Form Of Density Dependence Research Articles

Dynamic complexities in a mutual interference host–parasitoid model

The effect of parasitoid interference on a host–parasitoid model is investigated qualitatively. Mutual interference of parasitoids is a form of density dependence that is generally thought to have a stabilizing effect. Here we show that mutual interference not only can stabilize the dynamics but may strongly destabilize as well. Many forms of complex dynamics are observed, including pitchfork bifurcation with period-doubling cascade, attractor crises, chaotic bands with narrow or wide periodic windows, multiple attractors with fractal basins of attraction, intermittent chaos, and supertransient behavior. All these phenomena should be kept in mind when examining and interpreting the dynamics of populations.

Assessing uncertainty in ecological systems using global sensitivity analyses: a case example of simulated wolf reintroduction effects on elk

Often landmark conservation decisions are made despite an incomplete knowledge of system behavior and inexact predictions of how complex ecosystems will respond to management actions. For example, predicting the feasibility and likely effects of restoring top-level carnivores such as the gray wolf ( Canis lupus) to North American wilderness areas is hampered by incomplete knowledge of the predator-prey system processes and properties. In such cases, global sensitivity measures, such as Sobol’ indices, allow one to quantify the effect of these uncertainties on model predictions. Sobol’ indices are calculated by decomposing the variance in model predictions (due to parameter uncertainty) into main effects of model parameters and their higher order interactions. Model parameters with large sensitivity indices can then be identified for further study in order to improve predictive capabilities. Here, we illustrate the use of Sobol’ sensitivity indices to examine the effect of parameter uncertainty on the predicted decline of elk ( Cervus elaphus) population sizes following a hypothetical reintroduction of wolves to Olympic National Park, Washington, USA. The strength of density dependence acting on survival of adult elk and magnitude of predation were the most influential factors controlling elk population size following a simulated wolf reintroduction. In particular, the form of density dependence in natural survival rates and the per-capita predation rate together accounted for over 90% of variation in simulated elk population trends. Additional research on wolf predation rates on elk and natural compensations in prey populations is needed to reliably predict the outcome of predator–prey system behavior following wolf reintroductions.

Stable coexistence of an invasive plant and biocontrol agent: a parameterized coupled plant–herbivore model

Summary Coupled plant–herbivore models, allowing feedback from plant to herbivore populations and vice versa, enable us to predict the impact of biocontrol agents on their target weed populations; however, they are rarely used in biocontrol studies. We describe the population biology of the invasive plant Echium plantagineum and the weevil Mogulones larvatus, a biocontrol agent, in Australia. In order to understand the dynamics of this plant–herbivore system, a series of coupled models of increasing complexity was developed. A simple model was extended to include a seed bank, density‐dependent plant fecundity, competition between weevil larvae and plant tolerance of herbivory, where below a threshold plants could compensate for larval feeding. Parameters and functional forms were estimated from experimental and field data. The plant model, in the absence of the weevil, exhibited stable dynamics and provided a good quantitative description of field densities before the weevil was introduced. In the coupled plant–herbivore model, density dependence in both plant fecundity and weevil larval competition stabilized the dynamics. Without larval competition the model was unstable, and plant tolerance of herbivory exacerbated this instability. This was a result of a time delay in plant response to herbivore densities. Synthesis and applications. The coupled plant–herbivore model allowed us to predict whether stable coexistence of target plant and biocontrol agents was achievable at an acceptable level. We found this to be the case for the Echium–Mogulones system and believe that similar models would be of use when assessing new agents in this and other invasive plant biocontrol systems. Density dependence in new biocontrol agents should be assessed in order to determine whether it is likely to result in the aims of classical biocontrol: low, stable and sustainable populations of plant and herbivore. Further work should be done to characterize the strength of density dependence according to the niche occupied by the biocontrol agent, for example the strength and functional form of density dependence in stem borers may be quite different to that of defoliators.

