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A general framework for threshold parameters in population dynamics is developed using the concept of target reproduction numbers. This framework identifies reproduction numbers and other threshold parameters in the literature in terms of their roles in population control. The framework is applied to the analysis of single and multiple control strategies in ecology and epidemiology, and this provides new biological insights.

A key problem in environmental flow assessment is the explicit linking of the flow regime with ecological dynamics. We present a hybrid modeling approach to couple hydrodynamic and biological processes, focusing on the combined impact of spatial heterogeneity and temporal variability on population dynamics. Studying periodically alternating pool-riffle rivers that are subjected to seasonally varying flows, we obtain an invasion ratchet mechanism. We analyze the ratchet process for a caricature model and a hybrid physical-biological model. The water depth and current are derived from a hydrodynamic equation for variable stream bed water flows and these quantities feed into a reaction-diffusion-advection model that governs population dynamics of a river species. We establish the existence of spreading speeds and the invasion ratchet phenomenon, using a mixture of mathematical approximations and numerical computations. Finally, we illustrate the invasion ratchet phenomenon in a spatially two-dimensional hydraulic simulation model of a meandering river structure. Our hybrid modeling approach strengthens the ecological component of stream hydraulics and allows us to gain a mechanistic understanding as to how flow patterns affect population survival.

How growth, mortality, and dispersal in a species affect the species' spread and persistence constitutes a central problem in spatial ecology. We propose impulsive reaction-diffusion equation models for species with distinct reproductive and dispersal stages. These models can describe a seasonal birth pulse plus nonlinear mortality and dispersal throughout the year. Alternatively, they can describe seasonal harvesting, plus nonlinear birth and mortality as well as dispersal throughout the year. The population dynamics in the seasonal pulse is described by a discrete map that gives the density of the population at the end of a pulse as a possibly nonmonotone function of the density of the population at the beginning of the pulse. The dynamics in the dispersal stage is governed by a nonlinear reaction-diffusion equation in a bounded or unbounded domain. We develop a spatially explicit theoretical framework that links species vital rates (mortality or fecundity) and dispersal characteristics with species' spreading speeds, traveling wave speeds, as well as minimal domain size for species persistence. We provide an explicit formula for the spreading speed in terms of model parameters, and show that the spreading speed can be characterized as the slowest speed of a class of traveling wave solutions. We also give an explicit formula for the minimal domain size using model parameters. Our results show how the diffusion coefficient, and the combination of discrete- and continuous-time growth and mortality determine the spread and persistence dynamics of the population in a wide variety of ecological scenarios. Numerical simulations are presented to demonstrate the theoretical results.

The drift paradox asks how stream-dwelling organisms can persist, without being washed out, when they are continuously subject to the unidirectional stream flow. To date, mathematical analyses of the stream paradox have investigated the interplay of growth, drift and flow needed for species persistence under the assumption that the stream environment is temporally constant. However, in reality, streams are subject to major seasonal variations in environmental factors that govern population growth and dispersal. We consider the influence of such seasonal variations on the drift paradox, using a time-periodic integrodifferential equation model. We establish upstream and downstream spreading speeds under the assumption of periodically fluctuating environments, and also show the existence of periodic traveling waves. The sign of the upstream spreading speed then determines persistence. Fluctuating environments are characterized by seasonal correlations between the flow, transfer rates, diffusion and settling rates, and we investigate the effect of such correlations on the population spread and persistence. We also show how results in this paper can formally connect to those for autonomous integrodifferential equations, through the appropriate weighted averaging methods. Finally, for a specific dispersal function, we show that the upstream spreading speed is nonnegative if and only if the critical domain size exists in this temporally fluctuating environment.

A class of integral recursion models for the growth and spread of a synchronized single-species population is studied. It is well known that if there is no overcompensation in the fecundity function, the recursion has an asymptotic spreading speed c*, and that this speed can be characterized as the speed of the slowest non-constant traveling wave solution. A class of integral recursions with overcompensation which still have asymptotic spreading speeds can be found by using the ideas introduced by Thieme (J Reine Angew Math 306:94-121, 1979) for the study of space-time integral equation models for epidemics. The present work gives a large subclass of these models with overcompensation for which the spreading speed can still be characterized as the slowest speed of a non-constant traveling wave. To illustrate our results, we numerically simulate a series of traveling waves. The simulations indicate that, depending on the properties of the fecundity function, the tails of the waves may approach the carrying capacity monotonically, may approach the carrying capacity in an oscillatory manner, or may oscillate continually about the carrying capacity, with its values bounded above and below by computable positive numbers.

