Discrete-time Mathematical Models Research Articles

Qualitative analysis of a discrete-time phytoplankton\u2013zooplankton model with Holling type-II response and toxicity

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

A novel family of 1-D robust chaotic maps

Abstract Chaotic dynamics of various continuous and discrete-time mathematical models are used frequently in many practical applications. Many of these applications demand the chaotic behavior of the model to be robust. Therefore, it has been always a challenge to find mathematical models which exhibit robust chaotic dynamics. In the existing literature there exist a very few studies of robust chaos generators based on simple 1-D mathematical models. In this paper, we have proposed an infinite family consisting of simple one-dimensional piecewise smooth maps which can be effectively used to generate robust chaotic signals over a wide range of the parameter values.

Nonlinear Dynamics of Two Discrete-Time Versions of the Continuous-Time Brusselator Model

In this paper, we report results related with the dynamics of two discrete-time mathematical models, which are obtained from a same continuous-time Brusselator model consisting of two nonlinear first-order ordinary differential equations. Both discrete-time mathematical models are derived by integrating the set of ordinary differential equations, but using different methods. Such results are related, in each case, with parameter-spaces of the two-dimensional map which results from the respective discretization process. The parameter-spaces obtained using both maps are then compared, and we show that the occurrence of organized periodic structures embedded in a quasiperiodic region is verified in only one of the two cases. Bifurcation diagrams, Lyapunov exponents plots, and phase-space portraits are also used, to illustrate different dynamical behaviors in both discrete-time mathematical models.

A New Parametric Estimation Algorithm for Large-Scale Systems Described by State-Space Mathematical Models

This paper is concerned with the parametric estimation problem for large-scale systems composed of several interconnected systems, which are described by linear discrete-time state-space mathematical models with unknown parameters. A new recursive parametric estimation algorithm with a particular adaptive gain is proposed for estimating the interconnected system parameters. The stability analysis of the developed parametric estimation scheme is treated by using the Lyapunov method. The proposed recursive parametric estimation algorithm is applied to estimate the parameters of a large-scale systems composed of three interconnected systems. The obtained numerical simulation results show the good performance of this algorithm.

Discrete-time models for releases of sterile mosquitoes with Beverton–Holt-type of survivability

In this paper, we formulate discrete-time mathematical models for the interactive wild and sterile mosquitoes. Instead of the Ricker-type of nonlinearity for the survival functions, we assume the Beverton–Holt-type in these models. We consider three different strategies for the releases of sterile mosquitoes and investigate the model dynamics. Threshold values for the releases of sterile mosquitoes are derived for all of the models that determine whether the wild mosquitoes are wiped out or coexist with the sterile mosquitoes. Numerical examples are given to demonstrate the dynamics of the models.

Solution of classical evolutionary models in the limit when the diffusion approximation breaks down.

The discrete time mathematical models of evolution (the discrete time Eigen model, the Moran model, and the Wright-Fisher model) have many applications in complex biological systems. The discrete time Eigen model rather realistically describes the serial passage experiments in biology. Nevertheless, the dynamics of the discrete time Eigen model is solved in this paper. The 90% of results in population genetics are connected with the diffusion approximation of the Wright-Fisher and Moran models. We considered the discrete time Eigen model of asexual virus evolution and the Wright-Fisher model from population genetics. We look at the logarithm of probabilities and apply the Hamilton-Jacobi equation for the models. We define exact dynamics for the population distribution for the discrete time Eigen model. For the Wright-Fisher model, we express the exact steady state solution and fixation probability via the solution of some nonlocal equation then give the series expansion of the solution via degrees of selection and mutation rates. The diffusion theories result in the zeroth order approximation in our approach. The numeric confirms that our method works in the case of strong selection, whereas the diffusion method fails there. Although the diffusion method is exact for the mean first arrival time, it provides incorrect approximation for the dynamics of the tail of distribution.

