A general model of size-dependent population dynamics with nonlinear growth rate

We study a general model of size-dependent population dynamics with nonlinear growth rate. The existence of a local solution is shown by using Schauder's fixed point theorem. Uniqueness and continuous dependence on initial data are also established under additional conditions.     

A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients

This paper is devoted to the analysis of measure-valued solutions to a nonlinear structured population model given in the form of a nonlocal first-order hyperbolic problem on R⁺. We show global existence and Lipschitz continuity with respect to the model ingredients. In distinction to previous studies, where the L¹ norm was used, we apply the flat metric, similar to the Wasserstein W₁ distance. We argue that analysis using this metric, in addition to mathematical advantages, is consistent with intuitive understanding of empirical data. Lipschitz continuous dependence with respect to the model coefficients and initial data and the uniqueness of the weak solutions are shown under the assumption on the Lipschitz continuity of the kinetic functions. The proof of this result is based on the duality formula and the Gronwall-type argument.     

A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs

In this paper, a family of fourth-order Steffensen-type two-step methods is constructed to make progress in including Ren-Wu-Bi’s methods [H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206-210] and Liu-Zheng-Zhao’s method [Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensens method of fourth-order convergence and its applications, Appl. Math. Comput. 216 (2010) 1978-1983] as its special cases. Its error equation and asymptotic convergence constant are deduced. The family provides the opportunity to obtain derivative-free iterative methods varying in different rates and ranges of convergence. In the numerical examples, the family is not only compared with the related methods for solving nonlinear equations, but also applied in the solution of BVPs of nonlinear ODEs by the finite difference method and the multiple shooting method.     

Phase-Locked loops lock-in range in Frequency Modulated-Atomic Force Microscope nonlinear control system

Since the mid 1980s the Atomic Force Microscope is one the most powerful tools to perform surface investigation, and since 1995 Non-Contact AFM achieved true atomic resolution. The Frequency-Modulated Atomic Force Microscope (FM-AFM) operates in the dynamic mode, which means that the control system of the FM-AFM must force the microcantilever to oscillate with constant amplitude and frequency. However, tip-sample interaction forces cause modulations in the microcantilever motion. A Phase-Locked loop (PLL) is used to demodulate the tip-sample interaction forces from the microcantilever motion. The demodulated signal is used as the feedback signal to the control system, and to generate both topographic and dissipation images. As a consequence, a proper design of the PLL is vital to the FM-AFM performance. In this work, using bifurcation analysis, the lock-in range of the PLL is determined as a function of the frequency shift (Ω) of the microcantilever and of the other design parameters, providing a technique to properly design the PLL in the FM-AFM system.     

Bifurcation and chaos in an epidemic model with nonlinear incidence rates

This paper investigates a discrete-time epidemic model by qualitative analysis and numerical simulation. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation, Hopf bifurcation types and chaos. Also the largest Lyapunov exponents are numerically computed to confirm further the complexity of these dynamic behaviors. The obtained results show that discrete epidemic model can have rich dynamical behavior.     

A numerical approximation for Volterra’s population growth model with fractional order

This paper presents a numerical scheme for approximate solutions of the fractional Volterra’s model for population growth of a species in a closed system. In fact, the Bessel collocation method is extended by using the time-fractional derivative in the Caputo sense to give solutions for the mentioned model problem. In this extended of the method, a generalization of the Bessel functions of the first kind is used and its matrix form is constructed. And then, the matrix form based on the collocation points is formed for the each term of this model problem. Hence, the method converts the model problem into a system of nonlinear algebraic equations. We give some numerical applications to show efficiency and accuracy of the method. In applications, the reliability of the technique is demonstrated by the error function based on accuracy of the approximate solution.     

Existence theorems for some abstract nonlinear non-autonomous systems with delays

For some abstract classes of nonlinear non-autonomous systems with variable and state-dependent delays existence, non-existence and multiplicity of periodic solutions are discussed. To illustrate the efficiency of the method, we obtain some well-known results for applied systems as corollaries of our existence theorems.     

