Mechanisms for oscillations in a biological competition model

A four-dimensional dynamical system that models perceptual bistability in the brain is analyzed. Two variables represent the activity of two competing neural populations and they evolve in fast time; other two variables are slow and they are associated with an intrinsic negative feedback to each population. The external stimulus strength I is the bifurcation parameter. We construct the normal form and prove that oscillations occur in the system through supercritical Hopf bifurcations: as I decreases from large to moderate values a limit cycle is born; then it disappears for lower values of I. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resonance in Beverton–Holt population models with periodic and random coefficients

An increase in the mean population density in a fluctuating environment is known as resonance. Resonance has been observed in laboratory experiments and has been studied in discrete-time population models. We investigate this phenomenon in the Beverton–Holt model with either periodic or random variables for two biologically relevant coefficients: the intrinsic growth rate and the carrying capacity. Three types of resonance are defined: arithmetic, geometric and harmonic. Conditions are derived for each type of resonance in the case of period-2 coefficients and some results for period p>2. For period 2, regions in parameter space where each type of resonance occurs are shown to be subsets of each other. For the case of random coefficients with constant intrinsic growth rate, it is shown that the three types of resonance do not occur. Numerical examples illustrate resonance and attenuance (decrease in the mean population density) in the Beverton–Holt model when the coefficients are discrete random variables.     

From Individuals to Populations: A Symbolic Process Algebra Approach to Epidemiology

Is it possible to symbolically express and analyse an individual-based model of disease spread, including realistic population dynamics? This problem is addressed through the use of process algebra and a novel method for transforming process algebra into Mean Field Equations. A number of stochastic models of population growth are presented, exploring different representations based on alternative views of individual behaviour. The overall population dynamics in terms of mean field equations are derived using a formal and rigorous rewriting based method. These equations are easily compared with the traditionally used deterministic Ordinary Differential Equation models and allow evaluation of those ODE models, challenging their assumptions about system dynamics. The utility of our approach for epidemiology is confirmed by constructing a model combining population growth with disease spread and fitting it to data on HIV in the UK population. This work was supported by EPSRC through a Doctoral Training Grant (CM, from 2004–2007), and through System Dynamics from Individual Interactions: A process algebra approach to epidemiology (EP/E006280/1, all authors, 2007–2010).     

From behavioural to population level: Growth and competition

The aim of this work is to build models of population dynamics for growth and competition interaction by starting with detailed models at the individual level. At the individual level, we start with detailed models where the growth is described by linear terms. By considering individual interferences and by using aggregation methods, we show that the population level, different growth equation can emerge. We present an example of the emergence of logistic growth and an example of the emergence of logistic growth with Allee effect. Furthermore, in the case of two populations, we show that individual interferences can lead at the population level, to a model which has the same qualitative dynamics behaviour as the Lotka-Volterra competition model. Finally, we show that our model brings to light the effects of spatial heterogeneity on competition models. First, we find the stabilizing effects but also we show that destabilizing effects can occur.     

A mutualism-competition model characterizing competitors with mutualism at low density

A macroscopic model of two species is considered, in which mutualism is the dominant interaction when the species are at low density and competition is the dominant interaction when they are at high density. Our aim is to show that species using the same or similar resources can coexist without niche differentiation and that mutualism at low population density can lead to high production. The specific model is a novel combination of the Lotka–Volterra cooperative (mutualism) model and Lotka–Volterra competitive model. By comparing the dynamics of the specific system with those of the Lotka–Volterra competitive model, we demonstrate the mechanism by which the mutualism at low density promotes competitive coexistence by creating regions of mutualism that maintain coexistence. We also show situations in which high production occurs by (i) displaying regions of net mutualism in which the species with higher competitive ability (the superior) approaches a density larger than its carrying capacity when in isolation from the inferior species, and (ii) displaying regions of net mutualism in which both of the species approach densities larger than their carrying capacities, respectively. By comparing the dynamics of the specific system with those of the Lotka–Volterra mutualism model, we show that competition at high density promotes stability of the system.     

