We consider a general swarm model of self-propelling agents interacting through a pairwise potential in the presence of noise and communication time delay. Previous work has shown that a communication time delay in the swarm induces a pattern bifurcation that depends on the size of the coupling amplitude. We extend these results by completely unfolding the bifurcation structure of the mean field approximation. Our analysis reveals a direct correspondence between the different dynamical behaviors found in different regions of the coupling-time delay plane with the different classes of simulated coherent swarm patterns. We derive the spatiotemporal scales of the swarm structures, as well as demonstrate how the complicated interplay of coupling strength, time delay, noise intensity, and choice of initial conditions can affect the swarm. In particular, our studies show that for sufficiently large values of the coupling strength and/or the time delay, there is a noise intensity threshold that forces a transition of the swarm from a misaligned state into an aligned state. We show that this alignment transition exhibits hysteresis when the noise intensity is taken to be time dependent.

We consider a general model of self-propelling particles interacting through a pairwise attractive force in the presence of noise and communication time delay. Previous work by Erdmann et al.[Phys. Rev. E 71, 051904 (2005)] has shown that a large enough noise intensity will cause a translating swarm of individuals to transition to a rotating swarm with a stationary center of mass. We show that with the addition of a time delay, the model possesses a transition that depends on the size of the coupling amplitude. This transition is independent of the initial swarm state (traveling or rotating) and is characterized by the alignment of all of the individuals along with a swarm oscillation. By considering the mean field equations without noise, we show that the timedelay-induced transition is associated with a Hopf bifurcation. The analytical result yields good agreement with numerical computations of the value of the coupling parameter at the Hopf point.The collective motion of multiagent systems has long been observed in biological populations including bacterial colonies [1][2][3], slime molds [4,5], locusts [6], and fish [7]. However, mathematical studies of swarming behavior have been performed for only a few decades. In addition to providing examples of biological pattern formation, the information gained from these mathematical investigations has led to an increased ability to intelligently design and control man-made vehicles [8][9][10][11][12].Many types of mathematical models have been used to describe coherent swarms. One popular approach is based on a continuum approximation in which scalar and vector fields are used to describe all of the relevant quantities [6,[13][14][15][16]. Another popular approach is based on treating every biological or mechanical individual as a discrete particle [7,14,15,17,18]. Depending on the problem, these individual-based models may be deterministic or stochastic.Regardless of the type of swarm model being used, one can see the emergence of ordered swarm states from an initial disordered state where individual particles have random velocity directions [13,14,17]. These ordered states may be translational or rotational in motion, and they may be spatially distributed or localized in clusters.In particular, it is known that a localized swarm state may transition to a new dynamical region as the system parameters or the noise intensity is changed. For example, it has been shown in [18] that a planar model of self-propelling particles interacting via a harmonic attractive potential in the presence of noise possesses a noise-induced transition whereby the translational motion of the swarm breaks down into rotational motion.Another aspect of swarm modeling that has not yet been considered is the effect of time delayed interactions arising from finite communication times between individuals. Much attention has been given to the effects of time delays in the context of physiology [19], optics [20], neurons [21], lasers [22], and many other types of systems. The aim of this ...

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Although there are many random paths to an extinct state, there is an optimal path that maximizes the probability to extinction. In this paper, we show that the optimal path is associated with the dynamical systems idea of having maximum sensitive dependence to initial conditions. Using the equivalence between the sensitive dependence and the path to extinction, we show that the dynamical systems picture of extinction evolves naturally towards the optimal path in several stochastic models of epidemics.

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We consider a stochastic susceptible-exposed-infected-recovered (SEIR) epidemiological model. Through the use of a normal form coordinate transform, we are able to analytically derive the stochastic center manifold along with the associated, reduced set of stochastic evolution equations. The transformation correctly projects both the dynamics and the noise onto the center manifold. Therefore, the solution of this reduced stochastic dynamical system yields excellent agreement, both in amplitude and phase, with the solution of the original stochastic system for a temporal scale that is orders of magnitude longer than the typical relaxation time. This new method allows for improved time series prediction of the number of infectious cases when modeling the spread of disease in a population. Numerical solutions of the fluctuations of the SEIR model are considered in the infinite population limit using a Langevin equation approach, as well as in a finite population simulated as a Markov process.