Volterra Model Research Articles

A diffusive Lotka–Volterra model with Robin boundary condition and sign-changing growth rates in time-periodic environment

In this paper, a Lotka–Volterra model with Robin and free boundary conditions is considered in the heterogeneous time-periodic environment. We mainly consider the changes of local growth rates of native and invasive species that might be negative in some large regions. We study the spreading–vanishing dichotomy. When vanishing occurs, a native species cannot spread successfully as time goes to infinity. However, for an invasive species, in the long run, either it will go extinct or converge to the unique positive solution of time-periodic boundary value problem of logistic equation. When spreading occurs, both native and invasive species have upper and lower bounds. We also obtain the criteria for spreading and vanishing, and estimate of the asymptotic spreading speed.

Stationary distribution and extinction of a Lotka–Volterra model with distribute delay and nonlinear stochastic perturbations

This paper devotes to the study of a Lotka–Volterra model which has one prey and two predators with nonlinear stochastic perturbations and distributed delay. It is first proved that the autonomous system has a unique global and positive solution. Then, by constructing an appropriate stochastic Lyapunov function, we obtain the sufficient conditions which guarantee the existence of a stationary distribution of the positive solutions to the model. Furthermore, some sufficient conditions for extinction of the predator population is also established. Numerical simulations are provided to demonstrate the main results in the end.

Optimal resource allocation and adaptive robust control of technology innovation ecosystems based on cooperative game theory

This paper addresses a problem of optimal resource allocation for technology innovation ecosystem. Two layers goal are realized. First, an uncertain dynamical system is constructed with the Lotka–Volterra model for the description of the dynamics of the concerned innovation ecosystem, in which the growth rate and the symbiosis coefficient are specially treated as possibly (fast) time-varying uncertain parameters. Second, an adaptive robust resource allocation strategy is designed to drive the errors between the populations and the desired populations to be uniformly bounded and uniformly ultimately bounded, such that the first layer (basic) goal of resource allocation is achieved. Third, a two-player cooperative game is formulated for the seeking of the optimal control parameters, in which two (static) cost functions and a comprehensive performance index are constructed by performance analysis, and then the optimal control parameters (i.e., the Pareto-optimality) are obtained by minimizing the comprehensive performance index, such that the second layer (further) goal of optimal balance between the control inputs and the system performance is achieved. This work should be the first effort that addresses the optimal resource allocation and control for technology innovation ecosystem through a creative combination of dynamic modeling, control theory and game theory.

A polynomial multiple variance method for impulse response measurement

This manuscript describes a methodology for measuring the first-order Volterra kernel of a discrete-time nonlinear system, that is, the impulse response for small signal amplitudes of the nonlinear system. In the proposed approach, multiple linear identifications are performed using the same excitation signal multiplied by different gains, and the first-order Volterra kernel is obtained from a polynomial interpolation of the measured values. Any linear identification method proposed in the literature can be used with this approach. The proposed approach is a multiple variance (MV) method that, in contrast to all other MV methods, aims to estimate one of the kernels of the Volterra model, the first-order kernel, with high precision. The manuscript discusses the proposed methodology, determines the mean square deviation (MSD) due to noise of the measured coefficients, and the value of the optimal gains that minimize the MSD. Remarkably, it is shown in the manuscript that the optimal gains assume only a reduced set of values that depends on the order of nonlinearity. The optimal number of measurements is also determined. The conditions under which the proposed methodology is more convenient than the classical linear methods are discussed. The experimental results demonstrate the proposed methodology and its strengths.

Nonparametric identification based on Gaussian process regression for distributed parameter systems

This paper proposes a nonparametric identification method based on Gaussian process regression (GPR) for completely unknown nonlinear distributed parameter systems (DPSs). Inspired by linear parameter-varying (LPV) modelling approach, an interpolated spatio-temporal Volterra model is developed to represent the DPSs in nonparametric form, in which local Volterra models are interpreted as Gaussian processes. According to the empirical Bayesian approach, we design the third-order stable kernel structure used for embedding prior knowledge and derive the estimation of hyperparameters. The hyperparameters included in local weighting functions and kernel functions are determined by the maximum likelihood method. By utilising the nonparametric identification approach to avoid model structure selection, the proposed method can improve identification result for completely unknown distributed parameter systems. Finally, two case studies validate the effectiveness of the proposed identification method.

