How to scale up from animal movement decisions to spatiotemporal patterns: An approach via step selection.

Uncovering the mechanisms behind animal space use patterns is of vital importance for predictive ecology, thus conservation and management of ecosystems. Movement is a core driver of those patterns so understanding how movement mechanisms give rise to space use patterns has become an increasingly active area of research. This study focuses on a particular strand of research in this area, based around step selection analysis (SSA). SSA is a popular way of inferring drivers of movement decisions, but, perhaps less well appreciated, it also parametrises a model of animal movement. Of key interest is that this model can be propagated forwards in time to predict the space use patterns over broader spatial and temporal scales than those that pertain to the proximate movement decisions of animals. Here, we provide a guide for understanding and using the various existing techniques for scaling up step selection models to predict broad-scale space use patterns. We give practical guidance on when to use which technique, as well as specific examples together with code in R and Python. By pulling together various disparate techniques into one place, and providing code and instructions in simple examples, we hope to highlight the importance of these techniques and make them accessible to a wider range of ecologists, ultimately helping expand the usefulness of SSA.

Time scale theory on stability of explicit and implicit discrete epidemic models: applications to Swine flu outbreak.

Time scales theory has been in use since the 1980s with many applications. Only very recently, it has been used to describe within-host and between-hosts dynamics of infectious diseases. In this study, we present explicit and implicit discrete epidemic models motivated by the time scales modeling approach. We use these models to formulate the basic reproduction number, which determines whether an outbreak occurs or the disease dies out. We discuss the stability of the disease-free and endemic equilibrium points using the linearization method and Lyapunov function. Furthermore, we apply our models to swine flu outbreak data to demonstrate that the discrete models can accurately describe the epidemic dynamics. Our comparison analysis shows that the implicit discrete model can best describe the data regardless of the data frequency. In addition, we perform the sensitivity analysis on the key parameters of the models to study how these parameters impact the basic reproduction number.

Orthogonal Polynomials with Singularly Perturbed Freud Weights.

In this paper, we are concerned with polynomials that are orthogonal with respect to the singularly perturbed Freud weight functions. By using Chen and Ismail's ladder operator approach, we derive the difference equations and differential-difference equations satisfied by the recurrence coefficients. We also obtain the differential-difference equations and the second-order differential equations for the orthogonal polynomials, with the coefficients all expressed in terms of the recurrence coefficients.

Gene drives and population persistence vs elimination: The impact of spatial structure and inbreeding at low density.

Synthetic gene drive constructs are being developed to control disease vectors, invasive species, and other pest species. In a well-mixed random mating population a sufficiently strong gene drive is expected to eliminate a target population, but it is not clear whether the same is true when spatial processes play a role. In species with an appropriate biology it is possible that drive-induced reductions in density might lead to increased inbreeding, reducing the efficacy of drive, eventually leading to suppression rather than elimination, regardless of how strong the drive is. To investigate this question we analyse a series of explicitly solvable stochastic models considering a range of scenarios for the relative timing of mating, reproduction, and dispersal and analyse the impact of two different types of gene drive, a Driving Y chromosome and a homing construct targeting an essential gene. We find in all cases a sufficiently strong Driving Y will go to fixation and the population will be eliminated, except in the one life history scenario (reproduction and mating in patches followed by dispersal) where low density leads to increased inbreeding, in which case the population persists indefinitely, tending to either a stable equilibrium or a limit cycle. These dynamics arise because Driving Y males have reduced mating success, particularly at low densities, due to having fewer sisters to mate with. Increased inbreeding at low densities can also prevent a homing construct from eliminating a population. For both types of drive, if there is strong inbreeding depression, then the population cannot be rescued by inbreeding and it is eliminated. These results highlight the potentially critical role that low-density-induced inbreeding and inbreeding depression (and, by extension, other sources of Allee effects) can have on the eventual impact of a gene drive on a target population.

Derivation and Analysis of a Discrete Predator-Prey Model.

We derive a discrete predator-prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark-Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark-Sacker bifurcation as a function of the model parameters.

A Mean-Field Approximation of SIR Epidemics on an Erdös-Rényi Network Model.

