召集人：王皓（阿尔伯塔大学）、楼元（上海交通大学）、梁兴（中国科学技术大学）

时   间：2024.06.16—2024.06.22

Schedule of Frontiers in Mathematical Biosciences 2024

Titles and Abstracts

Theory of Stoichiometric Intraguild Predation: Algae, Ciliate, and Daphnia

高书飞 上海理工大学

Abstract: Consumers respond differently to external nutrient changes than producers, resulting in a mismatch in elemental composition between them and potentially having a significant impact on their interactions. To explore the responses of herbivores and omnivores to changes in elemental composition in producers, we develop a novel stoichiometric model with an intraguild predation structure. The model is validated using experimental data, and the results show that our model can well capture the growth dynamics of these three species. Theoretical and numerical analyses reveal that the model exhibits complex dynamics, including chaotic-like oscillations and multiple types of bifurcations, and undergoes long transients and regime shifts.

Bogdanov-Takens Bifurcation and Its Applications

黄继才 华中师范大学

Abstract: In this talk, we firstly recall Bogdanov-Takens bifurcation and degenerate Bogdanov-Takens bifurcation with codimension three. Secondly, we will provide some applications of Bogdanov-Takens bifurcation in some biological and epidemiological models.

Several Population-Toxicant PDE Models

黄启华 西南大学

Abstract: In this presentation, we will introduce several PDE models that describe the interactions between populations and toxicants in polluted aquatic environments. These models include size-structured population models affected by pollution, reaction-diffusion-advection models applicable to river ecosystems, and toxicant-taxis models relevant to lake ecosystems. Theoretical and numerical findings will be utilized to identify pivotal factors influencing population persistence and extinction, along with the spatial distributions of populations and toxicants.

Steady-State Bifurcation and Spike Pattern in the Klausmeier-Gray-Scott Model with Non-Diffusive Plants

蒋卫华 哈尔滨工业大学

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.

Propagation Phenomena in Nonautonomous Fisher-KPP Equations with Nonlocal Diffusion

金祝成 中国科学技术大学

Abstract: In this talk, we aim to understand and to describe the propagation phenomena generating from Fisher-KPP equations with nonlocal diffusion in general time heterogeneous media. The propagation phenomena can be mathematically quantified by two notions: traveling waves and spreading speeds. We first study generalized travelling waves for equations with general time heterogeneities both for the dispersal kernel and the reaction term. We investigate the existence and nonexistence of generalized travelling waves for such a problem. Under certain assumptions, we derive a sharp estimate for the average speed functions of generalized traveling waves. Next we show the exact spreading speeds for a nonlocal diffusion KPP equation equipped with compactly supported initial data. We use some ideas from the uniform persistence theory in dynamical systems.

Evolution of Dispersal and a Conjecture of Dockery et al.

King-Yeung Lam  Ohio State University

Abstract: In 1998, Dockery et al. showed that for two competing species which are identical except for their diffusion rate, the slower diffuser can exclude the faster counterpart regardless of initial data. It is then conjectured that the same result holds for N competing species, for any number N greater than or equal to 3. In this talk, we will discuss some recent progress using the tool of principal Floquet bundle. Our discussion includes the case when the coefficients are autonomous, time-periodic, or aperiodic. This is joint work with Robert S. Cantrell and Yuan Lou.

Biological Aggregations from Spatial Memory and Nonlocal Advection

柳迪 杭州师范大学

Abstract: In this talk, we investigate a nonlocal reaction-diffusion-advection model that integrates the spatial memory of previously visited locations and nonlocal detection in space, resulting in a coupled PDE-ODE system reflective of several multi-species models found in spatial ecology. Our study advances the mathematical understanding of such models by proving the existence and uniqueness of a global weak solution in one spatial dimension using an iterative approach. This is a joint work with Yurij Salmaniw, Jonathan R. Potts, Junping Shi and Hao Wang.

Dynamics of Several Modified Rosenzweig-MacArthur Models in Constant or Changing Environments

鲁敏 华中师范大学

Abstract: In this talk, we consider the dynamics and bifurcations of several modified Rosenzweig-MacArthur predator-prey models in constant or changing environments. By using the method of qualitative analysis and bifurcation theory, we studied the types and stability of equilibria, as well as the corresponding bifurcations in constant environments. Then, numerical simulations are given to illustrate the theoretical results. Finally, we observe some interesting phenomena under changing environments.

