Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population

Modelling the effect of telegraph noise in the SIRS epidemic model using Markovian switching

We discuss the effect of introducing telegraph noise, which is an example of an environmental noise, into the susceptible-infectious-recovered-susceptible (SIRS) model by examining the model using a finite-state Markov Chain (MC). First we start with a two-state MC and show that there exists a unique nonnegative solution and establish the conditions for extinction and persistence. We then explain how the results can be generalised to a finite-state MC. The results for the SIR (Susceptible-Infectious-Removed) model with Markovian Switching (MS) are a special case. Numerical simulations are produced to confirm our theoretical results

Differential Equations arising from Organising Principles in Biology

This workshop brought together experts in modeling and analysis of organising principles of multiscale biological systems such as cell assemblies, tissues and populations. We focused on questions arising in systems biology and medicine which are related to emergence, function and control of spatial and inter-individual heterogeneity in population dynamics. There were three main areas represented of differential equation models in mathematical biology. The first area involved the mathematical description of structured populations. The second area concerned invasion, pattern formation and collective dynamics. The third area treated the evolution and adaptation of populations, following the Darwinian paradigm. These problems led to differential equations, which frequently are non-trivial extensions of classical problems. The examples included but were not limited to transport-type equations with nonlocal boundary conditions, mixed ODE-reaction-diffusion models, nonlocal diffusion and cross-diffusion problems or kinetic equations

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

A general chemostat model with second-order Poisson jumps: asymptotic properties and application to industrial waste-water treatment

A chemostat is a laboratory device (of the bioreactor type) in which organisms (bacteria, phytoplankton) develop in a controlled manner. This paper studies the asymptotic properties of a chemostat model with generalized interference function and Poisson noise. Due to the complexity of abrupt and erratic fluctuations, we consider the effect of the second order Itô-Lévy processes. The dynamics of our perturbed system are determined by the value of the threshold parameter $\mathfrak{C}^{\star}_0$. If $\mathfrak {C}^{\star}_0$ is strictly positive, the stationarity and ergodicity properties of our model are verified (practical scenario). If $\mathfrak {C}^{\star}_0$ is strictly negative, the considered and modeled microorganism will disappear in an exponential manner. This research provides a comprehensive overview of the chemostat interaction under general assumptions that can be applied to various models in biology and ecology. In order to verify the reliability of our results, we probe the case of industrial waste-water treatment. It is concluded that higher order jumps possess a negative influence on the long-term behavior of microorganisms in the sense that they lead to complete extinction

Modelling wildebeest foraging processes and their interaction with zebra and lion in the Serengeti ecosystem

A Dissertation Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy in Mathematical and Computer Sciences and Engineering of the Nelson Mandela African Institution of Science and TechnologyAnimal movements and foraging processes for the migrating species, especially wildebeests and zebras, and the prey-predator interactions of these prey species with lions are ambiguous biological characteristics in the Serengeti ecosystem. These complex dynamics help animals to adapt and survive. To understand such dynamics, investigating factors that determine foraging efficiency and the prey-predator interaction is worth it. This dissertation presents deterministic mathematical models to examine wildebeest foraging processes and the prey-predator interaction of wildebeest, zebra, and lion populations. The first model studies the foraging processes of migrating wildebeests using the concepts of random walk and diffusive processes. The model was equipped with data collected from the Serengeti ecosystem from 18 GPS collared wildebeests and analysed in two spatial dimensions. The qualitative analysis of the model was performed, and the parameters that regulate foraging efficiency were calculated for both dry and wet seasons. Numerical simulations were performed, and the results show that directed movements can explain the great migration of wildebeests to different habitats. Wildebeests spread across different habitats to utilize the resources through diffusive trends. The mutual association between wildebeests and zebras was studied by developing the Lotka Volterra reaction-diffusion systems. This model was further modified to form the third model that includes the predation pressure from lions. The qualitative analyses of the models were carried out in two dimensions to determine points of equilibrium and the conditions for the stability and instability of the systems. The explicit Euler method was used to discretize the models and perform numerical simulations. The stability analyses of the models showed that wildebeests and zebras population growth approached their respective carrying capacities, and the absence of one prey species does not affect the existence of the other. The advection and diffusion parameters in the model produce Turing instabilities. Furthermore, the results show that both prey species are strongly affected by drought and predation pressure, especially from lions. Therefore, advection and diffusion of wildebeests and zebras are motivated by the search for better forage availability and avoidance of predators, while the predator's movement is motivated by capturing prey