The stochastic nature of epidemic dynamics on a network makes their direct study very challenging. One avenue to reduce the complexity is a mean-field approximation (or mean-field equation) of the dynamics; however, the classic mean-field equation has been shown to perform sub-optimally in many applications. Here, we adapt a recently developed mean-field equation for SIR epidemics on a network in continuous time to the discrete time case. With this new discrete mean-field approximation, this proof-of-concept study shows that, given the density of the network, there is a strong correspondence between the epidemics on an Erdös-Rényi network and a system of discrete equations. Through this connection, we developed a parameter fitting procedure that allowed us to use synthetic daily SIR data to approximate the underlying SIR epidemic parameters on the network. This procedure has improved accuracy in the estimation of the network epidemic parameters as the network density increases, and is extremely cheap computationally.

Solving the Integral Differential Equations with Delayed Argument by Using the DTM Method.

Recently, a lot of attention has been paid to the field of research connected with the wireless sensor network and industrial internet of things. The solutions found by theorists are next used in practice in such area as smart industries, smart devices, smart home, smart transportation and the like. Therefore, there is a need to look for some new techniques for solving the problems described by means of the appropriate equations, including differential equations, integral equations and integro-differential equations. The object of interests of this paper is the method dedicated for solving some integro-differential equations with a retarded (delayed) argument. The proposed procedure is based on the Taylor differential transformation which enables to transform the given integro-differential equation into a respective system of algebraic (nonlinear, very often) equations. The described method is efficient and relatively simple to use, however a high degree of generality and complexity of problems, defined by means of the discussed equations, makes impossible to obtain a general form of their solution and enforces an individual approach to each equation, which, however, does not diminish the benefits associated with its use.

Ecological features benefiting sustainable harvesters in socio-ecological systems: a case study of Swiftlets in Malaysia.

A major challenge in biodiversity management is overharvesting by unsustainable harvesters. If a scenario could be created where sustainable harvesters benefit more than the unsustainable ones, even in the short term, the issue of overharvesting would be solved. Everyone would then follow the lead of sustainable harvesters. However, creating such a scenario is not an easy task; the difficulty is intensified if the habitat is open access and there is no property rights system. Swiftlets in Sarawak, Malaysia, present a special case where sustainable harvesters are believed to be more beneficial than unsustainable harvesters. Edible nests built by adult Swiftlets are used as ingredients for a traditional luxurious soup in Chinese cuisine. A rise in nest prices has increased the instances of unsustainable harvesters wrongfully collecting nests along with the eggs and fledglings, which are then abandoned. Swiftlets live in caves and build nests on cave ceilings. It is known that Swiftlets escape from cave ceilings when these harvesters take the nests, never to return to the same place. This ecological feature appears to work as the Swiftlet's indirect punishment against unsustainable harvesters. This study constructs a stage-structured population model and examines the effect of property rights and the indirect punishment by Swiftlets on the population dynamics of the bird, and on the economic return of both sustainable and unsustainable harvesters. Our findings are as follows: the indirect punishment by Swiftlets provides sustainable harvesters a higher short-term return than unsustainable harvesters under the property rights system, as long as Swiftlets return to their original cave after escaping from the unsustainable harvesters. While previous studies regarding the management of the commons have stressed the importance of rules and regulations for sustainable harvesting without considering the ecological uniqueness of each species, this study suggests that ecological exploration and the discovery of ecological features are also essential for designing a sustainable framework.

A general theory of coexistence and extinction for stochastic ecological communities.

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Discrete-time population dynamics of spatially distributed semelparous two-sex populations.

Spatially distributed populations with two sexes may face the problem that males and females concentrate in different parts of the habitat and mating and reproduction does not happen sufficiently often for the population to persist. For simplicity, to explore the impact of sex-dependent dispersal on population survival, we consider a discrete-time model for a semelparous population where individuals reproduce only once in their life-time, during a very short reproduction season. The dispersal of females and males is modeled by Feller kernels and the mating by a homogeneous pair formation function. The spectral radius of a homogeneous operator is established as basic reproduction number of the population, [Formula: see text]. If [Formula: see text], the extinction state is locally stable, and if [Formula: see text] the population shows various degrees of persistence that depend on the irreducibility properties of the dispersal kernels. Special cases exhibit how sex-biased dispersal affects the persistence of the population.