Spatially explicit analyses unveil density dependence

Density-dependent processes are fundamental in the understanding of species population dynamics. Whereas the benefits of considering the spatial dimension in population biology are widely acknowledged, the implications of doing so for the statistical detection of spatial density dependence have not been examined. The outcome of traditional tests may therefore differ from those that include ecologically relevant locational information on both the prey species and natural enemy. Here, we explicitly incorporate spatial information on individual counts when testing for density dependence between an insect herbivore and its parasitoids. The spatially explicit approach used identified significant density dependence more frequently and in different instances than traditional methods. The form of density dependence detected also differed between methods. These results demonstrate that the explicit consideration of patch location in density-dependence analyses is likely to significantly alter current understanding of the prevalence and form of spatial density dependence in natural populations.

Dynamic complexities in a mutual interference host–parasitoid model

The effect of parasitoid interference on a host–parasitoid model is investigated qualitatively. Mutual interference of parasitoids is a form of density dependence that is generally thought to have a stabilizing effect. Here we show that mutual interference not only can stabilize the dynamics but may strongly destabilize as well. Many forms of complex dynamics are observed, including pitchfork bifurcation with period-doubling cascade, attractor crises, chaotic bands with narrow or wide periodic windows, multiple attractors with fractal basins of attraction, intermittent chaos, and supertransient behavior. All these phenomena should be kept in mind when examining and interpreting the dynamics of populations.

Sensitivity analysis of equilibrium in density‐dependent matrix population models

AbstractWe consider the effects of parameter perturbations on a density‐dependent population at equilibrium. Such perturbations change the dominant eigenvalue λ of the projection matrix evaluated at the equilibrium as well as the equilibrium itself. We show that, regardless of the functional form of density dependence, the sensitivity of λ is equal to the sensitivity of an effective equilibrium density , which is a weighted combination of the equilibrium stage densities. The weights measure the contributions of each stage to density dependence and their effects on demography. Thus, is in general more relevant than total density, which simply adds all stages regardless of their ecological properties. As log λ is the invasion exponent, our results show that successful invasion will increase , and that an evolutionary stable strategy will maximize . Our results imply that eigenvalue sensitivity analysis of a population projection matrix that is evaluated near equilibrium can give useful information about the sensitivity of the equilibrium population, even if no data on density dependence are available.

EFFICACY OF SIMPLE VIABILITY MODELS IN ECOLOGICAL RISK ASSESSMENT: DOES DENSITY DEPENDENCE MATTER?

One commonly used PVA (population viability analysis) approach applies a diffusion approximation (DA) of population growth to time series of abundance data to estimate population parameters and various metrics of extinction risk. The simplest versions of this PVA assume density-independent population growth, an assumption that is commonly called into question for populations experiencing self-limitation. Using time series data generated from simulations of populations limited by three commonly used forms of density dependence (ceiling, Beverton-Holt, and Ricker) we asked the question: “When do simple density-independent PVA models provide useful guidelines for prioritizing extinction risk despite density-dependence inherent in the underlying real populations?” Simple DA methods severely underestimated maximum growth rates (μmax) used to generate time series data for all three forms of density dependence. These methods also underestimated the intrinsic environmental variability in growth rates, or process error (σ2), for the ceiling model, but overestimated this parameter for the Beverton-Holt and Ricker models. Despite misestimation of the intrinsic parameters, the estimated probabilities of 50% and 75% declines were highly correlated with the observed probabilities for populations growing with a ceiling (coefficients of correlation, or R2 = 0.87–0.93). DA methods were less accurate for populations exhibiting more complex forms of density dependence (R2 = 0.61–0.79). Although correlations between observed and estimated risks were high, bias (e.g., over- and underestimation) was extensive. Estimated probabilities of 50% declines were typically much lower (overly optimistic) than observed probabilities of the same decline. By contrast, accuracy increased substantially for predictions of 75% decline, and the “optimistic” bias was replaced by conservative bias (overestimates of risk). Regardless of the form of density dependence, estimates of risk were least accurate when populations were recovering rapidly but were much more accurate when most needed by conservation practitioners: when the population fluctuated near its carrying capacity, recovered slowly to this abundance level, or declined toward extinction. Finally, when we classified risk in broad categories (e.g., extremely low, low, moderate, high, and extremely high), DA methods correctly or conservatively estimated the risk of a 75% decline for >85% of the parameter combinations, regardless of the form of density dependence followed by the real population.

