In this talk, I will report our recent research on the global dynamics of two-species competition reaction-diffusion systems on a time-varying domain under homogeneous Dirichlet boundary conditions. We show that the competitive exclusion holds true for asymptotically bounded, unbounded and periodic domains under appropriate conditions. We also apply these analytic results to a Lotka-Volterra competition model and then conduct some numerical simulations.
In this project, we propose a class of nonlocal dispersal models with time-varying spatial domains and fully characterize their asymptotic dynamics in asymptotically bounded, time-periodic and unbounded domains. The kernel is not necessarily symmetric or compactly supported, provoking anisotropic diffusion or convective effects. The nonlocal dispersal on time-varying domains induces some challenging problems such as the lack of regularizing effects of the associated semigroup and the time-dependent inherent coupling structure in kernel. Our developed techniques may apply to other nonlocal dispersal problems. We also conduct numerical simulations to illustrate our analytical results.
When a certain condition is satisfied, a reaction-diffusion equation has a spatially homogeneous periodic solution, i.e., a temporally periodic solution that does not depend on spatial variables. We analyse the orbital stability of this periodic solution. A sufficient condition is given for the homogeneity breaking instability, which is stated in terms of the manner of dependency of its temporal period on a certain parameter of the system.
In this talk, I will first prove the equivalence of two definitions for the measure of noncompactness in metric spaces. Then I will present an important result for Lipschitz maps, and give a characterization of a complete metric space in terms of the noncompactness measure.
We propose a West Nile virus transmission model in advective heterogeneous environments. We define the basic reproduction number R0 and characterize its asymptotic behavior with respect to the diffusion and advection coefficients. Our theoretical analysis establishes R0 as a critical threshold parameter that determines disease persistence or extinction. Numerical simulations provide quantitative insights into how key epidemiological parameters, particularly diffusion rates, advection coefficients, and vertical transmission efficiency, influence disease dynamics. Contrary to conventional expectations, our results reveal a non-monotonic relationship between R0 and diffusion rates, highlighting the complex interplay between spatial movement and disease transmission.
We investigate the dynamics and the asymptotic profiles of positive steady states of a specialist predator-prey model and a generalist predator-prey model in closed advective environments. The threshold dynamics are determined by the mortality rate of the specialist predators and the predation rate of the generalist predators, respectively. We demonstrate that, no matter how large advection and diffusion rates are, the specialist predators can successfully invade as long as the mortality rate is sufficiently small, while the generalist predators can take-over the prey if the predation rate is relatively large. Finally, we explore the impacts of diffusion and advection on the asymptotic profiles of positive steady states. Our results imply that large advection rates and small diffusion rates will lead to the concentration of the species at the downstream or the upstream. Moreover, if the diffusion rates are large, then the densities of the species will be more spatially homogeneous.
In this project, we investigate a partially degenerate Aedes aegypti population model in time-periodic advective environments. We first establish the threshold dynamics of the model in terms of the basic reproduction number. Then we study a partially degenerate periodic eigenvalue problem by reducing it to a scalar equation, and prove the existence of the principal eigenvalue with the help of the Krein-Rutman theorem. In the absence of advection, we also characterize the asymptotic behavior of the principal eigenvalue as the diffusion rate tends to zero or infinity. These results allow us to derive the asymptotic profiles of basic reproduction number with respect to diffusion and advection rates. Our theoretical analysis and numerical simulations indicate that advection, diffusion and temporal periodicity play a crucial role in determining mosquito population persistence or extinction.
In this talk, we consider a microorganism flocculation model with Michaelis-Menten functional response and time delay. The model has forward bifurcation and backward bifurcation. In the backward bifurcation case, a key parameter omega (omega <1) is obtained such that if R0 is in the closed interval [omega, 1], then the model also has microorganism equilibrium. For the forward bifurcation case (ODE model), we show that the microorganism-free equilibrium is globally asymptotically stable (GAS) if R0< 1; the microorganism-free equilibrium is globally attractive if R0=1; and the microorganism equilibrium is GAS if R0>1. For the backward bifurcation case (ODE model), we prove that the microorganism-free equilibrium is GAS if R0< omega; the model exhibits bistability if R0 is in the open interval (omega,1); and the microorganism equilibrium is GAS if R0> 1. If the Michaelis-Menten functional response becomes a bilinear case, the microorganism equilibrium is also GAS if R0=1. For the DDE model, we obtain sufficient conditions for the global asymptotic stability of the microorganism-free equilibrium and microorganism equilibrium, respectively.
We first recall some well-known results on the finite dimensional competitive ODE systems (Kolmogorov type systems). Then focusing on the competitive diffusion-advection systems, which are infinite dimensional dynamical systems, we report some recent development on the global dynamics, especially how to establish the nonexistence or uniqueness (and thus global stability) of positive equilibria.