Dynamic behaviors of a delay differential equation model of plankton allelopathy

AbstractIn this paper, we consider a modified delay differential equation model of the growth of n-species of plankton having competitive and allelopathic effects on each other. We first obtain the sufficient conditions which guarantee the permanence of the system. As a corollary, for periodic case, we obtain a set of delay-dependent condition which ensures the existence of at least one positive periodic solution of the system. After that, by means of a suitable Lyapunov functional, sufficient conditions are derived for the global attractivity of the system. For the two-dimensional case, under some suitable assumptions, we prove that one of the components will be driven to extinction while the other will stabilize at a certain solution of a logistic equation. Examples show the feasibility of the main results

Dynamic Behaviors of a Discrete Periodic Predator-Prey-Mutualist System

A nonautonomous discrete predator-prey-mutualist system is proposed and studied in this paper. Sufficient conditions which ensure the permanence and existence of a unique globally stable periodic solution are obtained. We also investigate the extinction property of the predator species; our results indicate that if the cooperative effect between the prey and mutualist species is large enough, then the predator species will be driven to extinction due to the lack of enough food. Two examples together with numerical simulations show the feasibility of the main results

Effect of cap rock thickness and permeability on geological storage of CO2: laboratory test and numerical simulation

Geological storage of CO2 is considered widely as an efficient method of mitigation of greenhouse gas emission. CO2 storage mechanism includes structural trapping, residual gas trapping, solubility trapping and mineral trapping. The shale cap rock acts as a seal for the storage when CO2 accumulates at the top of the reservoir. The injected CO2 may migrate through the cap rock under buoyancy force or pressure build-up which depends on the seal capacity of the cap rock. As a result, the effectiveness of containment of injected CO2 in the reservoir is largely dependent on the migration rate of CO2 through the cap rock. This paper investigates the effects of CO2 leakage through cap rock by a combination of experimental studies and numerical simulation. Firstly, experimental measurements on shale core samples collected from Australian cap rocks were conducted to determine properties, such as capillary pressure, pore size distribution and permeability. Based on the measured cap rock properties, the effect of thickness and permeability of cap rocks on CO2 leakage was studied using a commercial compositional simulator. Experimental results show that the permeabilities of the shale samples measured by transient pulse technique range from 60 to 300 nD; a non-Darcy calibration factor which equals the ratio of the measured permeability divided by 1000, is identified for samples with permeability lower than 1000 nD. Numerical simulation results show that the largest leakage of CO2 through the seal (cap cock) is about 7.0% with seal thickness of 3m and vertical permeability of 90 nD; both shale thickness and permeability affect the CO2 leakage significantly; with a given seal permeability, the leakage rate has a power relationship with shale thickness

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

On the Stability Property of the Infection-Free Equilibrium of a Viral Infection Model

The dynamics of a viral infection model with nonautonomous lytic immune response is studied from the perspective of dying out of the disease. With the help of the theory of exponential dichotomy of linear systems, we give a new proof about the global asymptotic stability of the infection-free equilibrium for the case R0=1. The result improves and complements one of the results of Wang et al. (2006)

Note on the Stability Property of a Cooperative System Incorporating Harvesting

The stability of a kind of cooperative model incorporating harvesting is revisited in this paper. By using an iterative method, the global attractivity of the interior equilibrium point of the system is investigated. We show that the condition which ensures the existence of a unique positive equilibrium is enough to ensure the global attractivity of the positive equilibrium. Our results significantly improve the corresponding results o

Uniqueness of Limit Cycles for a Class of Cubic Systems with Two Invariant Straight Lines

A class of cubic systems with two invariant straight lines dx/dt=y(1-x2),  dy/dt=-x+δy+nx2+mxy+ly2+bxy2. is studied. It is obtained that the focal quantities of O(0,0) are, W0=δ; if W0=0, then W1=m(n+l); if W0=W1=0, then W2=−nm(b+1); if W0=W1=W2=0, then O is a center, and it has been proved that the above mentioned cubic system has at most one limit cycle surrounding weak focal O(0,0). This paper also aims to solve the remaining issues in the work of Zheng and Xie (2009)

Extinction in Two-Species Nonlinear Discrete Competitive System

We propose a nonlinear discrete system of two species with the effect of toxic substances. By constructing a suitable Lyapunov-type function, we obtain the sufficient conditions which guarantee that one of the components will be driven to extinction while the other will be globally attractive with any positive solution of a discrete equation. Two examples together with their numerical simulations illustrate the feasibility of our main results. The results not only improve but also complement some known results