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Introduction In population dynamics, discrete-time dynamical systems have been used to describe interaction between ecological species. Comparing to continuous-time dynamical systems, discrete-time models are more suitable to describe populations with non overlapping generations. These models in general produce rich and complex dynamical behaviors. Among various population interaction, predator-prey models play a fundamental rule in mathematical ecology. The dynamics of predator-prey system is greatly depend on the implementation of the functional response, the availability of prey for predation. In this paper we consider a planar system which describes a predator-prey model. In order to reveal comprehensive dynamics of the system, we employee theoretical tools such as center manifold theorem along with numerical tools based on numerical continuation method. Material and methods Our analysis is based on theoretical and numerical techniques. We first determine all fixed points of the system and conditions under which these points may undergo different bifurcations. To reveal more dynamics of the system, we also use numerical bifurcation methods and numerical simulations, which further examine the obtained analytical results. Results and discussion For the resented discrete-time predator-prey system, we compute several bifurcation curves, all possible codimension-1 and codimension-2 bifurcations on thses curves along with their corresponding normal form coefficients. By branch switching technique and employing software package MatcontM, we compute stability boundaries for several cycles up to period 32. We also use numerical simulation, to compute basin of attraction and strange attractor emerging around a Neimark-Sacker bifurcation. Conclusion We can highlight the following results from this paper. Detection and location of all fixed points of a discrete-time predator pray system. Computing all possible codimension-1 and -2 bifurcation and their corresponding normal form coefficients which in turn reveal criticality of the bifurcation points and determine if extra bifurcation curves can emanate from each detected bifurcation. Computing orbits up to period 32 which determine stability thresholds for different cycles. Computing basin of attraction and strange of attractor which emerge around a Neimark-Sacker bifurcation by means of numerical simulation technique.

Autoassociative neural networks can be used for nonlinear processing and normalization of data. Because, firstly, they are able to learn and simulate complex nonlinear communications, and secondly, the communications can be learned through analyzing and distributing information on the neurons and weights and then combining the results of their processing. In this way, they actually make an interpolation between the input data and their communications. But these neural networks cannot model attractor dynamics that is obviously used in brain function. In this paper, the output of autoassociative neural network is connected to its input, and through recursive connections the ability of attractor behavior in these models is provided. This study showed that a recursive neuron with a logistic function forms two attractors, in its training point and its symmetry, but for it with a sigmoid nonlinear function can be formed an attractor in a certain range. In the experiments on face images, it was shown that the absorbance of the images to their attractors was improved from 52.67% to 87.27% by increasing the number of layers and the supervised layer-by-layer pre-training in order to adjust the attractors. [ABSTRACT FROM AUTHOR]

In this paper, first by using the constructions of probabilistic b-metric Menger space and probabilistic metric like Menger space, we introduce probabilistic b-metric like Menger space and present some examples of it. Then, we state and prove some fixed point theorems for teta-eta contraction multivalued mappings in these spaces. We show some fixed point results for generalized gamma-beta contraction type mappings on the probabilistic b-metric like Menger space. Finally, we present an example of usefulness and applicability of our results. ./files/site1/files/%D8%AE%D8%A7%D9%86%D9%87_%DA%AF%DB%8C%D8%B1.pdf

Let and be nonempty subsets of metric space . In this paper, first we introduce two types of non-cyclic contractions on Then we study the existence and uniqueness of an optimal pair of fixed points for such mappings by considering property UC, in metric spaces. Our main results are generalization of recent best proximity points theorems for the corresponding cyclic contractions due to Di Bari-Suzuki-Vetro and Felicit-Eldred.

In this paper, by using the viscosity iterative method and the hybrid steepest-descent method, we present a new algorithm for solving the variational inequality problem. The sequence generated by this algorithm is strong convergence to a common element of the set of common zero points of a finite family of inverse strongly monotone operators and the set of common fixed points of a finite family of demi-contractive mappings. Also, we prove that the sequence generated by this algorithm is strong convergence to a solution of a system of variational inequalities over the set of common fixed points of quasi-nonexpansive mappings and strict pseudo-contractive mappings in a Hilbert space. Finally, some applications of this results are present for solving the split common fixed point problem, which entails finding a point which belongs to the set of common fixed points of a finite family of of strict pseudo-contractive mappings in a Hilbert space such that its image under a linear transformation belongs to the set of common fixed points of a finite family of nonexpasive mappings in the image space../files/site1/files/63/1.pdf