Aphids, Ants and Ladybirds: a mathematical model predicting their population dynamics

The interaction between aphids, ants and ladybirds has been investigated from an ecological point of view since many decades, while there are no attempts to describe it from a mathematical point of view. This paper introduces a new mathematical model to describe the within-season population dynamics in an ecological patch of a system composed by aphids, ants and ladybirds, through a set of four differential equations. The proposed model is based on the Kindlmann and Dixon set of differential equations, focused on the prediction of the aphids-ladybirds population densities, that share a prey-predator relationship. The population of ants, in mutualistic relationship with aphids and in interspecific competition with ladybirds, is described according to the Holland and De Angelis mathematical model, in which the authors faced the problem of mutualistic interactions in general terms. The set of differential equations proposed here is discretized by means the Nonstandard Finite Difference scheme, successfully applied by Gabbriellini to the mutualistic model. The constructed finite-difference scheme is positivity-preserving and characterized by four nonhyperbolic steady-states, as highlighted by the phase-space and time-series analyses. Particular attention is dedicated to the steady-state most interesting from an ecological point of view, whose asymptotic stability is demonstrated via the Centre Manifold Theory. The model allows to numerically confirm that mutualistic relationship effectively influences the population dynamic, by increasing the peaks of the aphids and ants population densities. Nonetheless, it is showed that the asymptotical populations of aphids and ladybirds collapse for any initial condition, unlike that of ants that, after the peak, settle on a constant asymptotic value

The solution of some differential equations by nonstandard finite difference method

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2005Includes bibliographical references (leaves: 55-57)Text in English; Abstract: Turkish and Englishix, 66 leavesIn this thesis, the nonstandard finite difference method is applied to construct thenew finite difference equations for the first order nonlinear dynamic equation, second order singularly perturbed convection diffusion equation and nonlinear reaction diffusion partial differential equation The new discrete representation for the first order nonlinear dynamic equation allows us to obtain stable solutions for all step-sizes.For singularly perturbed convection diffusion equation, the error analysis reveals that the nonstandard finite difference representation gives the better results for any values of the perturbation parameters. Finally, the new discretization for the last equation is obtained.The lemma related to the positivity and boundedness conditions required for the nonstandard finite difference scheme is stated. Numerical simulations for all differential equarions are illustrated based on the parameters we considered

A nonstandard Euler-Maruyama scheme

We construct a nonstandard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behavior compared to the Euler-Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.Comment: Accepted in "Journal of Difference Equations and Applications", to appear, 201

Gutzwiller Wave-Function Solution for Anderson Lattice Model: Emerging Universal Regimes of Heavy Quasiparticle States

The recently proposed diagrammatic expansion (DE) technique for the full Gutzwiller wave function (GWF) is applied to the Anderson lattice model (ALM). This approach allows for a systematic evaluation of the expectation values with GWF in the finite dimensional systems. It introduces results extending in an essential manner those obtained by means of standard Gutzwiller Approximation (GA) scheme which is variationally exact only in infinite dimensions. Within the DE-GWF approach we discuss principal paramagnetic properties of ALM and their relevance to heavy fermion systems. We demonstrate the formation of an effective, narrow $f$-band originating from atomic $f$-electron states and subsequently interpret this behavior as a mutual intersite $f$-electron coherence; a combined effect of both the hybridization and the Coulomb repulsion. Such feature is absent on the level of GA which is equivalent to the zeroth order of our expansion. Formation of the hybridization- and electron-concentration-dependent narrow effective $f$-band rationalizes common assumption of such dispersion of $f$ levels in the phenomenological modeling of the band structure of CeCoIn$_5$. Moreover, we show that the emerging $f$-electron coherence leads in a natural manner to three physically distinct regimes within a single model, that are frequently discussed for 4$f$- or 5$f$- electron compounds as separate model situations. We identify these regimes as: (i) mixed-valence regime, (ii) Kondo-insulator border regime, and (iii) Kondo-lattice limit when the $f$-electron occupancy is very close to the $f$ electrons half-filling, $\langle\hat n_{f}\rangle\rightarrow1$. The non-Landau features of emerging correlated quantum liquid state are stressed.Comment: Submitted to Phys. Rev.

Nonstandard finite difference schemes with application to biological models

This paper deals with the construction of nonstandard finite difference methods for solving a specific Rosenzweig-MacArthur predator-prey model. The reorganization of the denominator of the discrete derivatives and nonlocal approximations of nonlinear terms are used in the design of new schemes. We establish that the proposed nonstandard finite difference methods are elementary stable and satisfy the positivity requirement. We provide some numerical comparisons to illustrate our results.Publisher's Versio

Numerical analysis of a nonlocal parabolic problem resulting from thermistor problem

We analyze the spatially semidiscrete piecewise linear finite element method for a nonlocal parabolic equation resulting from thermistor problem. Our approach is based on the properties of the elliptic projection defined by the bilinear form associated with the variational formulation of the finite element method. We assume minimal regularity of the exact solution that yields optimal order error estimate. The full discrete backward Euler method and the Crank-Nicolson-Galerkin scheme are also considered. Finally, a simple algorithm for solving the fully discrete problem is proposed