Predicting attenuant and resonant 2-cycles in periodically forced discrete-time two-species population models

Periodic environments may either enhance or suppress a population via resonant or attenuant cycles. We derive signature functions for predicting the responses of two competing populations to 2-periodic oscillations in six model parameters. Two of these parameters provide a non-trivial equilibrium and two provide the carrying capacities of each species in the absence of the other, but the remaining two are arbitrary and could be intrinsic growth rates. Each signature function is the sign of a weighted sum of the relative strengths of the oscillations of the perturbed parameters. Periodic environments are favourable for populations when the signature function is positive and are deleterious if the signature function is negative. We compute the signature functions of four classical, discrete-time two-species populations and determine regions in parameter space which are either favourable or detrimental to the populations. The six-parameter models include the Logistic, Ricker, Beverton–Holt, and Hassell models.     

Signature Function for Predicting Resonant and Attenuant Population 2-cycles

Populations are either enhanced via resonant cycles or suppressed via attenuant cycles by periodic environments. We develop a signature function for predicting the response of discretely reproducing populations to 2-periodic fluctuations of both a characteristic of the environment (carrying capacity), and a characteristic of the population (inherent growth rate). Our signature function is the sign of a weighted sum of the relative strengths of the oscillations of the carrying capacity and the demographic characteristic. Periodic environments are deleterious for populations when the signature function is negative. However, positive signature functions signal favorable environments. We compute the signature functions of six classical discrete-time single species population models, and use the functions to determine regions in parameter space that are either favorable or detrimental to the populations. The two-parameter classical models include the Ricker, Beverton-Holt, Logistic, and Maynard Smith models.     

A tale of two cycles – distinguishing quasi-cycles and limit cycles in finite predator–prey populations

Periodic predator – prey dynamics in constant environments are usually taken as indicative of deterministic limit cycles. It is known, however, that demographic stochasticity in finite populations can also give rise to regular population cycles, even when the corresponding deterministic models predict a stable equilibrium. Specifically, such quasi-cycles are expected in stochastic versions of deterministic models exhibiting equilibrium dynamics with weakly damped oscillations. The existence of quasi-cycles substantially expands the scope for natural patterns of periodic population oscillations caused by ecological interactions, thereby complicating the conclusive interpretation of such patterns. Here we show how to distinguish between quasi-cycles and noisy limit cycles based on observing changing population sizes in predator – prey populations. We start by confirming that both types of cycle can occur in the individual-based version of a widely used class of deterministic predator – prey model. We then show that it is feasible and straightforward to accurately distinguish between the two types of cycle through the combined analysis of autocorrelations and marginal distributions of population sizes. Finally, by confronting these results with real ecological time series, we demonstrate that by using our methods even short and imperfect time series allow quasi-cycles and limit cycles to be distinguished reliably.     

Demography and dispersal: invasion speeds and sensitivity analysis in periodic and stochastic environments

Invasion speeds can be calculated from matrix integrodifference equation models that incorporate stage-specific demography and dispersal. These models also permit the calculation of the sensitivity and elasticity of invasion speed to changes in demographic and dispersal parameters. Such calculations have been used to understand the factors determining invasion speed and to explore possible tactics to manage invasive species. In this paper, we extend these calculations to temporally varying environments. We present formulas for the invasion speed and its sensitivity and elasticity in both periodic and stochastic environments. Periodic models can describe seasonal variation within a year, or can be used to study the frequency of occurrence of events (e.g., floods, fires) on interannual time scales. Stochastic models can incorporate variances, covariances, and temporal autocorrelation of parameters. We show that the invasion speed is calculated from a growth rate which is in turn calculated from a periodic or stochastic product of moment-generating function matrices. We present a new formulation of sensitivity analysis, using matrix calculus, that applies equally to constant, periodic, and stochastic environments.     

A uniform framework for ordered restriction map problems.

Optical Mapping is an emerging technology for constructing ordered restriction maps of DNA molecules. The underlying computational problems for this technology have been studied and several models have been proposed in recent literature. Most of these propose combinatorial models; some of them also present statistical approaches. However, it is not a priori clear as to how these models relate to one another and to the underlying problem. We present a uniform framework for the restriction map problems where each of these various models is a specific instance of the basic framework. We achieve this by identifying two "signature" functions f() and g() that characterize the models. We identify the constraints these two functions must satisfy, thus opening up the possibility of exploring other plausible models. We show that for all of the combinatorial models proposed in literature, the signature functions are semi-algebraic. We also analyze a proposed statistical method in this framework and show that the signature functions are transcendental for this model. We also believe that this framework would provide useful guidelines for dealing with other inferencing problems arising in practice. Finally, we indicate the open problems by including a survey of the best known results for these problems.     

Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat

Recent research indicates that viruses are much more prevalent in aquatic environments than previously imagined. We derive a model of competition between two populations of bacteria for a single limiting nutrient in a chemostat where a virus is present. It is assumed that the virus can only infect one of the populations, the population that would be a more efficient consumer of the resource in a virus free environment, in order to determine whether introduction of a virus can result in coexistence of the competing populations. We also analyze the subsystem that results when the resistant competitor is absent. The model takes the form of an SIS epidemic model. Criteria for the global stability of the disease free and endemic steady states are obtained for both the subsystem as well as for the full competition model. However, for certain parameter ranges, bi-stability, and/or multiple periodic orbits is possible and both disease induced oscillations and competition induced oscillations are possible. It is proved that persistence of the vulnerable and resistant populations can occur, but only when the disease is endemic in the population. It is also shown that it is possible to have multiple attracting endemic steady states, oscillatory behavior involving Hopf, saddle-node, and homoclinic bifurcations, and a hysteresis effect. An explicit expression for the basic reproduction number for the epidemic is given in terms of biologically meaningful parameters. Mathematical tools that are used include Lyapunov functions, persistence theory, and bifurcation analysis.     

Optimal control problems arising in cell-cycle-specific cancer chemotherapy

We explore mathematical properties of models of cancer chemotherapy including cell-cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: 1 Periodic chemotherapy protocols are close to the optimum for a wide class of models and have additional favourable properties. 2 Two possible approaches, (a) to minimize the final count of malignant cells and the cumulative effect of the drug on normal cells, or (b) to maximize the final count of normal cells and the cumulative effect of the drug on malignant cells, lead to similar principles of optimization. From the mathematical viewpoint, the paper provides a catalogue of simplest mathematical models of cell-cycle dependent chemotherapy. They can be classified based on the number of compartments and types of drug action modelled. In all these models the optimal controls are complicated by the singular and periodic trajectories and multiple solutions. However, efficient numerical methods have been developed. In simpler cases, it is also possible to provide an exhaustive classification of solutions. We also discuss developments in estimation of cell cycle parameters and cell-cycle dependent drug action.     

An analysis of models describing predator-prey interaction

Mathematical models of the interaction between predator and host populations have been expressed as systems of nonlinear ordinary differential equations. Solutions of such systems may be periodic or aperiodic. Periodic, oscillatory solutions may depend on the initial conditions of the system or may be limit cycles. Aperiodic solutions can, but do not necessarily, exhibit oscillatory behavior. Therefore, it is important to characterize predatory-prey models on the basis of the possible types of solutions they may possess. This characterization can be accomplished using some well-known methods of nonlinear analysis. Examination of the system singular points and inspection of phase plane portraits have proved to be useful techniques for evaluating the effect of various modifications of early predator-prey models. Of particular interest is the existence of limit cycle oscillations in a model in which predator growth rate is a function of the concentration of prey.     

Potential landscape and probabilistic flux of a predator prey network

Predator-prey system, as an essential element of ecological dynamics, has been recently studied experimentally with synthetic biology. We developed a global probabilistic landscape and flux framework to explore a synthetic predator-prey network constructed with two Escherichia coli populations. We developed a self consistent mean field method to solve multidimensional problem and uncovered the potential landscape with Mexican hat ring valley shape for predator-prey oscillations. The landscape attracts the system down to the closed oscillation ring. The probability flux drives the coherent oscillations on the ring. Both the landscape and flux are essential for the stable and coherent oscillations. The landscape topography characterized by the barrier height from the top of Mexican hat to the closed ring valley provides a quantitative measure of global stability of system. The entropy production rate for the energy dissipation is less for smaller environmental fluctuations or perturbations. The global sensitivity analysis based on the landscape topography gives specific predictions for the effects of parameters on the stability and function of the system. This may provide some clues for the global stability, robustness, function and synthetic network design.     

Molecular variability elicits a tunable switch with discrete neuromodulatory response phenotypes

Conductance-based compartment modeling requires tuning of many parameters to fit the neuron model to target electrophysiological data. Automated parameter optimization via evolutionary algorithms (EAs) is a common approach to accomplish this task, using error functions to quantify differences between model and target. We present a three-stage EA optimization protocol for tuning ion channel conductances and kinetics in a generic neuron model with minimal manual intervention. We use the technique of Latin hypercube sampling in a new way, to choose weights for error functions automatically so that each function influences the parameter search to a similar degree. This protocol requires no specialized physiological data collection and is applicable to commonly-collected current clamp data and either single- or multi-objective optimization. We applied the protocol to two representative pyramidal neurons from layer 3 of the prefrontal cortex of rhesus monkeys, in which action potential firing rates are significantly higher in aged compared to young animals. Using an idealized dendritic topology and models with either 4 or 8 ion channels (10 or 23 free parameters respectively), we produced populations of parameter combinations fitting the target datasets in less than 80 hours of optimization each. Passive parameter differences between young and aged models were consistent with our prior results using simpler models and hand tuning. We analyzed parameter values among fits to a single neuron to facilitate refinement of the underlying model, and across fits to multiple neurons to show how our protocol will lead to predictions of parameter differences with aging in these neurons.     