Utility Maximization in Multivariate Volterra Models

This paper is concerned with portfolio selection for an investor with power utility in multiasset financial markets in a rough stochastic environment. We investigate Merton’s portfolio problem for different multivariate Volterra models, covering the rough Heston model. First we consider a class of multivariate affine Volterra models introduced in [E. Abi Jaber, E. Miller, and H. Pham, SIAM J. Financial Math.., 12 (2021), pp. 369–409]. Based on the classical Wishart model described in [N. Bäuerle and Z. Li, J. Appl. Probab., 50 (2013), pp. 1025–1043], we then introduce a new matrix-valued stochastic volatility model, where the volatility is driven by a Volterra–Wishart process. Due to the non-Markovianity of the underlying processes, the classical stochastic control approach cannot be applied in these settings. To overcome this issue, we provide a verification argument using calculus of convolutions and resolvents. The resulting optimal strategy can then be expressed explicitly in terms of the solution of a multivariate Riccati–Volterra equation. We thus extend the results obtained by Han and Wong to the multivariate case, avoiding restrictions on the correlation structure linked to the martingale distortion transformation used in [B. Han and H. Y. Wong, Finance Res. Lett., 39 (2021)]. We also provide existence and uniqueness theorems for the occurring Volterra processes and illustrate our results with a numerical study.

Modelling and Simulation of Competition for Students’ Population with Holling Type II Response

The increase in the country’s population attracted the establishment of more schools, both public and private schools, to cater for the increasing number of students. However, there have been dynamics of students’ population both in public and private schools through transfer from one category of school to the other, through completion of the learning period, and through dropout due to unknown reasons which have subjected both the public and private schools to compete in order to maintain a good number of students. In this work, a modified Lotka–Volterra model of schools and nonenrolled entities population in the education system is studied. Private schools and nonenrolled entities play the role of a predator in public schools. Again, public schools and nonenrolled entities play the role of predators in private schools. Holling type II functional responses have been integrated in the analysis of the Lotka–Volterra model. The equilibrium points are established and their stability are determined using the Routh–Hurwitz criterion and eigenvalue method. Global stability has been done for the positive equilibrium point. Bifurcation is also done around the positive equilibrium point. Finally, a graphical illustration of various parameter is derived to show their effect on schools when they are varied. It is revealed that the increase in parameters θ 2 , θ 3 , and η 3 greatly affects the schools population as they are the ones leading to predation in school. Therefore, proper strategies should be developed to focus on reducing the mentioned parameters to avoid leading schools’ population to extinct.

Autoregressive graph Volterra models and applications

Graph-based learning and estimation are fundamental problems in various applications involving power, social, and brain networks, to name a few. While learning pair-wise interactions in network data is a well-studied problem, discovering higher-order interactions among subsets of nodes is still not yet fully explored. To this end, encompassing and leveraging (non)linear structural equation models as well as vector autoregressions, this paper proposes autoregressive graph Volterra models (AGVMs) that can capture not only the connectivity between nodes but also higher-order interactions presented in the networked data. The proposed overarching model inherits the identifiability and expressibility of the Volterra series. Furthermore, two tailored algorithms based on the proposed AGVM are put forth for topology identification and link prediction in distribution grids and social networks, respectively. Real-data experiments on different real-world collaboration networks highlight the impact of higher-order interactions in our approach, yielding discernible differences relative to existing methods.

A Khatri-Rao product tensor network for efficient symmetric MIMO Volterra identification

The identification of symmetric tensor network MIMO Volterra models has been studied earlier via the computation of a Moore-Penrose pseudoinverse in tensor network form. The current state of the art requires the construction of a tensor network of a repeated Khatri-Rao product of a matrix with itself. This construction has a computational complexity that is dominated by one singular value decomposition (SVD) of an RI × IN matrix, where N is the number of measurements, I depends linearly on the number of inputs and input lags and R is the maximal tensor network rank. In this article, we prove an alternative method for constructing this tensor network without any computation whatsoever. The pseudoinverse can then be computed through an orthogonalization of the newly proposed tensor network. Furthermore, the proposed algorithm allows for the recursive identification of symmetric Volterra models of increasing degree D, which reduces the computation to one SVD of a RI × N matrix per step. Through numerical experiments we demonstrate how the proposed algorithm enables up to ten times faster identification of symmetric tensor network MIMO Volterra systems.