A new decentralized implicit adaptive regulator for large-scale systems described by discrete-time state-space mathematical models

In this paper, we treat the problem of decentralized implicit adaptive regulation for large-scale stochastic systems composed into a set of interconnected systems that are described by discrete-time state-space mathematical models with unknown parameters. The key idea in the decentralized regulation method is to design local regulator using only local information such that the state of each interconnected system is regulated to a certain constant reference signal. The main contribution is the proposition of a decentralized implicit adaptive regulator based on state-feedback strategy that can be applied to stochastic interconnected systems with unknown parameters. Furthermore, the practical implementation of the proposed decentralized implicit adaptive regulator can be made easily (low-cost implementation of the electronic components, short computation of the decentralized control law, etc.). A theorem is established and proved which gives sufficient stability conditions of the resulting closed-loop interconnected systems by using the Lyapunov method. An example of numerical simulation is treated to test the performance of the proposed decentralized implicit adaptive regulator.

Modelling releases of sterile mosquitoes with different strategies

To prevent the transmissions of malaria, dengue fever, or other mosquito-borne diseases, one effective weapon is the sterile insect technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population. To study the impact of the sterile insect technique on disease transmission, we formulate discrete-time mathematical models, based on difference equations, for the interactive dynamics of the wild and sterile mosquitoes, incorporating different strategies in releasing sterile mosquitoes. We investigate the model dynamics and compare the impact of the different release strategies. Numerical examples are given to demonstrate rich dynamical features of the models.

Interplay of population genetics and dynamics in the genetic control of mosquitoes

Some proposed genetics-based vector control methods aim to suppress or eliminate a mosquito population in a similar manner to the sterile insect technique. One approach under development in Anopheles mosquitoes uses homing endonuclease genes (HEGs)—selfish genetic elements (inherited at greater than Mendelian rate) that can spread rapidly through a population even if they reduce fitness. HEGs have potential to drive introduced traits through a population without large-scale sustained releases. The population genetics of HEG-based systems has been established using discrete-time mathematical models. However, several ecologically important aspects remain unexplored. We formulate a new continuous-time (overlapping generations) combined population dynamic and genetic model and apply it to a HEG that targets and knocks out a gene that is important for survival. We explore the effects of density dependence ranging from undercompensating to overcompensating larval competition, occurring before or after HEG fitness effects, and consider differences in competitive effect between genotypes (wild-type, heterozygotes and HEG homozygotes). We show that population outcomes—elimination, suppression or loss of the HEG—depend crucially on the interaction between these ecological aspects and genetics, and explain how the HEG fitness properties, the homing rate (drive) and the insect's life-history parameters influence those outcomes.

Green oak leaf roller moth dynamics: An application of discrete time mathematical models

Discrete mathematical model is selected for description of green oak leaf roller ( Tortrix viridana L.) population dynamics. Experimental data set from Rubtsov [Rubtsov, V.V., 1992. Models of Oscillation Processes in Forest Ecosystems, Forest Institute, Moscow, Russian Federation, p. 39 (in Russian)] is analyzed from the standpoint of phenomenological theory of forest insect population dynamics [Isaev, A.S., Khlebopros, R.G., Nedorezov, L.V., Kiselev, V.V., Kondakov, Yu. P., 1984. Dynamics of Forest Insects Populations, USSR, Nauka, Novosibirsk, p. 224 (in Russian); Isaev, A.S., Khlebopros, R.G., Nedorezov, L.V., Kiselev, V.V., Kondakov, Yu.P., Soukhovol'sky, V.G., 2001. Population Dynamics of Forest Insects, Russian Federation, Nauka, Moscow, p. 347 (in Russian)]. For checking hypotheses confidence for fitting of experimental time series, Moran–Ricker model, Maynard Smith–Slatkin model, and some others are used. Selection of statistical criteria for estimating parameter's values is discussed. In addition, probabilities of realizations of several dynamical regimes are estimated.