Separate seasons of infection and reproduction can lead to multi-year population cycles.

Many host-pathogen systems are characterized by a temporal order of disease transmission and host reproduction. For example, this can be due to pathogens infecting certain life cycle stages of insect hosts; transmission occurring during the aggregation of migratory birds; or plant diseases spreading between planting seasons. We develop a simple discrete-time epidemic model with density-dependent transmission and disease affecting host fecundity and survival. The model shows sustained multi-annual cycles in host population abundance and disease prevalence, both in the presence and absence of density dependence in host reproduction, for large horizontal transmissibility, imperfect vertical transmission, high virulence, and high reproductive capability. The multi-annual cycles emerge as invariant curves in a Neimark-Sacker bifurcation. They are caused by a carry-over effect, because the reproductive fitness of an individual can be reduced by virulent effects due to infection in an earlier season. As the infection process is density-dependent but shows an effect only in a later season, this produces delayed density dependence typical for second-order oscillations. The temporal separation between the infection and reproduction season is crucial in driving the cycles; if these processes occur simultaneously as in differential equation models, there are no sustained oscillations. Our model highlights the destabilizing effects of inter-seasonal feedbacks and is one of the simplest epidemic models that can generate population cycles.

Dynamic modeling of multivariate dimensions and their temporal relationships using latent processes: Application to Alzheimer's disease.

Alzheimer's disease gradually affects several components including the cerebral dimension with brain atrophies, the cognitive dimension with a decline in various functions, and the functional dimension with impairment in the daily living activities. Understanding how such dimensions interconnect is crucial for Alzheimer's disease research. However, it requires to simultaneously capture the dynamic and multidimensional aspects and to explore temporal relationships between dimensions. We propose an original dynamic structural model that accounts for all these features. The model defines dimensions as latent processes and combines a multivariate linear mixed model and a system of difference equations to model trajectories and temporal relationships between latent processes in finely discrete time. Dimensions are simultaneously related to their observed (possibly multivariate) markers through nonlinear equations of observation. Parameters are estimated in the maximum likelihood framework enjoying a closed form for the likelihood. We demonstrate in a simulation study that this dynamic model in discrete time benefits the same causal interpretation of temporal relationships as models defined in continuous time as long as the discretization step remains small. The model is then applied to the data of the Alzheimer's Disease Neuroimaging Initiative. Three longitudinal dimensions (cerebral anatomy, cognitive ability, and functional autonomy) measured by six markers are analyzed, and their temporal structure is contrasted between different clinical stages of Alzheimer's disease.

Continuous and discrete modeling of HIV-1 decline on therapy.

Mathematical models have shed light on the dynamics of HIV- 1 infection in vivo. In this paper, we generalize continuous mathematical models of drug therapy for HIV-1 by Perelson et al. (Science 271:1582-1586, 1996) and Perelson and Nelson (SIAM Rev 41:3-44, 1999) on time scales, i.e., a nonempty closed subset of real numbers in order to derive new discrete models that predict the total concentration of plasma virus as a function of time. One of our main goals is to compare discrete mathematical models with the continuous model in Perelson et al. (1996) where HIV infected patients were given protease inhibitors and sampled frequently thereafter. For the comparison, we use experimental data collected in Perelson et al. (1996) and estimate the parameters such as the virion clearance rate and the rate of loss of infected cells by fitting the total concentration of plasma virus to this data set. Our results show that discrete systems describe the best fit. In the previous models of this study, the efficacy of protease inhibitor is assumed to be perfect. Motivated by Perelson and Nelson (1999), we end the paper with a mathematical model of imperfect protease inhibitor and reverse transcriptase (RT) inhibitor combination therapy of HIV-1 infection on time scales with its stability analysis.

The effects of time valuation in cancer optimal therapies: a study of chronic myeloid leukemia.

