The asymptotic speed of propagation and traveling wave solutions are two important topics in the study of spatial dynamics of nonlinear evolutionary systems. In this talk, I will present a short survey on the recent progress and development about spreading speeds, minimal wave speeds, and traveling wave solutions of certain types of nonlinear evolutionary equations. I will also give an example to illustrate the application of the recently-developed theory of spreading speed and traveling wavefronts for monotone semiflows.
Based on the conception "pseudo-potential" of the incompressible plane flow, an exact solution to the Euler equation is given. With the KAM theory and the second order Melnikov function, it is proved that this solution describes infinitely many unsteady vortices distributed periodically on the whole plane and the Brownian motion appears along the border region separating different vortices.
In this talk, I will report our recent research on a periodic Ross-Macdonald type model with diffusion and advection. To study the possible impact of the mobility of humans and mosquitoes on malaria transmission, we establish the existence of the leftward and rightward spreading speeds and their coincidence with the minimum wave speeds in the left and right directions, respectively. For the model in a bounded domain, we also obtain a threshold result on the global attractivity of either zero or the positive periodic solution.
We apply the global Hopf bifurcation theory we developed recently for state-dependent delay differential equations, to investigate the global continuation with respect to a system parameter for slowly oscillating periodic solutions of state-dependent differential equations. To achieve this goal, we find sufficient conditions to obtain (a) upper and lower bound of the period of slowly oscillating periodic solutions in a connected bifurcation branch in the Fuller space, and (b) uniform boundedness of periodic solutions.
We consider a competition diffusion system with uniformly distributed delay. The complete analysis of the characteristic equation is given, and the stability of the constructed positive spatially non-homogeneous steady state solution is also obtained. Moreover, the occurrence of Hopf bifurcation near the steady state solution is proved by using the implicit function theorem with time delay as the bifurcation parameter. Finally, the formula determining the stability of the periodic solutions is given via center manifold theory.
The first part of this talk will deal with a brief review of numerical methods for solving Volterra integral equations of the first kind. Since such problems are ill-conditioned, numerical schemes based on higher-order quadrature formulas or on collocation using continuous piecewise polynomials are in general unstable, and one has to look for alternative approximation methods, such as discontinuous Galerkin (DG) methods.
In the second part I shall describe recent and ongoing joint work with P.J. Davies (University of Strathclyde) and D.B. Duncan (Heriot-Watt University) on DG and related discontinuous collocation methods for first-kind Volterra integral equations. These numerical methods are now quite well understood, except when the given Volterra integral equations possess highly oscillatory or weakly singular kernels. For such equations the convergence analysis of DG and collocation methods remains essentially open -- a great challenge for numerical analysts.
In this talk, after introducing some background about dynamical system, neural network and delay differential equation, we consider a ring neural network of identical elements with time delayed, nearest neighbor coupling. We give some global and local stability results and show how the presence of time delay allows for mode interactions leading to the coexistence of different oscillation patterns. The Hopf and equivariant Hopf bifurcations are analyzed. Regarding the coupling strengths as bifurcation parameters, we obtain codimension one bifurcation and the interaction of each critical bifurcations. Concrete formulae for the normal form coefficients are derived via the center manifold reduction that provide detailed information about the bifurcation and stability of various bifurcated solutions.
In this talk, I will first give a brief review on asymptotic speeds of spread (in short, spreading speeds), traveling waves and their global stability for biological evolution systems with spatial structure. Then I will present the mathematical theory and methods of monostable and bistable waves for monotone systems. Finally I will discuss their applications to various deterministic models on biological invasion and disease spread.
In this talk, a new approach based on a shooting method in a half line coupled with the technique of upper-lower solution pair is presented to study the existence and nonexistence of monotone waves for one form of the delayed Fisher equation that does not have the quasimonotonicity property. A necessary and sufficient condition is provided. This new method can be extended to investigate many other nonlocal and non-monotone delayed reaction diffusion equations.
This talk will review my current research directions in the design, analysis and implementation of numerical methods for differential equations. In particular, we will consider aspects of adaptive methods for PDEs including moving mesh, h-refinement and multi-rate methods; as well as natural marriages between these approaches and domain decomposition for parallel implementation. Along the way I will identify problems of (hopefully) mutual interest with the dynamical systems community with the hope of forging collaboration and student co-supervision.
Predator-prey interaction is a basic interspecies relation for ecological and social models. We review some recent results on global bifurcation of limit cycles, steady states in general predator-prey models, some with spatial structure, and some with strong Allee effect on the prey population. These results help to explain the rich spatial-temporal patterns which the system possesses.
In this talk, after recalling a brief history on the study of population models for a single species using differential equations, we discuss a class of population models with diffusion and delay effect. We will present some bifurcations results including steady state bifurcations and Hopf bifurcations. Further, we illustrate our results by analyzing two specific models: a food-limited model and a model with weak Allee effect.