It is well known that in many scalar models for the spread of a fitter phenotype or species into the territory of a less fit one, the asymptotic spreading speed can be characterized as the lowest speed of a suitable family of traveling waves of the model. Despite a general belief that multi-species (vector) models have the same property, we are unaware of any proof to support this belief. The present work establishes this result for a class of multi-species model of a kind studied by Lui [Biological growth and spread modeled by systems of recursions. I: Mathematical theory, Math. Biosci. 93 (1989) 269] and generalized by the authors [Weinberger et al., Analysis of the linear conjecture for spread in cooperative models, J. Math. Biol. 45 (2002) 183; Lewis et al., Spreading speeds and the linear conjecture for two-species competition models, J. Math. Biol. 45 (2002) 219]. Lui showed the existence of a single spreading speed c(*) for all species. For the systems in the two aforementioned studies by the authors, which include related continuous-time models such as reaction-diffusion systems, as well as some standard competition models, it sometimes happens that different species spread at different rates, so that there are a slowest speed c(*) and a fastest speed c(f)(*). It is shown here that, for a large class of such multi-species systems, the slowest spreading speed c(*) is always characterized as the slowest speed of a class of traveling wave solutions.

This paper is concerned with mathematical analysis of the 'critical domain-size' problem for structured populations. Space is introduced explicitly into matrix models for stage-structured populations. Movement of individuals is described by means of a dispersal kernel. The mathematical analysis investigates conditions for existence, stability and uniqueness of equilibrium solutions as well as some bifurcation behaviors. These mathematical results are linked to species persistence or extinction in connected habitats of different sizes or fragmented habitats; hence the framework is given for application of such models to ecology. Several approximations which reduce the complexity of integrodifference equations are given. A simple example is worked out to illustrate the analytical results and to compare the behavior of the integrodifference model to that of the approximations.

The discrete-time recursion system u_[n+1]=Q[u_n] with u_n(x) a vector of population distributions of species and Q an operator which models the growth, interaction, and migration of the species is considered. Previously known results are extended so that one can treat the local invasion of an equilibrium of cooperating species by a new species or mutant. It is found that, in general, the resulting change in the equilibrium density of each species spreads at its own asymptotic speed, with the speed of the invader the slowest of the speeds. Conditions on Q are given which insure that all species spread at the same asymptotic speed, and that this speed agrees with the more easily calculated speed of a linearized problem for the invader alone. If this is true we say that the recursion has a single speed and is linearly determinate. The conditions are such that they can be verified for a class of reaction-diffusion models.

The founding population in most new species introductions, or at the leading edge of an ongoing invasion, is likely to be small. Severe Allee effects-reductions in individual fitness at low population density-may then result in a failure of the species to colonize, even if the habitat could support a much larger population. Using a simulation model for plant populations that incorporates demography, mating systems, quantitative genetics, and pollinators, we show that Allee effects can potentially be overcome by transient hybridization with a resident species or an earlier colonizer. This mechanism does not require the invocation of adaptive changes usually attributed to invasions following hybridization. We verify our result in a case study of sequential invasions by two plant species where the outcrosser Cakile maritima has replaced an earlier, inbreeding, colonizer Cakile edentula (Brassicaceae). Observed historical rates of replacement are consistent with model predictions from hybrid-alleviated Allee effects in outcrossers, although other causes cannot be ruled out.

Migrations allow animals to track seasonal changes in resources, find mates, and avoid harsh climates, but these regular, longdistance movements also have implications for parasite dynamics and animal health. Migratory animals have been dubbed “superspreaders” of infection, but migration can also reduce parasite burdens within host populations via migratory escape from contaminated habitats and transmission hotspots, migratory recovery due to parasite mortality, and migratory culling of infected individuals. Here, we show that a single migratory host–macroparasite model can give rise to these different phenomena under different parametrizations, providing a unifying framework for a mechanistic understanding of the parasite dynamics of migratory animals. Importantly, our model includes the impact of parasite burden on host movement capability during migration, which can lead to “parasite-induced migratory stalling” due to a positive feedback between increasing parasite burdens and reduced movement. Our results provide general insight into the conditions leading to different health outcomes in migratory wildlife. Our approach lays the foundation for tactical models that can help understand, predict, and mitigate future changes of disease risk in migratory wildlife that may arise from shifting migratory patterns, loss of migratory behavior, or climate effects on parasite development, mortality, and transmission. [ABSTRACT FROM AUTHOR]