BACKGROUND: The mathematical design of optimal therapies to fight cancer is an important research field in today's Biomathematics and Biomedicine given its relevance to formulate patient-specific treatments. Until now, however, cancer optimal therapies have considered that malignancy exclusively depends on the drug concentration and the number of cancer cells, ignoring that the faster the cancer grows the worse the cancer is, and that early drug doses are more prejudicial. Here, we analyze how optimal therapies are affected when the time evolution of treated cancer is envisaged as an additional element determining malignancy, analyzing in detail the implications for imatinib-treated Chronic Myeloid Leukemia. METHODS: Taking as reference a mathematical model describing Chronic Myeloid Leukemia dynamics, we design an optimal therapy problem by modifying the usual malignancy objective function, unaware of any temporal dimension of cancer malignance. In particular, we introduce a time valuation factor capturing the increase of malignancy associated to the quick development of the disease and the persistent negative effects of initial drug doses. After assigning values to the parameters involved, we solve and simulate the model with and without the new time valuation factor, comparing the results for the drug doses and the evolution of the disease. RESULTS: Our computational simulations unequivocally show that the consideration of a time valuation factor capturing the higher malignancy associated with early growth of cancer and drug administration allows more efficient therapies to be designed. More specifically, when this time valuation factor is incorporated into the objective function, the optimal drug doses are lower, and do not involve medically relevant increases in the number of cancer cells or in the disease duration. CONCLUSIONS: In the light of our simulations and as biomedical evidence strongly suggests, the existence of a time valuation factor affecting malignancy in treated cancer cannot be ignored when designing cancer optimal therapies. Indeed, the consideration of a time valuation factor modulating malignancy results in significant gains of efficiency in the optimal therapy with relevant implications from the biomedical perspective, specially when designing patient-specific treatments.

An Improved Version of the Classical Banister Model to Predict Changes in Physical Condition.

In this paper, we formulate and provide the solutions to two new models to predict changes in physical condition by using the information of the training load of an individual. The first model is based on a functional differential equation, and the second one on an integral differential equation. Both models are an extension to the classical Banister model and allow to overcome its main drawback: the variations in physical condition are influenced by the training loads of the previous days and not only of the same day. Finally, it is illustrated how the first model works with a real example of the training process of a cyclist.

Persistence and extinction for stochastic ecological models with internal and external variables.

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research.

Biological control effects of non-reproductive host mortality caused by insect parasitoids.

As the rate of spread of invasive species increases, consumer-resource communities are often populated by a combination of exotic and native species at all trophic levels. In parasitoid-host communities, these novel associations may lead to disconnects between parasitoid preference and performance, and parasitoid oviposition may result in death of the parasitoid offspring, death of the host, or death of both. Despite their relevance for biological control risk and efficacy assessments, the direct and indirect population-level consequences of parasitoids attacking and killing their hosts without successfully reproducing (non-reproductive mortality) are poorly understood. Non-reproductive mortality induced by egg parasitoids (parasitoid-induced host egg abortion) may be particularly important for understanding the population dynamics of the invasive agricultural pest Halyomorpha halys (Hemiptera: Pentatomidae) and endemic stink bugs in North America, which are attacked by a suite of both native and introduced egg parasitoids. It is unclear, however, how various factors controlling parasitoid foraging and developmental success manifest at the population level. We constructed two related versions of a two-host-one-parasitoid model to evaluate the population-level consequences of non-reproductive host mortality. Egg abortion can result in strong negative or positive enemy-mediated indirect effects, taking the form of apparent competition, apparent parasitism, apparent amensalism, or apparent commensalism. For parasitoids limited in their reproductive output by the number of eggs they can produce, higher non-reproductive host mortality can reduce the strength of the positive indirect effect in cases of apparent parasitism, and it can reduce the negative indirect effect on the more suitable host in cases of apparent competition. For time-limited parasitoids, unsuitable hosts with high levels of non-reproductive parasitoid-induced mortality can be strongly suppressed in the presence of a suitable host, while the suitable host is only negligibly impacted (i.e., apparent amensalism). We evaluate these model-derived hypotheses within the context of H. halys and its native and nonnative parasitoids in North America, and discuss their application to risk assessment in biological control programs.