The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.

Study of static and thermal properties of symmetric and asymmetric nuclear matter using generalized density-dependent M3Y interaction

By using the most recent generalized form of the density-dependent nucleon-nucleon DDM3Y interaction, namely, CDM3Yn-Paris interaction, the basic static properties of symmetric and asymmetric nuclear matter such as binding energy per particle, pressure, velocity of sound, and compressibility are calculated. Also, at finite temperature, the thermal properties of nuclear matter are studied, such as the free energy, the pressure, the entropy, and the compressibility. In addition, a comparison using different density-dependent M3Y-Paris interaction (DDM3Y1 and BDM3Y1) is considered. The importance of using the density-dependent term in the M3Y-Paris interaction is to fulfill the saturation requirement for the nuclear matter because M3Y-Paris interaction has an attractive character. Thus, the nuclear matter generated with this interaction is unstable against collapse. This new version of the DDM3Y is the general one, and other previous density-dependent forms can be considered as a special case of this one. Therefore, all the explicit theoretical developments are based on the density-dependent CDM3Yn version. The results obtained are in good agreement with previous theoretical estimates.

Further explorations of Skyrme–Hartree–Fock–Bogoliubov mass formulas: I: Role of density dependence in pairing force

The HFB-2 mass formula is generalized to make the δ-function pairing force density-dependent. It is shown that the mass data rule out the simple model of a pairing force that vanishes completely in the nuclear interior. Consistency with the mass data is found for a fairly wide range of δ-function pairing forces with a partial weakening in the nuclear interior. In particular, the form of density dependence determined by the realistic nuclear-matter calculations of Garrido et al. is shown to be compatible with the mass data, 2135 measured masses being fitted with an rms error of 0.656 MeV. On this basis we construct a new mass table, HFB-3, running from one drip line to the other. Shell quenching at the neutron-drip line is now somewhat stronger than before, but otherwise the new mass formula does not differ in any conspicuous way from the HFB-2 mass formula.

POPULATION MODELS FOR MARINE RESERVE DESIGN: A RETROSPECTIVE AND PROSPECTIVE SYNTHESIS

We synthesize results from existing models of marine reserves to identify key theoretical issues that appear to be well understood, as well as issues in need of further exploration. Models of marine reserves are relatively new in the scientific literature; 32 of the 34 theoretical papers we reviewed were published after 1990. These models have focused primarily on questions concerning fishery management at the expense of other objectives such as conservation, scientific understanding, recreation, education, and tourism. Roughly one-third of the models analyze effects on cohorts while the remaining models have some form of complete population dynamics. Few models explicitly include larval dispersal. In a fisheries context, the primary conclusion drawn by many of the complete population models is that reserves increase yield when populations would otherwise be overfished. A second conclusion, resulting primarily from single-cohort models, is that reserves will provide fewer benefits for species with greater adult rates of movement. Although some models are beginning to yield information on the spatial configurations of reserves required for populations with specific dispersal distances to persist, it remains an aspect of reserve design in need of further analysis. Other outstanding issues include the effects of (1) particular forms of density dependence, (2) multispecies interactions, (3) fisher behavior, and (4) effects of concentrated fishing on habitat. Model results indicate that marine reserves could play a beneficial role in the protection of marine systems against overfishing. Additional modeling and analysis will greatly improve prospects for a better understanding of the potential of marine reserves for conserving biodiversity.

The Impact of Resource Limitation and the Phenology of Parasitoid Attack on the Duration of Insect Herbivore Outbreaks

Fluctuations in resource quality and quantity, and changes in mortality due to predators and parasites are thought to be of prime importance in the regular fluctuations of forest insects. We examine how food limitation and parasitoids with different phenologies of attack regulate the population cycles of insect hosts. Our analysis of the limit cycle of a model with a biologically realistic form of density dependence in the host yields two novel predictions. First, outbreaks will typically last for only 2 generations after parasitoids begin to reduce the host population below the maximum density. Second, host growth rate is important in determining cycle length only when parasitoids attack before the impacts of resource limitation affect the host. The robustness of these predictions are tested using a more general form of density dependence in the host, revealing that our predictions are valid as long as density dependence in the host is not too overcompensatory.