A cobweb model with gradient adjustment mechanism: nonlinear dynamics and multistability

A cobweb model, characterized by boundedly rational producers with a production adjustment mechanism based on the gradient rule, is described by a nonlinear discrete time dynamical system of the plane. Firms do not have a complete knowledge of the demand function and try to infer how the market will respond to their production changes by an empirical estimates of the marginal profits. Analytical conditions for local stability of the market equilibrium are provided, showing that the stability loss of the market equilibrium may give rise to chaotic dynamic as well. When memory is introduced in the production adjustment mechanism, a locally stabilizing effect is revealed as well as a globally qualitatively destabilizing role for memory. This is related to the occurrence of period doubling and Neimark–Sacker bifurcations, the latter being of supercritical nature as analytically proved. Endogenous fluctuations and multistability, with consequent loss of predictability in the long run dynamics, are observed.     

A semiparametric method for estimating nonlinear autoregressive model with dependent errors

The first-order nonlinear autoregressive model is considered and a semiparametric method is proposed to estimate regression function. In the presented model, dependent errors are defined as first-order autoregressive AR(1). The conditional least squares method is used for parametric estimation and the nonparametric kernel approach is applied to estimate regression adjustment. In this case, some asymptotic behaviors and simulated results for the semiparametric method are presented. Furthermore, the method is applied for the financial data in Iran’s Tejarat-Bank.     

The probabilistic properties of the nonlinear autoregressive model with conditional heteroskedasticity

1.IntroductionTakeseriesmodelswithconditionalheteroskedasticityarewidelyappliedinmanyareassuchasfinancialeconomicsandeconometrics.Consequently,thereisagrowingthetheoreticalinterestinthesemodels.Inthispaper,weareexclusivelyinterestedinthefollowingnonlinearautoregressivemodelwithconditionalheteroskedasticityat=rk(Xt~1,...Iac~p,01) eth(Xt~1,'IXo~~l02)(1.1)withthefollowingassumptions(i){Et}isasequenceofindependelltideticallydistributedrandomvariableswithacommonalmenteveryWherepositivedenistyi…     

Pattern formation (I): The Keller-Segel model

We investigate nonlinear dynamics near an unstable constant equilibrium in the classical Keller-Segel model. Given any general perturbation of magnitude δ, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes, over a time period of . Our result can be interpreted as a rigorous mathematical characterization for early pattern formation in the Keller-Segel model.     

An age structured population model in a spatially heterogeneous environment: Existence and uniqueness theory

This paper considers the existence and uniqueness of solutions of an age-structured population model in a spatially heterogeneous bounded environment. The model describes individual organism movement of diffusion and advection in the older life stage but assumes stationarity of individuals in the embryonic stage. The mathematical formulation is a nonlinear degenerate parabolic partial differential problem with non-locally integro-type initial-boundary conditions.     

On an epidemiological model with nonlinear infection incidence: Local and global perspective

Though bilinear and standard incident rates have been used frequently in classical epidemic models, nonlinearity in infection incidence is considered to be an useful modification due to its proximity with various realistic situations of disease propagation among ecological populations. In the present research, we consider an epidemiological model with SIS disease in the population. Infection is assumed to propagate following nonlinear incidence. First, we perform a stability and bifurcation analysis of the system around different equilibria from a local perspective. It is observed that the infected incidence fraction (p) plays the key role in controlling the disease dynamics. Then, using an iteration scheme and comparison argument, we investigate the global stability criteria of the model system around the endemic state and infer that the disease will persist among the species in the long run. Numerical simulation study is also carried out to illustrate and augment the analytical findings.     

Evolution of symbiosis in hermatypic corals: A model of the past,present, and future

This work can be considered a prequel to our previous paper on coral bleaching induced by global warming. We once again investigate, using Finsler geometry, dynamical energy budget theory and nonlinear modular mechanics, the origin of endosymbiosis, between reef-building corals and the algae. We assume their relationship starts out as entosymbiosis, with the algal organism living on the external surfaces of host coral exoskeleton, but with both gradually adapting to each other over evolutionary time-scales. Our main conclusion is that such an evolutionary conversion is possible and indeed is quite likely.     