Environmental quality and evolutionary potential: lessons from wild populations

An essential requirement to determine a population's potential for evolutionary change is to quantify the amount of genetic variability expressed for traits under selection. Early investigations in laboratory conditions showed that the magnitude of the genetic and environmental components of phenotypic variation can change with environmental conditions. However, there is no consensus as to how the expression of genetic variation is sensitive to different environmental conditions. Recently, the study of quantitative genetics in the wild has been revitalized by new pedigree analyses based on restricted maximum likelihood, resulting in a number of studies investigating these questions in wild populations. Experimental manipulation of environmental quality in the wild, as well as the use of naturally occurring favourable or stressful environments, has broadened the treatment of different taxa and traits. Here, we conduct a meta-analysis on recent studies comparing heritability in favourable versus unfavourable conditions in non-domestic and non-laboratory animals. The results provide evidence for increased heritability in more favourable conditions, significantly so for morphometric traits but not for traits more closely related to fitness. We discuss how these results are explained by underlying changes in variance components, and how they represent a major step in our understanding of evolutionary processes in wild populations. We also show how these trends contrast with the prevailing view resulting mainly from laboratory experiments on Drosophila. Finally, we underline the importance of taking into account the environmental variation in models predicting quantitative trait evolution.     

When can ecological speciation be detected with neutral loci?

It is not yet clear under what conditions empirical studies can reliably detect progress toward ecological speciation through the analysis of allelic variation at neutral loci. We use a simulation approach to investigate the range of parameter space under which such detection is, and is not, likely. We specifically test for the conditions under which divergent natural selection can cause a ‘generalized barrier to gene flow’ that is present across the genome. Our individual‐based numerical simulations focus on how population divergence at neutral loci varies in relation to recombination rate with a selected locus, divergent selection on that locus, migration rate and population size. We specifically test whether genetic differences at neutral markers are greater between populations in different environments than between populations in similar environments. We find that this expected signature of ecological speciation can be detected under part of the parameter space, most consistently when divergent selection is strong and migration is intermediate. By contrast, the expected signature of ecological speciation is not reliably detected when divergent selection is weak or migration is low or high. These findings provide insights into the strengths and weaknesses of using neutral markers to infer ecological speciation in natural systems.     

A spatially extended model for macroscopic spike-wave discharges

Spike-wave discharges are a distinctive feature of epileptic seizures. So far, they have not been reported in spatially extended neural field models. We study a space-independent version of the Amari neural field model with two competing inhibitory populations. We show that this competition leads to robust spike-wave dynamics if the inhibitory populations operate on different time-scales. The spike-wave oscillations present a fold/homoclinic type bursting. From this result we predict parameters of the extended Amari system where spike-wave oscillations produce a spatially homogeneous pattern. We propose this mechanism as a prototype of macroscopic epileptic spike-wave discharges. To our knowledge this is the first example of robust spike-wave patterns in a spatially extended neural field model.     

Why wait? Three mechanisms selecting for environment‐dependent developmental delays

Many species delay development unless particular environments or rare disturbance events occur. How can such a strategy be favoured over continued development? Typically, it is assumed that continued development (e.g. germination) is not advantageous in environments that have low juvenile/seedling survival (mechanism 1), either due to abiotic or competitive effects. However, it has not previously been shown how low early survival must be in order to favour environment‐specific developmental delays for long‐lived species. Using seed dormancy as an example of developmental delays, we identify a threshold level of seedling survival in ‘bad’ environments below which selection can favour germination that is limited to ‘good’ environments. This can be used to evaluate whether observed differences in seedling survival are sufficient to favour conditional germination. We also present mathematical models that demonstrate two other, often overlooked, mechanisms that can favour conditional germination in the absence of differences in seedling survival. Specifically, physiological trade‐offs can make it difficult to have germination rates that are equally high in all environments (mechanism 2). We show that such trade‐offs can either favour conditional germination or intermediate (mixed) strategies, depending on the trade‐off shape. Finally, germination in every year increases the likelihood that some individuals are killed in population‐scale disturbances before reproducing; it can thus be favourable to only germinate immediately after a disturbance (mechanism 3). We demonstrate how demographic data can be used to evaluate these selection pressures. By presenting these three mechanisms and the conditions that favour conditional germination in each case, we provide three hypotheses that can be tested as explanations for the evolution of environment‐dependent developmental delays.     

