Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III

In this paper we study a generalized Gause model with prey harvesting and a generalized Holling response function of type III: . The goal of our study is to give the bifurcation diagram of the model. For this we need to study saddle-node bifurcations, Hopf bifurcation of codimension 1 and 2, heteroclinic bifurcation, and nilpotent saddle bifurcation of codimension 2 and 3. The nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram. The Hopf bifurcation is studied by means of a generalized Liénard system, and for b=0 we discuss the potential integrability of the system. The nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4. But because it occurs with an invariant line, the effective highest codimension is 3. We develop normal forms (in which the invariant line is preserved) for studying of the nilpotent saddle bifurcation. For b=0, the reversibility of the nilpotent saddle is discussed. We study the type of the heteroclinic loop and its cyclicity. The phase portraits of the bifurcations diagram (partially conjectured via the results obtained) allow us to give a biological interpretation of the behavior of the two species.     

Explicit solutions of generalized Cauchy-Riemann systems using the transplant operator

In Kravchenko (2008) [8] it was shown that the tool introduced there and called the transplant operator transforms solutions of one Vekua equation into solutions of another Vekua equation, related to the first via a Schrödinger equation. In this paper we prove a fundamental property of this operator: it preserves the order of zeros and poles of generalized analytic functions and transforms formal powers of the first Vekua equation into formal powers of the same order for the second Vekua equation. This property allows us to obtain positive formal powers and a generating sequence of a “complicated” Vekua equation from positive formal powers and a generating sequence of a “simpler” Vekua equation. Similar results are obtained regarding the construction of Cauchy kernels. Elliptic and hyperbolic pseudoanalytic function theories are considered and examples are given to illustrate the procedure.     

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behavior of the s-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain Ω with smooth boundary ∂Ω. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on ∂Ω. It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order p for which

     

New additive results for the generalized Drazin inverse

In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of PQ=QP, we give explicit representations of the generalized Drazin inverse d(P+Q) in term of P, Pd, Q and Qd. We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.     

Generalized Navier-Stokes equations with initial data in local Q-type spaces

In this paper, we establish a link between Leray mollified solutions of the three-dimensional generalized Navier-Stokes equations and mild solutions for initial data in the adherence of the test functions for the norm of . This result applies to the usual incompressible Navier-Stokes equations.     

Bi-Sobolev mappings with differential matrices in Orlicz Zygmund classes

Bi-Sobolev mappings have been defined as those homeomorphisms such that f and f−1 belong to . We deduce regularity properties of the distortion of f from the regularity of the differential matrix Df−1 and conversely.     

On commutators of vector BMO functions and multilinear singular integrals with non-smooth kernels

Let T be a bounded multilinear operator on some product of Lq(Rn) spaces. Assume that T has a non-smooth associated kernel which satisfies certain weak regularity conditions but not regular enough to fall under the scope of the standard multilinear Calderón-Zygmund theory. The main aim of this paper is to establish a sufficient condition on the kernel of T so that the commutator of a vector BMO function and T is bounded on certain product Lp(Rn) spaces. We obtain boundedness of the commutator of and T by first proving certain pointwise estimates on the Fefferman-Stein sharp maximal operator. An important example of multilinear operators which satisfy our kernel conditions is the maximal mth order Calderón commutator.     

Navier-Stokes equations and nonlinear heat equations in modulation spaces with negative derivative indices

The Cauchy problems for Navier-Stokes equations and nonlinear heat equations are studied in modulation spaces . Though the case of the derivative index s=0 has been treated in our previous work, the case s≠0 is also treated in this paper. Our aim is to reveal the conditions of s, q and σ of for the existence of local and global solutions for initial data .     

Weak homogeneity in generalized function algebras

In this paper, weakly homogeneous generalized functions in the special Colombeau algebras are determined up to equality in the sense of generalized distributions. This yields characterizations that are formally similar to distribution theory. Further, we give several characterizations of equality in the sense of generalized distributions in these algebras (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms

By using the generalized continuation theorem, the existence of four positive periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response