Mountain pine beetle (MPB) outbreaks have caused major economic losses and ecological consequences in North American pine forests. Ecological and environmental factors impacting MPB life-history and stands susceptibility can help with the detection of MPB infested trees and thereby, improve control. Temperatures, water stress, host characteristics, and beetle pressure are among those ecological and environmental factors. They play different roles on MPB population dynamics at the various stages of an outbreak and these roles can be affected by intensive management. However, to make detailed connections between ecological and environmental variables and MPB outbreak phases, a deeper quantitative analysis on local scales is needed. Here, we used logistic regressions on a highly-detailed and georeferenced data set to determine the factors driving MPB infestations for the different phases of the current isolated MPB outbreak in Cypress Hills. While we showed that the roles of ecological and environmental factors in a forest intensively controlled for MPB are consistent with the literature for uncontrolled forests, we determined how these factors shifted through onset, peak, and collapse phases of the intensively controlled forest. MPB presence mostly depends on nearby beetle pressure, notably for the outbreak peak. However additional weather and host variables are necessary to achieve high predictive ability for MPB outbreak locations. Our results can help managers make appropriate decisions on where and how to focus their effort, depending on which phase the outbreak is in. [ABSTRACT FROM AUTHOR]

We study the evolutionary stability of dispersal strategies, including but not limited to those that can produce ideal free population distributions (that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium). The environment is assumed to be variable in space but constant in time. We assume that there is a separation of times scales, so that dispersal occurs on a fast timescale, evolution occurs on a slow timescale, and population dynamics and interactions occur on an intermediate timescale. Starting with advection–diffusion models for dispersal without population dynamics, we use the large time limits of profiles for population distributions together with the distribution of resources in the environment to calculate growth and interaction coefficients in logistic and Lotka–Volterra ordinary differential equations describing population dynamics. We then use a pairwise invasibility analysis approach motivated by adaptive dynamics to study the evolutionary and/or convergence stability of strategies determined by various assumptions about the advection and diffusion terms in the original advection–diffusion dispersal models. Among other results we find that those strategies which can produce an ideal free distribution are evolutionarily stable. [ABSTRACT FROM AUTHOR]

Tree-killing bark beetles are major disturbance agents affecting coniferous forest ecosystems. The role of environmental conditions on driving beetle outbreaks is becoming increasingly important as global climatic change alters environmental factors, such as drought stress, that, in turn, govern tree resistance. Furthermore, dynamics between beetles and trees are highly nonlinear, due to complex aggregation behaviors exhibited by beetles attacking trees. Models have a role to play in helping unravel the effects of variable tree resistance and beetle aggregation on bark beetle outbreaks. In this article we develop a new mathematical model for bark beetle outbreaks using an analogy with epidemiological models. Because the model operates on several distinct time scales, singular perturbation methods are used to simplify the model. The result is a dynamical system that tracks populations of uninfested and infested trees. A limiting case of the model is a discontinuous function of state variables, leading to solutions in the Filippov sense. The model assumes an extensive seed-bank so that tree recruitment is possible even if trees go extinct. Two scenarios are considered for immigration of new beetles. The first is a single tree stand with beetles immigrating from outside while the second considers two forest stands with beetle dispersal between them. For the seed-bank driven recruitment rate, when beetle immigration is low, the forest stand recovers to a beetle-free state. At high beetle immigration rates beetle populations approach an endemic equilibrium state. At intermediate immigration rates, the model predicts bistability as the forest can be in either of the two equilibrium states: a healthy forest, or a forest with an endemic beetle population. The model bistability leads to hysteresis. Interactions between two stands show how a less resistant stand of trees may provide an initial toe-hold for the invasion, which later leads to a regional beetle outbreak in the resistant stand. [ABSTRACT FROM AUTHOR]

