The Speakers

Title of the talk: Dynamical systems theory as laboratory for predator – prey systems.

Abstract: The main focus in this talk will be on predator – prey models describing biomass – herbivore interactions and  biomass - herbivore  - carnivore interactions. The properties of the proposed systems will be discussed, as well as possible extensions. The motivation for this study is the management issue of herbivores like rangifers versus keeping sustainable populations of carnivores like wolverines, lynx and golden eagles in the tundra and taiga region of northern Europe, Siberia and North America.  In case of time we will also discuss application of dynamical systems theory to certain  bioeconomic- and epidemiological models. Here the emphasis will be on the modelling of delay effects.

Title of the talk: Neural field models with and without microstructure 

Abstract: Four levels of neural field modeling will be reviewed. Conditions for existence and uniqueness of solutions of the level 4 neural field models, based on partial differential equations, will be formulated. Continuous dependence of the solutions on the spatiotemporal integration kernels, delays, firing rates, external inputs and measures describing microstructure will be examined. Connections between the Wilson-Cowan-type models (level 4) and the Hopfield network models (level 3) will be outlined. Existence of stationary solutions in some simplified models will be discussed. 

Title of the talk: Simple ecological model with complex dynamics: local and global bifurcation analysis 

Abstract: Simple mathematical models for two interacting species can exhibit a wide range of dynamics due to successive local and global bifurcations as system parameters change. The main objective of this talk is to present preliminary ideas regarding stability, local bifurcations, and global bifurcations in nonlinear dynamical systems. We will illustrate these concepts using a prey-predator model featuring a saturating functional response and a generalist predator. 

Title of the talk: Modeling of Infectious Diseases: A deterministic Approach

Abstract: Modeling infectious diseases plays a crucial role in understanding their dynamics and informing public health interventions. This talk presents a deterministic approach to modeling infectious diseases, focusing on the mathematical frameworks used to describe transmission dynamics and control strategies. The deterministic modeling approach assumes that the population is well-mixed and that parameters such as transmission rates remain constant over time. Various types of deterministic models, including compartmental models such as the susceptible-infected-recovered (SIR) model and its extensions, are discussed, highlighting their strengths and limitations. Furthermore, this talk explores how deterministic models are utilized to study disease outbreaks, evaluate the impact of interventions such as vaccination and social distancing, and forecast future disease trends.

Title of the talk: Delay Differential Equations in Population Models: Negative solutions and stability analysis

Abstract: My presentation will start with the ideas of delay differential equations. Through a simple example, we will observe how a discrete time delay could change system’s dynamics in comparison to the corresponding differential equation without time delay. In the second part, we’ll consider a stage-structured model exhibiting negative solutions. Some solutions even reach positive equilibrium in long run, but they are negative in a shorter time. We also show an example where some solutions may be attracted to a stable limit cycle with negative values in long run. The last part of the talk will discuss a variety of dynamics with respect to a single-time delay in a very specific predator-prey model.

 Title of the talk: Bifurcation analysis of dengue transmission models with explicit vector dynamics

Abstract: The investigation of epidemiological scenarios characterized by chaotic dynamics is crucial for understanding disease spread and improving disease control strategies. Motivated by dengue fever epidemiology, in this work, introducing a SIR-UV and the SIRSIR-UV model, which accounts for differences between primary and secondary infections and explicit dis- ease vector dynamics. Our analysis, employing nonlinear dynamics and bifurcation theory, provides key insights into how vectors contribute to the overall system dynamics. In this paper, the formalization of backward bifurcation using center manifold theory, computation of Hopf and global homoclinic bifurcation curves, and derivation of analytical expressions for transcritical and tangent bifurcations deepen the understanding. The observation of chaotic behavior with the inclusion of seasonal forcing in the vector population underscores the importance of considering external factors like climate in disease spread. Our findings align with those from previous models, emphasizing the significance of simplifying assumptions, such as implicit vector dynamics, when constructing models without vector control. This study brings significant insights to the mathematical modeling of vector-borne diseases, providing a manageable framework for exploring complex epidemiological scenarios and identifying key factors influencing disease spread. While the absence of strain structure may limit predictive power in certain scenarios, the SIRSIR-UV model serves as a starting point for understanding vector-borne infectious disease dynamics.