Pattern of variation in avian population growth rates.

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.

Population growth rate and its determinants: an overview.

We argue that population growth rate is the key unifying variable linking the various facets of population ecology. The importance of population growth rate lies partly in its central role in forecasting future population trends; indeed if the form of density dependence were constant and known, then the future population dynamics could to some degree be predicted. We argue that population growth rate is also central to our understanding of environmental stress: environmental stressors should be defined as factors which when first applied to a population reduce population growth rate. The joint action of such stressors determines an organism's ecological niche, which should be defined as the set of environmental conditions where population growth rate is greater than zero (where population growth rate = r = log(e)(N(t+1)/N(t))). While environmental stressors have negative effects on population growth rate, the same is true of population density, the case of negative linear effects corresponding to the well-known logistic equation. Following Sinclair, we recognize population regulation as occurring when population growth rate is negatively density dependent. Surprisingly, given its fundamental importance in population ecology, only 25 studies were discovered in the literature in which population growth rate has been plotted against population density. In 12 of these the effects of density were linear; in all but two of the remainder the relationship was concave viewed from above. Alternative approaches to establishing the determinants of population growth rate are reviewed, paying special attention to the demographic and mechanistic approaches. The effects of population density on population growth rate may act through their effects on food availability and associated effects on somatic growth, fecundity and survival, according to a 'numerical response', the evidence for which is briefly reviewed. Alternatively, there may be effects on population growth rate of population density in addition to those that arise through the partitioning of food between competitors; this is 'interference competition'. The distinction is illustrated using a replicated laboratory experiment on a marine copepod, Tisbe battagliae. Application of these approaches in conservation biology, ecotoxicology and human demography is briefly considered. We conclude that population regulation, density dependence, resource and interference competition, the effects of environmental stress and the form of the ecological niche, are all best defined and analysed in terms of population growth rate.

Population dynamics with a refuge: fractal basins and the suppression of chaos.

We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics.

RECRUITMENT LIMITATION, POPULATION REGULATION, AND LARVAL CONNECTIVITY IN REEF FISH METAPOPULATIONS

Two central debates in marine ecology concern the role of connectivity patterns via larval dispersal in structuring marine metapopulations and the relative importance of larval supply vs. events occurring during or after settlement in determining adult abundance. Both issues were examined using age-structured models and simulations of reef fish population dynamics at the regional scale of closed metapopulations and the local scale of individual reefs. Local populations on individual reefs were assumed to be at once both partially open and partially closed. Two sets of models over both spatial scales are presented. One set examines density-independent dynamics. In the other set, mortality in the first year depends on the density of the settling cohort, and the density dependence is compensatory. The sensitivities of local population dynamics to the rates of self-recruitment and external larval supply were predicted. If external larval supply is regular and there is little self-recruitment, then a local population can appear to be regulated without conventional forms of density dependence. Elsewhere, this process has been termed “recruitment regulation.” However, with increased self-recruitment a local population will grow without bound in the absence of regulatory density dependence. Metapopulation persistence requires a sufficiently strong linkage between stock and recruitment in some local population. In such a local population, individuals must contribute sufficiently many offspring to replace themselves locally in subsequent generations. Such a local population could either be partially closed or lie in a region of the metapopulation that is strongly interconnected. Metapopulation regulation requires that density dependence acts to curb population growth in local source populations but does not require that it acts in local sink populations. The density dependence need not be so strong as to prevent subsequent cohort sizes from correlating well with varying recruitment levels. Recruitment regulation alone cannot regulate a metapopulation. Two versions of the recruitment limitation hypothesis were examined. The first version states that varying recruitment levels are good predictors of subsequent population size; the second version states that postrecruitment demographic rates are density independent. The present models, along with a growing body of empirical data, support the first hypothesis but provide less support for the second one. Population regulation and the first form of recruitment limitation are not antithetical processes.

