Resident-invader dynamics of similar strategies in fluctuating environments

Journal of Mathematical Biology - We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of...     

Strategy selection in structured populations

Evolutionary game theory studies frequency dependent selection. The fitness of a strategy is not constant, but depends on the relative frequencies of strategies in the population. This type of evolutionary dynamics occurs in many settings of ecology, infectious disease dynamics, animal behavior and social interactions of humans. Traditionally evolutionary game dynamics are studied in well-mixed populations, where the interaction between any two individuals is equally likely. There have also been several approaches to study evolutionary games in structured populations. In this paper we present a simple result that holds for a large variety of population structures. We consider the game between two strategies, A and B, described by the payoff matrix . We study a mutation and selection process. For weak selection strategy A is favored over B if and only if σa+b>c+σd. This means the effect of population structure on strategy selection can be described by a single parameter, σ. We present the values of σ for various examples including the well-mixed population, games on graphs, games in phenotype space and games on sets. We give a proof for the existence of such a σ, which holds for all population structures and update rules that have certain (natural) properties. We assume weak selection, but allow any mutation rate. We discuss the relationship between σ and the critical benefit to cost ratio for the evolution of cooperation. The single parameter, σ, allows us to quantify the ability of a population structure to promote the evolution of cooperation or to choose efficient equilibria in coordination games.     

Density dependence,population persistence,and largely futile arguments

Wolda and Dennis (1993) suggest that no valid conclusions about population regulation can be drawn on the basis of statistical tests of density dependence in time series data of population abundance. They give some examples in which a population persists even if it is not regulated by a density-dependent process: a sequence of independent, identically distributed random variables, the numbers of the migrant moth Autographa gamma in Britain, annual rainfall data. We suggest that such time series data may show persistence because of a static constraint, which compels the numbers to remain within finite, positive limits, or to fit some prescribed distribution. But this mechanism can explain persistence in a biological population only when the population represents a sample from a regulated population (the case of A. gamma). We also comment on some suggestions made by Wolda and Dennis (1993) concerning the general value of statistical tests of density dependence, frequency of delayed versus non-delayed density dependence in natural populations, relative performance of different kinds of insect traps in sampling local populations, and the wider issue of how ecologists are likely to make progress in the study of population dynamics.     

Regular and chaotic behaviour of cardiac cells stimulated at frequencies between 2 and 20 Hz

We measured the non-linear dynamics of cardiac action potentials by varying the stimulation frequency from 2 to 20 Hz and obtained the following results: (i) When the fast Na⁺ current initiated the action potentials (fast action potentials) periodicity was maintained, i.e. the pattern of action potentials repeated after a finite number of stimuli. Chaotic sequences were absent. The transition from one sequence to the next occurred as a devil's staircase. (ii) When, however, the slow Ca²⁺ current initiated the action potentials (slow action potentials), we observed chaos, i.e. fully irregular behaviour, as well as bifurcations. (iii) Our results confirm the supposition that the global dynamics of cardiac cells can be well described by simple one-dimensional maps which predict these two kinds of behaviour.     

Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree

We present a general framework for modelling adaptive trait dynamics in which we integrate various concepts and techniques from modern ESS-theory. The concept of evolutionarily singular strategies is introduced as a generalization of the ESS-concept. We give a full classification of the singular strategies in terms of ESS-stability, convergence stability, the ability of the singular strategy to invade other populations if initially rare itself, and the possibility of protected dimorphisms occurring within the singular strategy's neighbourhood. Of particular interest is a type of singular strategy that is an evolutionary attractor from a great distance, but once in its neighbourhood a population becomes dimorphic and undergoes disruptive selection leading to evolutionary branching. Modelling the adaptive growth and branching of the evolutionary tree can thus be considered as a major application of the framework. A haploid version of Levene's soft selection model is developed as a specific example to demonstrate evolutionary dynamics and branching in monomorphic and polymorphic populations.     

