The interplay between multiple control mechanisms in a host-parasitoid system: a discrete-time stage-structured modelling approach.

We propose a discrete-time host-parasitoid model with stage structure in both species. For this model, we establish conditions for the existence and global stability of the extinction and parasitoid-free equilibria as well as conditions for the existence and local stability of an interior equilibrium and system persistence. We study the model numerically to examine how pesticide spraying may interact with natural enemies (parasitoids) to control the pest (host) species. We then extend the model to an impulsive difference system that incorporates both periodic pesticide spraying and augmentation of the natural enemies to suppress the pest population. For this system, we determine when the pest-eradication periodic solution is globally attracting. We also examine how varying the control measures (pesticide concentration, natural enemy augmentation and the frequency of applications) may lead to different pest outbreak or persistence outcomes when eradication does not occur.

High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures.

In this paper, we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy in the Bounded-Lipschitz norm is achieved by both schemes.

Modeling the invasion and establishment of a tick-borne pathogen.

We develop a discrete-time tick-host-pathogen model to describe the spread of a disease in a hard-bodied tick species. This model incorporates the developmental stages for a tick, the dependence of the tick life-cycle and disease transmission on host availability, and three sources of pathogen transmission. We first establish the global dynamics of the disease-free system. We then apply the model to two pathogens, Borellia burgdorferi and Anaplasma phagocytophila, using Ixodes ricinus as the tick species to study properties of the invasion and establishment of a disease numerically. In particular, we consider the basic reproduction number, which determines whether a disease can invade the tick-host system, as well as disease prevalence and time to establishment in the case of successful disease invasion. Using Monte Carlo simulations, we calculate the means of each of these disease metrics and their elasticities with respect to various model parameters. We find that increased tick survival may help enable disease invasion, decrease the time to disease establishment, and increase disease prevalence once established. In contrast, though disease invasion is sensitive to tick-to-host transmission and tick searching efficiencies, neither disease prevalence nor time to disease establishment is sensitive to these parameters. These differences emphasize the importance of developing approaches, such as the one highlighted here, that can be used to study disease dynamics beyond just pathogen invasion, including transitional and long-term dynamics.

A continuous-time mathematical model and discrete approximations for the aggregation of ß-Amyloid.

Alzheimer's disease is a degenerative disorder characterized by the loss of synapses and neurons from the brain, as well as the accumulation of amyloid-based neuritic plaques. While it remains a matter of contention whether ß-amyloid causes the neurodegeneration, ß-amyloid aggregation is associated with the disease progression. Therefore, gaining a clearer understanding of this aggregation may help to better understand the disease. We develop a continuous-time model for ß-amyloid aggregation using concepts from chemical kinetics and population dynamics. We show the model conserves mass and establish conditions for the existence and stability of equilibria. We also develop two discrete-time approximations to the model that are dynamically consistent. We show numerically that the continuous-time model produces sigmoidal growth, while the discrete-time approximations may exhibit oscillatory dynamics. Finally, sensitivity analysis reveals that aggregate concentration is most sensitive to parameters involved in monomer production and nucleation, suggesting the need for good estimates of such parameters.

Sensitivity equations for measure-valued solutions to transport equations.

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R})$:$$ ∂_t\mu_t + ∂_x(v(x) \mu_t) = 0.$$We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $$ ∂_t\mu^h_t + ∂_x(v^h(x)\mu^h_t) = 0.$$We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $∂artial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R})$ endowed with the dual norm $(C^{1,\alpha}(\mathbb{R}))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v=v[\mu_t](x)$.

Finite difference schemes for a structured population model in the space of measures.

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.

The trouble with surrogates in environmental risk assessment: a daphniid case study.

The use of indicator species to test for environmental stability and functioning is a widespread practice. In aquatic systems, several daphniids (Cladocera: Daphniidae) are commonly used as indicator species; registration of new pesticides are mandated by the Environmental Protection Agency to be accompanied by daphniid toxicity data. This reliance upon a few species to infer ecosystem health and function assumes similar responses to toxicants across species with potentially very different life histories and susceptibility. Incorporating lab-derived life-history data into a simple mathematical model, we explore the reliability of three different daphniid species as surrogates for each other by comparing their responses to reductions in survivorship and fecundity after simulated exposure to toxicants. Our results demonstrate that daphniid species' responses to toxicant exposure render them poor surrogates for one another, highlighting that caution should be exercised in using a surrogate approach to the use of indicator species in risk assessment.

Examining the effect of reoccurring disturbances on population persistence with application to marine mammals.

