Tonghua Zhang

Periodic solution of a prey-predator model with nonlinear state feedback control

Assume that when the number of pests reaches the certain threshold, pest management strategy will be taken to control pests. Based on this assumption, in this paper, we propose a pest management model with nonlinear state feedback control. We then analyze the dynamic behavior of the model. More precisely, we first investigate the singularity of the model by using method of qualitative analysis; secondly the existence of periodic solution of the model is studied by using successor functions and Poincare-Bendixson theorem; and then it is followed by the study of the stability of periodic solution; finally, an example with numerical simulations is given to illustrate our conclusions.

Stability analysis of a chemostat model with maintenance energy

Abstract In this paper, we dedicate ourselves to the study of a diffusive model for unstirred membrane reactors with maintenance energy and subject to a homogeneous Neumann boundary condition. It shows that the unique constant steady state is globally asymptotically stable when it exists. This result further implies the non-existence of any spatial patterns.

Bifurcation analysis of a mathematical model for genetic regulatory network with time delays

In this paper, we aim to investigate the dynamics of a gene regulatory network which is a time-delayed version of the model proposed by Elowitz and Leibler Nature 403 (2000) 335-338]. Based on the normal form theory and center-manifold reduction, Hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated. We also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network. It shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation. More precisely, (i) the amplitude increases with small time delay, while the change of amplitude is not sensitive to relatively large time delay. However, the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter, where amplitude always increases quickly and linearly with the transcriptional rate; (ii) the period of oscillation increases as the time delay increases, but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value, which implies that the robustness of period to the transcriptional rate variations occurs. Our numerical simulations also support the theoretical conclusions, namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks.

Turing instability and pattern induced by cross-diffusion in a predator-prey system with Allee effect

To have spatial patterns form, small Allee effect admits relatively large range of predator mortality rate.Found cross-diffusion is the key mechanism of spatial pattern formation.Obtained amplitude equations suggest both supercritical and subcritical bifurcation may occur.Weak Allee effect admits supercritical bifurcation, while strong one allows subcritical bifurcation. In this paper, we first propose a mathematical model for a spatial predator-prey system with Allee effect. And then by using the proposed model, we investigate the Turing instability and the phenomena of pattern formation. We show how cross-diffusion destabilizes the spatially uniform steady state. The method of multiple time scales is employed to derive the amplitude equations, which is the cubic Stuart-Landau equation in the supercritical case and the quintic in the subcritical case. Based on the amplitude equations, we obtain the asymptotic solutions of the model close to the onset of instability.

Dynamics analysis and control optimization of a pest management predator-prey model with an integrated control strategy

This work presents a new integrated pest management predator-prey model.It has shown the existence of the order-1 periodic orbit for the proposed model.It has verified the stability of the order-1 periodic orbit by geometric method.It has obtained optimal pest control level by an optimization problem. Pest management is a complex issue in real applications, and a practical program in pest control in general involves two pest thresholds, where the biological control and chemical control are activated respectively. Aiming at providing a good balance between the biological control and chemical control, this work presented an integrated pest management predator-prey model, where the yield of releases of predator and the strength of pesticide spraying are linearly dependent on the selected control level. Firstly, to determine the frequency of spraying chemical pesticide and releasing of predator, the existence of the order-1 periodic orbit of the proposed model is discussed by the successor function method. And then, to ensure a certain robustness of adopted control, the stability of the order-1 periodic orbit is verified by a stability criterion extracted for a general semi-continuous dynamical system. In addition, to minimize the total cost (i.e. culturing predators and spraying pesticide) in pest control, an optimization problem is formulated and the optimum pest control level is obtained. At last, to complement the theoretical results, the numerical simulations with a specific model are carried out step by step.

Spatio-temporal dynamics near the steady state of a planktonic system

Abstract The study of spatio-temporal behaviour of ecological systems is fundamentally important as it can provide deep understanding of species interaction and predict the effects of environmental changes. In this paper, we first propose a spatial model with prey taxis for planktonic systems, in which we also consider the herb behaviour in prey and effect of the hyperbolic mortality rate. Applying the homogeneous Neumann boundary condition to the model and using prey-tactic sensitivity coefficient as bifurcation parameter, we then detailedly analyse the stability and bifurcation of the steady state of the system: firstly, we carry out a study of the equilibrium bifurcation, showing the occurrence of fold bifurcation, Hopf bifurcation and the BT bifurcation; then by using an abstract bifurcation theory and taking prey-tactic sensitivity coefficient as the bifurcation parameter, we investigate the Turing–Hopf bifurcation, obtaining a branch of stable non-constant solutions bifurcating from the positive equilibrium, and our results show that prey-taxis can yield the occurrence of spatio-temporal patterns; finally, numerical simulations are carried out to illustrate our theoretical results, showing the existence of a periodic solution when the prey-tactic sensitivity coefficient is away from the critical value.

Hopf Bifurcation and Delay-Induced Turing Instability in a Diffusive lac Operon Model

In this paper, we investigate the dynamics of a lac operon model with delayed feedback and diffusion effect. If the system is without delay or the delay is small, the positive equilibrium is stable so that there are no spatial patterns formed; while the time delay is large enough the equilibrium becomes unstable so that rich spatiotemporal dynamics may occur. We have found that time delay can not only incur temporal oscillations but also induce imbalance in space. With different initial values, the system may have different spatial patterns, for instance, spirals with one head, four heads, nine heads, and even microspirals.

A new way of investigating the asymptotic behaviour of a stochastic SIS system with multiplicative noise

Abstract In this paper, we investigate a nonlinear stochastic SIS epidemic system with multiplicative noise. First, we transform the Ito’s integral into an equivalent Stratonovich integral. Then, by using the solution of Langevin equation and Ornstein–Uhlenbeck process, we prove that the system generates a random dynamical system which has a tempered compact random absorbing set, implying the condition for the extinction of the disease. Finally, the discussion and numerical simulation are given to demonstrate the obtained result.