Allee effects are an important component in the population dynamics of numerous species. Accounting for these Allee effects in population viability analyses generally requires estimates of low-density population growth rates, but such data are unavailable for most species and particularly difficult to obtain for large mammals. Here, we present a mechanistic modeling framework that allows estimating the expected low-density growth rates under a mate-finding Allee effect before the Allee effect occurs or can be observed. The approach relies on representing the mechanisms causing the Allee effect in a process-based model, which can be parameterized and validated from data on the mechanisms rather than data on population growth. We illustrate the approach using polar bears (Ursus maritimus), and estimate their expected low-density growth by linking a mating dynamics model to a matrix projection model. The Allee threshold, defined as the population density below which growth becomes negative, is shown to depend on age-structure, sex ratio, and the life history parameters determining reproduction and survival. The Allee threshold is thus both density- and frequency-dependent. Sensitivity analyses of the Allee threshold show that different combinations of the parameters determining reproduction and survival can lead to differing Allee thresholds, even if these differing combinations imply the same stable-stage population growth rate. The approach further shows how mate-limitation can induce long transient dynamics, even in populations that eventually grow to carrying capacity. Applying the models to the overharvested low-density polar bear population of Viscount Melville Sound, Canada, shows that a mate-finding Allee effect is a plausible mechanism for slow recovery of this population. Our approach is generalizable to any mating system and life cycle, and could aid proactive management and conservation strategies, for example, by providing a priori estimates of minimum conservation targets for rare species or minimum eradication targets for pests and invasive species. [ABSTRACT FROM AUTHOR]

Abstract: Opioid receptor agonists modulate both innate and adaptive immune responses. In this study, we examined the impact of long-term chronic morphine administration on the circulating T cell population dynamics in rhesus macaques. We found that the numbers of circulating Treg cells, and the functional activity of Th17 cells, were significantly increased with chronic morphine exposure. Our results also show that T cell populations with surface markers characteristic of gut-homing (CD161 and CCR6) and HIV-1 susceptibility (CCR5 and β7 integrin) were increased. These results represent the first detailed report of the impact of chronic morphine administration on circulating T cell dynamics. [Copyright &y& Elsevier]

The efficacy of direct control methods in bark beetle outbreaks is a disputed topic. While some studies report that control reduces tree mortality, others see little effect. Existing models, linking control rate to beetle population dynamics and tree infestations, give insights, but there is a need to take into account the environment spatial variability and its impact on beetle life cycle. Here, we use natural variability found in a carefully monitored and controlled infestation to simulate outbreak dynamics under different control effort and to explore the impact of control on outbreaks suppression and tree mortality. Our semi-empirical predictive model of the number of infested trees as a function of ecological and environmental variables is coupled to a simulation model for infestation dynamics. We show that even a little control can have a major impact on the number of infested trees after several years of sustained effort. However, a moderate control of 60% is required to reduce the beetle population on the long term. Furthermore, a control rate of 69%–83% is needed to achieve outbreak suppression in under 13 years depending on the abundance of incoming flights from outside sources. • Direct control impacts outbreak duration and host mortality even at a low level. • Moderate control is required to reduce pest population on the long-term. • A significant control rate is needed to achieve quick outbreak eradication. [ABSTRACT FROM AUTHOR]

Trade-offs between dispersal and reproduction are known to be important drivers of population dynamics, but their direct influence on the spreading speed of a population is not well understood. Using integrodifference equations, we develop a model that incorporates a dispersal–reproduction trade-off which allows for a variety of different shaped trade-off curves. We show there is a unique reproductive-dispersal allocation that gives the largest value for the spreading speed and calculate the sensitivities of the reproduction, dispersal, and trade-off shape parameters. Uncertainty in the model parameters affects the expected spread of the population and we calculate the optimal allocation of resources to dispersal that maximizes the expected spreading speed. Higher allocation to dispersal arises from uncertainty in the reproduction parameter or the shape of the reproduction trade-off curve. Lower allocation to dispersal arises from uncertainty in the shape of the dispersal trade-off curve, but does not come from uncertainty in the dispersal parameter. Our findings give insight into how parameter sensitivity and uncertainty influence the spreading speed of a population with a dispersal–reproduction trade-off. [ABSTRACT FROM AUTHOR]

