Fractional model for Middle East respiratory syndrome coronavirus on a complex heterogeneous network.

In this paper, we present a new fractional epidemiological model on a heterogeneous network to investigate Middle East respiratory syndrome (MERS-CoV), which is caused by a virus in the coronavirus family. We also consider the development of equations for the camel population, given that it is the primary animal source of the virus, as well as direct human interaction with this population. The model is configured in an SIS form for both the human population and the camel population. We study the equilibrium positions of the system and the conditions for the existence of each of them, as well as the local stability of each equilibrium position. Then, we provide some numerical examples that compare real data and numerical results.

PDE-guided reservoir computing for image denoising with small data.

While network-based techniques have shown outstanding performance in image denoising in the big data regime requiring massive datasets and expensive computation, mathematical understanding of their working principles is very limited. Not to mention, their relevance to traditional mathematical approaches has not attracted much attention. Therefore, we suggest how reservoir computing networks can be strengthened in combination with conventional partial differential equation (PDE) methods for image denoising, especially in the small data regime. Given image data, PDEs generate sequential datasets enhancing desired image features, which provide the network with a better guideline for training in reservoir computing. The proposed procedure, reservoir computing in collaboration with PDEs (RCPDE), offers a synergetic combination of data-driven network-based methods and mathematically well-established PDE methods. It turns out that RCPDE outperforms both the usual reservoir computing and existing PDE approaches in image denoising. Furthermore, RCPDE also excels deep neural networks such as a convolutional neural network both in quality and in time in the small data regime. We believe that RCPDE reveals the great potential of reservoir computing in collaboration with various mathematically justifiable dynamics for better performance as well as for better mathematical understanding.

Robust coexistence with alternative competition strategy in the spatial cyclic game of five species.

Alternative strategy is common in animal populations to promote reproductive fitness by obtaining resources. In spatial dynamics of cyclic competition, reproduction can occur when individuals obtain vacant rooms and, in this regard, empty sites should be resources for reproduction which can be induced by interspecific competition. In this paper, we study the role of alternative competition in the spatial system of cyclically competing five species by utilizing rock-paper-scissors-lizard-spock game. From Monte-Carlo simulations, we found that strong alternative competition can lead to the reemergence of coexistence of five species regardless of mobility, which is never reported in previous works under the symmetric competition structure. By investigating the coexistence probability, we also found that coexistence alternates by passing certain degrees of alternative competition in combination with mobility. In addition, we provided evidences in the opposite scenario by strengthening spontaneous competition, which exhibits the reemergence of coexistence similarly. Our findings may suggest more comprehensive perspectives to interpret mechanisms for biodiversity by alternative strategies in spatially extended systems than previously reported.

Multistability in the cyclic competition system.

Cyclically competition models have been successful to gain an insight of biodiversity mechanism in ecosystems. There are, however, still limitations to elucidate complex phenomena arising in real competition. In this paper, we report that a multistability occurs in a simple rock-paper-scissor cyclically competition model by assuming that intraspecific competition depends on the logistic growth of each species density. This complex stability is absent in any cyclically competition model, and we investigate how the proposed intraspecific competition affects biodiversity in the existing society of three species through macroscopic and microscopic approaches. When the system is multistable, we show basins of the asymptotically stable heteroclinic cycle and stable attractors to demonstrate how the survival state is determined by initial densities of three species. Also, we find that the multistability is associated with a subcritical Hopf bifurcation. This surprising finding will give an opportunity to interpret rich dynamical phenomena in ecosystems which may occur in cyclic competition systems with different types of interactions.

Emergence of unusual coexistence states in cyclic game systems.

Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature: biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and continuous spatiotemporal dynamical evolution from partial differential equations. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.