Predictive Modelling of Cyanobacterial Blooms in Lakes in Alberta, Canada

Russell Milne  University of Alberta

Abstract: Blooms of cyanobacteria (CB) can disrupt lake ecosystems by killing large numbers of lake organisms, including members of key fish species. Climate change is projected to exacerbate this problem by increasing water temperatures to be closer to those that facilitate maximum CB growth. This raises the question of how temperature increases, and the more severe CB blooms that result from them, will affect the viability and health of other lake species. To address this, we construct a stoichiometric model that includes common components of an Alberta lake food web (CB, algae, daphnia, yellow perch, walleye), and features the effects of oxygen depletion caused by CB blooms and uptake of microcystin secreted by CB. We fit this model using field data from CB observations in lakes in Alberta province, Canada, and simulate it under different warming scenarios to predict the magnitude and effects of CB blooms in the future.

Global Dynamics of the 3D Lotka-Volterra Competition Model with Seasonal Succession

牛磊 东华大学

Abstract: In this talk, we discuss the dynamics of 3D Lotka-Volterra competition model of differential equations with seasonal succession, which exhibits that populations experience an external periodically forced environment. We first establish an index formula on the carrying simplex for the associated Poincaré map of the model. We then classify the dynamics of the discrete dynamical systems induced by all the 3D Poincaré maps into 33 classes in terms of inequalities on parameters. There is no positive fixed point in classes 1-18 so that every orbit tends to certain boundary fixed point, while there exists a positive fixed point (but its uniqueness is unknown) for classes 19-33. We further present necessary and sufficient conditions for the (non)uniqueness of the positive fixed point when the model has identical intrinsic growth rate and death rate respectively, by which we completely classify the global dynamics into 37 dynamical classes. As a by-product, we obtain in which classes the positive fixed point is (non)unique. Especially, we find new phenomena and cases that possess a family of invariant closed curves on which all orbits are positive fixed points, or periodic orbits, or dense orbits, which implies the existence of infinitely many positive harmonic solutions, or subharmonic solutions, or quasiperiodic solutions, respectively. This is a joint work with Yi Wang and Xizhuang Xie.

Seasonal Disease Models of Blue Crab Population in the Chesapeake Bay

Junping Shi College of William & Mary

Abstract: The emergence of pathogens in marine systems affects us all by damaging fisheries and their supporting ecological communities. Climate change magnifies the disease spreading through direct and indirect effects on both the host and pathogen dynamics. A stage-structured epidemic model is constructed to study the impacts of density-dependent predation, cannibalism, fishing, and Hematodinium infection on the blue crab population in the Chesapeake Bay. It is shown that extinction, disease-free and disease-outbreak dynamics can occur under different parameter conditions, and the disease-outbreak could happen in different frequencies.

Local Perception and Learning Mechanisms in Resource-Consumer Dynamics

石青燕 江南大学

Abstract: Spatial memory is key in animal movement modeling, but it has been challenging to explicitly model learning to describe memory acquisition. In this paper, we study novel cognitive consumer-resource models with different consumer's learning mechanisms and investigate their dynamics. These models consist of two PDEs in composition with one ODE such that the spectrum of the corresponding linearized operator at a constant steady state is unclear. We describe the spectra of the linearized operators and analyze the eigenvalue problem to determine the stability of the constant steady state. We then perform bifurcation analysis by taking the memory-based diffusion rate as the bifurcation parameter. It is found that steady-state and Hopf bifurcations can both occur in these systems, and the bifurcation points are given so that the stability region can be determined. Moreover, rich spatial and spatiotemporal patterns can be generated in such systems via different types of bifurcation. Our effort establishes a new approach to tackle a hybrid model of PDE-ODE composition and provides a deeper understanding of cognitive movement-driven consumer-resource dynamics. This is a joint work with Yongli Song and Hao Wang.

Viral Infection Dynamics with Immune Chemokines and CTL Mobility

舒洪英 陕西师范大学

Abstract: We study a viral infection model incorporating both cell-to-cell infection and immune chemokines. Based on experimental results in the literature, we make a standing assumption that the cytotoxic T lymphocytes (CTL) will move toward the location with more infected cells,while the diffusion rate of CTL is a decreasing function of the density of infected cells.We first establish the global existence and ultimate boundedness of the solution via a priori energy stimates.We then define the basic reproduction number of viral infection R0 and prove (by the uniform persistence theory, Lyapunov function technique and LaSalle invariance principle) that the infection-free steady state E₀ is globally asymptotically stable if R₀ < 1. When R₀ > 1, then E0 becomes unstable, and another basic reproduction number of CTL response R₁ becomes the dynamic threshold in the sense that if R₁ < 1, then the CTL-inactivated steady state E₁ is globally asymptotically stable; and if R₁ > 1, then the immune response is uniform persistent and, under an additional technical condition the CTL-activated steady state E₂ is globally asymptotically stable. To establish the global stability results, we need to prove point dissipativity, obtain uniform persistence, construct suitable Lyapunov functions, and apply the LaSalle invariance principle.

