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A sofic shift is a shift space consisting of bi-infinite labels of paths from a labelled graph. Being a dynamical system, the distribution of its closed orbits may indicate the complexity of the shift. For this purpose, prime orbit and Mertens’ orbit counting functions are introduced as a way to describe the growth of the closed orbits. The asymptotic behaviours of these counting functions can be implied from the analyticity of the Artin–Mazur zeta function of the shift. Its zeta function is expressed implicitly in terms of several signed subset matrices. In this paper, we will prove the asymptotic behaviours of the counting functions for sofic shifts via their zeta function. This involves investigating the properties of the said matrices. Suprisingly, the proof simply uses some well-known facts about sofic shifts, especially on the minimal right-resolving presentations. Furthermore, we will demonstrate this result by revisiting the case for periodic-finite-type shifts, which are a particular type of sofic shifts. At the end, we will briefly discuss the application of our finding towards the finite group and homogeneous extensions of a sofic shift. [ABSTRACT FROM AUTHOR]

Refugia have been perceived as a major role in structuring species biodiversity, and understanding the impacts of this force in a community assembly with prey–predator species is a difficult task because refuge process can interact with different ecological components and may show counterintuitive effects. To understand this problem, we used a simple two-species model incorporating a functional response inspired by a Holling type-II equation and a prey refuge mechanism that depends on prey and predator population densities (i.e., density-dependent prey refuge). We then perform the co-dimension one and co-dimension two bifurcation analysis to examine steady states and its stability, together with the bifurcation points as different parameters change. As the capacity of prey refuge is varied, there occur critical values i.e., saddle-node and supercritical Hopf bifurcations. The interaction between these two co-dimension one bifurcations engenders distinct outcomes of ecological system such as coexistence of species, bistability phenomena and oscillatory dynamics. Additionally, we construct a parameter space diagram illustrating the dynamics of species interactions as prey refuge intensity and predation pressure vary; as the two saddle-node move nearer to one another, these bifurcations annihilate tangentially in a co-dimension two cusp bifurcation. We also realised several contrasting observations of refuge process on species biodiversity: for instance, while it is believed that some refuge processes (e.g., constant proportion of prey refuge) would result in exclusion of predator species, our findings show that density-dependent prey refuge is beneficial for both predator and prey species, and consequently, promotes the maintenance of species biodiversity. [ABSTRACT FROM AUTHOR]

In this work, we initiate the notion of a fuzzy cyclic (α, β)-admissibility to establish some fixed point results for contraction mappings involving a generalized simulation function in the class of fuzzy b-metric spaces. We give some illustrative examples to validate the new concepts and obtained results. At the end, we present an application on a Fredholm integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical scheme of the predictor–corrector type for solving nonlinear fractional initial value problems; the chosen fractional derivative is called the Atangana–Baleanu derivative defined in Caputo sense (ABC). This proposed method is based on Lagrangian quadratic polynomials to approximate the nonlinearity implied in the Volterra integral, which is obtained by reducing the given fractional differential equation via the properties of the ABC‐fractional derivative. Through this technique, we get corrector formula with high accuracy, which is implicit, as well as predictor formula, which is explicit and has the same precision order as the corrective formula. On the other hand, the so‐called memory term is computed only once for both prediction and correction phases, which indicates the low cost of the proposed method. Also, the error bound of the proposed numerical scheme is offered. Furthermore, numerical experiments are presented in order to assess the accuracy of the new method on two differential equations. Moreover, a case study is considered where the proposed predictor–corrector scheme is used to obtain approximate solutions of a coupled nonautonomous ABC‐fractional differential system. [ABSTRACT FROM AUTHOR]

Multiple interactions (e.g., mutualist-resource-competitor-exploiter interactions) type models are known to exhibit oscillatory behaviour as a result of their complexity. This large-amplitude oscillation often de-stabilises multispecies communities and increases the chances of species extinction. What mechanisms help species in a complex ecological system to persist? Some studies show that dispersal can stabilise an ecological community and permit multi-species coexistence. However, previous empirical and theoretical studies often focused on one- or two-species systems, and in real life, we have more than two-species coexisting together in nature. Here, we employ a (four-species) multiple interactions type model to investigate how competition interacts with other biotic factors and dispersal to shape multi-species communities. Our results reveal that dispersal has (de-)stabilising effects on the formation of multi-species communities, and this phenomenon shapes coexistence mechanisms of interacting species. These contrasting effects of dispersal can best be illustrated through its combined influences with the competition. To do this, we employ numerical simulation and bifurcation analysis techniques to track the stable and unstable attractors of the system. Results show the presence of Hopf bifurcations, transcritical bifurcations, period-doubling bifurcations and limit point bifurcations of cycles as we vary the competitive strength in the system. Furthermore, our bifurcation analysis findings show that stable coexistence of multiple species is possible for some threshold values of ecologically-relevant parameters in this complex system. Overall, we discover that the stability and coexistence mechanisms of multiple species depend greatly on the interplay between competition, other biotic components and dispersal in multi-species ecological systems. [ABSTRACT FROM AUTHOR]