Population dynamics and production of Acanthocyclops robustus (Sars) and Mesocyclops leuckarti (Claus) in Tjeukemeer

The population dynamics and production ecology of the two dominant copepod species, Acanthocyclops robustus and Mesocyclops leuckarti, in Tjeukemeer (The Netherlands) were studied for three successive years. Since copepods in Tjeukemeer show continuous recruitment, a population dynamics model INSTAR was developed and used to integrate field data on population density, population structure and fecundity and laboratory data on development rates and length-weight relationships. The pattern of size specific mortality indicated that both invertebrate and vertebrate predation were important in the regulation of population numbers.     

Action Potential Onset Dynamics and the Response Speed of Neuronal Populations

The result of computational operations performed at the single cell level are coded into sequences of action potentials (APs). In the cerebral cortex, due to its columnar organization, large number of neurons are involved in any individual processing task. It is therefore important to understand how the properties of coding at the level of neuronal populations are determined by the dynamics of single neuron AP generation. Here, we analyze how the AP generating mechanism determines the speed with which an ensemble of neurons can represent transient stochastic input signals. We analyze a generalization of the -neuron, the normal form of the dynamics of Type-I excitable membranes. Using a novel sparse matrix representation of the Fokker-Planck equation, which describes the ensemble dynamics, we calculate the transmission functions for small modulations of the mean current and noise noise amplitude. In the high-frequency limit the transmission function decays as ^–, where surprisingly depends on the phase _s at which APs are emitted. If at _s the dynamics is insensitive to external inputs, the transmission function decays as (i) ^–3 for the case of a modulation of a white noise input and as (ii) ^–2 for a modulation of the mean input current in the presence of a correlated and uncorrelated noise as well as (iii) in the case of a modulated amplitude of a correlated noise input. If the insensitivity condition is lifted, the transmission function always decays as ^–1, as in conductance based neuron models. In a physiologically plausible regime up to 1 kHz the typical response speed is, however, independent of the high-frequency limit and is set by the rapidness of the AP onset, as revealed by the full transmission function. In this regime modulations of the noise amplitude can be transmitted faithfully up to much higher frequencies than modulations in the mean input current. We finally show that the linear response approach used is valid for a large regime of stimulus amplitudes.     

Population dynamics and production of Daphnia hyalina Leydig and Daphnia cucullata Sars in Tjeukemeer

As part of a research programme on the food chains in Tjeukemeer, the Daphnia hyalina and Daphnia cucullata populations were studied for three successive years. To analyse the factors regulating the production of these two species, their population parameters (density, size distribution, fecundity) and population dynamics (birth rate, mortality rate) were studied and related to environmental factors. Since Daphnia in Tjeukemeer shows continuous recruitment, the population dynamics model INSTAR was developed and used to integrate field data with laboratory data on development rates and length-weight relationships. The dynamics of the Daphnia species are mainly regulated by temperature and fish predation, the latter affects both birth rate and mortality. Total annual Daphnia production was 3.1–6.9 g org. dry wt M^–2, and annual P/B ratio ranged from 25 to 40 for D. cucullata and from 45 to 49 for D. hyalina.     

Fixation probabilities in evolutionary game dynamics with a two-strategy game in finite diploid populations

Fixation processes in evolutionary game dynamics in finite diploid populations are investigated. Traditionally, frequency dependent evolutionary dynamics is modeled as deterministic replicator dynamics. This implies that the infinite size of the population is assumed implicitly. In nature, however, population sizes are finite. Recently, stochastic processes in finite populations have been introduced in order to study finite size effects in evolutionary game dynamics. One of the most significant studies on evolutionary dynamics in finite populations was carried out by Nowak et al. which describes “one-third law” [Nowak, et al., 2004. Emergence of cooperation and evolutionary stability in finite populations. Nature 428, 646-650]. It states that under weak selection, if the fitness of strategy α is greater than that of strategy β when α has a frequency , strategy α fixates in a β-population with selective advantage. In their study, it is assumed that the inheritance of strategies is asexual, i.e. the population is haploid. In this study, we apply their framework to a diploid population that plays a two-strategy game with two ESSs (a bistable game). The fixation probability of a mutant allele in this diploid population is derived. A “three-tenth law” for a completely recessive mutant allele and a “two-fifth law” for a completely dominant mutant allele are found; other cases are also discussed.     