Maximization of the total population in a reaction–diffusion model with logistic growth

This paper is concerned with a nonlinear optimization problem that naturally arises in population biology. We consider the effect of spatial heterogeneity on the total population of a biological species at a steady state, using a reaction–diffusion logistic model. Our objective is to maximize the total population when resources are distributed in the habitat to control the intrinsic growth rate, but the total amount of resources is limited. It is shown that under some conditions, any local maximizer must be of “bang–bang” type, which gives a partial answer to the conjecture addressed by Ding et al. (Nonlinear Anal Real World Appl 11(2):688–704, 2010). To this purpose, we compute the first and second variations of the total population. When the growth rate is not of bang–bang type, it is shown in some cases that the first variation becomes nonzero and hence the resource distribution is not a local maximizer. When the first variation becomes zero, we prove that the second variation is positive. These results implies that the bang–bang property is essential for the maximization of total population.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

Space‐time spectral collocation method for the one‐dimensional sine‐Gordon equation

In this article, we introduce a new space‐time spectral collocation method for solving the one‐dimensional sine‐Gordon equation. We apply a spectral collocation method for discretizing spatial derivatives, and then use the spectral collocation method for the time integration of the resulting nonlinear second‐order system of ordinary differential equations (ODE). Our formulation has high‐order accurate in both space and time. Optimal a priori error bounds are derived in the L²‐norm for the semidiscrete formulation. Numerical experiments show that our formulation have exponential rates of convergence in both space and time. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 670–690, 2015     

Investigations of interlaminar fracture toughness of laminated polymeric composites

The behavior of interlaminar fracture of fiber reinforced laminated polymeric composites has been investigated in modes I, II, and different mixed mode I/II ratios. The experimental investigations were carried out by using conventional beam specimens and the compound version of the CTS (compact tension shear) specimen. In this study, a compound version of the CTS specimen is used for the first time to determine the interlaminar fracture toughness of composites. In order to verify the results obtained by the CTS tests, conventional beam tests were also carried out. In the beam tests, specimens of double cantilever beam (DCB) and end notched flexure (ENF) were used to obtain the critical rates of the energy release for failure modes I and II. The CTS specimen is used to obtain different mixed mode ratios, from pure mode I to pure mode II, by varying the loading conditions. The highest mixed mode ratio obtained in the experiment was G _I /G _II =60. The data obtained from these tests were analyzed by the finite element method. The separated critical rates G _I and G _II of the energy release were calculated by using the modified virtual crack closure integral (MVCCI) method. The experimental investigations were performed on a unidirectional glass/epoxy composite. The results obtained by the beam and CTS tests were compared. It was found that the interlaminar fracture toughness G _IC ^init of mode I at crack initiation and the corresponding value G _II ^Cinit of mode II obtained by the conventional beam and the CTS tests were in rather good agreement. The experimental results of interlaminar fracture of mixed mode were used to obtain the parameters required for the failure criterion. The two different failure criteria were compared. The best correlation with the experimental data was obtained by using the failure criterion proposed by Wu in 1967 containing linear and quadratic terms of the rates of the energy release.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 307–322, May–June, 1998.     

On the depth of certain graded rings associated to an ideal

Let (R,m) be a local ring and Ian ideal of R. In this work we find conditions on Ithat allow us to describe simple relations among depth R(It), depth gr_I(R), depth S(I) and depth S(I/I ²). These relations are useful also from a practical point, of view since it is usually difficult to evaluate depth gr_I(R) and depth S(I/I ²) even with the help of a computer. Furthermore we study the class of ideals that satisfy one of the required conditions and we show that ideals generated by quadratic sequences are in this class     

Fracture mechanics analysis of debond growth in a single-fiber composite under cyclic loading

A model is developed to analyze the growth of a fiber/matrix debond along a broken fiber interface in a single-fiber composite subjected to tension-tension fatigue. The Paris law expressed in terms of debond growth and strain energy release rates is used. An analytical solution for the Mode II energy release rate G _II is obtained for long debonds, where the interface crack growth is self-similar. For short debonds, the interface crack interacts with the fiber break, and therefore a FEM modeling in combination with the virtual crack closure technique was performed to calculate the increase in G _II . Finally, the calculated G _II dependences are summarized in simple expressions that are used to simulate the debond growth in fatigue. A parametric study of the effect of Paris law parameters on the debond growth is performed.     

