Influence of prey refuge on predator–prey dynamics

A diffusive predator–prey system with Michaelis–Menten type functional response subject to prey refuge is considered. Bifurcation analysis of Hopf and Turing are carried out in detail. In particular, Turing domain is given in the two parameters space. The obtained results show that the refuges used by prey have great influence on the pattern formation of the populations. More specifically, as prey refuge being increased, spotted pattern and coexistence of spotted and stripe-like pattern emerge. It is also proved that the pattern is not dependent on the initial conditions, which means the pattern is controlled by the intrinsic mechanism.     

A delayed predator–prey model with strong Allee effect in prey population growth

In this paper, we consider a delayed predator-prey system with intraspecific competition among predator and a strong Allee effect in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium point and show that Hopf-bifurcation can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf-bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of Wu ({Trans. Am. Math. Soc.} 350:4799?C4838, 1998) for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.     

The study of a ratio-dependent predator–prey model with stage structure in the prey

In this paper, a ratio-dependent predator–prey model with stage structure in the prey is constructed and investigated. In the first part of this paper, some sufficient conditions for the existence and stability of three equilibriums are obtained. In the second part, we consider the effect of impulsive release of predator on the original system. A sufficient condition for the global asymptotical stability of the prey-eradication periodic solution is obtained. We also get the condition, under which the prey would never be eradicated, i.e., the impulsive system is permanent. At last, we give a brief discussion.

     

Global analysis of a Holling type II predator–prey model with a constant prey refuge

A global analysis of a Holling type II predator–prey model with a constant prey refuge is presented. Although this model has been much studied, the threshold condition for the global stability of the unique interior equilibrium and the uniqueness of its limit cycle have not been obtained to date, so far as we are aware. Here we provide a global qualitative analysis to determine the global dynamics of the model. In particular, a combination of the Bendixson–Dulac theorem and the Lyapunov function method was employed to judge the global stability of the equilibrium. The uniqueness theorem of a limit cycle for the Lineard system was used to show the existence and uniqueness of the limit cycle of the model. Further, the effects of prey refuges and parameter space on the threshold condition are discussed in the light of sensitivity analyses. Additional interesting topics based on the discontinuous (or Filippov) Gause predator–prey model are addressed in the discussion.     

Multiple crack propagation along the interface of a non-homogeneous composite subjected to anti-plane shear loading

The present work concerns with the multiple crack propagation along the interface of two bonded dissimilar strips. The crack faces are subjected to anti-plane shear traction. Galilean transformation is employed to reduce the problem to a quasi-static one. Then, using Fourier transforms and asymptotic analysis, the quasi-static problem is reduced to a pair of singular integral equations. That are solved numerically, using Gauss-Chebyshev integration formulae. The values of the dynamic stress intensity factors are obtained and compared with the previous similar works. Further, a parametric study is introduced to investigate the effect of crack growth rate, geometric and elastic characteristics of the composite on the values of dynamic stress intensity factors.     

Spatiotemporal dynamics of a predator–prey model

Spatial component of ecological interactions has been identified as an important factor in how ecological communities are shaped. In this paper, we consider a Holling?CTanner model with spatial diffusion. Choosing appropriate parameter values in parameter spaces, we obtain rich patterns, including spotted, black-eye, and labyrinthine patterns. The numerical results show that predator?Cprey system can exhibit complicated behavior.     

Stability and Hopf bifurcation for a delayed predator–prey model with stage structure for prey and Ivlev-type functional response

Nonlinear Dynamics - In this paper, we mainly investigate a delayed predator–prey model with stage structure for prey and Ivlev-type functional response. Four assumptions about this model are...     

Detection of coarse-grained unstable states of microscopic/stochastic systems: a timestepper-based iterative protocol

We address an iterative procedure that can be used to detect coarse-grained hyperbolic unstable equilibria (saddle points) of microscopic simulators when no equations at the macroscopic level are available. The scheme is based on the concept of coarse timestepping (Kevrekidis et al. in Commun. Math. Sci. 1(4):715–762, 2003) incorporating an adaptive mechanism based on the chord method allowing the location of coarse-grained saddle points directly. Ultimately, it can be used in a consecutive manner to trace the coarse-grained open-loop saddle-node bifurcation diagrams of complex dynamical systems and large-scale systems of ordinary and/or partial differential equations. We illustrate the procedure through two indicative examples including (i) a kinetic Monte Carlo simulation (kMC) of simple surface catalytic reactions and (ii) a simple agent-based model, a financial caricature which is used to simulate the dynamics of buying and selling of a large population of interacting individuals in the presence of mimesis. Both models exhibit coarse-grained regular turning points which give rise to branches of saddle points.     

Bifurcation and route-to-chaos analyses for Mathieu–Duffing oscillator by the incremental harmonic balance method

Bifurcations and route to chaos of the Mathieu–Duffing oscillator are investigated by the incremental harmonic balance (IHB) procedure. A new scheme for selecting the initial value conditions is presented for predicting the higher order periodic solutions. A series of period-doubling bifurcation points and the threshold value of the control parameter at the onset of chaos can be calculated by the present procedure. A sequence of period-doubling bifurcation points of the oscillator are identified and found to obey the universal scale law approximately. The bifurcation diagram and phase portraits obtained by the IHB method are presented to confirm the period-doubling route-to-chaos qualitatively. It can also be noted that the phase portraits and bifurcation points agree well with those obtained by numerical time-integration.     

