Shangbing Ai

Periodic solutions in a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor

Abstract. We obtain necessary and sufficient conditions on the existence of a unique positive equilibrium point and a set of sufficient conditions on the existence of periodic solutions for a 3-dimensional system which arises from a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat with an inhibitor. Our results improve the corresponding results obtained by Hsu, Luo, and Waltman [1].

Dynamics of Mosquitoes Populations with Different Strategies for Releasing Sterile Mosquitoes

To prevent the transmissions of malaria, dengue fever, or other mosquito-borne diseases, one of the effective weapons is the sterile insect technique in which sterile mosquitoes are released to reduce or eradicate the wild mosquito population. To study the impact of the sterile insect technique on disease transmission, we formulate continuous-time mathematical models for the interactive dynamics of the wild and sterile mosquitoes, incorporating different strategies in releasing sterile mosquitoes. We investigate the model dynamics and compare the impact of the different release strategies. Numerical examples are also given to demonstrate rich dynamical features of the models.

Traveling Waves in Spatial SIRS Models

We study traveling wavefront solutions for two reaction–diffusion systems, which are derived respectively as diffusion approximations to two nonlocal spatial SIRS models. These solutions characterize the propagating progress and speed of the spatial spread of underlying epidemic waves. For the first diffusion system, we find a lower bound for wave speeds and prove that the traveling waves exist for all speeds bigger than this bound. For the second diffusion system, we find the minimal wave speed and show that the traveling waves exist for all speeds bigger than or equal to the minimal speed. We further prove the uniqueness (up to translation) of these solutions for sufficiently large wave speeds. The existence of these solutions are proved by a shooting argument combining with LaSalle’s invariance principle, and their uniqueness by a geometric singular perturbation argument.

Mosquito-Stage-Structured Malaria Models and Their Global Dynamics

We formulate mosquito-stage-structured, continuous-time, compartmental, malaria models which include four distinct metamorphic stages of mosquitoes. We derive a formula for the reproductive number of infection and investigate the existence of endemic equilibria. We determine conditions under which the models undergo either forward or backward bifurcation. We carry out rigorous mathematical analysis on the model dynamics by proving the global stability of the infection-free equilibrium while a forward bifurcation occurs, and the global stability of the endemic equilibrium as the reproductive number is greater than one in certain cases. Our study can be also applied to other diseases such as Chikungunya fever.

Multi-bump solutions to Carrier's problem

Abstract We study the existence of well-known singularly perturbed BVP problem e2y″=1−y2−2b(1−x2)y, y(−1)=y(1)=0 introduced by G.F.xa0Carrier. In particular, we show that there exist multi-spike solutions, and the locations of interior spikes are clustered near x=0 and are separated by an amount of O(e|lne|), while only single spikes are allowed near the boundaries x=±1.

Layers and spikes in non-homogeneous bistable reaction-diffusion equations

We study e 2 u = f(u,x) = Au(1-u) (Φ-u), where A = A(u,x) > 0, Φ = Φ(x) ∈ (0,1), and e > 0 is sufficiently small, on an interval [0,L] with boundary conditions u = 0 at x = 0, L. All solutions with an e independent number of oscillations are analyzed. Existence of complicated patterns of layers and spikes is proved, and their Morse index is determined. It is observed that the results extend to f = A(u,x) (u - Φ-) (u - Φ) (u - Φ + ) with Φ-(x) < Φ(x) < Φ + (x) and also to an infinite interval.

Homoclinic Solutions to the Gray-Scott Model

An elementary proof is presented on the existence of homoclinic solutions to the Gray-Scott model studied by Hale, Peletier and Troy [1].

Travelling Wave Solutions in a Tissue Interaction Model for Skin Pattern Formation

We discuss the existence and the uniqueness of travelling wave solutions for a tissue interaction model on skin pattern formation proposed by Cruywagen and Murray. The geometric theory of singular perturbations is employed.

Traveling Waves in a Bioremediation Model

We study a bioremediation model that arises in restoring ground water and soil contaminated with organic pollutants. It describes an in situ bioredimedation scenario in which a sorbing substrate of contaminated soil is degraded by indigenous microorganisms in the presence of an injected nonsorbing electron acceptor. The model relates to the coupling of the advection, dispersion, and biological reaction simultaneously for the substrate, electron acceptor, and the total biomass by two advection-reaction-diffusion equations and an ODE. We establish the existence of traveling waves for the model with wider classes of kinetic functions. Our result generalizes previous results for this model which were established only for multiplicative Monod kinetics. In addition, the proof of our result, which is based on a dynamical systems approach, is simpler.