Cycles in cannibalistic egg-larval interactions

A model of a cannibalistic larval-egg interaction such as occurs in Tribolium is developed which leads to a system of nonlinear Volterra integral equations. I determine the local stability properties of the unique equilibrium point of the model. A Hopf bifurcation analysis shows that the model always undergoes a subcritical bifurcation when stability is lost. Numerical solutions confirm the presence of multiple attractors over a range of parameter values. The form of the cycles observed in the numerical solutions is analogous to that observed in laboratory populations of Tribolium.     

Convergence to equilibrium in a genetic model with differential viability between the sexes

A single locus, diallelic selection model with female and male viability differences is studied. If the variables are ratios of allele frequencies in each sex, a 2-dimensional difference equation describes the model. Because of the strong monotonicity of the resulting map, every initial genotypic structure converges to an equilibrium structure assuming that no equilibrium has eigenvalues on the unit circle.Partially supported by funds provided by a Science and Education Grant to the USDA-Forest Service, Southeastern Forest Experiment Station, Population Genetics of Forest Trees Research Unit, Raleigh, USASupported by a grant from the Max Kade Foundation, New York, USA     

The singularly perturbed Hodgkin-Huxley equations as a tool for the analysis of repetitive nerve activity

A qualitative analysis of the Hodgkin-Huxley model (Hodgkin and Huxley 1952), which closely mimics the ionic processes at a real nerve membrane, is performed by means of a singular perturbation theory. This was achieved by introducing a perturbation parameter that, if decreased, speeds up the fast variables of the Hodgkin-Huxley equations (membrane potential and sodium activation), whereas it does not affect the slow variables (sodium inactivation and potassium activation). In the most extreme case, if the perturbation parameter is set to zero, the original four-dimensional system degenerates to a system with only two differential equations. This degenerate system is easier to analyze and much more intuitive than the original Hodgkin-Huxley equations. It shows, like the original model, an infinite train of action potentials if stimulated by an input current in a suitable range. Additionally, explanations for the increased sensitivity to depolarizing current steps that precedes an action potential can be found by analysis of the degenerate system. Using the theory of Mishchenko and Rozov (1980) it is shown that the degenerate system does not only represent a simplification of the original Hodgkin-Huxley equations but also gives a valid approximation of the original model at least for stimulating currents that are constant within a suitable range.     

Statistical methods for model comparison in parameter estimation problems for distributed systems

In this note we outline some recent results on the development of a statistical testing methodology for inverse problems involving partial differential equation models. Applications to several problems from biology are presented. The statistical tests, which are in the spirit of analysis of variance (ANOVA), are based on asymptotic distributional results for estimators and residuals in a least squares approach.Research supported in part under grants NSF MCS 8504316, NASA NAG-1-517, and AFOSRF-49620-86-C-0111. Part of this research was carried out while the first author was a visiting scientist at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA, which is operated under NASA contracts NASI-18107 and NASI-18605     

Numerical simulation of thermal bone necrosis during cementation of femoral prostheses

The implant of a femoral prosthesis is a critical process because of the relatively high temperature values reached at the bone/cement interface during the cementation of the infibulum. In fact, the cement is actually a polymer that polymerizes in situ generating heat. Moreover, the conversion of monomer into polymer is never 100%; this is dangerous because of the toxicity of the monomer. In this paper, we present a 3-D axisymmetric mathematical model capable of taking into account both the geometry of the implant and the chemical/physical properties of the cement. This model, together with its numerical simulation, thus represents a useful tool to set up the optimal conditions for the new materials developed in this orthopaedic field. The real complex geometry is assumed to be a bone/cement/metallic system having cylindrical symmetry, thus allowing the model to be reduced to two space variables. The cementation process is described by the Fourier heat equation coupled with a suitable polymerization kinetics. The numerical approximation is accomplished by semi-implicit finite differences in time and finite elements in space with numerical quadrature. The full discrete scheme amounts to solve linear positive definite symmetric systems preceded by an elementwise algebraic computation. We present various numerical simulations which confirm some critical aspects of this orthopaedic fixing technique such as thermal bone necrosis and the presence of unreacted residual monomer.This work was partially supported by MURST (Fondi per la Ricerca Scientifica 40%) and CNR (IAN, Contract 880032601, and Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo, Sottoprogetto Calcolo Scientifico per Grandi Sistemi) of Italy     

Models for age structured populations with distributed maturation rates

In the use of age structured population models for agricultural applications such as the modeling of crop-pest interactions it is often essential that the model take into account the distribution in maturation rates present in some or all of the populations. The traditional method for incorporating distributed maturation rates into crop and pest models has been the so-called distributed delay method. In this paper we review the application of the distributed delay formalism to the McKendrick equation of an age structured population. We discuss the mathematical properties of the system of ordinary differential equations arising out of the distributed delay formalism. We then discuss an alternative method involving modification of the Leslie matrix.     

Global bifurcations in the FitzHugh-Nagumo equations for nerve wave propagation

The FitzHugh-Nagumo equations for action potential propagation along nerve axons and the corresponding ordinary differential equations for travelling waves are solved numerically. Above a critical value, a constant bias current can drive a wave-front solution. At the critical value, a global bifurcation occurs. As a result, the wave front switches into a pulse.Based on a thesis by one of the authors (H. F.).     

种间竞争的一个新的数学模型—对经典的Lotka—Vol—terra竞争方程的扩充

本文根据营养动力学理论,建立了一类种间竞争的新的数学模型:它是单种群增长的Cui-Lawson模型,在种间竞争上的推广。新的种间竞争模型克服了经典的种间竞争的Lotka-Volteira方程的局限与不足,具有更广泛和复杂的行为,并在特殊条件下以Lotka-Volterra竞争方程为其特例。因此,新的种间竞争的数学模型是更一般的解释性模型,是对经典的Lotka-Voterra竞争方程的扩充。     

大兴安岭林区针叶林的生长方程及火灾林木死亡率

本文用随机模拟的方法建立了北方针叶林的生长方程及其受到火灾侵害后的林木死亡率,定量地研究了火灾对这种林分变动的影响．     

Oscillatory reaction-diffusion equations on rings

We study the behavior of traveling waves in - systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of - waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the - system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the - system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order ²) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the - system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.