Competitive exclusion through reproductive interference

A simple differential equation model was developed to describe the competitive interaction that may occur between species through reproductive interference. The model has the form comparable to Volterra's competition equations, and the graphical analysis of the outcome of the two-species interaction based on its zero-growth isoclines proved that: (1) The possible outcome in this model, as in usual models of resource competition, is either stable coexistence of both species or gradual exclusion of one species by the other, depending critically upon the values of the activity overlapping coefficient c_ij; (2) but, for the same c_ij-values, competitive exclusion is much more ready to occur here than in resource competition; (3) and moreover, the final result of the competition is always dependent on the initial-condition due to its non-linear isoclines, i.e., even under the parameter condition that generally allows both species to coexist, an extreme bias in intial density to one species can readily cause subsequent complete exclusion of its counterparts. Thus, it may follow that the reproductive interference is likely to be working in nature as an efficient mechanism to bring about habitat partitioning in either time or space between some closely related species in insect communities, even though they inhabit heterogeneous habitats where resource competition rarely occurs so that they could otherwise attain steady coexistence.     

The random walk description for isotope exchange in a polypeptide

Isotope exchange in a polypeptide is considered from the point of view in which the boundary point between helix and coil regions of a polypeptide behaves like a weakly asymmetric random walker. We assume that the boundary point is reflected completely at the ends of a polypeptide. The equilibrium fraction of helix region is obtained under this assumption, and this is also confirmed by computer simulation. The experimental results of isotope exchange can be explained in this situation. On the other hand. the rate constant of exchange of a residue given by experiments can also be explained by another assumption, as considered before (M. Fujiwara and N. Saitô, Polym. J. 9 (1977) 625.), in which the nucleations of coil states take place in the helix region. Which of the two is of major importance is left to further studies.     

Age-dependent population diffusion with external constraint

We present a simple model for age dependent population diffusion when the dynamics is submitted to external constraints. Existence, uniqueness and dependence on the parameters of the solution are discussed.This work has been done within the framework of the cultural agreement between the Universities of Bordeaux and Rome     

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.     

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.