Evolutionary analysis of adaptive dynamics model under variation of noise environment

This paper proposes a stochastic model for the evolutionary adaptive dynamics of species subject to trait-dependent intrinsic growth rates and the influence of environmental noise. The aim of this paper is twofold: (a) mathematically we make an attempt to investigate the evolutionary adaptive dynamics for models with noises; (b) biologically we investigate how the noises in environment affect the evolutionary stability. We first investigate the extinction and permanence of the population in the presence of environmental noises. Combining evolutionary adaptive dynamics with stochastic dynamics, we then establish a fitness function with stochastic disturbance and obtain the evolutionary conditions for continuously stable strategy and evolutionary branching. Our study finds that under intense competition among species, increasing stochastic disturbance can lead to rapidly stable evolution towards smaller trait values, but there is an opposite effect under weak competition among species. This yields an interesting evolutionary threshold, beyond which any increasing stochastic disturbance can go against evolutionary branching and promote evolutionary stability. We then carry out the evolutionary analysis and numerical simulations to illustrate our theoretical results. Finally, for demonstrating the emergence of high-level polymorphism we perform long-term simulation of evolutionary dynamics.     

Spectral Flow in Fredholm Modules, Eta Invariants and the JLO Cocycle

We give a comprehensive account of an analytic approach to spectral flow along paths of self-adjoint Breuer–Fredholm operators in a type I or II von Neumann algebra N. The framework is that of odd unbounded-summable Breuer–Fredholm modules for a unital Banach *-algebra, A. In the type II case spectral flow is real-valued, has no topological definition as an intersection number and our formulae encompass all that is known. We borrow Ezra Getzlers idea (suggested by I. M. Singer) of considering spectral flow (and eta invariants) as the integral of a closed one-form on an affine space. Applications in both the types I and II cases include a general formula for the relative index of two projections, representing truncated eta functions as integrals of one forms and expressing spectral flow in terms of the JLO cocycle to give the pairing of the K-homology and K-theory of A.     

Dynamics of a generalized Ricker–Beverton–Holt competition model subject to Allee effects

In this article, we propose and study a generalized Ricker–Beverton–Holt competition model subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using the theory of monotone dynamics and the properties of critical curves for non-invertible maps, our analysis show that our model has relatively simple dynamics, i.e. almost every trajectory converges to a locally asymptotically stable equilibrium if the intensity of intra-specific competition intensity exceeds that of inter-specific competition. This equilibrium dynamics is also possible when the intensity of intra-specific competition intensity is less than that of inter-specific competition but under conditions that the maximum intrinsic growth rate of one species is not too large. The coexistence of two competing species occurs only if the system has four interior equilibria. We provide an approximation to the basins of the boundary attractors (i.e. the extinction of one or both species) where our results suggests that contest species are more prone to extinction than scramble ones are at low densities. In addition, in comparison to the dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the persistent attractor, whereas scramble competition models may have the extinction of both species as its only attractor under certain conditions, i.e. the essential extinction of two species due to strong Allee effects; (ii) Scramble competition models like Ricker type models can have much more complicated dynamical structure of interior attractors than contest ones like Beverton–Holt type models have; and (iii) Scramble competition models like Ricker type competition models may be more likely to promote the coexistence of two species at low and high densities under certain conditions: At low densities, weak Allee effects decrease the fitness of resident species so that the other species is able to invade at its low densities; While at high densities, scramble competition can bring the current high population density to a lower population density but is above the Allee threshold in the next season, which may rescue a species that has essential extinction caused by strong Allee effects. Our results may have potential to be useful for conservation biology: For example, if one endangered species is facing essential extinction due to strong Allee effects, then we may rescue this species by bringing another competing species subject to scramble competition and Allee effects under certain conditions.     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

Derivations into duals of ideals of Banach algebras

We introduce two notions of amenability for a Banach algebra A. LetI be a closed two-sided ideal inA, we sayA is I-weakly amenable if the first cohomology group ofA with coefficients in the dual space I* is zero; i.e.,H ¹(A, I*) = {0}, and,A is ideally amenable ifA isI-weakly amenable for every closed two-sided idealI inA. We relate these concepts to weak amenability of Banach algebras. We also show that ideal amenability is different from amenability and weak amenability. We study theI-weak amenability of a Banach algebraA for some special closed two-sided idealI.     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

Relatively weakly open sets in closed balls of Banach spaces,and the centralizer

Let f be a transcendental entire function and let I(f) denote the set of points that escape to infinity under iteration. We give conditions which ensure that, for certain functions, I(f) is connected. In particular, we show that I(f) is connected if f has order zero and sufficiently small growth or has order less than 1/2 and regular growth. This shows that, for these functions, Eremenko’s conjecture that I(f) has no bounded components is true. We also give a new criterion related to I(f) which is sufficient to ensure that f has no unbounded Fatou components.