On zero duality gap in surrogate constraint optimization: The case of rational-valued functions of constraints

This paper is concerned with the constrained optimization problem. A detailed discussion of surrogate constraints with zero duality gaps is presented. Readily available surrogate multipliers are considered that close the duality gaps where constraints are rational-valued. Through illustrative examples, the sources of duality gaps are examined in detail. While in the published literature, in many situations conclusions have been made about the existence of non-zero duality gaps, we show that taking advantage of full problem information can close the duality gaps. Overlooking such information can produce shortcomings in the research in which a non-zero duality gap is observed. We propose theorems to address the shortcomings and report results regarding implementation issues.     

A backward nonlinear heat equation: regularization with error estimates

In this article the authors consider a backward nonlinear heat equation. The uniqueness of the problem is proved and the problem is regularized by finite dimensional approximations. Error estimates in some particular cases are given.     

Allee effects in a discrete-time SIS epidemic model with infected newborns

We extend the framework of Rios-Soto et al. (Contemporary Mathematics, 2006, 410, 297) to include both compensatory (contest competition) and overcompensatory (scramble competition) population dynamics with and without the Allee effect. We compute the basic reproductive number ?₀, and use it to predict the (uniform) persistence or extinction of the infective population, where the population dynamics are compensatory and the Allee effect is either present or absent. We also explore the relationship between the demographic equation and the epidemic process, where the total population dynamics are overcompensatory. In particular, we show that the demographic dynamics drive both the susceptible and infective dynamics. This is in contrast to the recent observations of Franke and Yakubu, that the demographic dynamics can be chaotic while the infective dynamics are oscillatory and non-chaotic in periodically-forced SIS epidemic models (Mathematical Biosciences, 2006, 204, 68).     

A computational study of active set strategies in nonlinear programming with linear constraints

An essential part of many iterative methods for linearly constrained nonlinear programming problems is a procedure for determining those inequality constraints which will be active (that is, satisfied as equalities) at each iteration. We discuss experiments in which we used several strategies for identifying active constraints in conjunction with two well-known algorithms for linearly constrained optimization. The results indicate that in most cases a strategy which keeps the number of constraints in the active set as small as possible is computationally most efficient.Sponsored by the United States Army under Contract No. DAAG29-75-C-0024.     

The lifespan for nonlinear wave equations with multiple propagation speeds in 3D

In this paper, we consider the Cauchy problem for systems of nonlinear wave equations with multiple propagation speeds in three space dimensions. On the basis of some Klainerman–Sideris type weighted estimates, we get the lifespan of small amplitude solutions for systems with nonlinearities depending on both the unknown functions and their derivatives.     

A least-squares model specification test for a class of dynamic nonlinear economic models with systematically varying parameters

The problem of structural instability in estimated macroeconomic relationships has recently surfaced anew, due to a series of articles by the rational expectations school. A key explanation for the observed instability is said to be that the parameters appearing in modelled macroeconomic relationships, traditionally treated as constant or purely random, must realistically be viewed as the reflections of the demand and supply decisions of optimizing agents in the economy reacting rationally to changes in endogenous and exogenous variables, including changes induced by shifts in government policy decision rules. The present paper develops a least-squares measure for simultaneously testing the basic compatibility of prior dynamical, observational, and distributional model specifications against actual data for a class of dynamic nonlinear economic models with parameters explicitly modelled as nonlinear functions of endogenous and exogenous variables. Using invariant imbedding techniques, an algorithm is derived for sequentially updating the optimal least-squares estimates for parameters, endogenous variables, and squared residual modelling error sums as the duration of the process increases and new observations are obtained.Dedicated to R. BellmanThis work was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and the National Institutes of Health under Grant No. GM-3732-03. A previous version of this paper (Ref. 1) was presented at the Econometrics Modelling Research Seminar, Department of Economics, University of Southern California, Los Angeles, 1979. The authors are grateful to numerous colleagues for helpful comments.