Integrating individual movement behaviour into dispersal functions

Dispersal functions are an important tool for integrating dispersal into complex models of population and metapopulation dynamics. Most approaches in the literature are very simple, with the dispersal functions containing only one or two parameters which summarise all the effects of movement behaviour as for example different movement patterns or different perceptual abilities. The summarising nature of these parameters makes assessing the effect of one particular behavioural aspect difficult. We present a way of integrating movement behavioural parameters into a particular dispersal function in a simple way. Using a spatial individual-based simulation model for simulating different movement behaviours, we derive fitting functions for the functional relationship between the parameters of the dispersal function and several details of movement behaviour. This is done for three different movement patterns (loops, Archimedean spirals, random walk). Additionally, we provide measures which characterise the shape of the dispersal function and are interpretable in terms of landscape connectivity. This allows an ecological interpretation of the relationships found.     

Simple mathematical models for interacting wild and transgenic mosquito populations

Two discrete-time models for interacting populations of wild and genetically altered mosquito are presented, where the genetically altered mosquitoes are grouped into a single population without distinguishing their zygosity. The birth and death rates for both populations are density-dependent, and the mating rates between the mosquitoes are assumed to be either constant or proportional to the total populations for the two models, respectively. The existence and stability of the boundary and positive equilibria are investigated. In particular, it is shown that bifurcations from both boundary and positive equilibria can appear for the model with proportional mating rates. Stable equilibria, periodic-doubling bifurcations, aperiodic oscillations, and chaotic behavior are all illustrated by numerical simulations.     

Admixture facilitates adaptation from standing variation in the European aspen (Populus tremula L.), a widespread forest tree

Adaptation to new environments can start from new mutations or from standing variation already present in natural populations. Whether admixture constrains or facilitates adaptation from standing variation is largely unknown, especially in ecological keystone or foundation species. We examined patterns of neutral and adaptive population divergence in Populus tremula L., a widespread forest tree, using mapped molecular genetic markers. We detected the genetic signature of postglacial admixture between a Western and an Eastern lineage of P. tremula in Scandinavia, an area suspected to represent a zone of postglacial contact for many species of animals and plants. Stringent divergence‐based neutrality tests provided clear indications for locally varying selection at the European scale. Six of 12 polymorphisms under selection were located less than 1 kb away from the nearest gene predicted by the Populus trichocarpa genome sequence. Few of these loci exhibited a signature of ‘selective sweeps’ in diversity‐based tests, which is to be expected if adaptation occurs primarily from standing variation. In Scandinavia, admixture explained genomic patterns of ancestry and the nature of clinal variation and strength of selection for bud set, a phenological trait of great adaptive significance in temperate trees, measured in a common garden trial. Our data provide a hitherto missing direct link between past range shifts because of climatic oscillations, and levels of standing variation currently available for selection and adaptation in a terrestrial foundation species.     

Evolutionary and experimental analyses of inorganic phosphate transporter PiT family reveals two related signature sequences harboring highly conserved aspartic acids critical for sodium-dependent phosphate transport function of human PiT2

The mammalian members of the inorganic phosphate (P(i)) transporter (PiT) family, the type III sodium-dependent phosphate (NaP(i)) transporters PiT1 and PiT2, have been assigned housekeeping P(i) transport functions and are suggested to be involved in chondroblastic and osteoblastic mineralization and ectopic calcification. The PiT family members are conserved throughout all kingdoms and use either sodium (Na+) or proton (H+) gradients to transport P(i). Sequence logo analyses revealed that independent of their cation dependency these proteins harbor conserved signature sequences in their N- and C-terminal ends with the common core consensus sequence GANDVANA. With the exception of 10 proteins from extremophiles all 109 proteins analyzed carry an aspartic acid in one or both of the signature sequences. We changed either of the highly conserved aspartates, Asp28 and Asp506, in the N- and C-terminal signature sequences, respectively, of human PiT2 to asparagine and analyzed P(i) uptake function in Xenopus laevis oocytes. Both mutant proteins were expressed at the cell surface of the oocytes but exhibited knocked out NaP(i) transport function. Human PiT2 is also a retroviral receptor and we have previously shown that this function can be exploited as a control for proper processing and folding of mutant proteins. Both mutant transporters displayed wild-type receptor functions implying that their overall architecture is undisturbed. Thus the presence of an aspartic acid in either of the PiT family signature sequences is critical for the Na+-dependent P(i) transport function of human PiT2. The conservation of the aspartates among proteins using either Na+- or H+-gradients for P(i) transport suggests that they are involved in H+-dependent P(i) transport as well. Current results favor a membrane topology model in which the N- and C-terminal PiT family signature sequences are positioned in intra- and extracellular loops, respectively, suggesting that they are involved in related functions on either side of the membrane. The present data are in agreement with a possible role of the signature sequences in translocation of cations.