American Institute of Mathematical Sciences

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since temperature is known to have an effect on the length of certain delays.

We establish a framework to investigate approximate synchronization of coupled systems under general coupling schemes. The units comprising the coupled systems may be nonidentical and the coupling functions are nonlinear with delays. Both delay-dependent and delay-independent criteria for approximate synchronization are derived, based on an approach termed sequential contracting. It is explored and elucidated that the synchronization error, the distance between the asymptotic state and the synchronous set, decreases with decreasing difference between subsystems, difference between the row sums of connection matrix, and difference of coupling time delays between different units. This error vanishes when these factors decay to zero, and approximate synchronization becomes identical synchronization for the coupled system comprising identical subsystems and connection matrix with identical row sums, and with identical coupling delays. The application of the present theory to nonlinearly coupled heterogeneous FitzHugh-Nagumo neurons is illustrated. We extend the analysis to study approximate synchronization and asymptotic synchronization for coupled Lorenz systems and show that for some coupling schemes, the synchronization error decreases as the coupling strength increases, whereas in another case, the error remains at a substantial level for large coupling strength.

We recently derived a method, local orthogonal rectification (LOR), that provides a natural and useful geometric frame for analyzing dynamics relative to a base curve in the phase plane for two-dimensional systems of ODEs (Letson and Rubin, SIAM J. Appl. Dyn. Syst., 2018). This work extends LOR to apply to any embedded base manifold in a system of ODEs of arbitrary dimension and establishes a corresponding system of LOR equations for analyzing dynamics within the LOR frame, which maps naturally back to the original phase space. The LOR equations encode geometric properties of the underlying flow and remain valid, in general, beyond a local neighborhood of the embedded manifold. In addition to developing a general theory for LOR that makes use of a given normal frame, we show how to construct a normal frame that conveniently simplifies the computations involved in LOR. Finally, we illustrate the utility of LOR by showing that a blow-up transformation on the LOR equations provides a useful decomposition for studying trajectories' behavior relative to the embedded base manifold and by using LOR to identify canard behavior near a fold of a critical manifold in a two-timescale system.

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.

We consider a generational and continuous-time two-phase model of the cell cycle. The first model is given by a stochastic operator, and the second by a piecewise deterministic Markov process. In the second case we also introduce a stochastic semigroup which describes the evolution of densities of the process. We study long-time behaviour of these models. In particular we prove theorems on asymptotic stability and sweeping. We also show the relations between both models.

Tuberculosis (TB) is a leading cause of death from infectious disease. TB is caused mainly by a bacterium called Mycobacterium tuberculosis which often initiates in the respiratory tract. The interaction of macrophages and T cells plays an important role in the immune response during TB infection. Recent experimental results support that active TB infection may be induced by the dysfunction of Treg cell regulation that provides a balance between anti-TB T cell responses and pathology. To better understand the dynamics of TB infection and Treg cell regulation, we build a mathematical model using a system of differential equations that qualitatively and quantitatively characterizes the dynamics of macrophages, Th1 and Treg cells during TB infection. For sufficiently analyzing the interaction between immune response and bacterial infection, we separate our model into several simple subsystems for further steady state and stability studies. Using this system, we explore the conditions of parameters for three situations, recovery, latent disease and active disease, during TB infection. Our numerical simulations support that Th1 cells and Treg cells play critical roles in TB infection: Th1 cells inhibit the number of infected macrophages to reduce the chance of active disease; Treg cell regulation reduces the immune response to stabilize the dynamics of the system.

We consider a model of phenotypic evolution in populations with assortative mating of individuals. The model is given by a nonlinear operator acting on the space of probability measures and describes the relation between parental and offspring trait distributions. We study long-time behavior of trait distribution and show that it converges to a combination of Dirac measures. This result means that assortative mating can lead to a polymorphic population and sympatric speciation.

A new theorem on asymptotic stability of stochastic semigroups is given. This theorem is applied to a stochastic semigroup corresponding to Stein's neuronal model. Asymptotic properties of models with and without the refractory period are compared.