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We investigate a Leslie-type prey–predator system with an Allee effect to understand the dynamics of populations under stress. First, we determine stability conditions and conduct a Hopf bifurcation analysis using the Allee constant as a bifurcation parameter. At low densities, we observe that a weak Allee effect induces a supercritical Hopf bifurcation, while a strong effect leads to a subcritical one. Notably, a stability switch occurs, and the system exhibits multiple Hopf bifurcations as the Allee effect varies. Subsequently, we perform a sensitivity analysis to assess the robustness of the model to parameter variations. Additionally, together with the numerical examples, the FAST (Fourier amplitude sensitivity test) approach is employed to examine the sensitivity of the prey–predator system to all parameter values. This approach identifies the most influential factors among the input parameters on the output variable and evaluates the impact of single-parameter changes on the dynamics of the system. The combination of detailed bifurcation and sensitivity analysis bridges the gap between theoretical ecology and practical applications. Furthermore, the results underscore the importance of the Allee effect in maintaining the delicate balance between prey and predator populations and emphasize the necessity of considering complex ecological interactions to accurately model and understand these systems.
In this paper, we initiate the study of higher rank Baumslag–Solitar (BS) semigroups and their related C*-algebras. We focus on two rather interesting classes—one is related to products of odometers and the other is related to Furstenberg’s $\times p, \times q$ conjecture. For the former class, whose C*-algebras are studied in [32], we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting that encompasses (single-vertex) generalized BS semigroups. One of our main tools in this paper is from self-similar higher rank graphs and their C*-algebras.
For a connected Lie group G and an automorphism T of G, we consider the action of T on Sub$_G$, the compact space of closed subgroups of G endowed with the Chabauty topology. We study the action of T on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of G. We relate the distality of the T-action on Sub$^p_G$ with that of the T-action on G and characterise the same in terms of compactness of the closed subgroup generated by T in Aut$(G)$ when T acts distally on the maximal central torus and G is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$ and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group G acts distally on Sub$^p_G$ by conjugation if and only if G is either compact or is isomorphic to a direct product of a compact group and a vector group. Some of our results generalise those of Shah and Yadav.
Ergodic optimization aims to describe dynamically invariant probability measures that maximize the integral of a given function. For a wide class of intrinsically ergodic subshifts over a finite alphabet, we show that the space of continuous functions on the shift space contains two disjoint subsets: one is a dense $G_\delta $ set for which all maximizing measures have ‘relatively small’ entropy; the other is the set of functions having uncountably many, fully supported ergodic maximizing measures with ‘relatively large’ entropy. This result generalizes and unifies the results of Morris [Discrete Contin. Dyn. Syst.27 (2010), 383–388] and Shinoda [Nonlinearity31 (2018), 2192–2200] on symbolic dynamics, and applies to a wide class of intrinsically ergodic non-Markov symbolic dynamics without the Bowen specification property, including any transitive piecewise monotonic interval map, some coded shifts, and multidimensional $\beta $-transformations. Along with these examples of application, we provide an example of an intrinsically ergodic subshift with positive obstruction entropy to specification.
We prove the central limit theorem (CLT), the first-order Edgeworth expansion and a mixing local central limit theorem (MLCLT) for Birkhoff sums of a class of unbounded heavily oscillating observables over a family of full-branch piecewise $C^2$ expanding maps of the interval. As a corollary, we obtain the corresponding results for Boolean-type transformations on $\mathbb {R}$. The class of observables in the CLT and the MLCLT on $\mathbb {R}$ include the real part, the imaginary part and the absolute value of the Riemann zeta function. Thus obtained CLT and MLCLT for the Riemann zeta function are in the spirit of the results of Lifschitz & Weber [Sampling the Lindelöf hypothesis with the Cauchy random walk. Proc. Lond. Math. Soc. (3)98 (2009), 241–270] and Steuding [Sampling the Lindelöf hypothesis with an ergodic transformation. RIMS Kôkyûroku BessatsuB34 (2012), 361–381] who have proven the strong law of large numbers for sampling the Lindelöf hypothesis.
with the eikonal equation as a prototype. By the elementary row transformation of a matrix, we offer an affirmative answer to a question of Xu-Liu-Xuan in Xu et al. (J. Math. Anal. Appl.543 (2025), ID 128885, 21 p.).
We consider twist diffeomorphisms of the torus, $f:\mathrm {T^2\rightarrow T^2,}$ and their vertical rotation intervals, $\rho _V(\widehat {f})=[\rho _V^{-},\rho _V^{+}],$ where $\widehat {f}$ is a lift of f to the vertical annulus or cylinder. We show that $C^r$-generically, for any $r\geq 1$, both extremes of the rotation interval are rational and locally constant under $C^0$-perturbations of the map. Moreover, when f is area-preserving, $C^r$-generically, $\rho _V^{-}<\rho _V^{+}$. Also, for any twist map f, $\widehat {f}$ a lift of f to the cylinder, if $\rho _V^{-}<\rho _V^{+}=p/q$, then there are two possibilities: either $\widehat {f}^q(\bullet )-(0,p)$ maps a simple essential loop into the connected component of its complement which is below the loop, or it satisfies the curve intersection property. In the first case, $\rho _V^{+} \leq p/q$ in a $C^0$-neighborhood of $f,$ and in the second case, we show that $\rho _V^{+}(\widehat {f}+(0,t))>p/q$ for all $t>0$ (that is, the rotation interval is ready to grow). Finally, in the $C^r$-generic case, assuming that $\rho _V^{-}<\rho _V^{+}=p/q,$ we present some consequences of the existence of the free loop for $\widehat {f}^q(\bullet )-(0,p)$, related to the description and shape of the attractor–repeller pair that exists in the annulus. The case of a $C^r$-generic transitive twist diffeomorphism (if such a thing exists) is also investigated.