Dynamic behavior of stochastic predator-prey system

<abstract> <p>In this paper, a stochastic predator-prey system with mutual interference is studied, which provides guidance on creating appropriate biodegradable environments. By utilizing Mawhin's coincidence degree theorem and constructing a suitable Lyapunov function, a Volterra model with mutual interference is studied. A few sufficient conditions are obtained for existence, extinction and global asymptomatic stability of the positive solution of the model. Then we perform various numerical simulations to verify the stochastic and deterministic systems are global asymptotically stable. It is significant that such a model is firstly proposed with stochastic mutual interference.</p> </abstract>

Interactions between concerns for the environment and other sources of concern in 31 European countries

Understanding how different sources of concern interact in people’s mind is a question that has entertained generations of scholars. The finite-pool-of-worry (FPW) hypothesis states that humans have limited resources to worry, thus when they are worried about one issue they become less worried about other issues. Instead, the affect generalization theory (AGT) posits that an increased level of worry about one threat increases concerns about related threats. To this end, we adopt a Lotka–Volterra model to detect instances of AGT and FPW among worries for the environment, economy, safety, social issues and immigration in 31 European countries between 2012 and 2019 (Eurobarometer data). Consistently with AGT, we find that an increase in the concern for the environment often favors the growth of concerns for the economy. Meanwhile, consistently with FPW, an increase in the concerns for the economy and for other sources of worry, often pushes down concerns for the environment. Building on our results, we hypothesize the existence of a pyramid of worries. At the bottom of the pyramid lie worries like concerns for the economy, which generally predate other worries. Concerns for the environment lie at the very top of the pyramid as they are generally predated by other worries. Last, we find that AGT and FPW can coexist not only over time and across countries, but also as a result of an asymmetric interaction.

Game Analysis of Container Ports Co‐Competition and Coordinated Development in Guangdong‐Hong Kong‐Macao Greater Bay Area

The port groups of the Guangdong‐Hong Kong‐Macao Greater Bay Area faces a new competitive and cooperative game structure concerning global competition, making it is necessary to build a comprehensive port coordination and development mechanism. In this study, we apply the logistics model to conduct parameter estimation and evolution trend analysis on the development of the port group and highlight that the current container throughput of the Guangdong‐Hong Kong‐Macao Greater Bay Area port group will reach the maximum and the new impetus is needed to push it into the next round of development and evolution. Combined with the theory of ecological population, the Lotka–Volterra model of multiple groups is introduced to study competition and cooperation game, explore their interaction, and provide quantitative support for the formation of different central port groups in the Guangdong‐Hong Kong‐Macao Greater Bay Area. The study provides references for strengthening the construction of complementary resources and coordinated mechanism within the port group, realizing the differential development between ports in the Guangdong‐Hong Kong‐Macao Greater Bay Area.

Study on the Growth Driving Model of the Enterprise Innovation Community Based on the Lotka–Volterra Model: A Case Study of the Chinese Automobile Manufacturing Enterprise Community

The goal of this paper is to establish a feasible three‐species equilibrium model to analyze the symbiotic relationship of an automobile manufacturing community. In order to extend the Lotka–Volterra model to empirical analysis, this paper proposes an enterprise community symbiosis model based on a three‐dimensional Lotka–Volterra model, as the classical two‐dimensional Lotka–Volterra model has limited application scenarios. This paper takes the innovation assets of three communities of the Chinese automobile manufacturing industry as samples. The industrial community related to automobile manufacturing consists of the automobile manufacturing population, automobile parts population, and enterprise service industry population. The symbiosis system is empirically analyzed from two aspects: the balanced development of the three populations and the competitive evolution of the three populations. The stability of the model is tested by the data from information technology and the intelligent manufacturing community. In the process of dynamic simulation, the symbiotic relationship between automobile manufacturing‐related populations shows a significant “skew symbiotic relationship.” This paper reconstructs the “whole population symbiosis” optimization model as skew distribution is difficult to apply to support collaborative development. The symbiosis optimization under the equilibrium state of the three populations shows that the growth of the three automobile manufacturing industry populations has the possibility of equilibrium and reciprocity. The empirical analysis fully demonstrates the feasibility of this research paradigm. The evolution analysis of the symbiotic system shows that cooperative behavior is better than competitive strategy. The research paradigm proposed in this paper can better analyze the symbiosis mechanism of the enterprise community.