Spatiotemporal Dynamics Induced by Nonlocal Perception in the Resource-Consumer Model

宋永利  杭州师范大学

Abstract: Nonlocal perception is crucial to the mechanistic modeling of cognitive animal movement. We formulate a diffusive consumer-resource model with nonlocal perception on resource availability, where resource dynamics is explicitly modeled, to investigate the influence of nonlocal perception on stability and spatiotemporal patterns. For the finite domain, nonlocal perception described by two common types of resource detection function (spatial average or Green function) has no impact on the stability of the spatially homogeneous steady state. For the infinite domain, nonlocal perception described by the Laplacian or Gaussian detection function has no impact on stability either; however, the top-hat detection function can destabilize the spatially homogeneous steady state when the rate of perceptual movement is large and the detection scale belongs to an appropriate interval. Using the more realistic top-hat perception kernel, we investigate the influence of the detection scale, the perceptual movement rate and the resource’s carrying capacity on the spatiotemporal patterns and find the stripe spatial patterns, oscillatory patterns with different spatial profiles as well as spatiotemporal chaos.

药物毒理效应的普适性模型与数据分析

唐三一 山西大学

摘要: 大量实验数据或观察证实剂量---效应曲线存在复杂的量效关系，包括单调、U-型、倒U-型和钟型等。研究者依据各自的实验数据分别提出了众多的复杂函数曲线来拟合不同的量效关系，缺乏一般性，这使得应用相关模型具有较大挑战。为了克服上述困难，我们提出了具有普适性的药物毒理效应Ricker模型，理论上得到了模型存在药物毒理效应的复杂参数空间，且基于剂量相关参数的稳态解重构了药物毒理效应曲线。进一步考虑滞后效应、随机和空间等因素，发展模型、拓展模型的应用领域。同时，利用所发展的模型对害虫控制、中药制剂、肿瘤免疫治疗中扥众多类型的剂量---效应曲线进行拟合，发现所提出的模型能够对大量的实验数据进行更好的拟合，具有普适性。同时，也对过度补偿、药物毒理效应和水螅效应之间的内在逻辑关系、生物意义与模型结果等进行必要的讨论。

Dynamics of Population Models with Memory-Based Diffusions

王春程 哈尔滨工业大学

Abstract: Spatial memory has been considered significant in animal movement modeling. In this paper, we formulate a two-species interaction model by incorporating both random walk and spatial memory-based walk in their movement. The spatial memory-based walk, described by a chemotactic-like term, is derived by a modified Fick’s law involving a directed movement toward the gradient of the density distribution function at a past time. For the proposed model, local stability and bifurcations are studied at constant steady states. Unlike a classical reaction-diffusion equation, we show that the accumulation points of eigenvalues for the model will locate at a vertical line in the complex plane, which will make the model generate spatially inhomogeneous time-periodic patterns through Hopf bifurcation. As illustrations, we apply these results to competition and cooperative models with memory-based diffusion. For the competition model, it turns out that the outcomes are far more complicated than those of classic Lotka-Volterra reaction-diffusion models, due to the consideration of memory-based diffusion. In particular, the existence of periodic oscillations is proved under weak competition. Similar conclusions hold for the cooperative model.

Threshold Dynamics of a Mosquito-Borne Disease Model with Chemotaxis and Spatial Heterogeneity

王凯 广州大学

Abstract: In this talk, I will introduce a spatial chemotactic mosquito-borne disease model. We first proved the global existence and boundedness of solutions to guarantee the solvability of the model. Next, the threshold dynamics were obtained. Furthermore, we numerically explored the impacts of chemotactic effect, spatial heterogeneity and dispersal rates of infected individuals to provide a clear picture on disease severity. In particular, we find that the mosquito chemotaxis causes mild disease in some regions but severe in others, which suggests developing targeted strategies to control mosquitoes in specific locations and achieve a deep understanding on the chemotaxis.

