Seed dispersal in a patchy environment with global age dependence

A model of seed population dynamics proposed by S. A. Levin, A. Hastings, and D. Cohen is presented and analyzed. With the environment considered as a mosaic of patches, patch age is used along with time as an independent variable. Local dynamics depend not only on the local state, but also on the global environment via dispersal modelled by an integral over all patch ages. Basic technical properties of the time varying solutions are examined; necessary and sufficient conditions for nontrivial steady states are given; and general sufficient conditions for global asymptotic stability of these steady states are established. Primary tools of analysis include a hybrid Picard iteration, fixed point methods, monotonicity of solution structure, and upper and lower solutions for differential equations.This work was supported in part by National Science Foundation Grants MCS-7903497 and MCS-790349701     

Global stability of stationary solutions to a nonlinear diffusion equation in phytoplankton dynamics

We consider a nonlinear diffusion equation proposed by Shigesada and Okubo which describes phytoplankton growth dynamics with a selfs-hading effect.We show that the following alternative holds: Either (i) the trivial stationary solution which vanishes everywhere is a unique stationary solution and is globally stable, or (ii) the trivial solution is unstable and there exists a unique positive stationary solution which is globally stable. A criterion for the existence of positive stationary solutions is stated in terms of three parameters included in the equation.     

Global asymptotic stability for a vector disease model with spatial spread

Summary We analyze the global behaviour of a vector disease model which involves spatial spread and hereditary effects. This model can be applied to investigate growth and spread of malaria. No immunization is considered. We prove that, if the recovery rate is less than or equal to a threshold value, the disease dies out, otherwise the infectious people density tends to a homogeneous distribution. Our results follow using contracting convexes techniques and agree with the results given by K. L. Cooke for the model without diffusion.Work supported by C.N.R., Grant No. 79.00696.01.     

Existence and stability of periodic travelling wave solutions to Nagumo's nerve equation

Summary Nagumo's nerve conduction equation has a one-parameter family of spatially periodic travelling wave solutions. First, we prove the existence of these solutions by using a topological method. (There are some exceptional cases in which this method cannot be applied in showing the existence.) A periodic travelling wave solution corresponds to a closed orbit of a third-order dynamical system. The Poincaré index of the closed orbit is determined as a direct consequence of the proof of the existence. Second, we prove that the periodic travelling wave solution is unstable if the Poincaré index of the corresponding closed orbit is + 1. By using this result, together with the result of the author's previous paper, it is concluded that the slow periodic travelling wave solutions are always unstable. Third, we consider the stability of the fast periodic travelling wave solutions. We denote by L(c) the spatial period of the travelling wave solution with the propagation speed c. It is shown that the fast solution is unstable if its period is close to L_min, the minimum of L(c).     

Mutually synchronized relaxation oscillators as prototypes of oscillating systems in biology

Summary This paper discusses the analogy between phenomena in populations of coupled biological oscillators and the behaviour of systems of synchronized mathematical oscillators. Frequency entrainment in a set of coupled relaxation oscillators is investigated with perturbation methods. This analysis leads to quantitative results for entrainment and explains phenomena such as travelling waves in systems of spatially distributed oscillators.     

Shapes and functions of species–area curves (II): a review of new models and parameterizations

This paper has extended and updated my earlier list and analysis of candidate models used in theoretical modelling and empirical examination of species–area relationships (SARs). I have also reviewed trivariate models that can be applied to include a second independent variable (in addition to area) and discussed extensively the justifications for fitting curves to SARs and the choice of model. There is also a summary of the characteristics of several new candidate models, especially extended power models, logarithmic models and parameterizations of the negative-exponential family and the logistic family. I have, moreover, examined the characteristics and shapes of trivariate linear, logarithmic and power models, including combination variables and interaction terms. The choice of models according to best fit may conflict with problems of non-normality or heteroscedasticity. The need to compare parameter estimates between data sets should also affect model choice. With few data points and large scatter, models with few parameters are often preferable. With narrow-scale windows, even inflexible models such as the power model and the logarithmic model may produce good fits, whereas with wider-scale windows where inflexible models do not fit well, more flexible models such as the second persistence (P2) model and the cumulative Weibull distribution may be preferable. When extrapolations and expected shapes are important, one should consider models with expected shapes, e.g. the power model for sample area curves and the P2 model for isolate curves. The choice of trivariate models poses special challenges, which one can more effectively evaluate by inspecting graphical plots.     

