Vol. 8, No. 2 (2019): Regular Issue

This paper considers a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and nonlinear prey harvesting strategy subject to the zero-flux boundary conditions. To understand the dynamics of the considered system, we derive sufficient conditions for permanence analysis, local stability, global stability and Hopf bifurcation of interior equilibrium point. Further we also investigate the existence and non-existence of non-constant positive steady state solutions.

This paper considers a prey predator model with omnivore population and Holling type II response. First, we have studied the boundedness of the model system. The local and global stability of the equilibrium is investigated by analyzing the eigenvalues and constructing the appropriate Lyapunov functions respectively. The persistence of positive equilibrium is also discussed. The existence of Hopf bifurcation is investigated by analyzing the distribution of eigenvalues at the positive equilibrium point. By using the normal form theory and explicit formulae which determine the direction of bifurcating periodic solutions are derived. Some numerical simulations are carrying out, to check our theoretical results.

In this paper, we study the dynamical behaviours of a very special type of delay differential equation known as Lotka Volterra model (predator-prey model) via Katugampola fractional derivative. The existence and uniqueness results are obtained using the Krasnoselskii’s fixed point theorem. Moreover, we derived fractional Euler’s method for delay differential equations. Illustrative examples are solved to show the applicability of theory results.

Area of research related to prey-predator systems is an important topic. The concept of permanence is an important issues related to biological systems. In general permanence is considered as a combination of persistence and boundedness. Following this, this paper reviews few existing definitions of stochastic permanence. Following the existing definition, in this paper a new definition of permanence for stochastic biological systems has been proposed, which modifies the existing ones. The permanence of a general Lotka-Volterra stochastic two species cooperation, competition and predation systems is proved by new definition.

We present analytical solutions of the two-basin model of open-ocean deep convection. Originally suggested by Whitehead [Whitehead, J.A. (2000), Stratified convection with multiple states, Ocean Modelling, 2(3-4), 109-121], the model allows regimes with multiple steady states (multiple equilibria). We provide the full analytical description of the steady states for the particular case of constant surface heat flux from the ocean to the atmosphere, and explore analytically stability of the equilibria within the Lyapunov theory. The results show that, for this particular case, the steady state is unique and stable for all dynamic flow regimes. We also present analytical expressions for dependence of critical values of sea-surface heat flux, at which transitions between the dynamic regimes occur, on the model parameters.

The intention of this article is to exhibit the validity and applicability of a fractional sub-strips integral boundary condition on a nonlinear differential equation of fractional order. We propose convergent and stable methods to validate the existence and uniqueness of the stated problem. The scheme is also provided with an illustrative examples.

Development of nano-technology throws a new challenge to statistical physics. The classical statistical physics is underlaid by classical or quantum mechanics of closed systems, classical version of the Central Limit Theorem, the Markov model of Brownian motion, the linear approximation of the perturbation theory as a main mathematical tool. The nanotechnology has caused constructing new models: nonlocal in space and time dynamics, generalized Limit Theorem for the Levy-stable family as limit distributions, non-Markovian (subordinated) Levy-motion and instead of the Brownian one, self-similar (fractal) and non-local (integral and fractional differential) analysis. The article gives a short review of these concepts. The multiscale complex system is considered to exacute a selfsimilar motion obeying the Levy stable statistics, which generalized then to fractionally stable kind. The intermediate asymptotics is introduced on the base of truncated Lévy flight. This leads to the correspondence principle connecting classical and nano-statistics. Some numerical results are presented.

The paper is devoted to the investigation of regional observability with constraints on the state of semilinear parabolic systems. The purpose is to reconstruct the initial state between two prescribed functions only on an internal subregion ω of the system evolution domain Ω. The proofs use two approaches, the subdifferential and HUM approach. Finally, a numerical example is provided to verify the effectiveness of our theory results.

This paper is concerned with the controllability of nonlinear higher order fractional delay with impulses, which involved Caputo derivatives of any different orders. A necessary and sufficient condition for the controllability of linear fractional delay dynamical systems with impulses has proved, and a sufficient condition for the corresponding nonlinear systems has obtained. Examples has given to verify the results.

An invertible mapping between time series and networks was recently proposed. It can be used as a tool to figure out properties of the mapped time series. In the present work we use controlled artificial signals to numerically investigate how correlation properties of time series are mapped in the topological measures of associated networks. More specifically, we employ filtered uniform white noise and analyse how the autocorrelation sequence influences the average shortest-path length.