Qualitative Theory of Dynamical Systems最新文献

Ovadia and Rodriguez-Hertz (Neutralized local entropy and dimension bounds for invariant measures. arXiv:2302.10874v2) defined the neutralized Bowen open ball as

Yang et al. (Variational principle for neutralized Bowen topological entropy, arXiv:2303.01738v1) introduced the notion of neutralized Bowen topological entropy of subsets by replacing the usual Bowen ball by neutralized Bowen open ball. And they established variational principles for this notion. In this note, we extend this notion to the non-additive neutralized Bowen topological pressure and establish the variational principle for non-additive potentials with tempered distortion. Besides, we establish a Billingsley type theorem for non-additive neutralized Bowen topological pressure.

We study dynamics of a fast–slow Leslie–Gower predator–prey system with Allee effect and Holling Type II functional response. More specifically, we show some sufficient conditions to guarantee the existence of two positive equilibria of the system and their location, and then we further fully determine their dynamics. Based on geometric singular perturbation theory and the slow–fast normal form, we determine the associated bifurcation curve and observe canard explosion. Besides, we also find a homoclinic orbit to a saddle with slow and fast segments, in which, the stable and unstable manifolds of the saddle are connected under explicit parameters conditions.

In order to reveal the transmission dynamics of Avian influenza and explore effective control measures, we develop a non-local delayed reaction-diffusion model of Avian influenza with vaccination and multiple transmission routes in the heterogeneous spatial environment, taking into account the incubation period of Avian influenza in humans and poultry. Firstly, the well-posedness of model is obtained which includes the existence, uniform boundedness and the existence of global attractor. Further, the basic reproduction number ({mathcal {R}}_0 ) of this model is calculated by the definition of the spectral radius of the next generation operator, and its variational form is also derived. Further, the global dynamics of the model is established based on the biological significance of ({mathcal {R}}_0 ). To be more precise, if ({mathcal {R}}_0<1), the disease-free steady state is globally asymptotically stable (i.e., the disease is extinct), while if ({mathcal {R}}_0>1), the disease is uniformly persistent and model admits at least one endemic steady state. In addition, by constructing suitable Lyapunov functionals, we achieve the global asymptotic stability of the disease-free and endemic steady states of this model in spatially homogeneous. Finally, some numerical simulations illustrate the main theoretical results, and discuss the sensitivity of ({mathcal {R}}_0 ) on the model parameters and the influences of non-local delayed and diffusion rates on the transmission of Avian influenza. The theoretical results and numerical simulations show that prolonging the incubation period, controlling the movement of infected poultry, and regular disinfecting the environment are all effective ways to prevent Avian influenza outbreaks.

In this paper, we prove some existence theorems for elliptic boundary value problems within the p(x)-Laplacian on a variable Sobolev space. For this purpose, the main problem is transformed into a fixed point problem and then fixed point arguments such as Schaefer’s and Schauder’s theorems are used. Our approach involves fewer stringent assumptions on the nonlinearity function than the prior findings. An interesting example is presented to examine the validity of the theoretical findings.

Embarking on a captivating journey of discovery, our research centers around unraveling the fascinating interaction between fish and plankton. Employing a predator–prey model, we delve into the dynamic interplay of Daphnia and Algae, while scrutinizing the profound influence of fish as a top predator. Our analyses suggest that in most situations, the plankton should show transitions in response to predation pressure by fish. Thus, there exist two distinct stable states, one in which Daphnia is controlled by fish and phytoplankton biomass is high and another in which Daphnia is relatively unaffected by planktivores and algae are controlled by Daphnia. Switches from one regime to the other occur abruptly at a critical fish density. Beyond deterministic exploration, we delve into the influence of the Allee parameter, which significantly increases the number of Daphnia in a free state. To investigate the complete global dynamics of the deterministic model, we present a two-parametric bifurcation diagram. We have also analyze all possible local and global bifurcations that the system could go through. We explore the corresponding stochastic system and investigate the critical transitions with the presence of environmental noise. To understand the probabilistic mechanism behind noise-induced outbreaks, we utilize advanced techniques such as stochastic sensitivity functions and the confidence domain method. These tools grant us unique insights into the captivating world of noise-induced transitions—where stochastic trajectories skillfully navigate from one stable equilibrium point to another.

In this paper we prove that Picard iterations of BSDEs with globally Lipschitz continuous nonlinearities converge exponentially fast to the solution. Our main result in this paper is to establish a fundamental lemma to prove the global existence and uniqueness of an adapted solution to a singular backward stochastic nonlinear Volterra integral equation (for short, singular BSVIE) of order (alpha in (frac{1}{2},1)) under a weaker condition than Lipschitz one in Hilbert space.

This paper investigates the qualitative structures of a discrete ratio-dependent predator–prey model near a degenerate fixed point whose eigenvalues are (pm 1). By the normal form theory, Picard iteration and Takens’s theorem, this model is transformed into an ordinary differential system. Then the qualitative structures of this differential system near the highly degenerate equilibrium are analyzed with the blowing-up method, which yields the ones of the discrete model near the fixed point by the conjugacy between the discrete model and the time-one mapping of the vector field.

This article is about the examination of existence as well as uniqueness of solutions, Bielecki–Hyers–Ulam stability and Bielecki–Hyers–Ulam–Rassias stability of non–linear impulsive fractional Hammerstein integro–delay dynamic system and non–linear impulsive fractional mixed integro–dynamic system, in the context of time scales domain. The Banach contraction principle and Picard operator are the main tools that are applied to verify the existence along with uniqueness of solutions for both models. Also, Bielecki–Ulam’s type stability is obtained by utilizing Grönwall’s inequality on time scale. To overcome the hurdles in achieving desired outcomes, some assumptions are provided. At the end, the results are demonstrated with the help of examples.

Influenza remains one of the most widespread epidemics, characterized by serious pathogenicity and high lethality, posing a significant threat to public health. This paper focuses on an influenza A infection model that includes vaccination and asymptomatic patients. The deterministic model examines the existence and local asymptotic stability of equilibria. In light of the influence of environmental disruption on the spread of disease, we develop a stochastic model in which the transmission rate follows a log-normal Ornstein–Uhlenbeck process. To demonstrate the dynamic behavior of the stochastic model, we verify the existence and uniqueness of the global positive solution. The establishment of suitable Lyapunov functions allows for the determination of sufficient conditions for the stationary distribution and extinction of the disease. Furthermore, the expression of the local density function around the quasi-endemic equilibrium is represented. Eventually, numerical simulations are conducted to support theoretical results and explore the effect of environmental noise. Our findings indicate that high noise intensity can expedite the extinction of the disease, while low noise intensity can facilitate the disease reaching a stationary distribution. This information may be valuable in developing strategies for disease prevention and control.

This paper is the first to propose an allelopathic phytoplankton competition ODE model influenced by the fear effect based on natural biological phenomena. It is shown that the interplay of this fear effect and the allelopathic term cause rich dynamics in the proposed competition model, such as global stability, transcritical bifurcation, pitchfork bifurcation, and saddle-node bifurcation. We also consider the spatially explicit version of the model and prove analogous results. Numerical simulations verify the feasibility of the theoretical analysis. The results demonstrate that the primary cause of the extinction of non-toxic species is the fear of toxic species compared to toxins. Allelopathy only affects the density of non-toxic species. The discussion guides the conservation of species and the maintenance of biodiversity.