Plucking or cutting Gelidium sesquipedale? A demographic simulation of harvest impact using a population projection matrix model.

A matrix model describes the annual dynamics of a commercial (harvested by plucking) Gelidium sesquipedale population off Cape Espichel, Portugal. Vital rates were measured from a frond population divided into size classes; annual transition probabilities among them were calculated. Transition probabilities under harvest by cutting are derived by assuming that all harvested fronds are cut to the first size class, and none are plucked. Simulations of the annual population dynamics for harvest by both plucking and cutting are used to assess which harvest strategy will optimize yields. Assuming the same efficiency for both strategies, cutting fronds to 7 cm (as mechanical harvesters do) results in a higher population growth rate ( = 1.08 to 1.35) than occurs with the plucking technique ( = 0.85). Simulations of population recovery show the number of fronds in each size class available the next harvest season will be higher when cut than plucked. This model can also optimize yields by predicting the more efficient season opening, and harvesting cutting height.     

The dynamic properties of melittin in solution

Molecular dynamics simulations are described for the peptide melittin. The atomic trajectories are calculated both with normal potential energy functions and with additional distance restraints deduced from nuclear Overhauser effects observed in NMR experiments. The results are compared with NRM data on coupling constants and amide exchange rates and witt B-factors from X-ray crystallography. The observed correlations between experiment and molecular dynamics simulations suggest a relatively mobile C-terminus and relatively high flexibility around residue 11. It is noted that the high conformational variation around residue 11 is due in part to the presence of a proline at position 14 which results in a missing H-bond in the largely -helical structure. It is also noted that a proline is a common feature of many putative membrane spanning helices. A role for such prolines is suggested.     

Comparative ecology of Gammarus pulex (L.) and Asellus aquaticus (L.) I: population dynamics and microdistribution

Gammarus pulex and Asellus aquaticus generally occupy different zones in rivers; the former occurs in upper reaches but is replaced by the latter in lower reaches. Microdistribution and life-history patterns of G. pulex and A. aquaticus in sympatry and allopatry, were analyzed. Both species exhibited similar patterns of microhabitat selection, with larger individuals associated predominantly with large-sized substratum particles, and juveniles with weed. Coexisting populations of G. pulex and A. aquaticus had similar densities and population dynamics. Within each species, differences in population dynamics of allopatric and sympatric populations were observed. Although variation in population dynamics of G. pulex may be explained in terms of competition between the two species, the evidence is weak and equivocal. Differences in the dynamics of the two A. aquaticus populations were possibly a consequence of coal-mine and organic pollution, reducing the survival of offspring in the allopatric population.     

Optimal patch use and metapopulation dynamics

Summary We compared the metapopulation dynamics of predator—prey systems with (1) adaptive global dispersal, (2) adaptive local dispersal, (3) fixed global dispersal and (4) fixed local dispersal by predators. Adaptive dispersal was modelled using the marginal value theorem, such that predators departed patches when the instantaneous rate of prey capture was less than the long-term rate of prey capture averaged over all patches, scaled to the movement time between patches. Adaptive dispersal tended to stabilize metapopulation dynamics in a similar manner to conventional fixed dispersal models, but the temporal dynamics of adaptive dispersal models were more unpredictable than the smooth oscillations of fixed dispersal models. Moreover, fixed and adaptive dispersal models responded differently to spatial variation in patch productivity and the degree of compartmentalization of the system. For both adaptive dispersal and fixed dispersal models, localized (stepping-stone) dispersal was more strongly stabilizing than global (island) dispersal. Variation among predators in the probability of dispersal in relation to local prey density had a strong stabilizing influence on both within-patch and metapopulation dynamics. These results suggest that adaptive space use strategies by predators could have important implications for the dynamics of spatially heterogeneous trophic systems.     

