Note on the Persistence of a Nonautonomous Lotka-Volterra Competitive System with Infinite Delay and Feedback Controls

We study a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. We establish a series of criteria under which a part of n-species of the systems is driven to extinction while the remaining part of the species is persistent. Particularly, as a special case, a series of new sufficient conditions on the persistence for all species of system are obtained. Several examples together with their numerical simulations show the feasibility of our main results

Dynamic Behaviors of a Discrete Lotka-Volterra Competition System with Infinite Delays and Single Feedback Control

A nonautonomous discrete two-species Lotka-Volterra competition system with infinite delays and single feedback control is considered in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that if the the discrete Lotka-Volterra competitive system with infinite delays and without feedback control is permanent, then, by choosing some suitable feedback control variable, the permanent species will be driven to extinction. That is, the feedback control variable, which represents the biological control or some harvesting procedure, is the unstable factor of the system. Such a finding overturns the previous scholars’ recognition on feedback control variables

Chaos to Permanence - Through Control Theory

Work by Cushing et al. [18] and Kot et al. [60] demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species [35]. We utilize present chaotic behavior and a control algorithm based on [66, 72] to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from [30], a ratio-dependent one-prey, two-predator model from [35] and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model [67] and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Chaos to Permanence-Through Control Theory

Work by Cushing et al. \cite{Cushing} and Kot et al. \cite{Kot} demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species \cite{EVG}. We utilize present chaotic behavior and a control algorithm based on \cite{Vincent97,Vincent2001} to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from \cite{Harvesting}, a ratio-dependent one-prey, two-predator model from \cite{EVG} and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model \cite{Upad} and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved

Permanence and almost periodic solution of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales

In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model is obtained. Furthermore, by means of the almost periodic functional hull theory on time scales and Lyapunov functional, some criteria are obtained for the existence, uniqueness and global attractivity of almost periodic solutions of the model. Our results complement and extend some scientific work in recent years. Finally, an example is given to illustrate the main results.Comment: 31page

Extending Feynman's Formalisms for Modelling Human Joint Action Coordination

The recently developed Life-Space-Foam approach to goal-directed human action deals with individual actor dynamics. This paper applies the model to characterize the dynamics of co-action by two or more actors. This dynamics is modelled by: (i) a two-term joint action (including cognitive/motivatonal potential and kinetic energy), and (ii) its associated adaptive path integral, representing an infinite--dimensional neural network. Its feedback adaptation loop has been derived from Bernstein's concepts of sensory corrections loop in human motor control and Brooks' subsumption architectures in robotics. Potential applications of the proposed model in human--robot interaction research are discussed. Keywords: Psycho--physics, human joint action, path integralsComment: 6 pages, Late

Nonlinear Stochastic Systems And Controls: Lotka-Volterra Type Models, Permanence And Extinction, Optimal Harvesting Strategies, And Numerical Methods For Systems Under Partial Observations

This dissertation focuses on a class of stochastic models formulated using stochastic differential equations with regime switching represented by a continuous-time Markov chain, which also known as hybrid switching diffusion processes. Our motivations for studying such processes in this dissertation stem from emerging and existing applications in biological systems, ecosystems, financial engineering, modeling, analysis, and control and optimization of stochastic systems under the influence of random environments, with complete observations or partial observations. The first part is concerned with Lotka-Volterra models with white noise and regime switching represented by a continuous-time Markov chain. Different from the existing literature, the Markov chain is hidden and canonly be observed in a Gaussian white noise in our work. We use a Wonham filter to estimate the Markov chain from the observable evolution of the given process, and convert the original system to a completely observable one. We then establish the regularity, positivity, stochastic boundedness, and sample path continuity of the solution. Moreover, stochastic permanence and extinction using feedback controls are investigated. The second part develops optimal harvest strategies for Lotka-Volterra systems so as to establish economically, ecologically, and environmentally reasonable strategies for populations subject to the risk of extinction. The underlying systems are controlled regime-switching diffusions that belong to the class of singular control problems. We construct upper bounds for the value functions, prove the finiteness of the harvesting value, and derive properties of the value functions. Then we construct explicit chattering harvesting strategies and the corresponding lower bounds for the value functions by using the idea of harvesting only one species at a time. We further show that this is a reasonable candidate for the best lower bound that one can expect. In the last part, we study optimal harvesting problems for a general systems in the case that the Markov chain is hidden and can only be observed in a Gaussian white noise. The Wonham filter is employed to convert the original problem to a completely observable one. Then we treat the resulting optimal control problem. Because the problem is virtually impossible to solve in closed form, our main effort is devoted to developing numerical approximation algorithms. To approximate the value function and optimal strategies, Markov chain approximation methods are used to construct a discrete-time controlled Markov chain. Convergence of the algorithm is proved by weak convergence method and suitable scaling