Abstract In this paper, we develop a methodology for constructively proving the existence of saddle-node bifurcations on unbounded domains. In particular, we combine several previous results in order to formulate a Newton-Kantorovich argument to obtain our results. We demonstrate our approach on a class of 1D activator-inhibitor models. The first ingredient is for us to prove the existence of stationary localized 1D solutions. Our approach relies on having an approximate, compactly supported solution, $\overline{\mathbf{u}}$. We construct this solution using Fourier series and then define a well-chosen fixed point map that is contracting on a neighborhood around $\overline{\mathbf{u}}$. For this matter, we construct an approximate inverse of the linearization around $\overline{\mathbf{u}}$ and establish sufficient conditions under which the contraction is achieved. The second ingredient is to derive an augmented zero finding problem whose solutions correspond to a saddle-node bifurcations. Extending the methodology for proving the existence of localized solutions to this augmented problem, we are able to constructively obtain the existence of saddle-nodes on unbounded domains. The third ingredient is for us to verify that we obtain a non-degenerate saddle-node bifurcation. To do so, we control the spectrum of the linearization around the solution to the augmented zero finding problem. This allows us to rule out a degenerate Hopf bifurcation, as well as controlling exactly the number of unstable directions at the bifurcation. Quantitative explicit estimates as well as the requisite codes for the rigorous numerics are provided.
Abstract In this paper, a two-dimensional Leslie-Gower stochastic predator-prey system with jumps is investigated. Path-wise estimation, stochastic ultimate boundedness, existence and uniqueness of the global positive solution, and other properties of the solution are analyzed. Next, the conditions required for the system in question to become extinct are given. Numerical simulations are elaborated based on the Euler scheme and the Fokker-Planck approach. Systems’ parameters analysis is investigated and the obtained results demonstrate that population systems attributes can be altered via Lévy noise.
In this paper, we investigate a parabolic-elliptic predator-prey chemotaxis-fluid system with indirect pursuit-evasion interaction and prescribed signal concentration on the boundary: {n(t )+ u & centerdot; del(n) = Delta n - chi V & centerdot; (nVv) + n(a(1 )- b(1)n + r(1)c), x is an element of Omega, t > 0, c(t )+ u & centerdot; del c = Delta c + xi V & centerdot; (cV w) + c(a(2 )- b(2)c - r(2)n), x is an element of Omega, t > 0, u & centerdot; del v = Delta v - v + c, x is an element of Omega, t > 0, (*) u & centerdot; del w = Delta w - w + n, x is an element of Omega, t > 0, u(t) = Delta u + del P + (n + c)V phi, del & centerdot; u = 0, x is an element of Omega, t > 0, partial derivative n/partial derivative nu = partial derivative c/partial derivative nu = 0, v = v(*)(x, t), w = w(*)(x,t), u = 0, x is an element of Omega, t > 0, where Omega subset of R-3 is a smoothly bounded domains, chi > 0, xi > 0, kappa is an element of R, a(i) is an element of R, b(i )> 0, r(i) > 0 (i = 1, 2). Additionally, we impose a non-negative prescribed concentration (v(*), w(*)) is an element of (C-2,C-1 (Omega(sic) & times; [0, infinity)))(2) for the signal chemical on the boundary. It is shown that system (*) admits a globally uniformly bounded classical solution for any suitably regular initial data. Furthermore, by constructing an appropriate Lyapunov functional, we show that every solution converges to a spatially homogeneous equilibrium as time tends to infinity.
Artificial forces are matched to solutions in the method of manufactured solutions. Yet, dominant source terms might then mask barely tested nonlinear terms in verifications. Furthermore, many geophysical models do frequently not provide for manufactured forces. A known alternative approach is to find domain shapes that render fluid dynamic systems tractable. Manufactured geometry has, though, not yet been developed into a generalized system as has been the method of manufactured solutions. Manufacturing geometry has for this work been systematically generalized and formulated. That is, tractability for arbitrary hypersurface shapes, via the addition of spatial dimensions, is enabled without having to deduce shapes a priori. As a result, manufactured bathymetries, or topographies render momentum transport equations tractable without any additional forces. Verifications of numerical solvers with analytical solutions, thus, become compatible with existing geophysical models. Standing wave solutions provide stable bathymetries, whereas dynamic solutions yield dynamic bathymetries. The method is demonstrated for the incompressible Euler equation, the Euler equation without advection, the viscous shallow water equation (SWE)-configuration of the Navier-Stokes equations, and the SWEs without advection. If the formulation of the resulting domain shape is not compact, tractability is retained by numerically integrating the domain geometry. A use case to verify numerical schemes is demonstrated, comparing the numerical algorithm of a new ocean model with analytical solutions. In principle, the method is applicable to arbitrary unstructured meshes by introducing the spatial dimension $n+1$.
The present study deals with the computation of the electromagnetic vector fields, being scattered by a penetrable spherical cavity that is situated beneath the Earth's surface and needs to be identified in real-life applications. Aiming to solve the important direct problem, the excitation source is modelled as a time-harmonic magnetic dipole, randomly oriented in three-dimensional space and positioned on the Earth's surface, operating in the low-frequency regime. In the context of low-frequency analysis, the incident, external and internal fields are expanded in terms of the low angular frequency, which is the same for each medium, namely, the conductive Earth and the lossless air cavity. The static zeroth-order Rayleigh term, along with the first three dynamic contributions, yield an accurate approximation of the solution. Higher-order terms are considered as insignificant within the low-frequency regime and are consequently omitted. Accordingly, Maxwell's equations are reformulated into a finite set of interdependent elliptic partial differential equations, constrained by transmission conditions on the cavity interface and by asymptotic decay conditions as the spatial variable tends towards infinity, that is, to regions well beyond the domain of observation. From this point onward, the 3D scattering boundary value problems are solved incrementally, with the determination of the unknown constant coefficients, leading either to explicit expressions or to infinite systems of linear algebraic equations. These can be solved by employing standard cut-off techniques. Subsequently, we proceed to the visualization of the resulting fields, providing their graphical representation, which reveals the expected physical behaviour and provides an efficient analytical tool for an inversion scheme.
