Special Session 23: New trends in pattern formations and dynamics for dissipative systems and related topics

Coexistence of two strongly competitive species in a reaction-advection-diffusion system

Inkyung Ahn Korea University Korea Co-Author(s):    Wonhyung Choi

Abstract: The main focus of this article is to investigate the behavior of two strongly competitive species in a spatially heterogeneous environment using a Lotka-Volterra-type reaction-advection-diffusion model. The model assumes that one species diffuses at a constant rate while the other species moves toward a more favorable environment through constant diffusion and directional movement. The study finds that no stable coexistence can be guaranteed when both species disperse randomly. In contrast, stable coexistence between the two species is possible when one of the species exhibits advection-diffusion. The study also reveals the existence of unstable coexistence imposed by bistability in a strongly competitive system, regardless of the diffusion type. The study concludes that the species moving toward a better environment has a competitive advantage, allowing them to survive even when their population density is initially low. Finally, the study identifies the unique globally asymptotically stable coexistence steady states of the system at high advection rates.

A free boundary model for super invaders

Yihong Du University of New England Australia Co-Author(s):

Abstract: In this talk I will report some recent progress on a free boundary model, where the free boundary conditions are deduced from the biological assumption that the species expands or shrinks its population range through its members at the range boundary keeping the population density there at a preferred level. It turns out that this strategy would make the species a super invader: No matter whether the growth function f(u) is monostable, bistable (strong Allee) or of combustion type (weak Allee), the species always spreads successfully.

Pulse dynamics on a star-shaped metric graph with different widths

Shin-Ichiro Ei Josai University Japan Co-Author(s):    Ryota Asami, Haruki Shimatani

Abstract: The motion of pulses is considered on a star-shaped metric graph composed of paths with different widths. We derive the equation that describes the motion of a pulse based on these widths. As applications, we demonstrate the motion towards the junction for pulses described by the FitzHugh-Nagumo equation and the Gierer-Meinhardt equation.

Pattern formation in IGP-communities with anti-predator behavior

Gaetana Gambino University of Palermo, Department of Mathematics and Computer Science Italy Co-Author(s):

Abstract: A wide variety of predatory relationships are possible in ecological communities and ecosystems. Intraguild predation, or IGP, represents a combination of predation and competition in which species rely on the same prey resources and benefit from preying on each other. In this talk we shall describe the spatiotemporal dynamics of a three-species reaction-diffusion system, in which the IGP local interaction is of Lotka-Volterra type and the IG-Prey exhibits anti-predator behavior, dispersing along local gradients in predator density. We first show that the local dynamics support the bistability of the spatially homogeneous coexistence equilibrium with oscillations due to a subcritical Hopf bifurcation. We demonstrate that the predator avoidance strategy ignites cross-diffusion-driven Turing instability, leading to the emergence of stationary patterns. Via weakly nonlinear analysis, we derive asymptotic profiles of emergent stationary patterns, revealing that anti-predator behavior can account for the IG-prey and IG-predator segregation patterns observed in the ecology literature. We also prove that the predator avoidance strategy can stabilize coexistence states in IGP communities beyond the conditions imposed by the corresponding spatially homogeneous model. Finally, we investigate the dynamics near a codimension-two Turing-Hopf point, reproducing the time-oscillatory inhomogeneous structures supported in this regime.

Some blow-up problems in delay differential equations

Tetsuya Ishiwata Shibaura Institute of Technology Japan Co-Author(s):    Yukihiko Nakata

Abstract: It is well known that time lags or histories sometimes cause instability or oscillation. In this talk, we focus on delay differential equations and discuss the effects of time delay for such instabilities from the viewpoint of a finite time blow-up of the solutions. We also show some numerical examples and give our observations.

On hot spots conjecture for domain with n-axes of symmetry

Yi Li John Jay College of Criminal Justice, CUNY USA Co-Author(s):    Dr. Hongbin Chen

Abstract: In this talk, we prove the hot spots conjecture for rotationally symmetric domains in $\mathbb{R}^{n}$ by the continuity method. More precisely, we show that the odd Neumann eigenfunction in $x_{n}$ associated with lowest nonzero eigenvalue is a Morse function on the boundary, which has exactly two critical points and is monotone in the direction from its minimum point to its maximum point. As a consequence, we prove that the Jerison and Nadirashvili`s conjecture 8.3 holds true for rotationally symmetric domains and are also able to obtain a sharp lower bound for the Neumann eigenvalue. We will also discuss some recent results on n-axes symmetry or hyperbolic drum type domains.

Role of chemotaxis in some SIS PDE epidemic model with singular sensitivity

Yuan Lou Shanghai Jiao Tong University Peoples Rep of China Co-Author(s):    Rachidi Salako, Youshan Tao, Shenggao Zhou

Abstract: We consider a repulsive chemotaxis SIS epidemic model with logarithmic sensitivity and mass-action transmission. The global existence and boundedness of smooth solutions to the corresponding no-flux initial boundary value problem in the spatially one-dimensional setting are established. Furthermore, the asymptotic analysis of the steady states reveal that the susceptible populations move to low-risk regions, whereas the infected individuals become spatially homogeneously distributed when the repulsive-taxis coefficient goes to infinity. This talk is based on joint work with R. Salako, Y.S. Tao and S.G. Zhou.

Singular limit of mathematical models related to controlling invasive alien species

Harunori Monobe Osaka Metropolitan University Japan Co-Author(s):    H. Izuhara, C.-H. Wu and S. Iwasaki

Abstract: We consider the dynamics of some Lotka-Volterra competition reaction-diffusion systems with the effect of controlling species. Our purpose in this talk is to investigate how the species behave depending on the effect of controlling species is large. In particular, we study the singular limit of the model and show that some free boundary problems appears. This is a joint work with H. Izuhara, C.-H. Wu and S. Iwasaki.

