Volterra Model Research Articles

Investigating the development and interaction of bus-metro based on Lotka–Volterra models: Evidence from seven central cities in China

Bus and metro are the two most important public transport modes in many metropolises in China, and they both have experienced rapid growth and meanwhile coexisted for decades. However, little is known on how the metro and bus interacted with each other during their rapid growths. This study was proposed to investigate the growth and interaction of bus-metro from the macro perspective. The passenger volume data for metro and bus were collected from seven central cities to represent the development of the two public transport modes, and the Logistic model and Lotka–Volterra model were employed to model the growth as well as the interaction of bus-metro, respectively. The modeling results show that the development of bus conforms to the Logistic model (i.e. S-shaped curve), while the bus-metro interaction conforms to the Lotka–Volterra model with interaction modes of competition (Shanghai city from 2000–2009, Shanghai city from 2009–2018, Guangzhou city from 2009–2017, Nanjing city from 2008–2018), and mutualism (Guangzhou city from 2000–2009). The further analysis indicates that urban characteristics and policies determine the interaction, and the parameters of the Lotka–Volterra model could be used to judge the bus-metro interaction type.

Qualitative analysis of a degenerate fixed point of a discrete predator–prey model with cooperative hunting

This paper investigates the qualitative properties near a degenerate fixed point of a discrete predator–prey model with cooperative hunting derived from the Lotka–Volterra model where the eigenvalues of the corresponding linear operator are ±1. Applying the theory of the normal form and the Takens's theorem, we change the problem of this discrete model into the one of an ordinary differential system. With the technique of desingularization to blow up the degenerate equilibrium of the ordinary differential system, we obtain its qualitative properties. Utilizing the conjugacy between the discrete model and the time‐one mapping of the vector field, we obtain the qualitative structures of this discrete model.

A general theory of coexistence and extinction for stochastic ecological communities.

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Symbiosis Evolution of Science Communication Ecosystem Based on Social Media: A Lotka–Volterra Model-Based Simulation

Social media has become an important way for science communication. Some scholars have examined how to help scientists engage with social media from operational training, policy guidance, and social media services improving. The main contribution of this study is to construct a symbiosis evolution model of science communication ecosystem (SCE) between scientists and social media platforms based on the symbiosis theory and the Lotka–Volterra model to discuss the evolution of their symbiotic patterns and population size under different symbiosis coefficients. The results indicate that (1) the size of the symbiosis coefficients determines the equilibrium outcome of the symbiosis evolution of scientists and social media platforms; (2) scientists and social media platforms can promote each other’s population size under the mutualism pattern, which can achieve sustainable science communication; (3) “1 + 1 > 2” effect can only be achieved under the symmetric mutualism pattern and the growth of scientists and social media platforms is more stable and sufficient than that of other patterns. The findings will provide additional perspectives for promoting the sustainable development of science communication based on social media.

Novel explicit breath wave and numerical solutions of an Atangana conformable fractional Lotka–Volterra model

This research studies novel analytical solutions of an Atangana conformable fractional Lotka–Volterra (LV) model system arising in ecology by means of three systematic schemes (the extended simplest equation method, modified Kudryashov method, and the sech–tanh expansion method). Then we use the obtained solutions to evaluate the boundary and initial conditions that helps to apply the B-spline collection numerical schemes to evaluate the value of absolute error. The investigation aims also to explain the accuracy of the analytical solutions by evaluating absolute value of errors between exact and numerically obtained solutions. In order to better explain the obtained solutions, some relevant sketches are given in three different types. The novelty of the work done is attempted to explain by comparing our results with some available and relevant results. Further, relevant strengths, usefulness, practical applications, and the methods’ ability for applying in various nonlinear evolution equations have been explored.

Tractable models of ecological assembly

AbstractEcological assembly is a fundamental and yet poorly understood process. Three main obstacles hinder the development of a theory of assembly, and when these issues are sidestepped by making strong assumptions, one can build an assembly graph in which nodes are ecological communities and edges are invasions shifting their composition. The graph can then be analysed directly, without the need to consider dynamics. To showcase this framework, we build and analyse assembly graphs for the competitive Lotka–Volterra model, showing that in these cases sequential assembly (in which species invade a community one at a time) can reach the same configurations found when starting the system with all species present at different initial conditions. We discuss how our results can advance our understanding of assembly both from an empirical and a theoretical point of view, informing the study of ecological restoration and the design of ecological communities.

The need for alternative plant species interaction models

Abstract Aims The limitations of classical Lotka–Volterra models for analyzing and interpreting competitive interactions among plant species have become increasingly clear in recent years. Three of the problems that have been identified are (i) the absence of frequency-dependence, which is important for long-term coexistence of species, (ii) the need to take unmeasured (often unmeasurable) variables influencing individual performance into account (e.g. spatial variation in soil nutrients or pathogens) and (iii) the need to separate measurement error from biological variation. Methods We modified the classical Lotka–Volterra competition models to address these limitations. We fitted eight alternative models to pin-point cover data on Festuca ovina and Agrostis capillaris over 3 years in an herbaceous plant community in Denmark. A Bayesian modeling framework was used to ascertain whether the model amendments improve the performance of the models and increase their ability to predict community dynamics and to test hypotheses. Important Findings Inclusion of frequency-dependence and measurement error, but not unmeasured variables, improved model performance greatly. Our results emphasize the importance of comparing alternative models in quantitative studies of plant community dynamics. Only by considering possible alternative models can we identify the forces driving community assembly and change, and improve our ability to predict the behavior of plant communities.

Positivity preserving truncated Euler–Maruyama Method for stochastic Lotka–Volterra competition model

The well-known stochastic Lotka–Volterra model for interacting multi-species in ecology has some typical features: highly nonlinear, positive solution and multi-dimensional. The known numerical methods including the tamed/truncated Euler–Maruyama (EM) applied to it do not preserve its positivity. The aim of this paper is to modify the truncated EM to establish a new positive preserving truncated EM (PPTEM). To simplify the proof as well as to make our theory more understandable, we will first develop a nonnegative preserving truncated EM (NPTEM) and then establish the PPTEM. Of course, we should point out that the NPTEM has its own right as many SDE models in applications have their nonnegative solutions.

Competing or collaborating, with no symmetrical behaviour: Leadership opportunities and winning strategies under stability

In this paper, a new dynamic mathematical model describing leadership emergence or disappearance in agent based networks is proposed. Through a generalised Verhulst–Lotka–Volterra model, a triad of agents operates in a market where each agent detains a quota. The triad is composed of a leader, who leads communication, and two followers. Communications flows both ways from leader to followers and vice versa. Competition, collaboration and cheating are allowed. Stability solutions are investigated analytically through a fixed point analysis. Various solutions exist depending on the type of behavioural interactions. Results show that communication counts: survival of the leader is a condition for stability. All configurations with the leader out of the market are unstable. Conversely, the two followers position is highly difficult. The three agents cannot all survive unless they behave under mutual collaboration and in very special conditions. For followers, cheating the leader, especially if the leader is collaborating, can be a disaster. By the way, collaboration with the leader may not always ensure market survival. However, this can be a strategy to survive and even share the leadership, in particular when the other agent cheats (or is cheated by) the leader. Cheating is a cause of instability. In fact, only a few cases reach stability: this occurs when cheating comes from the leader and the leader always wins. The leader may be interested in cheating if she does not want to share the leadership with a follower, that is to get the monopoly of the market.

