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In spatially structured microbial communities, clonal growth of stationary cells passively generates clusters of related individuals. This can lead to stable cooperation without the need for recognition mechanisms. However, recent research suggests that some biofilm-forming microbes may have mechanisms of kin recognition. To explore this unexpected observation, we studied the effects of different types of cooperation in a microbial colony using spatially explicit, agent-based simulations of two interacting strains. We found scenarios that favor a form of kin recognition in spatially structured microbial communities. In the presence of a "cheater" strain, a strain with greenbeard cooperation was able to increase in frequency more than a strain with obligate cooperation. This effect was most noticeable in high density colonies and when the cooperators were not as abundant as the cheaters. We also studied whether a polychromatic greenbeard, in which cells only cooperate with their own type, could provide a numerical benefit beyond a simple, binary greenbeard. We found the greatest benefit to a polychromatic greenbeard when cooperation is highly effective. These results suggest that in some ecological scenarios, recognition mechanisms may be beneficial even in spatially structured communities.

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Spatial self-organization, a common feature of multi-species communities, can provide important insights into ecosystem structure and resilience. As environmental conditions gradually worsen (e.g., resource depletion, erosion intensified by storms, drought), some ecological systems collapse to an irreversible state once a tipping point is reached. Spatial patterning may be one way for them to cope with such changes. We use a mathematical model to describe self-organization of an eroding marsh shoreline based on three-way interactions between sediment volume and two ecosystem engineers - smooth cordgrass Spartina alterniflora and ribbed mussels Geukensia demissa. Our model indicates that scale-dependent interactions between multiple ecosystem engineers drive the self-organization of eroding marsh edges and regulate the spatial scale of shoreline morphology. Spatial self-organization of the marsh edge increases the system's productivity, allows it to withstand erosion, and delays degradation that otherwise would occur in the absence of strong species interactions. Further, changes in wavelength and variance of the spatial patterns give insight into marsh recession. Finally, we find that the presence of mussels in the system modulates the spatial scale of the patterns, generates patterns with shorter wavelengths, and allows the system to tolerate a greater level of erosion. Although previous studies suggest that self-organization can emerge from local interactions and can result in increased ecosystem persistence and stability in various ecosystems, our findings extend these concepts to coastal salt marshes, emphasizing the importance of the ecosystem engineers, smooth cordgrass and ribbed mussels, and demonstrating the potential value of self-organization for ecosystem management and restoration.

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There was a mistake in the Matlab code we used to generate time series solutions of our model, Eqs. (16)-(18). The corrected text below replaces one paragraph on p. 7, and the figures below replace Figs. 4-5 on p. 8. There is no qualitative change to our results. However, there is a quantitative change in the initial dead oyster shell volume B(0) needed for reef survival. The corrected threshold B(0), about 0.40 m3 per m2 of sea floor, is more consistent with a recently experimentally estimated threshold of 0.30 m (Colden, Latour, and Lipcius, Mar Ecol Prog Ser 582: 1-13, 2017) than was our old incorrect threshold of about 0.12 m3.

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Smooth cordgrass Spartina alterniflora is a grass species commonly found in tidal marshes. It is an ecosystem engineer, capable of modifying the structure of its surrounding environment through various feedbacks. The scale-dependent feedback between marsh grass and sediment volume is particularly of interest. Locally, the marsh vegetation attenuates hydrodynamic energy, enhancing sediment accretion and promoting further vegetation growth. In turn, the diverted water flow promotes the formation of erosion troughs over longer distances. This scale-dependent feedback may explain the characteristic spatially varying marsh shoreline, commonly observed in nature. We propose a mathematical framework to model grass-sediment dynamics as a system of reaction-diffusion equations with an additional nonlocal term quantifying the short-range positive and long-range negative grass-sediment interactions. We use a Mexican-hat kernel function to model this scale-dependent feedback. We perform a steady state biharmonic approximation of our system and derive conditions for the emergence of spatial patterns, corresponding to a spatially varying marsh shoreline. We find that the emergence of such patterns depends on the spatial scale and strength of the scale-dependent feedback, specified by the width and amplitude of the Mexican-hat kernel function.

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During an epidemic, individual nodes in a network may adapt their connections to reduce the chance of infection. A common form of adaption is avoidance rewiring, where a noninfected node breaks a connection to an infected neighbor and forms a new connection to another noninfected node. Here we explore the effects of such adaptivity on stochastic fluctuations in the susceptible-infected-susceptible model, focusing on the largest fluctuations that result in extinction of infection. Using techniques from large-deviation theory, combined with a measurement of heterogeneity in the susceptible degree distribution at the endemic state, we are able to predict and analyze large fluctuations and extinction in adaptive networks. We find that in the limit of small rewiring there is a sharp exponential reduction in mean extinction times compared to the case of zero adaption. Furthermore, we find an exponential enhancement in the probability of large fluctuations with increased rewiring rate, even when holding the average number of infected nodes constant.