Segregation pattern in a mass conserved reaction-diffusion system from a model of asymmetric cell division

Yoshihisa Morita Ryukoku Joint Research Center Sci & Tech Japan Co-Author(s):

Abstract: We study a four-component reaction-diffusion system with mass conservation in a bounded domain with the Neumann boundary condition. This system models the segregation pattern during the maintenance phase of asymmetric cell devision. Utilizing the mass conservation, we reduce the stationary problem of the system to a two-component elliptic system with nonlocal terms, formulating it as the Euler-Lagrange equation of an energy functional. In this talk we focus on the existence of equilibrium solutions with segregation pattern in a cylindrical domain. This result is based on the joint work with Prof. Y. Oshita (Okayama Univ.).

Reaction-diffusion type modeling of the self-propelled motion.

Masaharu Nagayama Hokkadio University Japan Co-Author(s):    Natsume Motohashi, Hiroyuki Kitahata, Yasuaki Kobayashi, Ken-Ichi Nakamura, Koya Sakakibara, Keisuke Takasao, Harunori Monobe

Abstract: A mathematical model was developed that included self-propelled objects in motion with deformation, such as droplets, and self-propelled objects without deformation, such as camphor. This study represents the self-propelled object by a volume-conserving Phase-Field equation derived from the $L^2$ gradient flow. The self-propelled object shape during motion is successfully controlled depending on the parameters included in the model equations. Moreover, adding a spatially inhomogeneous function to the potential term made it possible to represent the self-propelled object motion in elliptical and dumbbell shapes.

The speed of bistable traveling fronts in the Lotka-Volterra competition-diffusion system

Ken-Ichi Nakamura Meiji University Japan Co-Author(s):    Toshiko Ogiwara

Abstract: We consider front propagation in the classical 2-species Lotka-Volterra competition-diffusion system under strong competition conditions. The system has a unique traveling front solution (up to translation) connecting two stable states. The sign of the front speed gives us significant information about which species prevails over the other, and identifying the sign is still a challenging problem. In this talk, we give some new results on the sign of the speed of bistable traveling fronts based on comparison arguments. The results determine the propagation direction of the front for a much broader range of parameters than previous results.

Forced waves for an epidemic model of West-Nile virus with climate change effect

Toshiko Ogiwara Josai University Japan Co-Author(s):    Jong-Shenq Guo, Wonhyung Choi, Chin-Chin Wu

Abstract: In this talk, we deal with the existence of forced waves for an epidemic model of West-Nile virus in a shifting environment. Here a forced wave is a traveling wave whose wave speed is the same as the environmental shifting speed. The forced waves we constructed have the property that the waves tend to the positive endemic state of the epidemic model as the time tends to infinity. The derivation of these forced waves relies on a careful construction of a suitable lower solution with the help of Schauder`s fixed point theorem.

Cytokine-induced coherent structures in a reaction-diffusion-chemotaxis model of Multiple Sclerosis

Rossella Rizzo University of Palermo Italy Co-Author(s):    Francesco Gargano, Maria Carmela Lombardo, Marco Sammartino, Vincenzo Sciacca

Abstract: In this work, we develop a model for the evolution of the Multiple Sclerosis pathology that considers the modulatory influence of cytokines on the activation rate of macrophages. Our starting point is the reaction-diffusion-chemotaxis model proposed in (Lombardo, Barresi, Bilotta, Gargano, Pantano, Sammartino, J.Math.Biol. (2017)), and we modify the macrophage activation mechanism. We explore the hypothesis, e.g., Lassmann, (2018), that cytokines mediate the activation mechanism. Through a weakly nonlinear analysis, we characterize the chemotaxis-driven Turing instability and construct the stationary patterns that emerge from this instability. Using biologically relevant parameter values, we show that the asymptotic solutions of our model system reproduce the concentric demyelinating rings, confluent plaques, and preactive lesions observed in Bal\`{o} sclerosis and type III Multiple Sclerosis. Furthermore, we explore the initiation and progression of demyelinated plaques through extensive numerical simulations on two-dimensional domains. Our findings reveal that the alternative scenario proposed here results in a less aggressive pathology and slower disease progression. Under the appropriate regularity conditions on the initial data, we prove the existence of a unique global solution to our proposed system. This study provides insights into the role of cytokines in the pathogenesis of Multiple Sclerosis and offers potential avenues for therapeutic interventions.

Stability of traveling waves in non-cooperative systems with nonlocal dispersal of equal diffusivities

Masahiko Shimojo Tokyo Metropolitan University Japan Co-Author(s):    Jong-Shenq Guo

Abstract: We explore the stability of traveling waves in a class of non-cooperative reaction-diffusion systems with nonlocal dispersal, when the diffusivities are equal. Our stability criterion for traveling waves is based on measuring initial perturbations by a weighted relative entropy. To demonstrate the practical applications of this theory, we will present several examples from ecology and epidemiology.

The dynamics of the coupled reaction-diffusion Lengyel-Epstein system with two layers modeling CIMA chemical reactions

Fengqi Yi Dalian University of Technology Peoples Rep of China Co-Author(s):    Qidong Wu

Abstract: In this talk, I will report our recent works on the dynamics of the two-coupled reaction-diffusion Lengyel-Epstein system incorporating distributed-delay couplings in activators. For one hand, we are mainly concerned with the existence and stability of the non-constant steady state solutions for the one-layered diffusive Lengyel-Epstein system. It is found that as the system parameter changes, the stability of the nonconstant steady state solutions of the one-layered system changes. This leads to the emergence of the periodic solutions via Hopf bifurcations. On the other hand, we are interested in studying Turing instability of the symmertric non-constant steady state solutions driven by inter-reactor diffusions.