Existence and stability of periodic solution of a Lotka–Volterra predator–prey model with state dependent impulsive effects

AbstractAccording to biological and chemical control strategy for pest, we investigate the dynamic behavior of a Lotka–Volterra predator–prey state-dependent impulsive system by releasing natural enemies and spraying pesticide at different thresholds. By using Poincaré map and the properties of the Lambert W function, we prove that the sufficient conditions for the existence and stability of semi-trivial solution and positive periodic solution. Numerical simulations are carried out to illustrate the feasibility of our main results

Time Delayed Stage-Structured Predator-Prey Model with Birth Pulse and Pest Control Tactics

Normally, chemical pesticides kill not only pests but also their natural enemies. In order to better control the pests, two-time delayed stage-structured predator-prey models with birth pulse and pest control tactics are proposed and analyzed by using impulsive differential equations in present work. The stability threshold conditions for the mature prey-eradication periodic solutions of two models are derived, respectively. The effects of key parameters including killing efficiency rate, pulse period, the maximum birth effort per unit of time of natural enemy, and maturation time of prey on the threshold values are discussed in more detail. By comparing the two threshold values of mature prey-extinction, we provide the fact that the second control tactic is more effective than the first control method

Nonlinear dynamics of plankton ecosystem with impulsive control and environmental fluctuations

It is well known that the density of plankton populations always increases and decreases or keeps invariant for a long time, and the variation of plankton density is an important factor influencing the real aquatic environments, why do these situations occur? It is an interesting topic which has become the common interest for many researchers. As the basis of the food webs in oceans, lakes, and reservoirs, plankton plays a significant role in the material circulation and energy flow for real aquatic ecosystems that have a great effect on the economic and social values. Planktonic blooms can occur in some environments, however, and the direct or indirect adverse effects of planktonic blooms on real aquatic ecosystems, such as water quality, water landscape, aquaculture development, are sometimes catastrophic, and thus planktonic blooms have become a challenging and intractable problem worldwide in recent years. Therefore, to understand these effects so that some necessary measures can be taken, it is important and meaningful to investigate the dynamic growth mechanism of plankton and reveal the dynamics mechanisms of formation and disappearance of planktonic blooms. To this end, based on the background of the ecological environments in the subtropical lakes and reservoirs, this dissertation research takes mainly the planktonic algae as the research objective to model the mechanisms of plankton growth and evolution. In this dissertation, some theories related to population dynamics, impulsive control dynamics, stochastic dynamics, as well as the methods of dynamic modeling, dynamic analysis and experimental simulation, are applied to reveal the effects of some key biological factors on the dynamics mechanisms of the spatial-temporal distribution of plankton and the termination of planktonic blooms, and to predict the dynamics evolutionary processes of plankton growth. The main results are as follows: Firstly, to discuss the prevention and control strategies on planktonic blooms, an impulsive reaction-diffusion hybrid system was developed. On the one hand, the dynamic analysis showed that impulsive control can significantly influence the dynamics of the system, including the ultimate boundedness, extinction, permanence, and the existence and uniqueness of positive periodic solution of the system. On the other hand, some experimental simulations were preformed to reveal that impulsive control can lead to the extinction and permanence of population directly. More precisely, the prey and intermediate predator populations can coexist at any time and location of their inhabited domain, while the top predator population undergoes extinction when the impulsive control parameter exceeds some a critical value, which can provide some key arguments to control population survival by means of some reaction-diffusion impulsive hybrid systems in the real life. Additionally, a heterogeneous environment can affect the spatial distribution of plankton and change the temporal-spatial oscillation of plankton distribution. All results are expected to be helpful in the study of dynamic complex of ecosystems. Secondly, a stochastic phytoplankton-zooplankton system with toxic phytoplankton was proposed and the effects of environmental stochasticity and toxin-producing phytoplankton (TPP) on the dynamics mechanisms of the termination of planktonic blooms were discussed. The research illustrated that white noise can aggravate the stochastic oscillation of plankton density and a high-level intensity of white noise can accelerate the extinction of plankton and may be advantageous for the disappearance of harmful phytoplankton, which imply that the white noise can help control the biomass of plankton and provide a guide for the termination of planktonic blooms. Additionally, some experimental simulations were carried out to reveal that the increasing toxin liberation rate released by TPP can increase the survival chance of phytoplankton population and reduce the biomass of zooplankton population, but the combined effects of those two toxin liberation rates on the changes in plankton are stronger than that of controlling any one of the two TPP. All results suggest that both white noise and TPP can play an important role in controlling planktonic blooms. Thirdly, we established a stochastic phytoplankton-toxic producing phytoplankton-zooplankton system under regime switching and investigated how the white noise, regime switching and TPP affect the dynamics mechanisms of planktonic blooms. The dynamical analysis indicated that both white noise and toxins released by TPP are disadvantageous to the development of plankton and may increase the risk of plankton extinction. Also, a series of experimental simulations were carried out to verify the correctness of the dynamical analysis and further reveal the effects of the white noise, regime switching and TPP on the dynamics mechanisms of the termination of planktonic blooms. On the one hand, the numerical study revealed that the system can switch from one state to another due to regime shift, and further indicated that the regime switching can balance the different survival states of plankton density and decrease the risk of plankton extinction when the density of white noise are particularly weak. On the other hand, an increase in the toxin liberation rate can increase the survival chance of phytoplankton but reduce the biomass of zooplankton, which implies that the presence of toxic phytoplankton may have a positive effect on the termination of planktonic blooms. These results may provide some insightful understanding on the dynamics of phytoplankton-zooplankton systems in randomly disturbed aquatic environments. Finally, a stochastic non-autonomous phytoplankton-zooplankton system involving TPP and impulsive perturbations was studied, where the white noise, impulsive perturbations and TPP are incorporated into the system to simulate the natural aquatic ecological phenomena. The dynamical analysis revealed some key threshold conditions that ensure the existence and uniqueness of a global positive solution, plankton extinction and persistence in the mean. In particular, we determined if there is a positive periodic solution for the system when the toxin liberation rate reaches a critical value. Some experimental simulations also revealed that both white noise and impulsive control parameter can directly influence the plankton extinction and persistence in the mean. Significantly, enhancing the toxin liberation rate released by TPP increases the possibility of phytoplankton survival but reduces the zooplankton biomass. All these results can improve our understanding of the dynamics of complex of aquatic ecosystems in a fluctuating environment