A macro‐ecological approach to predation density‐dependence

Predation often deviates from the law of mass action: many micro‐ and meso‐scale experiments have shown that consumption saturates with resource abundance, and decreases due to interference between consumers. But does this observation hold at macro‐ecological scales, spanning many species and orders of magnitude in biomass? If so, what are its consequences for large‐scale ecological patterns and dynamics?We perform a meta‐analysis of predator–prey pairs of mammals, birds and reptiles, and show that total predation rates appear to increase, not as the product of predator and prey densities following the Lotka–Volterra (mass action) model, but rather as the square root of that product. This suggests a phenomenological power‐law expression of the effective cross‐ecosystem predation rate. We discuss whether the same power‐law may hold dynamically within an ecosystem, and assuming that it does, we explore its consequences in a simple food chain model. The empirical exponents fall close to the boundary between regimes of donor and consumer limitation. Exponents on this boundary are singular in multiple ways. First, they maximize predator abundance and some stability metrics. Second, they create proportionality relations between biomass and productivity, both within and between trophic levels. These intuitive relations do not hold in general in mass action models, yet they are widely observed empirically.These results provide evidence of mechanisms limiting predation across multiple ecological scales. Some of this evidence was previously associated with donor control, but we show that it supports a wider range of possibilities, including forms of consumer control. As limiting consumption counter‐intuitively allows larger populations, it is worthwhile to reconsider whether the observed predation rates arise from microscopic mechanisms, or could hint at selective pressure at the population level.

Optimization of isolated power systems with renewables and storage batteries based on nonlinear Volterra models for the specially protected natural area of lake Baikal

Lake Baikal conservation as UNESCO World Heritage property, national and international environmental laws dictate specific requirements and rules for the operation of energy facilities in the specially protected areas of Lake Baikal. Such restrictions have given particular strength to penetration of the renewables and storage systems. This article presents the universal method for optimizing the structure and installed capacity of isolated power systems using the example of a real tourist base. On the basis of the pool of mathematical models and control algorithms for power equipment the unified model of isolated power system was formed. The choice of mathematical models is made in accordance to the problem specifics and the climatic indicators. For optimal control of the processes of charging and discharging and determining the operating modes of the storage batteries the model based on nonlinear integral Volterra equations was used, which takes into account the nonlinear dependence of efficiency on the state of charge. In order to solve the problem of optimizing the installed capacity of the equipment the brute-force method was used. The results showed that the optimal use is polycrystalline solar panels with a total capacity of 60 kW and storage batteries with a capacity of 336 kWh.

Microbial Interactions as Drivers of a Nitrification Process in a Chemostat.

This article deals with the inclusion of microbial ecology measurements such as abundances of operational taxonomic units in bioprocess modelling. The first part presents the mathematical analysis of a model that may be framed within the class of Lotka–Volterra models fitted to experimental data in a chemostat setting where a nitrification process was operated for over 500 days. The limitations and the insights of such an approach are discussed. In the second part, the use of an optimal tracking technique (developed within the framework of control theory) for the integration of data from genetic sequencing in chemostat models is presented. The optimal tracking revisits the data used in the aforementioned chemostat setting. The resulting model is an explanatory model, not a predictive one, it is able to reconstruct the different forms of nitrogen in the reactor by using the abundances of the operational taxonomic units, providing some insights into the growth rate of microbes in a complex community.

Further Study on Dynamics for a Fractional-Order Competitor-Competitor-Mutualist Lotka–Volterra System

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.

Memory cross Volterra model for Doherty power amplifier with group delay mismatch

Memory cross Volterra model for Doherty power amplifier with group delay mismatch

An Improved Robust Predictive Control Approach Based on Generalized 3rd Order S-PARAFAC Volterra Model Applied to a 2-DoF Helicopter System

This paper proposes a generalized robust predictive control approach based on 3rd order S-PARAFAC Volterra Model. The main idea is to use the predictive control law based on the latter model with taking into account the parameter and operation uncertainties resulting from the measurement noise and the robust identification technique named Unknown But Bounded Error (UBBE). One of the advantages is the convexity of the objective function with respect to the parameter uncertainty set. The min-max optimization problem becomes thus simpler, by minimizing the objective function in the worst-case only over the set of uncertain models. This work proposes a new generalized nonlinear robust control algorithm dedicated for uncertain processes. In this algorithm we optimize a quadratic criterion and take into account the physical and geometrical constraints due to parameter uncertainties. The performances of this work are illustrated and validated on a benchmark as a continuous stirred-tank reactor system (CSTR) and on an experimental 2-DoF helicopter system.

Turnover Modulates the Need for a Cost of Resistance in Adaptive Therapy.

