regime. We consider the retrieval process as consisting of the synchronized firing activity of the neurons coding for the same memory pattern. The retrieval dynamics is investigated for multiple patterns, with particular attention to the case of overlapping memories. In this case, we hypothesize a dynamical nontransitive mechanism based on synchronization, that allows for a shared feature to participate in multiple memory representations. The problem of the choice of a cognitive plausible time-scale for the retrieval analysis is investigated by analyzing the information that can be inferred from finite-time analyses. Different types of indicators are proposed in order to evaluate the temporal dynamics of the neurons engaged in the retrieval process. We interpret the simulation results as suggestive of a role for chaotic dynamics in allowing for flexible composition of elementary meaningful units in memory representations.     

Species synchrony and its drivers: neutral and nonneutral community dynamics in fluctuating environments

Independent species fluctuations are commonly used as a null hypothesis to test the role of competition and niche differences between species in community stability. This hypothesis, however, is unrealistic because it ignores the forces that contribute to synchronization of population dynamics. Here we present a mechanistic neutral model that describes the dynamics of a community of equivalent species under the joint influence of density dependence, environmental forcing, and demographic stochasticity. We also introduce a new standardized measure of species synchrony in multispecies communities. We show that the per capita population growth rates of equivalent species are strongly synchronized, especially when endogenous population dynamics are cyclic or chaotic, while their long-term fluctuations in population sizes are desynchronized by ecological drift. We then generalize our model to nonneutral dynamics by incorporating temporal and nontemporal forms of niche differentiation. Niche differentiation consistently decreases the synchrony of species per capita population growth rates, while its effects on the synchrony of population sizes are more complex. Comparing the observed synchrony of species per capita population growth rates with that predicted by the neutral model potentially provides a simple test of deterministic asynchrony in a community.     

Effects of density dependent sex allocation on the dynamics of a simultaneous hermaphroditic population: Modelling and analysis

In this work we present a mathematical model describing the dynamics of a population where sex allocation remains flexible throughout adult life and so can be adjusted to current environmental conditions. We consider that the fractions of immature individuals acquiring male and female sexual roles are density dependent through nonlinear functions of a weighted total population size. The main goal of this work is to understand the role of life-history parameters on the stabilization or destabilization of the population dynamics.The model turns out to be a nonlinear discrete model which is analysed by studying the existence of fixed points as well as their stability conditions in terms of model parameters. The existence of more complex asymptotic behaviours of system solutions is shown by means of numerical simulations.Females have larger fertility rate than males. On the other hand, increasing population density favours immature individuals adopting the male role. A positive equilibrium of the system exists whenever fertility and survival rates of one of the sexual roles, if shared by all adults, allow population growing while the opposite happens with the other sexual role. In terms of the female inherent net reproductive number, η_F, it is shown that the positive equilibria are stable when η_F is larger and closed to 1 while for larger values of η_F a certain asymptotic assumption on the investment rate in the female function implies that the population density is permanent. Depending on the other parameters values, the asymptotic behaviour of solutions becomes more complex, even chaotic. In this setting the stabilization/destabilization effects of the abruptness rate in density dependence, of the survival rates and of the competition coefficients are analysed.