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In a previous paper (Feng et al., J. Differential Equations (2023)), we studied a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting, in which we assumed that the harvesting quantity was relatively large (0 < κ = l G c E < 1) and obtained a length threshold of the closed season T ¯ ⁎ depending on the harvesting parameter and the seasonal fluctuation period. It was shown that the origin is globally asymptotically stable if and only if T ¯ ≤ T ¯ ⁎ , and there exists a unique globally asymptotically stable T -periodic solution if and only if T ¯ > T ¯ ⁎. In this paper, we continue to investigate the periodic dynamics of this model when the harvesting quantity is relatively small (κ = l G c E ≥ 1). By finding another smaller length threshold T ¯ ⁎ ∈ (0 , T ¯ ⁎) , we determine the number of periodic solutions and study their stability, which imply the occurrence of bifurcation of periodic solutions. Our results demonstrate that designing the closed harvesting season properly can prevent the species from extinction. Moreover, by comparing with the continuous harvesting model and combining the results in our previous paper and this article, we provide a complete understanding on the global dynamics for the periodic switching model. Some numerical simulations are also carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue for the Camassa-Holm equation y ″ = 1 4 y + λ m (t) y , with the Neumann-Dirichlet boundary conditions, where potential m admits to change sign. We first study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations. Then based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for this indefinite Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u ″ (t) + A u (t) + b (t) u ′ (t) = 0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b (t) ∼ (1 + t) − β (− 1 < β < 1). In the case b (t) ≡ 1 , it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v ′ (t) + A v (t) = F n (t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b (t) v ′ (t) + A v (t) = 0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n -th order profile u ˜ n is given via v ′ (t) + m n (t) A v (t) = F n (t) which coefficient m n (time-scale) differs each other. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h , it has been shown that for any M > 1 there exists s M ∈ R such that the equation x ″ + f (x) x ′ + h (t , x) = s has two positive T -periodic solutions u 1 (⋅ ; s) and u 2 (⋅ ; s) satisfying min ⁡ { u 1 (t ; s) : t ∈ [ 0 , T ] } > M and min ⁡ { u 2 (t ; s) : t ∈ [ 0 , T ] } < 1 / M for every s < s M. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T -periodic solutions of indefinite differential equations. • Establish a refinement result of the Ambrosetti-Prodi type in the absence of anti-coercivity condition. • Investigate the dynamic behaviors of positive periodic solutions to a generalized Liénard equation. • Provide a new approach to study the existence of positive periodic solutions for indefinite differential equations. • Find a relationship between the strict lower functions and Leray-Schauder degree. [ABSTRACT FROM AUTHOR]

The main aim of this article is to establish an effective criterion for a positive parametric topological entropy to differential equations with multivalued nonautonomous impulses on tori. The crucial tool for this goal consists in developing an appropriate Ivanov-like inequality for the lower estimate of a new kind of entropy by means of the logarithm of asymptotic Nielsen numbers for the compositions of a one-parameter family of multivalued admissible maps. This inequality is then applied to impulsive differential equations on tori via the associated Poincaré translation operators along their trajectories. The obtained results generalize in a significant way those in our recent papers, especially in the one with the same title (whence the indication by II), into a multivalued setting. Another nontrivial generalization can be regarded with respect to their simple reduction into a "nonparametric" case. [ABSTRACT FROM AUTHOR]

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in L ∞. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in L ∞. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

This note is to fix some small gaps in the proofs of some of the results in our paper [R.B. Salako and W. Shen, Journal of Differential Equations 262 (2017) 5635–5690]. [ABSTRACT FROM AUTHOR]

This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path ω with Hurst index H ∈ (1 3 , 1 2 ]. We first construct the approximation ω δ of ω by probabilistic arguments, and then use the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and the approximative system generate a continuous random dynamical systems φ and φ δ. As a consequence of the Wong-Zakai approximation of the solution, φ δ converges to φ as δ → 0. [ABSTRACT FROM AUTHOR]

We deal with the following fractional Choquard equation (− Δ) s u + V (x) u = (I μ ⁎ | u | 2 μ , s ⁎ ) | u | 2 μ , s ⁎ − 2 u , x ∈ R N , where I μ (x) is the Riesz potential, s ∈ (0 , 1) , 2 s < N ≠ 4 s , 0 < μ < min ⁡ { N , 4 s } and 2 μ , s ⁎ = 2 N − μ N − 2 s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V (x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V (x) vanishes at infinity. [ABSTRACT FROM AUTHOR]