Molecular dynamics simulations of ribonuclease T1

Molecular dynamics simulations in vacuum and with a water sphere around the active site were performed on the 2GMP-RNase T1 complex. The presence of water led to the maintenance of the 2-GMP-RNase T1 interactions as compared to the X-ray structure, including the hydrogen bonds implicated in the enzyme-inhibitor recognition process. The sidechain of His92 in the molecular dynamics water simulation, however, hydrogen bonds directly to the phosphate of 2GMP in contrast to the X-ray structure but in support of the role of that residue in the enzyme's catalytic mechanism. Fluctuations of activesite residues are not strongly influenced by water, possibly owing to the exclusion of water by the bound 2GMP, which did show an increase in mobility. Analysis of the 2GMP-RNase T1 interactions versus time reveal an equilibrium fluctuation in the presence of water, leading to a less favorable 2GMP-RNase T1 interaction energy, suggesting a possible relationship between picosecond fluctuations and inhibitor dissociation occurring in the millisecond time domain.Abbreviations RNase T1 Ribonuclease T1 (EC.3.1.27.3) - 2GMP Guanosine-2-monophosphate - SBS Stochastic Bondary Simulation - VS Vacuum Simulation - MD Molecular Dynamics     

Effects of noise on some dynamical models in ecology

We investigate effects of random perturbations on the dynamics of one-dimensional maps (single species difference equations) and of finite dimensional flows (differential equations for n species). In particular, we study the effects of noise on the invariant measure, on the correlation dimension of the attractor, and on the possibility of detecting the nonlinear deterministic component by applying reconstruction techniques to the time series of population abundances. We conclude that adding noise to maps with a stable fixed-point obscures the underlying determinism. This turns out not to be the case for systems exhibiting complex periodic or chaotic motion, whose essential properties are more robust. In some cases, adding noise reveals deterministic structure which otherwise could not be observed. Simulations suggest that similar results hold for flows whose attractor is almost two-dimensional.     

How to analyze long-term insect population dynamics under climate change: 50-year data of three insect pests in paddy fields

We can precisely predict the future dynamics of populations only if we know the underlying mechanism of population dynamics. Long-term data are important for the elucidation of such mechanisms. In this article we analyze the 50-year dynamics of annual light-trap catches of three insect pest species living in paddy fields in Japan: the rice stem borer, Chilo suppressalis (Walker) (Lepidoptera: Pyralidae); the green rice leafhopper, Nephotettix cincticeps (Uhler) (Hemiptera: Deltocephalidae); and the small brown planthopper, Laodelphax striatellus (Fallén) (Hemiptera: Delphacidae). We separate the long-term dynamics into two components by using locally weighted scatterplot smoothing: (1) the underlying dynamics of populations, and (2) the influence of the past changes in the environment. The former component is analyzed by response surface analysis and vector autoregression to evaluate the nonlinearity of density-dependence and the inter-specific influence of density, respectively. On the basis of these analyses, we perform the state-space model analyses. The state-space model selected by Akaikes information criterion indicates that the observed number of light-trap catches of C. suppressalis and N. cincticeps in summer increases with increasing temperatures in the previous winter. It also indicates that the influence of temperature is not carried over to the next year. We utilize the selected model to predict the impact of global warming on these species, by substituting the temperature predicted by a general circulation model.     

Characterizing the symmetric equilibrium of multi-strain host-pathogen systems in the presence of cross immunity

We investigate the population dynamics of host-pathogen systems in which the pathogen has a potentially arbitrary number of antigenically distinct strains interacting via cross-immunity. The interior equilibrium configuration of the symmetric multiple strain SIR model with cross-immunity is characterized. We develop an efficient iterative method for numerically solving the equilibrium equation together with a number of informative analytical approximations to the full solution. Equilibrium properties are studied as a function of the number of strains, reproduction number, infectious period, and cross immunity profile. We establish that the prevalence in the system increases monotonically with the number of strains and the reduction in cross immunity. Moreover, we demonstrate the existence of a phase transition separating high prevalence and low prevalence parameter regions, with the critical point being defined by R₀1, where is the level of cross-immunity and R₀ is the reproduction number. Above the threshold, prevalence saturates with increasing numbers of strains as a result of the inclusion of prohibition of co-infection in the model. Below the threshold, prevalence saturates much more rapidly as the number of strains increases - indicating that when cross-protection is sufficiently intense, the selective advantage for a pathogen to increase its diversity is substantially less than in the threshold region. Similarly, there is limited benefit to increased transmissibility (or decreased cross-immunity) both for the high and low diversity pathogen systems compared with systems at the threshold R₀1 where small increase in transmissibility can result in significant increase in prevalence.     