We develop a two-state Markov chain to describe the effect of reoccurring disturbances on a population that is modeled by discrete-time matrix model. The environment is described by three parameters that define the magnitude of impact of a disturbance, the average duration of impact of a disturbance, and the average time between disturbances. We derive an approximation for the stochastic growth rate in order to examine how these three parameters affect population growth. From this approximation, we calculate the sensitivity and elasticity of the growth rate with respect to the environmental parameters. We show that the average duration of impact of a disturbance and the average time between disturbances contribute equally to the stochastic growth rate. We also show that the elasticity of the stochastic growth rate is more sensitive to changes in the magnitude of impact than to changes in either the average duration of impact of a disturbance or the average time between disturbances. These conclusions hold irrespective of the population under consideration. We then provide an application of the model formulation to examine how disturbances, such as oil spills, may affect a sperm whale population. The model results suggest that, in oder to mitigate the impact of disturbances, management strategies should focus on reducing the magnitude of impact. Meanwhile, if it is more feasible to reduce either the duration of impact or the time between impacts, managers should focus on whichever is easier to obtain. In addition, when applied to a sperm whale population, our model shows that the probability of extinction can dramatically increase when disturbance frequency increases but is not greatly impacted by the assumption that all disturbances have the same magnitude.

The evolution of toxicant resistance in daphniids and its role on surrogate species.

Prolonged exposure to a disturbance such as a toxicant has the potential to result in rapid evolution to toxicant resistance in many short-lived species such as daphniids. This evolution may allow a population to persist at higher levels of the toxicant than is possible without evolution. Here we apply evolutionary game theory to a Leslie matrix model for a daphniid population to obtain a Darwinian model that couples population dynamics with the dynamics of an evolving trait. We use the Darwinian model to consider how the evolution of resistance to the lethal or sublethal effects of a disturbance may change the population dynamics. In particular, we determine the conditions under which a daphniid population can persist by evolving toxicant resistance. We then consider the implications of this evolution in terms of the use of daphniids as surrogate species. We show for three species of daphniids that evolution of toxicant resistance means that one species may persist while another does not. These results suggest that toxicant studies that do not consider the potential of a species (or its surrogate) to develop toxicant resistance may not accurately predict the long term persistence of the species.

Sublethal Effects in Pest Management: A Surrogate Species Perspective on Fruit Fly Control.

Tephritid fruit flies are economically important orchard pests globally. While much effort has focused on controlling individual species with a combination of pesticides and biological control, less attention has been paid to managing assemblages of species. Although several tephritid species may co-occur in orchards/cultivated areas, especially in mixed-cropping schemes, their responses to pesticides may be highly variable. Furthermore, predictive efforts about toxicant effects are generally based on acute toxicity, with little or no regard to long-term population effects. Using a simple matrix model parameterized with life history data, we quantified the responses of several tephritid species to the sublethal effects of a toxicant acting on fecundity. Using a critical threshold to determine levels of fecundity reduction below which species are driven to local extinction, we determined that threshold levels vary widely for the three tephritid species. In particular, Bactrocera dorsalis was the most robust of the three species, followed by Ceratitis capitata, and then B. cucurbitae, suggesting individual species responses should be taken into account when planning for area-wide pest control. The rank-order of susceptibility contrasts with results from several field/lab studies testing the same species, suggesting that considering a combination of life history traits and individual species susceptibility is necessary for understanding population responses of species assemblages to toxicant exposure.

Analysis of lethal and sublethal impacts of environmental disasters on sperm whales using stochastic modeling.

Mathematical models are essential for combining data from multiple sources to quantify population endpoints. This is especially true for species, such as marine mammals, for which data on vital rates are difficult to obtain. Since the effects of an environmental disaster are not fixed, we develop time-varying (nonautonomous) matrix population models that account for the eventual recovery of the environment to the pre-disaster state. We use these models to investigate how lethal and sublethal impacts (in the form of reductions in the survival and fecundity, respectively) affect the population's recovery process. We explore two scenarios of the environmental recovery process and include the effect of demographic stochasticity. Our results provide insights into the relationship between the magnitude of the disaster, the duration of the disaster, and the probability that the population recovers to pre-disaster levels or a biologically relevant threshold level. To illustrate this modeling methodology, we provide an application to a sperm whale population. This application was motivated by the 2010 Deepwater Horizon oil rig explosion in the Gulf of Mexico that has impacted a wide variety of species populations including oysters, fish, corals, and whales.

Competitive exclusion and coexistence in a two-strain pathogen model with diffusion.

We consider a two-strain pathogen model described by a system of reaction-diffusion equations. We define a basic reproduction number R0 and show that when the model parameters are constant (spatially homogeneous), if R0 >1 then one strain will outcompete the other strain and drive it to extinction, but if R0 ≤ 1 then the disease-free equilibrium is globally attractive. When we assume that the diffusion rates are equal while the transmission and recovery rates are heterogeneous, then there are two possible outcomes under the condition R0 < 1: 1) Competitive exclusion where one strain dies out. 2) Coexistence between the two strains. Thus, spatial heterogeneity promotes coexistence.

Passive Acoustic Monitoring of the Environmental Impact of Oil Exploration on Marine Mammals in the Gulf of Mexico.

The Gulf of Mexico is a region densely populated by marine mammals that must adapt to living in a highly active industrial environment. This paper presents a new approach to quantifying the anthropogenic impact on the marine mammal population. The results for sperm and beaked whales of a case study of regional population dynamics trends after the Deepwater Horizon oil spill, derived from passive acoustic-monitoring data gathered before and after the spill in the vicinity of the accident, are presented.

Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model.

We study a quasilinear hierarchically size-structured population model presented in [4]. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. In [4] we showed that solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this paper we develop a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method is provided. We also develop a high resolution second order scheme to compute the measure-valued solution of the model and perform a comparative study between the two schemes.

Preface to '' Modeling with measures ''.

Different communities met in the research workshop ``Modeling with Measures" that took place at the Lorentz Center (Leiden, The Netherlands) during 26th--30th of August 2013. They were groups of researchers active in the following fields.

Competitive exclusion and coexistence in an n-species Ricker model.

We analyse a discrete-time Ricker competition model with n competing species and give sufficient conditions, which depend on the competition coefficients only, for one species to survive (not necessarily at an equilibrium) and to drive all the other species to extinction. Our results complement and extend similar existing results from the literature. For the model reduced to three species ([Formula: see text]), we also investigate various scenarios under which all species coexist, in the sense that each species is robustly uniformly persistent. We provide a few numerical simulations to illustrate that coexistence does not necessarily mean convergence to the interior equilibrium, and that the interior dynamics can be quite complex.

A second-order high-resolution finite difference scheme for a size-structured model for the spread of Mycobacterium marinum.

We present a second-order high-resolution finite difference scheme to approximate the solution of a mathematical model of the transmission dynamics of Mycobacterium marinum (Mm) in an aquatic environment. This work extends the numerical theory and continues the preliminary studies on the model first developed in Ackleh et al. [Structured models for the spread of Mycobacterium marinum: foundations for a numerical approximation scheme, Math. Biosci. Eng. 11 (2014), pp. 679-721]. Numerical simulations demonstrating the accuracy of the method are presented, and we compare this scheme to the first-order scheme developed in Ackleh et al. [Structured models for the spread of Mycobacterium marinum: foundations for a numerical approximation scheme, Math. Biosci. Eng. 11 (2014), pp. 679-721] to show that the first-order method requires significantly more computational time to provide solutions with a similar accuracy. We also demonstrated that the model can be a tool to understand surprising or nonintuitive phenomena regarding competitive advantage in the context of biologically realistic growth, birth and death rates.

Finite difference approximations for a size-structured population model with distributed states in the recruitment.

We consider a size-structured population model where individuals may be recruited into the population at different sizes. First- and second-order finite difference schemes are developed to approximate the solution of the model. The convergence of the approximations to a unique weak solution is proved. We then show that as the distribution of the new recruits become concentrated at the smallest size, the weak solution of the distributed states-at-birth model converges to the weak solution of the classical Gurtin-McCamy-type size-structured model in the weak* topology. Numerical simulations are provided to demonstrate the achievement of the desired accuracy of the two methods for smooth solutions as well as the superior performance of the second-order method in resolving solution-discontinuities. Finally, we provide an example where supercritical Hopf-bifurcation occurs in the limiting single state-at-birth model and we apply the second-order numerical scheme to show that such bifurcation also occurs in the distributed model.

Deconstructing the surrogate species concept: a life history approach to the protection of ecosystem services.

The use of the surrogate species concept is widespread in environmental risk assessment and in efforts to protect species that provide ecosystem services, yet there are no standard protocols for the choice of surrogates. Surrogates are often chosen on the basis of convenience or vague resemblances in physiology or life history to species of concern. Furthermore, our ability to predict how species of concern will fare when subjected to disturbances such as environmental contaminants or toxicants is often based on woefully misleading comparisons of static toxicity tests. Here we present an alternative approach that features a simple mathematical model parameterized with life history data applied to an assemblage of species that provide an important ecosystem service: a suite of parasitoid wasps that provide biological control of agricultural pests. Our results indicate that these parasitoid wasp species have different population responses to toxic insult--that is, we cannot predict how all four species will react to pesticide exposure simply by extrapolating from the response of any one species. Furthermore, sensitivity analysis of survivorship and reproduction demonstrates that the life stage most sensitive to pesticide disturbance varies among species. Taken together, our results suggest that the ability to predict the fate of a suite of species using the response of just one species (the surrogate species concept) is widely variable and potentially misleading.

Robust uniform persistence and competitive exclusion in a nonautonomous multi-strain SIR epidemic model with disease-induced mortality.

A nonautonomous version of the SIR epidemic model in Ackleh and Allen (2003) is considered, for competition of [Formula: see text] infection strains in a host population. The model assumes total cross immunity, mass action incidence, density-dependent host mortality and disease-induced mortality. Sufficient conditions for the robust uniform persistence of the total population, as well as of the susceptible and infected subpopulations, are given. The first two forms of persistence depend entirely on the rate at which the population grows from the extinction state, respectively the rate at which the disease is vertically transmitted to offspring. We also discuss the competitive exclusion among the [Formula: see text] infection strains, namely when a single infection strain survives and all the others go extinct. Numerical simulations are also presented, to account for the situations not covered by the analytical results. These simulations suggest that the nonautonomous nature of the model combined with the disease induced mortality allow for many strains to coexist. The theoretical approach developed here is general enough to apply to other nonautonomous epidemic models.