Pattern Formation,Long-Term Transients,and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator–prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing–Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system’s dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.     

On the sensitive dependence on initial conditions of the dynamics of networks of spiking neurons

We have previously formulated an abstract dynamical system for networks of spiking neurons and derived a formal result that identifies the criterion for its dynamics, without inputs, to be “sensitive to initial conditions”. Since formal results are applicable only to the extent to which their assumptions are valid, we begin this article by demonstrating that the assumptions are indeed reasonable for a wide range of networks, particularly those that lack overarching structure. A notable aspect of the criterion is the finding that sensitivity does not necessarily arise from randomness of connectivity or of connection strengths, in networks. The criterion guides us to cases that decouple these aspects: we present two instructive examples of networks, one with random connectivity and connection strengths, yet whose dynamics is insensitive, and another with structured connectivity and connection strengths, yet whose dynamics is sensitive. We then argue based on the criterion and the gross electrophysiology of the cortex that the dynamics of cortical networks ought to be almost surely sensitive under conditions typically found there. We supplement this with two examples of networks modeling cortical columns with widely differing qualitative dynamics, yet with both exhibiting sensitive dependence. Next, we use the criterion to construct a network that undergoes bifurcation from sensitive dynamics to insensitive dynamics when the value of a control parameter is varied. Finally, we extend the formal result to networks driven by stationary input spike trains, deriving a superior criterion than previously reported. Action Editor: John Rinzel     

A mathematical model for the phase of sexual reproduction in monogonont rotifers

Recently, the optimal sex allocation in monogonont rotifers is studied in [1], and, as a closely related question, the relative frequencies of the relevant types of mictic females. The authors focus on the evolution of the age at which young mictic females lose their fertilization susceptibility and they address the threshold age of fertilization that maximizes resting egg production. Assuming that a stationary population is achieved, with stable age distribution, they obtain their results, without knowing the stationary population. Our aim is to study this problem in the framework of the theory of nonlinear age-dependent population dynamics developed by G. F. Webb in [13], which is more appropriate from the mathematical point of view and permits to us to obtain analytically the stationary population and consequently it is analytically shown that a threshold age of fertilization equal to the age of maturation is not an ESS, despite the fact that then the production of resting eggs is maximum, which has been obtained by simulation in [1]. Received: 30 June 1998 / Revised version: 17 May 1999 / Published on line: 5 May 2000     

Demographic traits of understory trees and population dynamics of aPicea-Abies forest in Taisetsuzan National Park, northern Japan

The size structure transition matrices ofPicea jezoensis, Picea glehnii andAbies sachalinensis of a sub-boreal forest in Hokkaido, northern Japan were constructed based on the demography of each species (Picea jezoensis andPicea glehnii were dealt with together asPicea) during a 4-year period. Two types of matrices, density-independent and density-dependent population dynamics models, were investigated for evaluating the ‘waiting pattern’ betweenPicea spp. andA. sachalinensis. For the density-dependent model, it was assumed that the demographic traits of understory trees, the recruitment rate, the understory mortality rate and the transition probability from the understory to canopy stages, were regulated by the one-sided competitive effect of canopy trees. The observed size structure ofPicea was almost consistent with the stationary size structure obtained in both the density-independent and the density-dependent models, whereas the observed size structure ofA. sachalinensis was not realized in the two models. The effects of both the transition probability from the understory to canopy stages and the recruitment rate on the dynamics of canopy trees were investigated. ForPicea, two parameters—recruitment rate (e _i ) and transition probability from the understory to canopy stages-exponentially affected the dynamics of canopy trees. In contrast, forAbies sachalinensis, the two parameters affected linearly the dynamics of canopy trees. In conclusion, the population dynamics ofPicea andA. sachalinensis was determined by the parameters of the recruitment rate and the transition probability from the understory to canopy stages, relating to waiting patterns of understory trees for future gap formation. InPicea, the demographic parameters of understory trees intensively regulated the dynamics of canopy trees if compared withA. sachalinensis, suggesting that the performance of understory trees plays a key role in the population dynamics ofPicea. This reflects the growth pattern of understory trees in the regeneration of the two species.     