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A new method is proposed to infer unobserved epidemic subpopulations by exploiting the synchronization properties of multistrain epidemic models. A model for dengue fever is driven by simulated data from secondary infective populations. Primary infective populations in the driven system synchronize to the correct values from the driver system. Most hospital cases of dengue are secondary infections, so this method provides a way to deduce unobserved primary infection levels. We derive center manifold equations that relate the driven system to the driver system and thus motivate the use of synchronization to predict unobserved primary infectives. Synchronization stability between primary and secondary infections is demonstrated through numerical measurements of conditional Lyapunov exponents and through time series simulations.

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During an epidemic, people may adapt or alter their social contacts to avoid infection. Various adaptation mechanisms have been studied previously. Recently, a new adaptation mechanism was presented in [1], where susceptible nodes temporarily deactivate their links to infected neighbors and reactivate when their neighbors recover. Considering the same adaptation mechanism on a scale-free network, we find that the topology of the subnetwork consisting of active links is fundamentally different from the original network topology. We predict the scaling exponent of the active degree distribution and derive mean-field equations by using improved moment closure approximations based on the conditional distribution of active degree given the total degree. These mean field equations show better agreement with numerical simulation results than the standard mean field equations based on a homogeneity assumption.

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When an epidemic spreads in a population, individuals may adaptively change the structure of their social contact network to reduce risk of infection. Here we study the spread of an epidemic on an adaptive network with community structure. We model two communities with different average degrees. The disease model is susceptible-infected-susceptible (SIS), and adaptation is rewiring of links between susceptibles and infectives. Locations of rewired links are selected so that the community structure will be preserved if susceptible-infective links are homogeneously distributed. The bifurcation structure is obtained, and a mean field model is developed that accurately predicts the steady-state behavior of the system. In a static network, weakly connected heterogeneous communities can have significantly different infection levels. In contrast, adaptation promotes similar infection levels and alters the network structure so that communities have more similar average degrees. We estimate the time for network restructuring to allow infection incursion from one community to another and show that it is inversely proportional to the number of cross-links between communities. In extremely heterogeneous systems, periodic oscillations in infection level can occur due to repeated infection incursions.

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Disease spread in a society depends on the topology of the network of social contacts. Moreover, individuals may respond to the epidemic by adapting their contacts to reduce the risk of infection, thus changing the network structure and affecting future disease spread. We propose an adaptation mechanism where healthy individuals may choose to temporarily deactivate their contacts with sick individuals, allowing reactivation once both individuals are healthy. We develop a mean-field description of this system and find two distinct regimes: slow network dynamics, where the adaptation mechanism simply reduces the effective number of contacts per individual, and fast network dynamics, where more efficient adaptation reduces the spread of disease by targeting dangerous connections. Analysis of the bifurcation structure is supported by numerical simulations of disease spread on an adaptive network. The system displays a single parameter-dependent stable steady state and non-monotonic dependence of connectivity on link deactivation rate.

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Adaptive social networks, in which nodes and network structure coevolve, are often described using a mean-field system of equations for the density of node and link types. These equations constitute an open system due to dependence on higher-order topological structures. We propose a new approach to moment closure based on the analytical description of the system in an asymptotic regime. We apply the proposed approach to two examples of adaptive networks: recruitment to a cause model and adaptive epidemic model. We show a good agreement between the improved mean-field prediction and simulations of the full network system.

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We model recruitment in adaptive social networks in the presence of birth and death processes. Recruitment is characterized by nodes changing their status to that of the recruiting class as a result of contact with recruiting nodes. Only a susceptible subset of nodes can be recruited. The recruiting individuals may adapt their connections in order to improve recruitment capabilities, thus changing the network structure adaptively. We derive a mean field theory to predict the dependence of the growth threshold of the recruiting class on the adaptation parameter. Furthermore, we investigate the effect of adaptation on the recruitment level, as well as on network topology. The theoretical predictions are compared with direct simulations of the full system. We identify two parameter regimes with qualitatively different bifurcation diagrams depending on whether nodes become susceptible frequently (multiple times in their lifetime) or rarely (much less than once per lifetime).