Highlights • Functional responses from predator prey theory can be used for pest management. • Managers, surveying and removing pests, take the role of predators foraging for pests. • Survey costs affect the slope of the functional response curve. • Removal costs cause the curve to saturate. • Type II and III responses can arise, depending on the search strategies employed. Abstract In this study, we explore how the functional response framework can be implemented in pest management. Here, managers take the role of predators foraging on pests and facing monetary costs for survey and control in a spatial domain where the pest distribution and control strategy do not have to be random. To investigate this framework quantitatively, we simulated various management processes on different pest spatial distributions using a spatially-explicit individual-based model and Monte-Carlo simulations, and also confirmed some of the results analytically. By graphing the number of pests controlled versus pest density, we obtained management functional response curves. Whether the management functional response was shaped like a type I, type II or type III functional response depended on the management costs and the search area. However, the management spatial strategy and the pest spatial distribution had little effect on the functional response. We applied our model to the management of mountain pine beetle epidemic in Cypress Hills, Saskatchewan, Canada, with simulations matching the real number of attacked trees controlled by managers. We showed how to make an analogy between functional responses in predator–prey interactions and in human–pest interactions and thereby, apply insights from the functional response framework to pest management. [ABSTRACT FROM AUTHOR]

Predators and prey may be simultaneously exposed to environmental toxins, but one may be more susceptible than the other. To study the effects of environmental toxins on food web dynamics, we develop a toxin-dependent predator–prey model that combines both direct and indirect toxic effects on two trophic levels. The direct effects of toxins typically reduce organism abundance by increasing mortality or reducing fecundity. Such direct effects, therefore, alter both bottom-up food availability and top-down predatory ability. However, the indirect effects, when mediated through predator–prey interactions, may lead to counterintuitive effects. This study investigates how the balance of the classical predator–prey dynamics changes as a function of environmental toxin levels. While high toxin concentrations are shown to be harmful to both species, possibly leading to extirpation of both species, intermediate toxin concentrations may affect predators disproportionately through biomagnification, leading to reduced abundance of predators and increased abundance of the prey. This counterintuitive effect significantly increases biomass at the lower trophic level. Environmental toxins may also reduce population variability by preventing populations from fluctuating around a coexistence equilibrium. Finally, environmental toxins may induce bistable dynamics, in which different initial population levels produce different long-term outcomes. Since our toxin-dependent predator–prey model is general, the theory developed here not only provides a sound foundation for population or community effects of toxicity, but also could be used to help develop management strategies to preserve and restore the integrity of contaminated habitats. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the effect of parasitoid phenology on host–parasitoid population cycles. Recent experimental research has shown that parasitized hosts can continue to interact with their unparasitized counterparts through competition. Parasitoid phenology, in particular the timing of emergence from the host, determines the duration of this competition. We construct a discrete-time host–parasitoid model in which within-generation dynamics associated with parasitoid timing is explicitly incorporated. We found that late-emerging parasitoids induce less severe, but more frequent, host outbreaks, independent of the choice of competition model. The competition experienced by the parasitized host reduces the parasitoids’ numerical response to changes in host numbers, preventing the ‘boom-bust’ dynamics associated with more efficient parasitoids. We tested our findings against experimental data for the forest tent caterpillar (Malacosoma disstria Hübner) system, where a large number of consecutive years at a high host density is synonymous with severe forest damage. [Copyright &y& Elsevier]

Abstract: Mathematical models have been widely applied to perform chemical risk assessments on biological populations for a variety of ecotoxicological processes. In this paper, by introducing a dose-dependent mortality rate function, we formulate a toxin-dependent aquatic population model that integrates mortality as toxin effect in addition to considering the effects of toxin on growth and recruitment. The model describes the direct effect of toxin on population by treating the concentration of toxin in the environment as a parameter. The model is more convenient to connect with data than traditional differential equation models that describe the interaction between toxin and population. We analyze the positive invariant region and the stability of boundary and interior steady states. The model is connected to experimental data via model parametrization. In particular, we consider the toxic effects of mercury on rainbow trout (Oncorhynchus mykiss) and obtain an appropriate range for each model parameter. The parameter estimates are then used to illustrate the long-time behavior of the population under investigation. The numerical results provide threshold values of toxin concentration in the environment to keep the population from extirpation. The findings are consistent with surface water quality guidelines. It may be appropriate to apply our model to other species and other chemicals of interest to consider guideline development. [Copyright &y& Elsevier]