Functional response of the Holling type II is incorporated into a predator-prey model with predators using hawk-dove tactics to consider combination effects of nonlinear functional response and individual tactics. By mathematical analysis, it is shown that the model undergoes a sequence of bifurcations including saddle-node bifurcation, supercritical Hopf bifurcation and homoclinic bifurcation. New phenomena are found that include the bistable coexistence of prey and predators in the form of a stable limit cycle and a stable positive equilibrium, the bistable coexistence of prey and predators in a large stable limit cycle that encloses three positive equilibria and a stable positive equilibrium within the cycle, and the bistable coexistence of two stable limit cycles.
Malaria is one of the most important parasitic infections in humans, and more than two billion people are at risk every year. To understand how the spatial heterogeneity and extrinsic incubation period of the parasite within the mosquito affect the dynamics of malaria epidemiology, we propose a nonlocal and time-delayed reaction-diffusion model. We then introduce the basic reproduction ratio for this model and show that it serves as a threshold parameter that predicts whether malaria will spread. A sufficient condition is obtained to guarantee that the disease will stabilize at a positive steady state eventually in the case where all the parameters are spatially independent. Further, we use two vaccination programs to simulate the efficiency of spatial control strategies. If time permits, I will also mention our more recent work on the global dynamics of an extended system, which incorporates a vector-bias term into this model. This talk is based on joint works with Drs. Yijun Lou and Zhiting Xu.
In this talk, I will report our recent research on a periodic reaction and diffusion competition model, which describes the propagation of two competing species in the bad and good seasons. The existence and global stability of time-periodic bistable traveling waves are established for such a system under appropriate conditions. The methods involve the upper and lower solutions, spreading speeds of monostable systems, and the monotone semiflows approach. This talk is based on the joint work with Dr. Xiaoqiang Zhao.
It is known since the early 1960's that the products of genes can regulate the expression of other genes, or themselves. This has led to the notion of gene regulatory network, for whose different models have been proposed in the last decades. Among those, a particular class of piecewise affine differential equations has attracted various researches. One the most appealing aspect of this class of models is its underlying discrete structure, which facilitates both algorithmic and mathematical analysis. In this talk one will show how this fact applies to the description of orbits in these systems. In particular, one will describe a coarse, but general bound on their topological entropy. Then, some results about periodic orbits will be explained, as well as a recent construction method, similar to a horseshoe. Some elements for a proof that systems obtained by this method are chaotic will be provided.
In this talk, I will report our recent research on the Hopf bifurcation in a delay-coupled system of neural oscillators. After investigating the eigenvalue problem of the corresponding characteristic equation of the system, we establish the codimension 1 Hopf bifurcations under appropriate conditions. Meanwhile, without solving the equation, we obtain the form of the bifurcated periodic solutions by appealing to the representation theory of Lie group. Finally, we studied the bifurcation directions and stabilities via the center manifold theory and normal form approach. This talk is based on my joint work with Dr. S. Guo.
A disease transmission model of SEIRS type with distributed delays in latent and temporary immune periods is presented. With general probability distributions in both of these periods, we address the threshold property of the reproductive number $R_0$ and the dynamical properties of the disease-free/endemic equilibrium points present in the model. More specifically, we a. show the dependence of $R_0$ on the probability distribution in the latent period and the independence of $R_0$ from the distribution in the temporary immunity, b. prove that the disease free equilibrium is always globally asymptotically stable when $R_0<1$, and c. establish that an endemic equilibrium exists when $R_0>1$ with different possible stability properties,according to the choice of probability functions in the latent and temporary immune periods. In particular, the endemic steady state is at least locally asymptotically stable if the probability distribution in the temporary immunity is a decreasing exponential function when the duration of the latency stage is fixed or exponentially decreasing. It may become oscillatory under certain conditions when there exists a constant delay in the temporary immunity period. Numerical simulations are given to verify the theoretical predictions.
In this talk, I will report our recent research on a diffusive logistic equation with a free boundary and seasonal succession, which is formulated to investigate the spreading of a new or invasive species, where the free boundary represents the expanding front and the time periodicity accounts for the effect of the bad and good seasons. The condition to determine whether the species spatially spreads to infinity or vanishes at a finite space interval is derived, and when the spreading happens, the asymptotic spreading speed of the species is also given. Our results reveal the effect of seasonal succession on the dynamical behavior of the spreading of the single species. This talk is based on the joint work with Dr. Xiaoqiang Zhao.
In this talk, I will report my recent research on the existence of small amplitude traveling wave train solutions and two kinds of traveling wave solutions for a diffusive predator-prey system with general Holling type functional response. This problem reduces to the existence of periodic orbits, point-to-point connection, and point-to-periodic orbit connection for the associated wave profile equation. I will also discuss the minimal wave speed and biological invasion for such a model.