A search for maximum species abundances in ecological communities under conditional diversity optimization

We study a multispecies community of autotrophic microorganisms which grow in a batch culture regime with several perfectly complementary resources. A basic hypothesis is that a stationary phase of the polyculture corresponds to a maximum diversity under the constraints having the meaning of matter conservation laws. The corresponding conditional extremum problem is studied in detail. It is shown that a unique solution to this problem—a “species structure formula”—adequately describes the experimental data. We prove a number of strict statements concerning the domain of definition and maxima of the obtained solutions. These statements find an adequate interpretation as limitation principles in ecology and in the problems of community structure control.     

Dynamics of IAA and cytokinins in flower tissues of transgenic tobacco mutant plants with mutant phenotype

A comparative analysis of IAA and total cytokinin contents at various developmental stages of floral morphogenesis (I–XII) in transgenic tobacco (Nicotiana tabacum L.) plants with mutant phenotypes was performed. Disturbances in flower structure were correlated with changes in the content and dynamics of these phytohormones. In plants with the phenotype “wavy corolla”, IAA content increased by three times as compared with control at stage VIII; in plants with the phenotype “longostyly”, IAA content increased gradually from VIII to XII stage and sharply (by nine times) at stage XII. In plants with the phenotype “wavy corolla”, total content of cytokinins was lower than in control at all developmental stages (except for stage I), whereas their dynamics coincided with that in control flowers. In plants with the phenotype “longostyly”, cytokinin content was twice lower than in control at early developmental stages, whereas it oscillated at later stages, approaching to control value at stage XII. It is supposed that differences observed could result from T-DNA insertion into plant own genes, which led to changes in the content and dynamics of phytohormones in developing flowers and production of anomalies in the flower structure.     

Fast inactivating potassium current in cultured hippocampal neurons

Using a voltage-clamp whole-cell technique, we studied transmembrane currents in hippocampal neurons after their long-lasting cultivation. Based on the activational characteristics and data on pharmacological sensitivity, we isolated and described an A-type voltage-activated fast inactivating potassium current (FIPC). This transient FIPC was activated at −50… −40 mV and was rather sensitive to 4-aminopyridine (4-AP). Extracellular application of 5 mM 4-AP decreased the FIPC amplitude by 75%, while application of 10 mM tetraethylammonium evoked no significant changes in it. Kinetics of FIPC activation could be described by one exponent in the fourth degree. With variations of the membrane potential from −40 to 30 mV, the activation time constant varied from 2.8 to 1.5 msec. Inactivation kinetics was described by one exponent with the time constant varying from 37 msec at −45 mV to 18 msec at 40 mV. Stationary activation and inactivation curves could be described by Boltzmann's equation; a half value of the level of stationary inactivation was reached at −80 mV, while stationary activation was attained at −25 mV. Kinetics of deinactivation (from stationary inactivation) was monoexponential with the time constant of 41 msec. It is supposed that FIPC through the membrane of hippocampal neurons is provided by the type Kv4.2 potassium channels.     

The impact of population dynamics on Y-chromosome microsatellite polymorphism. Mathematical modeling

In this article, we use mathematical modeling to study the impact of population dynamics on Y-chromosome STR-polymorphism accumulation in two independently evolving populations, namely, on the changes in genetic distance between the populations. Comparative analysis using two definitions of genetic distance—(δμ)² and ASD—shows that, in contrast to (δμ)², ASD is almost linearly dependent on time (except for sparse stationary populations, where deviations are observed). When the population numbers undergo oscillations, ASD proves to be smaller than that for stationary populations.     

How small are small mutation rates?

We consider evolutionary game dynamics in a finite population of size N. When mutations are rare, the population is monomorphic most of the time. Occasionally a mutation arises. It can either reach fixation or go extinct. The evolutionary dynamics of the process under small mutation rates can be approximated by an embedded Markov chain on the pure states. Here we analyze how small the mutation rate should be to make the embedded Markov chain a good approximation by calculating the difference between the real stationary distribution and the approximated one. While for a coexistence game, where the best reply to any strategy is the opposite strategy, it is necessary that the mutation rate μ is less than N ^−1/2exp[−N] to ensure that the approximation is good, for all other games, it is sufficient if the mutation rate is smaller than (N ln N)⁻¹. Our results also hold for a wide class of imitation processes under arbitrary selection intensity.     