We study the geometry induced on the local orbit spaces of Killing vector fields on (Riemannian) $G$-manifolds, with an emphasis on the cases $G=\textrm {Spin}(7)$ and $G=G_2$. Along the way, we classify the harmonic morphisms with one-dimensional fibres from $G_2$-manifolds to Einstein manifolds.
We consider local escape rates and hitting time statistics for unimodal interval maps of Misiurewicz–Thurston type. We prove that for any point z in the interval, there is a local escape rate and hitting time statistics that is one of three types. While it is key that we cover all points z, the particular interest here is when z is periodic and in the postcritical orbit that yields the third part of the trichotomy. We also prove generalized asymptotic escape rates of the form first shown by Bruin, Demers and Todd.
In 1954, B. H. Neumann discovered that if $G$ is a group in which all conjugacy classes have finite cardinality at most $m$, then the derived group $G'$ is finite of $m$-bounded order. In 2018, G. Dierings and P. Shumyatsky showed that if $|x^G| \le m$ for any commutator $x\in G$, then the second derived group $G''$ is finite and has $m$-bounded order. This paper deals with finite groups in which $|x^G|\le m$ whenever $x\in G$ is a commutator of prime power order. The main result is that $G''$ has $m$-bounded order.
We establish a one-to-one correspondence between Kähler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non-linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\mathcal{T})$, where ${\mathcal{T}}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms.
Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a Kähler metric. In particular, we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing–Yano tensor, which is a necessary condition for a metric to be conformal to Kähler. This gives an invariant characterization of algebraically special Riemannian metrics of type D in dimensions higher than four.
The results on Γ-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the Γ-limit, which is then used to study deterministic and stochastic homogenization problems for this type of functional.
Using bi-contact geometry, we define a new type of Dehn surgery on an Anosov flow with orientable weak invariant foliations. The Anosovity of the new flow is strictly connected to contact geometry and the Reeb dynamics of the defining bi-contact structure. This approach gives new insights into the properties of the flows produced by Goodman surgery and clarifies under which conditions Goodman’s construction yields an Anosov flow. Our main application gives a necessary and sufficient condition to generate a contact Anosov flow by Foulon–Hasselblatt Legendrian surgery on a geodesic flow. In particular, we show that this is possible if and only if the surgery is performed along a simple closed geodesic. As a corollary, we have that any positive skewed $\mathbb {R}$-covered Anosov flow obtained by a single surgery on a closed orbit of a geodesic flow is orbit equivalent to a positive contact Anosov flow.
Seasonal changes and cyclical human activities (such as periodic fishing bans, Wolbachia-based mosquito population control, and school term breaks) have significant impacts on population dynamics. We propose a general switching dynamical model to describe these periodic changes. The existence, uniqueness and stability of positive periodic solutions are thoroughly investigated. The results are stated in terms of an introduced threshold value. To demonstrate their practicability, the obtained results are applied to two biological situations.
Our goal is to show that both the fast and slow versions of the triangle map (a type of multi-dimensional continued fraction algorithm) in dimension n are ergodic, resolving a conjecture of Messaoudi, Noguiera, and Schweiger [Ergodic properties of triangle partitions. Monatsh. Math.157 (2009), 283–299]. This particular type of higher dimensional multi-dimensional continued fraction algorithm has recently been linked to the study of partition numbers, with the result that the underlying dynamics has combinatorial implications.
and discuss generalized weighted Hardy-type inequalities associated with the measure $d\mu=e^{v(x)}dx$. As an application, we obtain several Liouville-type results for positive solutions of the non-linear elliptic problem with singular lower order term
where Ω is a bounded or an unbounded exterior domain in ${\mathbb{R}}^N$, $N \gt p \gt 1$, $B+p-1 \gt 0$, as well as of the non-autonomous quasilinear elliptic problem
We solve generalizations of Hubbard’s twisted rabbit problem for analogs of the rabbit polynomial of degree $d\geq 2$. The twisted rabbit problem asks: when a certain quadratic polynomial, called the Douady rabbit polynomial, is twisted by a cyclic subgroup of a mapping class group, to which polynomial is the resulting map equivalent (as a function of the power of the generator)? The solution to the original quadratic twisted rabbit problem, given by Bartholdi and Nekrashevych, depended on the 4-adic expansion of the power of the mapping class by which we twist. In this paper, we provide a solution to a degree-d generalization that depends on the $d^2$-adic expansion of the power of the mapping class element by which we twist.
We extend a result of Lopes and Thieullen [Sub-actions for Anosov flows. Ergod. Th. & Dynam. Sys.25(2) (2005), 605–628] on sub-actions for smooth Anosov flows to the setting of geodesic flow on locally CAT($-1$) spaces. This allows us to use arguments originally due to Croke and Dairbekov to prove a volume rigidity theorem for some interesting locally CAT($-1$) spaces, including quotients of Fuchsian buildings and surface amalgams.