Exact solutions of the SEIR epidemic model are derived, and various properties of solutions are obtained directly from the exact solution. In this paper Abel differential equations play an important role in establishing the exact solution of SEIR differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an Abel differential equation. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

In this paper the half-linear differential equation with one-dimensional p -Laplacian (p = α + 1), (| u ′ | α sgn u ′) ′ = α (λ α + 1 + b (t)) | u | α sgn u , t ≥ t 0 , is considered, where α and λ are positive constants. It is proved that if the function b (t) is absolutely integrable on [ t 0 , ∞) , then the above equation has two nonoscillatory solutions u + (t) and u − (t) such that u ± (t) ∼ c e ± λ t and u ± ′ (t) ∼ ± c λ e ± λ t (t → ∞), and moreover, any nontrivial solution u (t) of the above equation satisfies either u (t) ∼ c e λ t , u ′ (t) ∼ c λ e λ t (t → ∞) or u (t) ∼ c e − λ t , u ′ (t) ∼ − c λ e − λ t (t → ∞). Here, c is a nonzero constant. A weaker result is also shown under the weaker condition that b (t) is conditionally integrable on [ t 0 , ∞). [ABSTRACT FROM AUTHOR]

• The normal form formula of double-Hopf bifurcation is generated for general differential equations with the nonlocality. • The formula is applicable to functional differential equations and partial differential equations with or without the nonlocality. • The algorithm is favourable to reveal complex dynamics, e.g. periodic, quasi-periodic and multi-periodic oscillations. • Stable spatially nonhomogeneous periodic and quasi-periodic solutions are found in a nonlocal predator-prey model. This paper is concerned with the calculation of the normal form on the center manifold up to the third-order term at the double-Hopf singularity for general partial functional differential equations with the nonlocal effect. We derive explicit formulas of the normal form, that can be applied for both functional differential equations and partial differential equations with or without nonlocal effects in a bounded spatial domain. This provides an effective tool to establish existences of multi-periodic and quasi-periodic oscillations of a double-Hopf singularity for such equations. As an example, the Holling-Tanner predator-prey model with the nonlocal intraspecific competition of prey is considered. It turns out that double-Hopf bifurcation will occur because of the nonlocal interaction. Many spatio-temporal dynamics are found, including stable spatially homogeneous or nonhomogeneous periodic solutions, and stable spatially nonhomogeneous quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain results on topological properties of flows for nonnegative time in the framework of generalized ODEs. We define the concept of generalized semiflow and we present some recursive properties as minimality and recurrence. Using correspondence theorems, we translate our results to measure differential equations which are known to be special cases of generalized ODEs. Some examples are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the averaging principle for a fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index H (1 / 3 < H ≤ 1 / 2). Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the L 1 -sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized ODEs. Secondly, we use the well known correspondence between solutions of generalized ODEs and solutions of measure differential equations, obtaining analogues results for the last equations. Finally, we apply some of these results for dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

This paper considers compound operators on a general vector space. Explicit representations for these operators are obtained in various contexts with emphasis on the compound differential equations corresponding to a given linear differential equation. Applications to estimation of eigenvalues of self-adjoint boundary value problems and of the codimension of the stable manifold of a linear differential equation in a Banach space are given. [ABSTRACT FROM AUTHOR]

In this paper we aim to find the highest number of critical periods in a class of planar systems of polynomial differential equations for fixed degree having a center. We fix our attention to lower bounds of local criticality for low degree planar polynomial centers. The main technique is the study of perturbations of reversible holomorphic (isochronous) centers, inside the reversible centers class. More concretely, we study the Taylor developments of the period constants with respect to the perturbation parameters. First, we see that there are systems of degree 3 ≤ n ≤ 16 for which up to first order at least (n 2 + n − 4) / 2 critical periods bifurcate from the center. Second, we improve this number for centers with degree from 3 to 9. In particular, we obtain 6 and 10 critical periods for cubic and quartic degree systems, respectively. [ABSTRACT FROM AUTHOR]