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Native oyster populations in Chesapeake Bay have been the focus of three decades of restoration attempts, which have generally failed to rebuild the populations and oyster reef structure. Recent restoration successes and field experiments indicate that high-relief reefs persist, likely due to elevated reef height which offsets heavy sedimentation and promotes oyster survival, disease resistance and growth, in contrast to low-relief reefs which degrade in just a few years. These findings suggest the existence of alternative stable states in oyster reef populations. We developed a mathematical model consisting of three differential equations that represent volumes of live oysters, dead oyster shells (=accreting reef), and sediment. Bifurcation analysis and numerical simulations demonstrated that multiple nonnegative equilibria can exist for live oyster, accreting reef and sediment volume at an ecologically reasonable range of parameter values; the initial height of oyster reefs determined which equilibrium was reached. This investigation thus provides a conceptual framework for alternative stable states in native oyster populations, and can be used as a tool to improve the likelihood of success in restoration efforts.

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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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Real networks consisting of social contacts do not possess static connections. That is, social connections may be time dependent due to a variety of individual behavioral decisions based on current network connections. Examples of adaptive networks occur in epidemics, where information about infectious individuals may change the rewiring of healthy people, or in the recruitment of individuals to a cause or fad, where rewiring may optimize recruitment of susceptible individuals. In this paper, we will review some of the dynamical properties of adaptive networks, and show how they predict novel phenomena as well as yield insight into new controls. The applications will be control of epidemic outbreaks and terrorist recruitment modeling.

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Extinction of an epidemic or a species is a rare event that occurs due to a large, rare stochastic fluctuation. Although the extinction process is dynamically unstable, it follows an optimal path that maximizes the probability of extinction. We show that the optimal path is also directly related to the finite-time Lyapunov exponents of the underlying dynamical system in that the optimal path displays maximum sensitivity to initial conditions. We consider several stochastic epidemic models, and examine the extinction process in a dynamical systems framework. Using the dynamics of the finite-time Lyapunov exponents as a constructive tool, we demonstrate that the dynamical systems viewpoint of extinction evolves naturally toward the optimal path.

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We study the effect of migration between coupled populations, or patches, on the stability properties of multistrain disease dynamics. The epidemic model used in this work displays a Hopf bifurcation to oscillations in a single, well-mixed population. It is shown numerically that migration between two non-identical patches stabilizes the endemic steady state, delaying the onset of large amplitude outbreaks and reducing the total number of infections. This result is motivated by analyzing generic Hopf bifurcations with different frequencies and with diffusive coupling between them. Stabilization of the steady state is again seen, indicating that our observation in the full multistrain model is based on qualitative characteristics of the dynamics rather than on details of the disease model.

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We study vaccine control for disease spread on an adaptive network modeling disease avoidance behavior. Control is implemented by adding Poisson-distributed vaccination of susceptibles. We show that vaccine control is much more effective in adaptive networks than in static networks due to feedback interaction between the adaptive network rewiring and the vaccine application. When compared to extinction rates in static social networks, we find that the amount of vaccine resources required to sustain similar rates of extinction are as much as two orders of magnitude lower in adaptive networks.

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The idea behind adaptive behavioral epidemiology is that groups and individuals respond to the knowledge of a disease threat by changing their habits to avoid interactions with those who are contagious. Network-based models take this adaptive behavior into account by allowing the network to "rewire" its connections.

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This paper examines the interplay of the effect of cross immunity and antibody-dependent enhancement (ADE) in multistrain diseases. Motivated by dengue fever, we study a model for the spreading of epidemics in a population with multistrain interactions mediated by both partial temporary cross immunity and ADE. Although ADE models have previously been observed to cause chaotic outbreaks, we show analytically that weak cross immunity has a stabilizing effect on the system. That is, the onset of disease fluctuations requires a larger value of ADE with small cross immunity than without. However, strong cross immunity is shown numerically to cause oscillations and chaotic outbreaks even for low values of ADE.

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The effect of time delay on nonlinear oscillators is an important problem in the study of dynamical systems. The dynamics of an erbium-doped fiber ring laser with an extra loop providing time-delayed feedback is studied experimentally by measuring the intensity of the laser. The delay time for the feedback is varied from approximately 0.3 to approximately 900 times the cavity round-trip time, over four orders of magnitude, by changing the length of fiber in the delay line. Depending on the delay, we observe either regular oscillations or complex dynamics. The size of the fluctuations increases for delays long compared with the round-trip time of the laser cavity. The complexity of the fluctuations is quantified by creating spatiotemporal representations of the time series and performing a Karhunen-Loève decomposition. The complexity increases with increasing delay time. The experiment is extended by mutually coupling two fiber ring lasers together. The delay time for the mutual coupling is varied from approximately 0.2 to approximately 600 times the cavity round-trip time, over four orders of magnitude again. In this case the fluctuations are generally larger than the single laser case. The complexity of the dynamics for the mutually coupled system is less at short delays and larger at long delays when compared to the uncoupled case. The width of the optical spectra of the coupled lasers also narrows.