Adaptive therapy seeks to exploit intratumoral competition to avoid, or at least delay, the emergence of therapy resistance in cancer. Motivated by promising results in prostate cancer, there is growing interest in extending this approach to other neoplasms. As such, it is urgent to understand the characteristics of a cancer that determine whether or not it will respond well to adaptive therapy. A plausible candidate for such a selection criterion is the fitness cost of resistance. In this article, we study a general, but simple, mathematical model to investigate whether the presence of a cost is necessary for adaptive therapy to extend the time to progression beyond that of a standard-of-care continuous therapy. Tumor cells were divided into sensitive and resistant populations and we model their competition using a system of two ordinary differential equations based on the Lotka-Volterra model. For tumors close to their environmental carrying capacity, a cost was not required. However, for tumors growing far from carrying capacity, a cost may be required to see meaningful gains. Notably, it is important to consider cell turnover in the tumor, and we discuss its role in modulating the impact of a resistance cost. To conclude, we present evidence for the predicted cost-turnover interplay in data from 67 patients with prostate cancer undergoing intermittent androgen deprivation therapy. Our work helps to clarify under which circumstances adaptive therapy may be beneficial and suggests that turnover may play an unexpectedly important role in the decision-making process. SIGNIFICANCE: Tumor cell turnover modulates the speed of selection against drug resistance by amplifying the effects of competition and resistance costs; as such, turnover is an important factor in resistance management via adaptive therapy.See related commentary by Strobl et al., p. 811.

Modeling and protecting global ecological networks

With the rapid urbanization worldwide and ever-increasing impacts of human activities since at least 200 years ago, we are now facing a harsh situation of our biosphere. Building a global-level network model on ecological systems is of great importance, which would be able to provide us predictive and quantitative responses to human activities, leading to viable suggestions to policymakers. In this paper, we propose a multi-layer model for the global ecological network, where a number of local networks are connected via long-range interactions associated with migrant species, which can be induced by human activities or natural migration of wildlife, and each local network is generated by a trophic-level-based stochastic model. Predator–prey dynamics is described by a networked Lotka–Volterra model that accounts for the self-suppression effects on basal species, and the negative feedback loops. Impacts of human activities are modeled by investigating the quantitative changes of biodiversity under certain protecting strategies. We reveal that the global ecological network is organized in a clustered small-world manner, with in-degree distribution more heterogeneous than out-degree distribution. Protecting endangered species, popular preys and predicted-to-be-extinct species is more effective than randomly selected species or influential predators. Protecting after entering the fast extinction stage is more effective than at the beginning for some high trophic level species.

Markowitz Portfolio Selection for Multivariate Affine and Quadratic Volterra Models

This paper concerns portfolio selection with multiple assets under rough covariance matrix. We investigate the continuous-time Markowitz mean-variance problem for a multivariate class of affine and quadratic Volterra models. In this incomplete non-Markovian and nonsemimartingale market framework with unbounded random coefficients, the optimal portfolio strategy is expressed by means of a Riccati backward stochastic differential equation (BSDE). In the case of affine Volterra models, we derive explicit solutions to this BSDE in terms of multidimensional Riccati--Volterra equations. This framework includes multivariate rough Heston models and extends the results of Han and Wong [Appl. Math. Optim. (2020)]. In the quadratic case, we obtain new analytic formulae for the the Riccati BSDE and we establish their link with infinite-dimensional Riccati equations. This covers rough Stein--Stein and Wishart type covariance models. Numerical results on a two-dimensional rough Stein--Stein model illustrate the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategy. In particular for positively correlated assets, we find that the optimal strategy in our model is a “buy rough sell smooth” one.

Modeling of Lithium-ion Batteries via Tensor-Network-Based Volterra Model

Modeling of Lithium-ion Batteries via Tensor-Network-Based Volterra Model

Identification of the Oculo-Motor System in the Form Volterra Model Based on Eye-Tracking Data

Instrumental computing and software tools have been developed for constructing a nonlinear dynamic model of the human oculo-motor system (OMS) based on the data of input-output experiments using test visual stimulus and innovative technology. Volterra model in the form of multidimensional transition functions of the 1st, 2nd and 3rd orders, taking into account the inertial and nonlinear properties of the OMS was used as the identification tool. Eye-tracking data developed in the Matlab environment are tested on real datasets from an experimental study of OMS.