Analysis of the power law logistic population model with slowly varying coefficients

We apply a multiscale method to construct general analytic approximations for the solution of a power law logistic model, where the model parameters vary slowly in time. Such approximations are a useful alternative to numerical solutions and are applicable to a range of parameter values. We consider two situations—positive growth rates, when the population tends to a slowly varying limiting state; and negative growth rates, where the population tends to zero in infinite time. The behavior of the population when a transition between these situations occurs is also considered. These approximations are shown to give excellent agreement with the numerical solutions of test cases. Copyright © 2012 John Wiley & Sons, Ltd.     

Transitions in a logistic population model incorporating an Allee effect

In some species, the population may decline to zero; that is, the species becomes extinct if the population falls below a given threshold. This phenomenon is well known as an Allee effect. In most Allee models, the model parameters are constants, and the population tends either to a nonzero limiting state (survival) or to zero (extinction). However, when environmental changes occur, these parameters may be slowly varying functions of time. Then, application of multitiming techniques allows us to construct approximations to the evolving population in cases where the population survives to a slowly varying surviving state and those where the population declines to zero. Here, we investigate the solution of a logistic population model exhibiting an Allee effect, when the carrying capacity and the limiting density interchange roles, via a transition point. We combine multiscaling analysis with local asymptotic analysis at the transition point to obtain an overall expression for the evolution of the population. We show that this shows excellent agreement with the results of numerical computations. Copyright © 2015 John Wiley & Sons, Ltd.     

Survival to extinction in a slowly varying harvested logistic population model

This work considers a harvested logistic population for which birth rate, carrying capacity and harvesting rate all vary slowly with time. Asymptotic results from earlier work, obtained using a multiscaling technique, are combined to construct approximate expressions for the evolving population for the situation where the population initially survives to a slowly varying limiting state, but then, due to increasing harvesting, is reduced to extinction in finite time. These results are shown to give very good agreement with those obtained from numerical computation.     

Multiscaling analysis of a slowly varying anaerobic digestion model

We construct the slowly varying limiting state solutions to a nonlinear dynamical system for anaerobic digestion with Monod-based kinetics involving slowly varying model parameters arising from slow environmental variation. The advantage of these approximate solutions over numerical solutions is their applicability to a wide range of parameter values. We use these limiting state solutions to develop analytic approximations to the full nonlinear system by applying a multiscaling technique. The approximate solutions are shown to compare favorably with numerical solutions.     

Tumor evasion from immune control: Strategies of a MISS to become a MASS

We biologically describe the phenomenon of the evasion of tumors from immune surveillance where tumor cells, initially constrained to exist in a microscopic steady state (MISS) elaborate strategies to evade from the immune control and to reach a macroscopic steady state (MASS). We, then, describe “evasion” as a long term loss of equilibrium in a framework of prey–predator-like models with adiabatic varying parameters, whose changes reflect the evolutionary adaptation of the tumor in a “hostile” environment by means of the elaboration of new strategies of survival. Similarities and differences between the present work and the interesting seminal paper [Kuznetsov VA, Knott GD. Modeling tumor regrowth and immunotherapy. Math Comput Model 2001;33:1275–87] are discussed. We also propose and study a model of clonal resistance to the immune control with slowly varying adaptive mutation parameter.     

Convex additively slowly varying functions

We study the problem of subtraction of slowly varying functions. It is well-known that the difference of two slowly varying functions need not be slowly varying and we look for some additional conditions which guarantee the slow variation of the difference. To this end we consider all possible decompositions L=F+G of a given increasing convex additively slowly varying function L into a sum of two increasing convex functions F and G. We characterize the class of functions L for which in every such decomposition the summands are necessarily additively slowly varying. The class OΠ₂⁺ we obtain is related to the well-known class OΠ_g where, instead of first order differences as in OΠ_g, we have second order differences.     