Chemical Organisation Theory

Complex dynamical reaction networks consisting of many components that interact and produce each other are difficult to understand, especially, when new component types may appear and present component types may vanish completely. Inspired by Fontana and Buss (Bull. Math. Biol., 56, 1–64) we outline a theory to deal with such systems. The theory consists of two parts. The first part introduces the concept of a chemical organisation as a closed and self-maintaining set of components. This concept allows to map a complex (reaction) network to the set of organisations, providing a new view on the system’s structure. The second part connects dynamics with the set of organisations, which allows to map a movement of the system in state space to a movement in the set of organisations. The relevancy of our theory is underlined by a theorem that says that given a differential equation describing the chemical dynamics of the network, then every stationary state is an instance of an organisation. For demonstration, the theory is applied to a small model of HIV-immune system interaction by Wodarz and Nowak (Proc. Natl. Acad. USA, 96, 14464–14469) and to a large model of the sugar metabolism of E. Coli by Puchalka and Kierzek (Biophys. J., 86, 1357–1372). In both cases organisations where uncovered, which could be related to functions. Both authors contributed equally.     

Genome inhomogeneity is determined mainly by WW and SS dinucleotides

According to the hypothesis of the modular structure of DNA,genomes consist of modules of various nature which may differin statistical characteristics. Statistical analysis helps inrevealing the differences in statistical characteristics andpredicting the modular structure. In this connection the questionabout the contribution of each word of length l (l-tuple) tothe inhomogeneity of genetic text arises. The notion of stationary(i.e. relatively evenly distributed over a genome) versus non-stationaryl-tuples has been introduced previously. In this paper, thedinucleotide distributions for all long sequences from GenBankwere analyzed and it was shown that non-stationary dinucleotidesare closely associated with polyW and polyS tracts (W denotes‘weak’ nucleotides A or T, while S stands for the‘strong’ nucleotides G or C). Thus, genome inhomogeneityis shown to be determined mainly by AA, TT, GG, CC, AT, TA,GC and CG dinucleotides. It has been demonstrated that neither‘codon usage’ nor the ‘isochore model’can account for this phenomenon.     

Peculiarities of the beam-plasma discharge development in high-pressure gas and prospects of its application

The dynamics of a stationary beam-plasma discharge at the neutral gas pressure of 10⁻³-1 Torr is investigated. Three discharge modes have been revealed: two of them are characterized by novel properties and have not been studied as of yet. Dissipative instabilities with different increments have been found. The generation of ion flows has been studied and the experiments on “dry” cleaning of the metal specimen surfaces using them have been carried out. The pressure ranges in a neutral gas at which the discharge can be efficiently applied in nonequilibrium plasma chemistry are given.     

Critical conditions for phytoplankton blooms

We motivate and analyse a reaction—advection—diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton—water—light-system characterized by the same dimensionless parameters will show the same dynamics.     

Structure and dynamics of one-dimensional ionic solutions in biological transmembrane channels.

The structure and dynamics of solvated alkali metal cations in transmembrane channels are treated using the molecular dynamics simulation technique. The simulations are based on a modified Fischer-Brickmann model (Fischer, W., and J. Brickmann, 1983, Biophys. Chem., 18:323-337) for gramicidin A-type channels. The trajectories of all particles in the channel as well as two-dimensional pair correlation functions are analyzed. It is found from the analysis of the stationary simulation state that one-dimensional solvation complexes are formed and that the number of water molecules in the channel varies for different alkali metal cations.     

