1,234 research outputs found
Asymptotic properties of stochastic population dynamics
In this paper we stochastically perturb the classical Lotka{Volterra model x_ (t) = diag(x1(t); ; xn(t))[b + Ax(t)] into the stochastic dierential equation dx(t) = diag(x1(t); ; xn(t))[(b + Ax(t))dt + dw(t)]: The main aim is to study the asymptotic properties of the solution. It is known (see e.g. [3, 20]) if the noise is too large then the population may become extinct with probability one. Our main aim here is to nd out what happens if the noise is relatively small. In this paper we will establish some new asymptotic properties for the moments as well as for the sample paths of the solution. In particular, we will discuss the limit of the average in time of the sample paths
On the asymptotic stability and numerical analysis of solutions to nonlinear stochastic differential equations with jumps
This paper is concerned with the stability and numerical analysis of solution to highly nonlinear stochastic differential equations with jumps. By the Itô formula, stochastic inequality and semi-martingale convergence theorem, we study the asymptotic stability in the pth moment and almost sure exponential stability of solutions under the local Lipschitz condition and nonlinear growth condition. On the other hand, we also show the convergence in probability of numerical schemes under nonlinear growth condition. Finally, an example is provided to illustrate the theor
Almost sure stability with general decay rate of neutral stochastic pantograph equations with Markovian switching
This paper focuses on the general decay stability of nonlinear neutral stochastic pantograph equations with Markovian switching (NSPEwMSs). Under the local Lipschitz condition and non-linear growth condition, the existence and almost sure stability with general decay of the solution for NSPEwMSs are investigated. By means of M-matrix theory, some sufficient conditions on the general decay stability are also established for NSPEwMSs
Generalised criteria on delay dependent stability of highly nonlinear hybrid stochastic systems
Our recent paper [2] is the first to establish delay dependent criteria for highly nonlinear hybrid stochastic differential delay equations (SDDEs) (by highly nonlinear we mean the coefficients of the SDDEs do not have to satisfy the linear growth condition). This is an important breakthrough in the stability study as all existing delay stability criteria before could only be applied to delay equations where their coefficients are either linear or nonlin- ear but bounded by linear functions (namely, satisfy the linear growth condition). In this continuation, we will point out one restrictive condition imposed in our earlier paper [2]. We will then develop our ideas and methods there in order to remove this restrictive condition so that our improved results cover a much wider class of hybrid SDDEs
Razumikhin-type theorem for stochastic functional differential systems via vector Lyapunov function
This paper is concerned with input-to-state stability of SFDSs. By using stochastic analysis techniques, Razumikhin techniques and vector Lyapunov function method, vector Razumikhin-type theorem has been established on input-to-state stability for SFDSs. Novel sufficient criteria on the pth moment exponential input-to-state stability are obtained by the established vector Razumikhin-type theorem. When input is zero, an improved criterion on exponential stability is obtained. Two examples are provided to demonstrate validity of the obtained results
STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS
Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system.
Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time. Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with applications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investigated. The impact of incorporating time-delays and random noise in such class of differential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunting cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stability of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochastic epidemic SIRC model with time-delay for spread of COVID-19 among population. The basic reproduction number ℛs0 for the stochastic model which is smaller than ℛ0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research.
The main findings, theoretically and numerically, show that time-delays and random noise have a significant impact in the dynamics of ecological and biological systems. They also have an important role in ecological balance and environmental stability of living organisms. A small scale of white noise can promote the survival of population; While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms
A simple method for detecting chaos in nature
Chaos, or exponential sensitivity to small perturbations, appears everywhere
in nature. Moreover, chaos is predicted to play diverse functional roles in
living systems. A method for detecting chaos from empirical measurements should
therefore be a key component of the biologist's toolkit. But, classic
chaos-detection tools are highly sensitive to measurement noise and break down
for common edge cases, making it difficult to detect chaos in domains, like
biology, where measurements are noisy. However, newer tools promise to overcome
these limitations. Here, we combine several such tools into an automated
processing pipeline, and show that our pipeline can detect the presence (or
absence) of chaos in noisy recordings, even for difficult edge cases. As a
first-pass application of our pipeline, we show that heart rate variability is
not chaotic as some have proposed, and instead reflects a stochastic process in
both health and disease. Our tool is easy-to-use and freely available
Minimal models of invasion and clonal selection in cancer
In this thesis we develop minimal models of the relationship between
motility, growth, and evolution of cancer cells. We utilise simple simulations
of a population of individual cells in space to examine how changes in
mechanical properties of invasive cells and their surroundings can affect the
speed of cell migration. We also find that the growth rate of large lesions
depends weakly on the migration speed of escaping cells, and has stronger and
more complex dependencies on the rates of other stochastic processes in the
model, namely the rate at which cells transition to being motile and the
reverse rate at which cells cease to be motile.
To examine how the rates of growth and evolution of an ensemble of cancerous
lesions depends on their geometry and underlying fitness landscape, we develop
an analytical framework in which the spatial structure is coarse grained and
the cancer treated as a continuously growing system with stochastic migration
events. Both approaches conclude that the whole ensemble can undergo
migration-driven exponential growth regardless of the dependence of size on
time of individual lesions, and that the relationship between growth rate and
rate of migration is determined by the geometrical constraints of individual
lesions. We also find that linear fitness landscapes result in
faster-than-exponential growth of the ensemble, and we can determine the
expected number of driver mutations present in several important cases of the
model.
Finally, we study data from a clinical study of the effectiveness of a new
low-dose combined chemotherapy. This enables us to test some important
hypotheses about the growth rate of pancreatic cancers and the speed with which
evolution occurs in reality. Despite this, we find that the frequency of
resistant mutants is far too high to be explained without resorting to novel
mechanisms of cross-resistance to multiple drugs
- …