Sensitivity analysis for expected utility maximization in incomplete Brownian market models

We examine the issue of sensitivity with respect to model parameters for the problem of utility maximization from final wealth in an incomplete Samuelson model and mainly, but not exclusively, for utility functions of positive-power type. The method consists in moving the parameters through change of measure, which we call a weak perturbation, decoupling the usual wealth equation from the varying parameters. By rewriting the maximization problem in terms of a convex-analytical support function of a weakly-compact set, crucially leveraging on the work (Backhoff and Fontbona in SIAM J Financ Math 7(1):70–103, 2016), the previous formulation let us prove the Hadamard directional differentiability of the value function with respect to the market price of risk, the drift and interest rate parameters, as well as for volatility matrices under a stability condition on their Kernel, and derive explicit expressions for the directional derivatives. We contrast our proposed weak perturbations against what we call strong perturbations, where the wealth equation is directly influenced by the changing parameters. Contrary to conventional wisdom, we find that both points of view generally yield different sensitivities unless e.g. if initial parameters and their perturbations are deterministic.     

Weighted moments for a supercritical branching process in a varying or random environment

Let W be the limit of the normalized population size of a supercritical branching process in a varying or random environment. By an elementary method, we find sufficient conditions under which W has finite weighted moments of the form EWpl(W), where p > 1, l 0 is a concave or slowly varying function.     

Numerical approximation of a metastable system

Using the Becker-Döring cluster equations as an example,we highlight some of the problems that can arise in the numericalapproximation of dynamical systems with slowly varying solutions.We describe the Becker-Döring model, summarize some ofits properties and construct a numerical approximation whichallows accurate and efficient computation of solutions in thelong, slowly varying metastable phase. We use the approximationto obtain test results and discuss the clear relationship betweenthem and equilibrium solutions of the Becker-Döring equations.     

Parameter estimation for time varying nonlinear circuit from state analysis and simulation

This paper develops a parameter estimation technique for a nonlinear circuit. The nonlinear circuit is represented by a state space model and perturbation theory is applied to obtain the approximate analytical solution for the state vector. The state model is assumed to be slowly time varying so that the parameter vector is constant over different time slots. The expressions obtained for the state vector are matched with the noisy data using the gradient algorithm and hence the parameter vector is estimated. Simulations are based on discretization of the state space model using MATLAB.     

Pricing perpetual American options under multiscale stochastic elasticity of variance

This paper studies pricing the perpetual American options under a constant elasticity of variance type of underlying asset price model where the constant elasticity is replaced by a fast mean-reverting Ornstein–Ulenbeck process and a slowly varying diffusion process. By using a multiscale asymptotic analysis, we find the impact of the stochastic elasticity of variance on the option prices and the optimal exercise prices with respect to model parameters. Our results enhance the existing option price structures in view of flexibility and applicability through the market prices of elasticity risk.     

Asymptotic estimates for the canonical products with zeros of a particular form

We obtain sharp asymptotic estimates on the whole complex plane for the canonical products with zeros of the form λn = s-nαl(n), where α > 0 and l(t) is a slowly varying function.     

Extending satisficing control strategy to slowly varying nonlinear systems

Based on the satisficing control strategy, a novel approach to design a stabilizing control law for nonlinear time varying systems with slowly varying parameters (slowly varying systems) is presented. The satisficing control strategy has been originally introduced for time-invariant systems; however, this technique does not have any stability proof for time varying systems. In this paper, first, a parametric version of the satisficing control strategy is developed. Then, by considering the time as a frozen parameter, the parametric satisficing control strategy is utilized. Finally, a theorem is presented which suggested a stabilizing satisficing control law for the slowly varying control systems. Moreover, in this theorem, the maximum admissible rate of change of the system dynamics is evaluated. The efficiency of the proposed approach is demonstrated by a computer simulation.     

