Threshold dynamics of a stochastic mathematical model for Wolbachia infections

A stochastic mathematical model is proposed to study how environmental heterogeneity and the augmentation of mosquitoes with Wolbachia bacteria affect the outcomes of dengue disease. The existence and uniqueness of the positive solutions of the system are studied. Then the V-geometrically ergodicity and stochastic ultimate boundedness are investigated. Further, threshold conditions for successful population replacement are derived and the existence of a unique ergodic steady-state distribution of the system is explored. The results show that the ratio of infected to uninfected mosquitoes has a great influence on population replacement. Moreover, environmental noise plays a significant role in control of dengue fever

Research on pattern dynamics of a class of predator-prey model with interval biological coefficients for capture

Due to factors such as climate change, natural disasters, and deforestation, most measurement processes and initial data may have errors. Therefore, models with imprecise parameters are more realistic. This paper constructed a new predator-prey model with an interval biological coefficient by using the interval number as the model parameter. First, the stability of the solution of the fractional order model without a diffusion term and the Hopf bifurcation of the fractional order α were analyzed theoretically. Then, taking the diffusion coefficient of prey as the key parameter, the Turing stability at the equilibrium point was discussed. The amplitude equation near the threshold of the Turing instability was given by using the weak nonlinear analysis method, and different mode selections were classified by using the amplitude equation. Finally, we numerically proved that the dispersal rate of the prey population suppressed the spatiotemporal chaos of the model

Analysis of a stochastic delay competition system driven by Lévy noise under regime switching

This paper is concerned with a stochastic delay competition system driven by Lévy noise under regime switching. Both the existence and uniqueness of the global positive solution are examined. By comparison theorem, sufficient conditions for extinction and non-persistence in the mean are obtained. Some discussions are made to demonstrate that the different environment factors have significant impacts on extinction. Furthermore, we show that the global positive solution is stochastically ultimate boundedness under some conditions, and an important asymptotic property of system is given. In the end, numerical simulations are carried out to illustrate our main results

Limiting Behavior of Stochastic Processes Involving Martingale Structures

A stochastic process is given by a family of random variables indexed by elements of a set. We have considered stochastic processes of three different types, each involving an associated martingale structure. Martingale is a sequence of random variables for which the conditional expectation at a certain time point given the entire past is given by the present value of the sequence. Martingales possess nice theoretical properties with wide applicability. We have exploited martingale tools and techniques to derive the limiting results related to the stochastic processes. The processes we have considered are given below. An evolutionary urn scheme based on the rock-paper-scissors game The random multiplicative cascade model for intermittent processes A set-indexed partial sum process with dependent increments The evolutionary urn scheme based on the rock-paper-scissors game is known to model species interactions in ecological systems. Therefore its limiting behavior is of interest to ecologists to understand the long term species composition of a certain ecological system. We have considered a generalization of the process to accommodate more than three species. Simulations in the general set up suggest interesting phenomena that are counter-intuitive when compared to the three-player case. The second chapter of this thesis is motivated by data sets with variable intermittency, which makes it diffcult to use standard modeling tools. It has been observed that a special class of multiplicative models, namely the Random Multiplicative Cascade models reproduce some characteristics of the data. We have derived theoretical results related to the multiplicative cascade models under a missing data set up. We have applied the method to the daily stock volume data of Tesla. Also, we have proposed a change point detection method for intermittent time series. This can possibly be extended to spatial processes as well. The last chapter of the thesis is related to a set indexed partial sum process, with martingale differences as its increments. We have derived the weak limit of the system under the Lindeberg type condition and the metric entropy integrability condition. In spite of a common martingale structure underlying each of these three processes, they are fundamentally different. Therefore the methods to derive the limiting properties are unique to each process. For example, in the case of the rock-paper-scissors urn scheme, the key idea behind the derivation of almost sure limit is noticing a connection between sub-martingale structures within the game and a well known convergence theorem of polynomial sequence. However, for the random multiplicative cascade model, the main challenge lies in deriving asymptotic theory on a tree structure. In the third chapter, we have used probabilistic tools and techniques like generic chaining, symmetrization, and truncation to derive weak limit of the set indexed partial sum process