Impact of resource distributions on the competition of species in stream environment.

Our earlier work in Nguyen et al. (Maximizing metapopulation growth rate and biomass in stream networks. arXiv preprint arXiv:2306.05555 , 2023) shows that concentrating resources on the upstream end tends to maximize the total biomass in a metapopulation model for a stream species. In this paper, we continue our research direction by further considering a Lotka-Volterra competition patch model for two stream species. We show that the species whose resource allocations maximize the total biomass has the competitive advantage.

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.

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.

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.

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.

A bifurcation theorem for evolutionary matrix models with multiple traits.

One fundamental question in biology is population extinction and persistence, i.e., stability/instability of the extinction equilibrium and of non-extinction equilibria. In the case of nonlinear matrix models for structured populations, a bifurcation theorem answers this question when the projection matrix is primitive by showing the existence of a continuum of positive equilibria that bifurcates from the extinction equilibrium as the inherent population growth rate passes through 1. This theorem also characterizes the stability properties of the bifurcating equilibria by relating them to the direction of bifurcation, which is forward (backward) if, near the bifurcation point, the positive equilibria exist for inherent growth rates greater (less) than 1. In this paper we consider an evolutionary game theoretic version of a general nonlinear matrix model that includes the dynamics of a vector of mean phenotypic traits subject to natural selection. We extend the fundamental bifurcation theorem to this evolutionary model. We apply the results to an evolutionary version of a Ricker model with an added Allee component. This application illustrates the theoretical results and, in addition, several other interesting dynamic phenomena, such as backward bifurcation induced strong Allee effects.

A juvenile-adult population model: climate change, cannibalism, reproductive synchrony, and strong Allee effects.

We study a discrete time, structured population dynamic model that is motivated by recent field observations concerning certain life history strategies of colonial-nesting gulls, specifically the glaucous-winged gull (Larus glaucescens). The model focuses on mechanisms hypothesized to play key roles in a population's response to degraded environment resources, namely, increased cannibalism and adjustments in reproductive timing. We explore the dynamic consequences of these mechanics using a juvenile-adult structure model. Mathematically, the model is unusual in that it involves a high co-dimension bifurcation at [Formula: see text] which, in turn, leads to a dynamic dichotomy between equilibrium states and synchronized oscillatory states. We give diagnostic criteria that determine which dynamic is stable. We also explore strong Allee effects caused by positive feedback mechanisms in the model and the possible consequence that a cannibalistic population can survive when a non-cannibalistic population cannot.