Vortex Motion and the Geometric Phase. Part II. Slowly Varying Spiral Structures

Summary. We derive formulas for the long time evolution of passive interfaces in three ``canonical' incompressible, inviscid, two-dimensional flow models. The point vortex models, introduced in Part 1 [1] are (i) a ``restricted' three-vortex problem, (ii) a vortex and a particle in a closed circular domain, and (iii) a particle in the flowfield of a mixing layer model undergoing a vortex pairing instability. In each configuration, it was shown in Part 1 that the passive particle exhibits a geometric or Hannay-Berry phase over long time periods induced by the slowly varying periodic background field. In this paper we show how the formula for the evolution of a passive interface driven by the dynamics of the vortices inherits this geometric phase effect. The interface wraps into a spiral formation around the ``parent' vortex, with a slowly varying component induced by the farfield vorticity. The length formula for the long time growth of the slowly rotating spiral decomposes into a ``dynamic' part and a ``geometric' part. The dynamic part is the length in the ``unperturbed' system—i.e., in the absence of the background field—and the geometric part is the contribution of the geometric phase θ _g for a passive particle in the flow. We derive the following simple formula for an interface along a smooth curve joining two arbitrary particles labelled A and B . Define as the difference in interface lengths between the ``unperturbed' system and the ``perturbed' slowly varying system at the end of the long time period T . In each case, , where ξ is the radial coordinate parametrizing the interface at t=0 . Received September 4, 1997; second revision received January 16, 1998; accepted January 27, 1998     

An analytical study of the sensitivity of a discrete stochastic model for sales dynamics

We consider a discrete model for sales dynamics in the case of a stochastic model of the market. The model includes “fast” and “slow” components of the market situation described by a stochastic process of “white noise” type and the correlated stochastic process. By using an integral representation of the main characteristics of the Kalman filter, we obtain expressions for stochastic parameters of additional errors of the estimate that arise in the case where the characteristics of noises are inexact. We make an asymptotical analysis of these expressions and give recommendations for the price-forming strategy in the case of uncertainty of the market situation. Bibliography: 2 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 110–116.     

Global analysis of a model of differential susceptibility induced by genetics

This paper is concerned with global analysis of an SIS epidemiological model in a population of varying size with two dissimilar groups of susceptible individuals. We prove that this system has no periodic solutions and use the Poincaré index theorem to determine the number of rest points and their stability properties. It has been shown that multiple equilibria (bistability) occurs for suitable values of parameters. We also give some numerical examples of all possible bifurcations of this system.     

Fast-slow dynamics in Logistic models with slowly varying parameters

Fast-slow behaviors in the Logistic models with slowly varying parameters are revealed by using singular perturbation method. We first rewrite the Logistic models with slowly varying parameters in the form of singularly perturbed systems and separate their fast and slow limits. Then we apply matching to obtain the approximate solutions, which are explicit and analytical and compare very well with the numerically integrated ones. More importantly, we prove the uniform validity of the approximate solutions rigorously and give the error estimate between the approximate solutions and the exact solutions via the way of upper and lower solutions.     

Scaling properties of Hausdorff and packing measures

Let . Let be a continuous increasing function defined on , for which and is a decreasing function of t. Let be a norm on , and let , , denote the corresponding metric, and Hausdorff and packing measures, respectively. We characterize those functions such that the corresponding Hausdorff or packing measure scales with exponent by showing it must be of the form , where L is slowly varying. We also show that for continuous increasing functions and defined on , for which , is either trivially true or false: we show that if , then for a constant c, where is the Lebesgue measure on . Received June 17, 2000 / Accepted September 6, 2000 / Published online March 12, 2001     

Robust Adaptive Identification of Fuzzy Systems with Uncertain Data

This study presents a method of adaptive identification of parameters describing Sugeno fuzzy inference system in presence of bounded disturbances while maintaining the readability and interpretability of the fuzzy model during and after identification. This method do not require any a priori knowledge of a bound on the disturbance and noise and of a bound on the unknown parameters values. The method can be used for the robust and adaptive identification of slowly time varying nonlinear systems using fuzzy inference systems. The suggested method was used to build a fuzzy expert system that approximates the functional relationship between physical fitness and some of the measurable physiological parameters by their real measurements and opinion (human-experiences) of a medical expert.