Dynamic model based nonlinear tracking control of a planar parallel manipulator

Based on the dynamic model, a novel nonlinear tracking controller is developed to overcome the nonlinear dynamics and friction of a planar parallel manipulator. The dynamic model is formulated in the active joint space, and the active joint friction is described with the Coulomb + viscous friction model. A nonlinear tracking controller is designed to eliminate the tracking error by using the power function. The nonlinear tracking controller is proven to guarantee asymptotic convergence to zero of both the tracking error and error rate with the Barbalat’s lemma. The trajectory tracking experiment of the proposed controller is implemented on an actual five-bar planar parallel manipulator both at the low-speed and high-speed motion. Moreover, the control performances of the proposed controller are compared with the results of the augmented PD (APD) controller.     

An Exact Derivation from Three-Dimensions of the Linear Equations for the Bending of an Elastically Monoclinic Plate and Associated Three-Dimensional Solutions

The exact linear three-dimensional equations for a elastically monoclinic (13 constant) plate of constant thickness are reduced without approximation to a single 4th order differential equation for a thickness-weighted normal displacement plus two auxiliary equations for weighted thickness integrals of a stress function and the normal strain. The 4th order equation is of the same form as in classical (Kirchhoff) theory except the unknown is not the midsurface normal displacement. Assuming a solution of these plate equations, we construct so-called modified Saint-Venant solutions—“modified” because they involve non-zero body and surface loads. That is, solutions of the exact three-dimensional elasticity equations that exhibit no boundary layers and that are subject to a special set of body and surface loads that leave the analogous plate loads arbitrary.     

The Strong Ellipticity Condition Under Changes in the Current and Reference Configuration

In this note we study the condition of strong ellipticity under changes in the current and reference configuration for the finite hyperelastostatic case. The outcome is that strong ellipticity is preserved provided one adjusts the vectors used in the definition of this condition accordingly.     

Representation of discrete sequences with N-dimensional iterated function systems in tensor form

Iterated Function System (IFS) models have been used to represent discrete sequences where the attractor of the IFS is self-affine or piecewise self-affine in R ² or R ³ (R is the set of real numbers). In this paper, the piecewise hidden-variable fractal model is extended from R ³ to R ⁿ (n is an integer greater than 3), which is called the multi-dimensional piecewise hidden variable fractal model. This new model uses a “mapping partial derivative” and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the hidden variables. Therefore the result is very general. Moreover, the piecewise hidden-variable fractal model in tensor form is more terse than in the usual matrix form.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Bifurcation analysis of a diffusive ratio-dependent predator–prey model

In this paper, a ratio-dependent predator–prey model with diffusion is considered. The stability of the positive constant equilibrium, Turing instability, and the existence of Hopf and steady state bifurcations are studied. Necessary and sufficient conditions for the stability of the positive constant equilibrium are explicitly obtained. Spatially heterogeneous steady states with different spatial patterns are determined. By calculating the normal form on the center manifold, the formulas determining the direction and the stability of Hopf bifurcations are explicitly derived. For the steady state bifurcation, the normal form shows the possibility of pitchfork bifurcation and can be used to determine the stability of spatially inhomogeneous steady states. Some numerical simulations are carried out to illustrate and expand our theoretical results, in which, both spatially homogeneous and heterogeneous periodic solutions are observed. The numerical simulations also show the coexistence of two spatially inhomogeneous steady states, confirming the theoretical prediction.     

Stabilizing control of Hopf bifurcation in the Hodgkin–Huxley model via washout filter with linear control term

It is a significant issue to control bifurcation because many neuronal diseases have close relevance to bifurcation of neuron system. Some studies have been done on bifurcation control in the Hodgkin–Huxley (HH) model, but there is no clear mathematical criterion for bifurcation stabilization. In this paper, according to Routh–Hurwitz stability criterion, we employ linear control term of washout filter-aided dynamic feedback controller to stabilize bifurcation of the HH model. As a result, we can deduce linear control gain based on the criterion, and simulation shows the method is effective for making the HH model stable. The controller designs described here are achieved by electrical stimulus, so it may have potential applications in the diagnosis and therapy of dynamical diseases.     

Painlevé integrability, similarity reductions, new soliton and soliton-like similarity solutions for the (2+1)-dimensional BKP equation

The (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation of B-type (BKP) is hereby investigated. New soliton solutions and soliton-like similarity solutions are constructed for the (2+1)-dimensional BKP equation. The similarity solutions are not travelling wave solutions when the arbitrary functions involved are chosen appropriately. Painlevé test shows that there are two solution branches, one of which has the resonance ?2. And four similarity reductions for the BKP equation are given out through nontrivial variable transformations. Moreover, abundant soliton behaviour modes of the solutions, such as soliton fusion and soliton reflection, are discussed in detail.     

Continuously Dislocated Elastic Bodies with a Neo-Hookean Like Energy Subjected to Anti-plane Shear

In this paper, we consider a materially uniform but inhomogeneous body and we are interested in three particular cases of inhomogeneities corresponding to three distinct distributions of dislocations. The field of defects enters the equilibrium equations through the components of the tensor field describing the relaxation procedure. We examine what form should these components take in order for the material to admit states of anti-plane shear. The results obtained in this paper hold for a class of materials that obey a specific form for the stored energy function. In the special case of no dislocations, this class falls under the well known class of Neo-Hookean materials.      

Asymptotic Analysis of Shell-like Inclusions with High Rigidity

We study the problem of an elastic shell-like inclusion with high rigidity in a three-dimensional domain by means of the asymptotic expansion method. The analysis is carried out in a general framework of curvilinear coordinates. After defining a small real adimensional parameter ε, we characterize the limit problems when the rigidity of the inclusion has order of magnitude \frac1e\frac{1}{\varepsilon } and \frac1e³\frac{1}{\varepsilon^{3}} with respect to the rigidities of the surrounding bodies. Moreover, we prove the strong convergence of the solution of the initial three-dimensional problem towards the solution of the simplified limit problem.