Thomas M. Michelitsch

Predator–prey type models are commonly used to describe the dynamic interplay of two interacting species, the predator and the prey. Contrary to earlier studies where delay terms represent the lag between birth and maturity of the particular species, we here analyze a delayed predator–prey interaction. Delayed interaction can be associated with the occurrence of an additional intermediate species, generated from the prey and consumed by the predator. In an epidemiological context, this scenario can be connected with a compartment model, where the susceptible compartment plays the role of the prey, the intermediate compartment corresponds to the latency (incubation) period, and the infected and infectious state is identified with the predator. We consider this setting with and without spatial diffusion. The diffusion corresponds to random motions of the microscopic agents. In the case without diffusion and for a sharp delay time (sojourn time in the intermediate state), a Hopf instability emerges when the delay time exceeds a critical threshold. A weakly nonlinear expansion around this critical point leads to a Hopf normal form allowing to determine a stable long-time limit cycle in an explicit form. Including diffusion into the model, we derive the Ginzburg–Landau normal form, which leads to an approximate phase diffusion equation with stable turbulent phase solutions. We derive analytically the coefficients of the Ginzburg–Landau equation and compute the parameter regions where phase turbulence and weak chaos are expected. As new features, defect mediated patterns such as spirals, but also weakly turbulent structures emerge, resulting from the complex interplay of delay and diffusion. In addition to mathematical biology, our model has various interdisciplinary applications in the kinetics of certain chemical reactions, social science, finance, economy, and others.

We consider a mortal random walker evolving with discrete time on a network, where transitions follow a degree-biased Markovian navigation strategy. The walker starts with a random initial budget T1∈N and must maintain a strictly positive budget to remain alive. Each step incurs a unit cost, decrementing the budget by one; the walker perishes (is ruined) upon depletion of the budget. However, when the walker reaches designated target nodes, the budget is renewed by an independent and identically distributed (IID) copy of its initial value. The degree bias is tuned to either favor or disfavor visits to these target nodes. Our model exhibits connections with stochastic resetting. The evolution of the budget can be interpreted as a deterministic drift on the integer line toward negative values, where the walker is intermittently reset to positive IID random positions and dies at the first hit of the origin. The first part of the paper focuses on the target-hitting statistics of an immortal Markovian walker. We analyze the target-hitting counting process (THCP) for an arbitrary set of target nodes. In the special case where a single target node coincides with the starting node, the THCP reduces to a renewal counting process. We establish connections with classical results from the literature. Within this framework, the second part of the paper addresses the dynamics of the evanescent walker. We derive analytical results for arbitrary configurations of target nodes, including the evanescent propagator matrix, the survival probability, the mean residence time on a set of nodes during the walker’s lifetime, and the expected lifetime itself. Additionally, we compute the expected number of target hits (i.e., budget renewals) in a lifetime of the walker and related distributions. We explore both analytically and numerically a set of characteristic scenarios, including a forager scenario, in which frequent encounters with target nodes extend the walker’s lifetime, and a detrimental scenario, where such encounters instead reduce it. Finally, we identify a neutral scenario in which frequent visits to target nodes have no effect on the walker’s lifetime. Our analytical results are validated through random walk simulations.

We analyze a Predator-Prey model with delay. In an epidemiological context, the delay can be attributed to the latency period of the disease. We consider this setting without and with diffusive terms. Diffusion corresponds to random spatial motions of the microscopic agents. In the case without diffusion and for a sharp delay time, a Hopf instability exists when the delay time exceeds a critical threshold. A weakly nonlinear expansion around this critical point leads to a Hopf normal form allowing to determine a stable long time limit cycle in explicit form. Including diffusion into the model, we derive the Ginzburg-Landau normal form which leads to an approximate phase diffusion equation with stable turbulent phase solutions. As new features, defect mediated patterns such as spirals or targets, but also weakly turbulent structures emerge from the interplay of delay and diffusion. Besides mathematical biology, our model has applications in the kinetics of certain chemical reactions, social science, and economy.

We consider a generalized SEIS (susceptible, exposed, infectious, and susceptible) model where individuals are divided into three compartments: S (healthy and susceptible), E (infected but not just infectious, or exposed), and I (infectious). Finite waiting times in the compartments yield a system of delay-differential or memory equations and may exhibit oscillatory (Hopf) instabilities of the otherwise stationary endemic state, leading normally to regular oscillations in the form of an attractive limit cycle in the phase space spanned by the compartment rates. In the present paper, our aim is to demonstrate that in the dynamics of delayed SEIS models, persistent chaotic attractors can bifurcate from these limit cycles and become accessible if the nonlinear interaction terms fulfill certain basic requirements, which to our knowledge were not addressed in the literature so far. Computing the largest Lyapunov exponent, we show that chaotic behavior exists in a wide parameter range. Finally, we discuss a more general system and show that a sudden falloff of the infection rate with respect to increasing infection number may be responsible for the emergence of chaotic time evolution. Such a falloff can describe mitigation measures, such as wearing masques, individual isolation, or vaccination. The model may have a wide range of interdisciplinary applications beyond epidemic spreading, for instance, in the kinetics of certain chemical reactions.

We consider a discrete-time Markovian random walk with resets on a connected undirected network. The resets, in which the walker is relocated to randomly chosen nodes, are governed by an independent discrete-time renewal process. Some nodes of the network are target nodes, and we focus on the statistics of first hitting of these nodes. In the non-Markov case of the renewal process, we consider both light- and fat-tailed inter-reset distributions. We derive the propagator matrix in terms of discrete backward recurrence time probability density functions, and in the light-tailed case, we show the existence of a non-equilibrium steady state. In order to tackle the non-Markov scenario, we derive a defective propagator matrix, which describes an auxiliary walk characterized by killing the walker as soon as it hits target nodes. This propagator provides the information on the mean first passage statistics to the target nodes. We establish sufficient conditions for ergodicity of the walk under resetting. Furthermore, we discuss a generic resetting mechanism for which the walk is non-ergodic. Finally, we analyze inter-reset time distributions with infinite mean where we focus on the Sibuya case. We apply these results to study the mean first passage times for Markovian and non-Markovian (Sibuya) renewal resetting protocols in realizations of Watts–Strogatz and Barabási–Albert random graphs. We show nontrivial behavior of the dependence of the mean first passage time on the proportions of the relocation nodes, target nodes, and of the resetting rates. It turns out that, in the large-world case of the Watts–Strogatz graph, the efficiency of a random searcher particularly benefits from the presence of resets.

The propagation of an acoustic wave generated by a δ-pulse unit point force in a three dimensional anisotropic medium is considered. A general expression for the retarded 3 × 3 elastodynamic displacement Green's function is derived for arbitrary anisotropic symmetry. The initial integration problem of four integrations is reduced to only one ϕ-integration problem combined with the problem of solving three algebraic equations. The evaluation of expressions of the Green's function for the behavior at infinity yields the radiation condition and describes the acoustic wave propagation and gives the possible distinguished propagation directions from the source point for outgoing acoustic waves (high symmetry directions of the crystal beside other distinguished directions). For high symmetry directions of the crystal this condition yields three outgoing singular acoustic waves, one with longitudinal and two with transverse polarization. For elastic isotropy of the medium we obtain the result known from literature [17] which was derived there by using other techniques. For hexagonal symmetry of the medium we derive closed-form expressions of the farfield waves for the high symmetry directions (i.e. for propagation directions parallel and in the plane perpendicular to the hexagonal c-axis).