Evolutionary Path and Sustainable Optimization of an Innovation Ecosystem for a High-Tech Enterprise Based on Empirical Evidence from Hubei Province

This study focuses on the evolutionary path and sustainable optimization of an innovation ecosystem for a high-tech enterprise. To analyze the formation and structure of such an ecosystem, this study applies the Lotka–Volterra model to argue that cooperative symbiosis is an effective path for system evolution. It establishes a system dynamics model to analyze the system feedback mechanism and takes the development of an innovation ecosystem for a high-tech enterprise in Hubei Province as an example to discuss the dynamic factors that affect the system’s evolution and to design the sustainable optimization path of the system. The empirical results show that the system’s innovation input can only promote the positive effects of system evolution to a certain extent, and that regulatory elements such as policies and infrastructure can impact governance effectiveness and promote system optimization and sustainable development.

Exploration of China–ASEAN Trade Relations in the Context of Sustainable Economic Development—Based on the Lotka–Volterra Model

The concept of sustainable development as a consensus is being recognized and practiced by the international community, and how to promote China–ASEAN trade relations and stable and sustainable economic development is an important guarantee for the coordinated economic development of the region. This paper studies the trade relationship between China and the ten ASEAN countries by using the theory of ecological population evolution and determines the ecological trade relationship between China and the ten ASEAN countries by establishing the Lotka–Volterra model of the evolution of two populations—the relationship of competition, the relationship of partial benefit, and the relationship of mutually beneficial symbiosis—so as to analyze the degree of trade friction between the two sides and propose corresponding strategies for coordinated development of the regional economy. According to the results of the population evolution competition model, there is a competitive relationship between China, Brunei, and Vietnam, which may cause trade friction. China has a trade-biased relationship with Indonesia, Cambodia, Malaysia, and Singapore, which is less likely to cause trade friction. China’s trade with Laos, Myanmar, the Philippines, and Thailand is mutually beneficial. The possibility of trade friction is low, and the trade relationship between China and ASEAN as a whole is also mutually beneficial. On this basis, in order to further enhance the trade relations between PRC and the ten ASEAN countries and promote the coordinated development of the regional economy, this paper puts forward the policy suggestions of improving infrastructure, strengthening independent innovation, seeking cooperation areas, and actively investing in foreign direct investment.

Diagnosis and monitoring of a person's psychophysiological state based on multidimensional transient functions of the oculo-motor system

An important scientific and practical problem has been solved, which consists in the creation of methods and tools for nonlinear dynamic identification "input-output" of the human oculomotor system (OMS). Identification is carried out based on Volterra models in the form of multidimensional transient functions (n-dimensional integrals from Volterra kernels) using innovative eye tracking technology. As test signals, visual stimuli are used, which are displayed on the monitor screen at different distances from the starting position, which formally corresponds to the action of step signals with different amplitudes at the OMS input. Experimental studies of the OMS of individuals were carried out using the Tobii Pro TX300 eye tracker. The transient functions of the first and diagonal intersections of the transient functions of the second and third order were determined based on the data of oculographic studies. The models differ from the known ones in that they provide the possibility of modeling the OMS in each interval of input signals beyond the radius of convergence of the Volterra series. The obtained multidimensional functions are used as a source of primary data in the implementation of intelligent information technology for the diagnosis and monitoring of human psychophysiological states. A set of heuristic features are proposed, which are determined using integral and differential transformations of the obtained transient functions of the OMS. The informativeness of individual signs and all their possible combinations in pairs according to the indicator of the probability of correct recognition was studied. The research results were obtained using the construction of Bayesian classifiers in different spaces of the proposed features. Figs.: 2. Refs.: 15 titles.
 Keywords: psychophysiological state, diagnosis, monitoring, oculo-motor system, identification, Volterra model, multidimensional transient functions, test visual stimuli, eye-tracking technology

Analysis of the dynamical behavior of solutions for a class of hybrid generalized Lotka–Volterra models

In the paper, the generalized Lotka–Volterra systems with switching are investigated. It is assumed that functions included in the right parts of the considered systems satisfy nonlinear constraints of the power type. Conditions are found for the switching law that provide the desired features of the dynamical behavior of solutions of such systems. In particular, the problems to guarantee the asymptotic stability of a given equilibrium position with the required region of attraction, the ultimate boundedness of solutions, the global asymptotic stability are studied. To solve these problems, the method of splitting the phase space of the studied systems into several parts is used. In each of these parts, some partial estimates for the chosen Lyapunov function are constructed and conditions are established to ensure the transition of solutions from one part to another. Moreover, different Lyapunov functions can be used in different parts of the phase space.

Evolutionary game analysis of big data discriminatory pricing diffusion based on the supervision of relevant interest parties

AbstractThe use of big data technology is significant for the market competition of two‐side enterprises. However, big data technology is likely to become a tool for discriminatory pricing, causing damage to the benefit‐relevant parties. This paper constructs a four party evolutionary game model between two‐side enterprises, governments, suppliers, and consumers. Lotka–Volterra model is introduced to explore the evolutionary impact of the supervisory behaviors of the benefit‐relevant parties on the diffusion of big data discriminatory pricing (BDDP) in two‐side enterprises. Using MATLAB simulation tools, the evolution game and diffusion evolution model are mathematically deduced, and the analogue simulation is used for correlation analysis. This paper can help two‐side enterprises, governments, suppliers, and consumers better understand the diffusion law of BDDP and more accurately predict the development of it. At the same time, it provides a quantitative theoretical basis for the reasonable decision‐making of governments, suppliers, and consumers.

Competition of alliances in a cyclically dominant eight-species population

In a diverse population, where many species are present, competitors can fight for surviving at individual and collective levels. In particular, species, which would beat each other individually, may form a specific alliance that ensures them stable coexistence against the invasion of an external species. Our principal goal is to identify those general features of a formation which determine its vitality. Therefore, we here study a traditional Lotka–Volterra model of eight-species where two four-species cycles can fight for space. Beside these formations, there are other solutions which may emerge when invasion rates are varied. The complete range of parameters is explored and we find that in most of the cases those alliances prevail which are formed by equally strong members. Interestingly, there are regions where the symmetry is broken and the system is dominated by a solution formed by seven species. Our work also highlights that serious finite-size effects may emerge which prevent observing the valid solution in a small system.

Ідентифікація окуло-моторної системи людини на основі ряду Вольтерри: застосування в системі захисту інформації

The information technology of biometric identification of a person has re-ceived further development due to the use as a source of primary data of infor-mation models of the oculo-motor system (OMS) of the «input-output» type based on the Volterra series. Eye-tracking technology is used to build models. Experimental studies of the OMS of two respondents were carried out. Based on the data obtained with the Tobii Pro TX300 eye-tracker, the transient func-tions of the first, second and third orders of the OMS when applying the Volter-ra series model were determined. This makes it possible to increase the accura-cy of OMS modeling and, as a result, to increase the reliability of recognition in the space of the proposed heuristic features, which are determined using inte-gral and differential transformations of multidimensional transient functions of OMS, which greatly simplifies the identification of features and the practical implementation of the Bayesian classifier.A high variability of the transient functions of the second and third or-ders for two respondents was revealed. Thus, it seems appropriate to use multidimensional transient functions for biometric identification.A set of heuristic features are proposed, which are determined on the basis of multidimensional transient functions obtained from eye-tracking data. The informativeness of individual features and their combinations in pairswas investigated. Two-dimensional feature spaces with the maximum value of the probability of correct recognition indicator when solving the problem of biometric identification of a person were found (Pmax=0.974). The research results were obtained using the construction of Bayesian clas-sifiers in different spaces of the proposed features by means of machine learning based on the data of the formed trainingsamples