Distance decay of community dynamics in rocky intertidal sessile assemblages evaluated by transition matrix models

It is well known that the similarity in species composition between two communities decays with the geographic distance that separates them. It is thus likely that the similarity in the dynamics of two communities also decays with distance, because the distance–decay relationship is fundamental in nature. However, the distance–decay relationships of community dynamics have not yet been revealed. We used transition matrix models to evaluate distance–decay relationships of seasonal community dynamics (from spring to summer) in rocky intertidal sessile assemblages along the Pacific coast of Japan between 31°N and 43°N. We evaluated the distance–decay relationships of whole-community dynamics and of three dynamics-related components—recruitment, disturbance, and species interaction (competition and facilitation)—for communities separated by distances ranging from several meters to thousands of kilometers. The similarity of the recruitment dynamics among communities declined rapidly with distance within the fine spatial scale, but only moderately within larger scales. The similarity of the disturbance dynamics was independent of distance, and the similarity of species interaction declined slightly with increasing distance. The similarity of whole-community dynamics declined rapidly with distance at a fine spatial scale and moderately at larger scales. The fact that the distance–decay relationship of whole-community dynamics was similar to that of recruitment may suggest that recruitment processes are the most important determinant of spatial variability of community dynamics at our study sites during the study period.     

A stochastic model for head lice infections

We investigate the dynamics of head lice infections in schools, by consideringa model for endemic infection based on a stochastic SIS (susceptible-infected-susceptible) epidemic model, with the addition of an external source of infection. We deduce a range of properties of our model, including the length of a single outbreak of infection. We use the stationary distribution of the number of infected individuals, in conjunction with data from a recent study carried out in Welsh schools on the prevalence of head lice infections, and employ maximum likelihood methods to obtain estimates of the model parameters. A complication is that, for each school, only a sample of the pupils was checked for infection. Our likelihood function takes account of the missing data by incorporating a hypergeometric sampling element. We arrive at estimates of the ratios of the “within school” and “external source” transmission rates to the recovery rate and use these to obtain estimates for various quantities of interest.      

Volterra-Verhulst prey-predator systems with time dependent coefficients: Diffusion type approximation and periodic solutions

In treating the Volterra-Verhulst prey-predator system with time dependent coefficients, we ask how far this deterministic system represents or approximates the dynamics of the population evolving in a realistic environment which is stochastic in nature. We consider a stochastic system withsmall Gaussian noise type fluctuations. It is shown that the higher moments of the deviation of the deterministic system from the stochastic approach zero as the strength δ of the perturbation decays to zero. For any δ>0 and allT>0, ε>0, the sample population paths that stay within ε distance from the deterministic path during [0,T] form a collection of positive probability. In comparing the stationary distributions of the two systems, we show that the weak limits of those of the stochastic system form a subset of those of the deterministic system. This is in analogy with a result of May connected with the stability of the two systems. Plant and rodent populations possess periodic parameters andexhibit periodic behaivor. We establish theoretically this periodicity under periodicity conditions on the coefficients and perturbing random forces. We also establish a central limit property for the prey-predator system.     

A life‐history perspective on the demographic drivers of structured population dynamics in changing environments

Current understanding of life‐history evolution and how demographic parameters contribute to population dynamics across species is largely based on assumptions of either constant environments or stationary environmental variation. Meanwhile, species are faced with non‐stationary environmental conditions (changing mean, variance, or both) created by climate and landscape change. To close the gap between contemporary reality and demographic theory, we develop a set of transient life table response experiments (LTREs) for decomposing realised population growth rates into contributions from specific vital rates and components of population structure. Using transient LTREs in a theoretical framework, we reveal that established concepts in population biology will require revision because of reliance on approaches that do not address the influence of unstable population structure on population growth and mean fitness. Going forward, transient LTREs will enhance understanding of demography and improve the explanatory power of models used to understand ecological and evolutionary dynamics.     

Effect of predator presence on parental behavior of convict cichlid <Emphasis Type="Italic">Amatitlania nigrofasciata</Emphasis> (Perciformes,Cichlidae)

Results of study of the effect of presence of model predator on parental care in convict cichlid Amatitlania nigrofasciata are presented. Main differences in parental functions of male and female in the family group and their transformation in the case of a potential threat for the clutch and litter are considered. The dynamics of parental behavior throughout the period of parental care and the effect of the predator’s presence on this dynamics are analyzed. It is shown that predator’s presence exerts a complex transforming effect on the entire structure of parental behavior in the family group of the convict cichlid. Male behavior is distinguished by highest lability in the period after passing of juveniles to free swimming.