Electrostatics of the FeS clusters in respiratory complex I

Respiratory complex I couples the transfer of electrons from NADH to ubiquinone and the translocation of protons across the mitochondrial membrane. A detailed understanding of the midpoint reduction potentials (E_m) of each redox center and the factors which influence those potentials are critical in the elucidation of the mechanism of electron transfer in this enzyme. We present accurate electrostatic interaction energies for the iron-sulfur (FeS) clusters of complex I to facilitate the development of models and the interpretation of experiments in connection to electron transfer (ET) in this enzyme. To calculate redox titration curves for the FeS clusters it is necessary to include interactions between clusters, which in turn can be used to refine E_m values and validate spectroscopic assignments of each cluster. Calculated titration curves for clusters N4, N5, and N6a are discussed. Furthermore, we present some initial findings on the electrostatics of the redox centers of complex I under the influence of externally applied membrane potentials. A means of determining the location of the FeS cofactors within the holo-complex based on electrostatic arguments is proposed. A simple electrostatic model of the protein/membrane system is examined to illustrate the viability of our hypothesis.     

Modelling and analyzing density-dependent population processes: Comparison between wild and laboratory strains of the bean weevil,Callosobruchus chinensis (L.)

Three models were constructed for analyzing the population characteristics ofC. chinensis on stored beans; model A describing the whole reproductive process with a single equation, model B describing the three age-specific processes (oviposition, egg survival and larval survival) with separate equations, and model C which describes all these processes not for the whole habitat but for the individual beans comprizing it. The logit equation was employed here as a common basis to describe the density-response relationship involved. All three models showed very good fit to the experimental data obtained for both laboratory and wild strains of the weevil. The parameter values characterizing the population dynamics were, however, widely different between the two strains; the laboratory one which had been reared for some 500 generations showed significantly higher reproductive capacity, less sensitive and gentler response to crowding in both adult and egg stages, and more uniform egg distribution among individual beans, as compared with the wild strain newly introduced. Sensitivity analyses using these models suggested that these changes in population characteristics have been attained by the process of domestication or adaptation to stable laboratory conditions through a long period of time. This process seemed in effect to have optimized the population's performances in the laboratory environment. Evolutionary significance of such optimization was discussed with reference to the selection pressure which may have acted upon individuals.     

Complex Genetic Effects on Early Vegetative Development Shape Resource Allocation Differences Between Arabidopsis lyrata Populations

Costs of reproduction due to resource allocation trade-offs have long been recognized as key forces in life history evolution, but little is known about their functional or genetic basis. Arabidopsis lyrata, a perennial relative of the annual model plant A. thaliana with a wide climatic distribution, has populations that are strongly diverged in resource allocation. In this study, we evaluated the genetic and functional basis for variation in resource allocation in a reciprocal transplant experiment, using four A. lyrata populations and F₂ progeny from a cross between North Carolina (NC) and Norway parents, which had the most divergent resource allocation patterns. Local alleles at quantitative trait loci (QTL) at a North Carolina field site increased reproductive output while reducing vegetative growth. These QTL had little overlap with flowering date QTL. Structural equation models incorporating QTL genotypes and traits indicated that resource allocation differences result primarily from QTL effects on early vegetative growth patterns, with cascading effects on later vegetative and reproductive development. At a Norway field site, North Carolina alleles at some of the same QTL regions reduced survival and reproductive output components, but these effects were not associated with resource allocation trade-offs in the Norway environment. Our results indicate that resource allocation in perennial plants may involve important adaptive mechanisms largely independent of flowering time. Moreover, the contributions of resource allocation QTL to local adaptation appear to result from their effects on developmental timing and its interaction with environmental constraints, and not from simple models of reproductive costs.