Further Mathematical Modelling of Mating Sex Ratios & Male Strategies with Special Relevance to Human Life History.

Influential models of male reproductive strategies have often ignored the importance of mate guarding, focusing instead on trade-offs between fitness gained through care for dependants in a pair bond versus fitness from continued competition for additional mates. Here we follow suggestions that mate guarding is a distinct alternative strategy that plays a crucial role, with special relevance to the evolution of our own lineage. Human pair bonding may have evolved in concert with the evolution of our grandmothering life history, which entails a shift to male-biased sex ratios in the fertile ages. As that sex ratio becomes more male biased, payoffs for mate-guarding increase due to partner scarcity. We present an ordinary differential equation model of mutually exclusive strategies (dependant care, multiple mating, and mate guarding), calculate steady-state frequencies and perform bifurcation analysis on parameters of care and guarding efficiency. Mate guarding triumphs over alternate strategies when populations are male biased, and guarding is fully efficient. When guarding does not ensure complete certainty of paternity, and multiple maters are able to gain some paternity from guarders, multiple mating can coexist with guarding. At female-biased sex ratios, multiple mating takes over, unless the benefit of care to the number of surviving offspring produced by the mates of carers is large.

Mutualisms impact species' range expansion speeds and spatial distributions.

Species engage in mutually beneficial interspecific interactions (mutualisms) that shape their population dynamics in ecological communities. Species engaged in mutualisms vary greatly in their degree of dependence on their partner from complete dependence (e.g., yucca and yucca moth mutualism) to low dependence (e.g., generalist bee with multiple plant species). While current empirical studies show that, in mutualisms, partner dependence can alter the speed of a species' range expansion, there is no theory that provides conditions when expansion is sped up or slowed down. To address this, we built a spatially explicit model incorporating the population dynamics of two dispersing species interacting mutualistically. We explored how mutualisms impacted range expansion across a gradient of dependence (from complete independence to obligacy) between the two species. We then studied the conditions in which the magnitude of the mutualistic benefits could hinder versus enhance the speed of range expansion. We showed that either complete dependence, no dependence, or intermediate degree of dependence on a mutualist partner can lead to the greatest speeds of a focal species' range expansion based on the magnitude of benefits exchanged between partner species in the mutualism. We then showed how different degrees of dependence between species could alter the spatial distribution of the range expanding populations. Finally, we identified the conditions under which mutualistic interactions can turn exploitative across space, leading to the formation of a species' range limits. Our work highlights how couching mutualisms and mutualist dependence in a spatial context can provide insights about species range expansions, limits, and ultimately their distributions.

Bridging instrumental and visual perception with improved color difference equations: A multi-center study.

OBJECTIVES: This multicenter study aimed to evaluate visual-instrumental agreement of six color measurement devices and optimize three color difference equations using a dataset of visual color differences (∆V) from expert observers. METHODS: A total of 154 expert observers from 16 sites across 5 countries participated, providing visual scaling on 26 sample pairs of artificial teeth using magnitude estimation. Three color difference equations (ΔE*ab, ∆E00, and CAM16-UCS) were tested. Optimization of all three equations was performed using device-specific weights, and the standardized residual sum of squares (STRESS) index was used to evaluate visual-instrumental agreement. RESULTS: The ΔE*ab formula exhibited STRESS values from 18 to 40, with visual-instrumental agreement between 60 % and 82 %. The ∆E00 formula showed STRESS values from 26 to 32, representing visual-instrumental agreement of 68 % to 74 %. CAM16-UCS demonstrated STRESS values from 32 - 39, with visual-instrumental agreement between 61-68 %. Following optimization, STRESS values decreased for all three formulas, with ΔE' demonstrating average visual-instrumental agreement of 79 % and ∆E00 of 78 %. CAM16-UCS showed average visual-instrumental agreement of 76 % post optimization. SIGNIFICANCE: Optimization of color difference equations notably improved visual-instrumental agreement, overshadowing device performance. The optimzed ΔE' formula demonstrated the best overall performance combining computational simplicty with outstanding visual-instrumental agreement.