The Northern Yellowstone Elk: Density Dependence and Climatic Conditions

We analyzed a time series of estimates of elk (Cervus elaphus) numbers on the northern Yellowstone winter range from 1964 to 1979 and 1986 to 1995 using a variety of discrete time stochastic population dynamic models. These models included adjustments for density, an increase in the area of winter range used by elk, lagged effects of the weather covariates of spring precipitation, snow depth and winter temperature, and the impacts of the 1988 drought and fires. An information-criteria-based model-selection process strongly supported evidence of density dependence. The best model, a Ricker model, distinguished between the 2 time periods. The bulk of the difference between the 2 periods is attributed to an increase in the amount of winter range used by elk. Inclusion of the covariates spring precipitation and spring precipitation squared greatly improved the model fit. We detected a short-lived increase in elk population growth rate following the 1988 drought and fires. Fertility and survivorship of adults appeared to have different density-dependent forms that together result in a biphasic relationship between population growth rate and density. This study confirms the presence of density-dependent regulation in the northern Yellowstone elk herd, and enhances our understanding of population dynamics of these ungulates.

Effects of Releasing Maladapted Individuals: A Demographic‐Evolutionary Model

A model of the joint dynamics of change in population size N and evolution in a quantitative trait z, as a result of a general form of density dependence, local stabilizing selection, and immigration of individuals deviating from the local optimum, is analyzed. For weak selection and migration, a reduction in total equilibrium population size below the initial level without immigration, K, is shown to occur if the immigrants deviates more than square root of 8 = 2.83 genetic standard deviations from the optimum and if the rate of migration m is sufficiently large relative to the strength of stabilizing selection s. For the Lotka-Volterra form of density dependence, two additional equilibria are shown to exist below K, provided that the strength of selection is large relative to the strength of density dependence. Reintroduction of an initially extinct population is possible if the immigrants are not too maladapted and if the genetic variance is sufficiently large. For a simplified version of the model corresponding to competition between similar species or different haplotypes, the equilibrium population size is always exactly at K if m < Ksz1(2) and is above K otherwise, which shows the importance of including recombination in the model.

Characterizing population vulnerability for 758 species

We investigate relationships between life history traits and the character of population dynamics as revealed by time series data. Our classification of time series is according to ‘extinction category,’ where we identify three classes of populations: (i) weakly varying populations with such high growth rates that long‐term persistence is likely (unless some extreme catastrophe occurs); (ii) populations with such low growth rates that average population size must be large to buffer them against extinction in a variable environment; and (iii) highly variable populations that fluctuate so dramatically that dispersal or some other refuge mechanism is likely to be key to their avoidance of extinction. Using 1941 time series representing 758 species from the Global Population Dynamics Database, we find that, depending on the form of density dependence one assumes, between 46 and 90% of species exhibit dynamics that are so variable that even large carrying capacities could not buffer them against extinction on a 100‐year time horizon. The fact that such a large proportion of population dynamics are so locally variable vindicates the growing realization that dispersal, habitat connectedness, and large‐scale processes are key to local persistence. Furthermore, for mammals, simply by knowing body size, age at first reproduction, and average number of offspring we could correctly predict extinction categories for 83% of species (60 of 72).

FOOD AS A MECHANISM OF DENSITY-DEPENDENT REGULATION OF BREEDING NUMBERS IN THE MERLIN,FALCO COLUMBARIUS

Population regulation of Merlins (Falco columbarius) was studied during 16 years in a small area supporting up to 11 breeding pairs, and during seven years in an expanded area where breeding numbers varied between 28 and 38 pairs. During the study period, the previously regular 4-yr cyclicity in rodent abundance disappeared, for unknown reasons. The functional form of density dependence agreed with a version of the Ricker equation, which showed that there was direct density dependence in the number of Merlin pairs attempting to breed. Moreover, in seasons with high small-rodent density, local population density of Merlins increased, and breeding territories were established in areas that, in other years, did not sustain breeders. Overwinter loss of males was directly related to breeding population size in the preceding year, as well as to small-rodent density. In contrast, overwinter loss of females appeared to be independent of breeding population size and food abundance in the preceding year.