Stoichiometric Microplastics Models in Natural and Laboratory Environments

Tianxu Wang University of Alberta

Abstract: Microplastics pose a severe threat to marine ecosystems; however, relevant mathematical modeling and analysis are lacking. This study formulates two stoichiometric producer-grazer models to investigate the interactive effects of microplastics, nutrients, and light on population dynamics under different settings. One model incorporates optimal microplastic uptake and foraging behavior based on nutrient availability for natural settings, while the other model does not include foraging in laboratory settings. We establish the well-posedness of the models and examine their long-term behaviors. Our results reveal that in natural environments, producers and grazers exhibit higher sensitivity to microplastics, and the system may demonstrate bistability or tristability. Moreover, the influences of microplastics, nutrients, and light intensity are highly intertwined. The presence of microplastics amplifies the constraints on grazer growth related to food quality and quantity imposed by extreme light intensities, while elevated phosphorus input enhances the system’s resistance to intense light conditions. Furthermore, higher environmental microplastic levels do not always imply elevated microplastic body burdens in organisms, as organisms are also influenced by nutrients and light. We also find that grazers are more vulnerable to microplastics compared to producers. If producers can utilize microplastics for growth, the system displays significantly greater resilience to microplastics.

传染病基本再生数的计算和应用

王稳地 西南大学

Abstract: 首先介绍传染病基本再生数的生物学意义，接着讨论传染病动力学模型基本再生数的数学计算方法，这里着重分析再生数与主特征值的联系，后面通过理论分析和数值计算方法给出时间和空间异质性对基本再生数的影响，展示基本再生数在揭示传染病传播机理方面的重要作用。

Discrete Inverse Method for Extracting Disease Transmission Rates from Accessible Infection Data

Xiunan Wang University of Tennessee at Chattanooga

Abstract: In this presentation, we introduce a data-driven inverse method leveraging discretization of differential equation models to estimate time-varying transmission rates for infectious diseases. Our approach, applied to three distinct classes of diseases (seasonal, nonseasonal periodic, and aperiodic), reveals valuable insights from pandemic or epidemic incidence data. The method is intuitive, facilitating rapid implementation even with multi-year datasets. Integrated with machine learning, it offers a tool for forecasting disease dynamics based on future conditions such as weather, policy decisions, and human mobility trends. Our findings provide actionable guidance for public health authorities.

关于具时滞和对流的单种群模型的分支分析

魏俊杰 哈尔滨工业大学

摘要: 本报告将主要介绍刻画单种群增长规律的具有时滞和对流的反应扩散方程的分支分析，包括非常值稳态解的存在性，在非常值稳态解处以时滞为分支参数的Hopf分支的存在性及分支周期解的性质分析等。

Linking Bifurcation Analysis of Holling-Tanner Model with Generalist Predator to a Changing Environment

向创 南京师范大学

Abstract: In this talk, we first rigorously analyze Holling-Tanner model with generalist predators who have alternative food sources, and then discuss transient dynamics via a changing environment. For a constant environment, we shown that the highest codimension of a nilpotent cusp is 3, and the model can undergo degenerate Bogdanov-Takens bifurcation of codimension 3. Moreover, we show that a center-type equilibrium is a weak focus with order at most 2, and the model can exhibit Hopf bifurcation of codimension 2. Our results indicate that generalist predators can cause not only richer dynamics and bifurcations, but also the extinction of prey for some positive initial densities. In a changing environment, the populations start along one stable state but can track unstable states or oscillations when the system crosses a bifurcation point, and then tend to another stable state or oscillations. This tracking on transient dynamics predicts regime shifts under environmental changes. Finally, we focus on a periodic environment and find that the populations converge to a periodic solution or an invariant torus depending on both the initial environmental capacity and the amplitude of periodic fluctuation.

Multi-Scale Models and Applications in HIV/AIDS Infections

肖燕妮 西安交通大学

Abstract: Coupling the models in various scales becomes challenging. In this talk I shall briefly give an introduction to modelling approach at population/individual level. I shall give an idea of developing a multi-scale model that nests the within-host viral dynamics into the between-host transmission model, and some thoughts on how to link transmission dynamics to viral dynamics via modelling behavior changes. Then I illustrate our main results with applications in HIV/AIDS infections. Finally I shall give concluding remarks and challenges on multi-scale modelling approach.  

Threshold Behavior and Exponential Ergodicity of an SIR Epidemic Model: the Impact of Random Jamming and Hospital Capacity

原三领 上海理工大学

Abstract: This article uses hospital capacity to determine the treatment rate for an infectious disease. To examine the impact of random jamming and hospital capacity on the spread of the disease, we propose a stochastic SIR model with nonlinear treatment rate and degenerate diffusion. Our findings demonstrate that the disease's persistence or eradication depends on the basic reproduction number. If <1, the disease is eradicated with a probability of 1, while>1 results in the disease being almost surely strongly stochastically permanent. We also demonstrate that if>1, the Markov process has a unique stationary distribution and is exponentially ergodic. Additionally, we identify a critical capacity which determines the minimum hospital capacity required.

Stochastic Generalized Kolmogorov Systems with Small Diffusion

周保全 东北师范大学

Abstract: This talk presents the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. We establishe a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this talk introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology.