Related papers: Completely Integrable Replicator Dynamics Associat…
In this paper we construct a family of integrable reductions of the dressing chain, described in its Lotka-Volterra form. For each $k,n\in\mathbb N$ with $n\geqslant 2k+1$ we obtain a Lotka-Volterra system $\hbox{LV}_b(n,k)$ on $\mathbb…
In this paper, we consider the almost periodic dynamics of a multispecies Lotka-Volterra mutualism system with time varying delays on time scales. By establishing some dynamic inequalities on time scales, a permanence result for the model…
We consider a family of non-autonomous reaction-diffusion equations with almost periodic, rapidly oscillating principal part and nonlinear interactions. As the frequency of the oscillations tends to infinity, we prove that the solutions of…
While generic competitive systems exhibit mixtures of hierarchy and cycles, real-world systems are predominantly hierarchical. We demonstrate and extend a mechanism for hierarchy; systems with similar agents approach perfect hierarchy in…
In this letter, we deal with evolutionary game theoretic learning processes for population games on networks with dynamically evolving communities. Specifically, we propose a novel mathematical framework in which a deterministic,…
We consider an hierarchy of integrable 1+2-dimensional equations related to Lie algebra of the vector fields on the line. The solutions in quadratures are constructed depending on $n$ arbitrary functions of one argument. The most…
Global dynamical behaviors of the competitive Lotka-Volterra system even in 3-dimension are not fully understood. The Lyapunov function can provide us such knowledge once it is constructed. In this paper, we construct explicitly the…
We classify integrable third order equations in 2+1 dimensions which generalize the examples of Kadomtsev-Petviashvili, Veselov-Novikov and Harry Dym equations. Our approach is based on the observation that dispersionless limits of…
We consider a system of differential equations with nonlinear Steklov boundary conditions, related to the fractional problem $$(-\Delta)^s u_i = f_i(x,u_i) - \beta u_i^p \sum_{j\neq i} a_{ij} u_j^p,$$ where $i = i,\dots, k$, $s\in(0,1)$,…
The Lotka-Volterra predator-prey model still represents the paradigm for the description of the competition in population dynamics. Despite its extreme simplicity, it does not admit an analytical solution, and for this reason, numerical…
We present $\frac{m^{2}}{4}+\frac{m}{2}+\frac{1-\left(-1\right)^{m}}{8}$ homogeneous $(3m-2)$-parameter families of Liouville integrable $(2m)$- and $(2m-1)$-dimensional Lotka-Volterra systems. We also study inhomogeneous versions of these…
An equivalence between an integro-differential operator M and an evolution operator Ln is determined. From this equivalence the fundamental solution of Ln is estimated in terms of the fundamental solution related to the third-order operator…
We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes…
Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations…
We consider here the class of fully-nonlinear symmetry-integrable third-order evolution equations in 1+1 dimensions that were proposed recently in the journal Open Communications in Nonlinear Mathematical Physics, vol. 2, 216--228 (2022).…
Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie…
We construct a new family of infinite-dimensional quasi-graded Lie algebras on hyperelliptic curves. We show that constructed algebras possess infinite number of invariant functions and admit a decomposition into the direct sum of two…
We examine the algebraic complete integrability of Lotka-Volterra equations in three dimensions. We restrict our attention to Lotka-Volterra systems defined by a skew symmetric matrix. We obtain a complete classification of such systems.…
Based on the structure of Casimir elements associated with general Hopf algebras there are constructed Liouville-Arnold integrable flows related with naturally induced Poisson structures on arbitrary co-algebra and their deformations. Some…
The three equations named in the title are examples of infinite-dimensional completely integrable Hamiltonian systems, and are related to each other via simple geometric constructions. In this paper, these interrelationships are further…