Infection dynamics of different Wolbachia-types within one host population

Wolbachia are widespread intracellular symbionts of arthropods which are known to cause several reproductive manipulations in their hosts, the commonest of which being cytoplasmic incompatibility (CI), male killing (MK), and the induction of parthenogenesis (PI). Strains of endosymbionts inducing one of these effects can be referred to as `Wolbachia-types'. Here, we try to ascertain whether two of these Wolbachia-types can stably coexist within one population. We investigate this question by means of two discrete-time mathematical models which describe the dynamics of an infection of a host population with either CI- and MK- or CI- and PI-Wolbachia. We derive analytical solutions for two special cases of each model showing that stable coexistence of the respective Wolbachia-types is not possible if no doubly infected individuals occur within the population and that stable coexistence is possible when doubly infected hosts do exist and transmission of the endosymbionts is perfect. Moreover, we show that a population infected with either CI- or MK-Wolbachia at equilibrium can resist invasion of the respective other Wolbachia-type as a single infection. In contrast, a population infected with CI-Wolbachia can be invaded by PI-Wolbachia as a single infection with the CI-Wolbachia going extinct. Computer simulations confirmed these findings for the general models. We discuss our results with respect to the prevalence of the Wolbachia-types considered here and the emergence of PI- from CI-Wolbachia.

Fuzzy-Neural Model for Nonlinear Systems Identification

A state space representation of both continuous and discrete time mathematical models of Recurrent Neural Networks (RNN) are given in two layer Jordan canonical architecture, and a new improved Back-Propagation (BP) type learning method, is proposed. Some topology improvements, are suggested. The proposed RTNN model is linear in small and nonlinear in large, which permits to apply all well known state- and output linear systems design methods. The obtained RTNN model is incorporated in a rule-based fuzzy system, giving the possibility to approximate and to identify a complex nonlinear plants. Simulation results of nonlinear systems identification by the proposed fuzzy-neural system, using RNN BP learning, are given.

Identification and control of flexible civil structures with time delays

The identification of flexible civil structures as discrete-time mathematical models in a ready form for computer control is shown by using a simple identification technique and neglecting the noise problem. The use of a control mechanism for the purpose of modeling and also for the control of the flexible structure justifies the noise-free consideration. The effect of time delay during a control process on the dynamics and stability of the structural systems is shown in illustrative examples. The design of a feedback control law to compensate for the detrimental effect of time delay is given with an application to control the response of a flexible frame structure using a tendon control mechanism.

Adaptive Control for a Robot ARM and a Robot Drive System with Elasticity

Abstract First, this paper presents an application of a Model Reference Adaptive Control System (MRACS) to the position control, and the force feedback control of a flexible robot arm. Second, this paper investigates the application of an MRACS to the speed control, and the position control of a DC servo system, and a Direct-Drive(DD) servo system with a flexible transmission element. The flexible robot arm is simply modelled as a flexible cantilever beam. The servo systems with a flexible transmission element are treated as a servo mechanism with a one degree of freedom oscillatory load Using discrete-time mathematical models as a lumped parameter system, the discrete MRACS is designed for each discrete model Experimental results have shown that the control performance is good and the MRACS is effective.

Adaptive control of DC servo system and DD servo system with a flexible transmission element. (Speed control and position control).

This paper describes the application of a Model Reference Adaptive Control System (MRACS) to the speed control and the position control of a DC servo system and a Direct-Drive (DD) servo system with a flexible transmission element. The servo systems are simply modelled as a servo mechanism with a one degree of freedom oscillatroy load. After inducing the discrete-time mathematical models of the servo systems for the speed and position as a lumped parameter system, the discrete MRACS with a simple algorithm and structure is designed for each discrete model. A set of experiments has been carried out to examine the effectiveness of the MRACS. A rotary encoder was used to measure the speed and the position of loads in the experiments. Experimental results have shown that the control performance is good and the constructed MRACS is effective.

Chemical noise and measurement error processes. Continuous and discrete time models

Industrial and measurement fault events are often connected with sudden changes (jumps) in some measured value. In this paper new continuous and discrete time mathematical models are proposed for chemical measurement error and noise processes which are suited to describe jumps, as well as drift and white noise components in the form of stochastic differential equations. The most simple discrete time stationary model has an autoregressive form with non-normally (mixed) distributed white noise sequence, therefore the most important problems connected with jump detection can be simply formulated and solved. The properties of the proposed models and algorithms are demonstrated on real industrial noise processes obtained from recording of strip columns in a great petrochemical plant. The proposed models and algorithms are of importance in case of intelligent automated control and monitoring system for fault detection.