The transmembrane domain of Neu in a lipid bilayer: molecular dynamics simulations

The results of full-atom molecular dynamics simulations of the transmembrane domains (TMDs) of both native, and Glu⁶⁶⁴-mutant (either protonated or unprotonated) Neu in an explicit fully hydrated dimyristoylphosphatidylcholine (DMPC) lipid bilayer are presented. For the native TMD peptide, a 10.05 ns trajectory was collected, while for the mutant TMD peptides 5.05 ns trajectories were collected for each. The peptides in all three simulations display stable predominantly -helical hydrogen bonding throughout the trajectories. The only significant exception occurs near the C-terminal end of the native and unprotonated mutant TMDs just outside the level of the lipid headgroups, where -helical hydrogen bonding develops, introducing a kink in the backbone structure. However, there is no indication of the formation of a bulge within the hydrophobic region of either native or mutant peptides. Over the course of the simulation of the mutant peptide, it is found that a significant number of water molecules penetrate the hydrophobic region of the surrounding lipid molecules, effectively hydrating Glu⁶⁶⁴. If the energy cost of such water penetration is significant enough, this may be a factor in the enhanced dimerization affinity of Glu⁶⁶⁴-mutant Neu.     

Population structure and dynamics of an evergreen shade herb, Ainsliaea apiculata (Asteraceae), with special reference to herbivore effects

The population structure and dynamics of Ainsliaea apiculata, a forest understory evergreen herb widely distributed in Japan, was examined in a Chamaecyparis obtusa forest in Ibaraki Prefecture, central Japan (36°51N, 140°33E; 750 m a.s.l.). The mean population growth rate () calculated from the transition matrices for 4 years was 0.69 per year, predicting that the population size will decrease remarkably. There was a significant positive correlation between the survival of old leaves and the growth of new shoots in the following year. The shoots, especially new leaves, were damaged severely by herbivores (caterpillars of Leioptilus sp.). The survival rate of leaves formed in the previous spring to the next spring was remarkably low (41–54%). The growth of new shoots depended mainly on the reserves contained in old shoots, especially those in old leaves. New shoots of A. apiculata began to develop in spring, even though they were formed in autumn of the previous year. A defoliation experiment also showed that the removal of old shoots at the beginning of the growing season significantly inhibited the growth of new shoots. Damage to old shoots by herbivores severely influenced the growth and population dynamics of A. apiculata.     

Alternate-site isotopic labeling of ribonucleotides for NMR studies of ribose conformational dynamics in RNA

Heteronuclear NMR spin relaxation studies of conformational dynamics are coming into increasing use to help understand the functions of ribozymes and other RNAs. Due to strong magnetic interactions within the ribose ring, however, these studies have thus far largely been limited to ¹³C and ¹⁵N resonances on the nucleotide base side chains. We report here the application of the alternate-site ¹³C isotopic labeling scheme, pioneered by LeMaster for relaxation studies of amino acid side chains, to nucleic acid systems. We have used different strains of E. coli to prepare mononucleotides containing ¹³C label in one of two patterns: Either C1′ or C2′ in addition to C4′, termed (1′/2′,4′) labeling, or nearly complete labeling at the C2′ and C4′ sites only, termed (2′,4′) labeling. These patterns provide isolated H spin systems on the labeled carbon atoms and thus allow spin relaxation studies without interference from scalar or dipolar coupling. Using relaxation studies of AMP dissolved in glycerol at varying temperature to produce systems with correlation times characteristic of different size RNAs, we demonstrate the removal of errors due to interaction in T ₁ measurements of larger nucleic acids and in T _1ρ measurements in RNA molecules. By extending the applicability of spin relaxation measurements to backbone ribose groups, this technology should greatly improve the flexibility and completeness of NMR analyses of conformational dynamics in RNA.