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Dynamical Models of Biology and Medicine

Mathematical and computational modeling approaches in biological and medical research are experiencing rapid growth globally. This Special Issue Book intends to scratch the surface of this exciting phenomenon. The subject areas covered involve general mathematical methods and their applications in biology and medicine, with an emphasis on work related to mathematical and computational modeling of the complex dynamics observed in biological and medical research. Fourteen rigorously reviewed papers were included in this Special Issue. These papers cover several timely topics relating to classical population biology, fundamental biology, and modern medicine. While the authors of these papers dealt with very different modeling questions, they were all motivated by specific applications in biology and medicine and employed innovative mathematical and computational methods to study the complex dynamics of their models. We hope that these papers detail case studies that will inspire many additional mathematical modeling efforts in biology and medicin

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

AFIT School of Engineering Contributions to Air Force Research and Technology. Calendar Year 1971

This report contains abstracts of Master of Science theses and Doctoral Dissertations completed during the 1971 calendar year at the School of Engineering, Air Force Institute of Technology

Qualitative Theory of Switched Integro-differential Equations with Applications

Switched systems, which are a type of hybrid system, evolve according to a mixture of continuous/discrete dynamics and experience abrupt changes based on a switching rule. Many real-world phenomena found in branches of applied math, computer science, and engineering are naturally modelled by hybrid systems. The main focus of the present thesis is on hybrid impulsive systems with distributed delays (HISD). That is, studying the qualitative behaviour of switched integro-differential systems with impulses. Important applications of impulsive systems can be found in stabilizing control (e.g. using impulsive control in combination with switching control) and epidemiology (e.g. pulse vaccination control strategies), both of which are studied in this work. In order to ensure the models are well-posed, some fundamental theory is developed for systems with bounded or unbounded time-delays. Results on existence, uniqueness, and continuation of solutions are established. As solutions of HISD are generally not known explicitly, a stability analysis is performed by extending the current theoretical approaches in the switched systems literature (e.g. Halanay-like inequalities and Razumikhin-type conditions). Since a major field of research in hybrid systems theory involves applying hybrid control to problems, contributions are made by extending current results on stabilization by state-dependent switching and impulsive control for unstable systems of integro-differential equations. The analytic results found are applied to epidemic models with time-varying parameters (e.g. due to changes in host behaviour). In particular, we propose a switched model of Chikungunya disease and study its long-term behaviour in order to develop threshold conditions guaranteeing disease eradication. As a sequel to this, we look at the stability of a more general vector-borne disease model under various vaccination schemes. Epidemic models with general nonlinear incidence rates and age-dependent population mixing are also investigated. Throughout the thesis, computational methods are used to illustrate the theoretical results found

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods