Special Session 79: Delayed Reaction-Diffusion Equations and Applications

Bogdanov-Takens bifurcation and multi-peak spatiotemporal staggered periodic patterns in a nonlocal Holling-Tanner predator-prey model

Xun Cao Harbin Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: In this talk, we will discuss spatiotemporal dynamics of a reaction-diffusion Holling-Tanner predator-prey model with nonlocal prey competition involving purely spatial heat kernel. Firstly, the first bifurcation curve is mathematically described, that is a piecewise smooth parameter curve of dividing the stability and instability of the coexistence equilibrium. The concepts of Turing/Hopf instability are extended to the higher codimension bifurcation instability, because the non-smooth points of the first bifurcation curve can be Bogdanov-Takens/Turing-Hopf/Hopf-Hopf instability point. Then, utilizing normal form method, spatiotemporal dynamics near $Z_2$ symmetric Bogdanov-Takens singularity are theoretically and numerically studied, including the stable coexistence of a pair of steady states with the shape of $\cos(\frac{2x}{l})$ and a spatiotemporal staggered periodic solution with the shape of $\cos(\omega t)\cos(\frac{2x}{l})$. It is found that the larger the spatial size of a habitat is, the more complex the distributions of a species can be, while too narrow or wide range of nonlocal interactions inhibit the formations of complex spatiotemporal patterns.

Steady-state bifurcation and spike pattern in the Klausmeier-Gray-Scott model with non-diffusive plants

Weihua Jiang Harbin Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: We studied the Klausmeier-Gray-Scott model with non-diffusive plants, which is a coupled ODE-PDE system. We first established the critical conditions for instability of the constant steady state in general coupled ODE-PDE activator-inhibitor systems. In addition, the local structure of the nonconstant steady state and the condition to determine the local bifurcation direction were obtained. Secondly, for the model with non-diffusive plants, the Turing bifurcation was subcritical and the nonconstant steady-state bifurcation solutions were unstable. Finally, we investigated the spatial pattern of the model with slowly diffusive plants to understand the formation of the spike pattern of the model with non-diffusive plants.

Global dynamics and asymptotic spreading of a diffusive age-structured model in spatially periodic media

Hao Kang Tianjin University Peoples Rep of China Co-Author(s):    Hao Kang and Shuang Liu

Abstract: The paper is concerned with the persistence and spatial propagation of populations with age structure in spatially periodic media. We first provide a complete characterization of the global dynamics for the problem via investigating the existence, uniqueness and global stability of the nontrivial equilibrium. This leads to a necessary and sufficient condition for populations to survive, in term of the principal eigenvalue of the associated linearized problem with periodic condition. We next establish the spatial propagation dynamics for the problem and derive the formula for the asymptotic speed of spreading. The result suggests that the propagating fronts of populations are uniform for all age groups with a common spreading speed. Our approach is to develop the theory of generalized principal eigenvalues and the homogenization method via overcoming some new challenges arising from the nonlocal age boundary condition.

Accelerating propagation in the periodic Fisher-KPP equation with nonlocal dispersal

NA LI Harbin institute of Technology Peoples Rep of China Co-Author(s):

Abstract: This talk is devoted to the problem how the tail of the nonlocal dispersal kernel influences the propagation speed of the Fisher-KPP equation in time periodic environment. When the nonlocal dispersal kernel is exponentially bounded, applying the monotone dynamical system theory, we prove that the solution level set is asymptotically linear in time. When the nonlocal dispersal kernel is exponentially unbounded, the solution level set propagates with an infinite asymptotic speed. Further, based on a heaviness characterization for the kernel tail, we establish the fine estimates of the fundamental solution and then determine sharp bounds for the solution level set by constructing subtle upper and lower solutions. The bounds are expressed in terms of the decay of the nonlocal dispersal kernel.

Lattice-based stochastic models motivate non-linear diffusion descriptions of memory-based dispersal

Yifei Li Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Matthew J Simpson, Chuncheng Wang

Abstract: The role of memory and cognition in the movement of individuals (e.g. animals) within a population, is thought to play an important role in population dispersal. In response, there has been increasing interest in incorporating spatial memory effects into classical partial differential equation (PDE) models of animal dispersal. However, the specific detail of the transport terms, such as diffusion and advection terms, that ought to be incorporated into PDE models to accurately reflect the memory effect remains unclear. To bridge this gap, we propose a straightforward lattice-based model where the movement of individuals depends on both on crowding effects and the historic distribution within the simulation. The advantage of working with the individual-based model is that it is straightforward to propose and implement memory effects within the simulation in a way that is more intuitive than proposing extensions of classical PDE models. Through deriving the continuum limit description of our stochastic model we obtain a novel nonlinear diffusion equation which encompasses memory-based diffusion terms. In this talk I will show the relationship between memory-based diffusion and the individual-based movement mechanisms that depend upon memory effects.

A reaction-diffusion model with spatially inhomogeneous delays

Yijun Lou The Hong Kong Polytechnic University Hong Kong Co-Author(s):

Abstract: Motivated by population growth in a heterogeneous environment, this talk presents a reaction-diffusion model with spatially dependent parameters. In particular, a term for spatially uneven maturation durations is included in the model, which puts the current investigation among the very few studies on reaction-diffusion systems with spatially dependent delays. Rigorous analysis is performed, including the well-posedness of the model, the basic reproduction ratio formulation and long-term behavior of solutions. Under mild assumptions on model parameters, extinction of the species is predicted when the basic reproduction ratio is less than one. When the birth rate is an increasing function and the basic reproduction ratio is greater than one, uniqueness and global attractivity of a positive equilibrium can be established with the help of a novel functional phase space. Permanence of the species is shown when the birth function is in a unimodal form and the basic reproduction ratio is greater than one. The synthesized approach proposed here is applicable to broader contexts of studies on the impact of spatial heterogeneity on population dynamics, in particular, when the delayed feedbacks are involved and the response time is spatially varying.

Dynamics of reaction diffusion equations with memory-based diffusions

Chuncheng Wang Harbin Institute of Technology Peoples Rep of China Co-Author(s):    Chuncheng Wang

Abstract: In this talk, a class of reaction diffusion equations with memory-based diffusions is considered.The principle of linearized stability and the theory of normal form is established. In addition, the global boundedness of solutions is also proved. These are joint works with Junping Shi, Hao Wang, Yanhui Fan and Xuanyu Liu.

Spatial dynamics for time-periodic partially degenerate reaction-diffusion systems

Shi-Liang Wu Xidian University Peoples Rep of China Co-Author(s):

Abstract: Partially degenerate reaction-diffusion systems (i.e., reaction-diffusion systems with some but not all diffusion coefficients being zeros) arises from may particular fields, such as biology and epidemiology. In this talk, we introduce some results on spatial dynamics for time-periodic partially degenerate reaction-diffusion systems.

Global Dynamics of a Time-delayed Nonlocal Reaction-Diffusion Model of Within-host Viral Infections

Xiao-Qiang Zhao Canada Co-Author(s):    Zhimin Li and Xiao-Qiang Zhao

Abstract: In this talk, I will report our recent research on a time-delayed nonlocal reaction-diffusion model of within-host viral infections. We introduce the basic reproduction number R0 and show that the infection-free steady state is globally asymptotically stable when R0 is less than or equals one, while the disease is uniformly persistent when R0 is greater than one. In the case where all coefficients and reaction terms are spatially homogeneous, we obtain an explicit formula of R0 and the global attractivity of the positive constant steady state. Numerically, we illustrate the analytical results and investigate the impact of drugs on curtailing the spread of the virus.