Table of contents

After introducing some of the basic definitions and results from the theory of groupoid and Lie algebroid, we investigate the discrete Lagrangian mechanics from the viewpoint of groupoid theory and give the connection between groupoids variation and the methods of the first and second discrete variational principles.

The purpose of the present paper is to study the entropy h_s(Φ) of a quantum dynamical systems Φ = (L,s,ϕ), where s is a bayessian state on an orthomodular lattice L. Having introduced the notion of entropy h_s(ϕ,) of partition of a Boolean algebra B with respect to a state s and a state preserving homomorphism ϕ, we prove a few results on that, define the entropy of a dynamical system h_s(Φ), and show its invariance. The concept of sufficient families is also given and we establish that h_s(Φ) comes out to be equal to the supremum of h_s(ϕ,), where varies over any sufficient family. The present theory has then been extended to the quantum dynamical system (L,s,ϕ), which as an effect of the theory of commutators and Bell inequalities can equivalently be replaced by the dynamical system (B,s₀,ϕ), where B is a Boolean algebra and s₀ is a state on B.

In this paper, a non-isospectral differential-difference Kadomtsev–Petviashvilli equation (n-DΔKPE) is presented. Then, the Casoratian solutions of the n-DΔKPE are obtained by generalizing Casoratian conditions of the non-isospectral DΔKPE, single-soliton solution is also derived by using Hiorta's method.

It is shown that the Kronecker product can be applied to constructing new discrete integrable coupling system of soliton equation hierarchy in this paper. A direct application to the fractional cubic Volterra lattice spectral problem leads to a novel integrable coupling system of soliton equation hierarchy. It is also indicated that the study of discrete integrable couplings by using the Kronecker product is an efficient and straightforward method. This method can be used generally.

Based upon the covariant prolongation structures theory, we construct the sl(2,R) × R(ρ) prolongation structure for Konno–Asai–Kakuhata equation. By taking two and one-dimensional prolongation spaces, we obtain the inverse scattering equations given by Konno et al. and the corresponding Riccati equation. The Bäcklund transformations are also presented.

In this paper, we directly extend the applications of the Adomian decomposition method to investigate the complex KdV equation. By choosing different forms of wave functions as the initial values, three new types of realistic numerical solutions: numerical positon, negaton solution, and particularly the numerical analytical complexiton solution are obtained, which can rapidly converge to the exact ones obtained by Lou et al. Numerical simulation figures are used to illustrate the efficiency and accuracy of the proposed method.

We study solutions of the nonlinear Schrödinger equation (NLSE) and higher-order nonlinear Schrödinger equation (HONLSE) with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method (GLGRM), the abundant solutions of NLSE and HONLSE are obtained.

Two basic Darboux transformations of a spectral problem associated with a classical Boussinesq–Burgers equation are presented in this letter. They are used to generate new solutions of the classical Boussinesq–Burgers equation.

In this paper, to construct exact solution of nonlinear partial differential equation, an easy-to-use approach is proposed. By means of the transformation of the independent variables and the travelling wave transformation, the partial differential equation is reduced to an ordinary differential equation. To solve the ordinary differential equation, we assume the soliton solution in the explicit expression and obtain the travelling wave solution. By the transformation back to the original independent variables, the soliton solution of the original partial differential equation is derived. We investigate the short wave model for the Camassa–Holm equation and the Degasperis–Procesi equation respectively. One-cusp soliton solution of the Camassa–Holm equation is obtained. One-loop soliton solution of the Degasperis–Procesi equation is also obtained, the approximation of which in a closed form can be obtained firstly by the Adomian decomposition method. The obtained results in a parametric form coincide perfectly with those given in the present reference. This illustrates the efficiency and reliability of our approach.

We present the DKP oscillator model of spins 0 and 1, in a noncommutative space. In the case of spin 0, the equation is reduced to Klein–Gordon oscillator type, the wave functions are then deduced and compared with the DKP spinless particle subjected to the interaction of a constant magnetic field. For the case of spin 1, the problem is equivalent with the behavior of the DKP equation of spin 1 in a commutative space describing the movement of a vectorial boson subjected to the action of a constant magnetic field with additional correction which depends on the noncommutativity parameter.

By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.

A rotational parameter R_θ has been introduced to complex wavelet transform (CWT). The rotational CWT (RCWT) corresponds to a matrix element ⟨ψ|U₂(θ;μ;k)|F⟩ in the context of quantum mechanics, where U₂(θ;μ;k) is a two-mode rotational displacing-squeezing operator in the ⟨η| representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac's representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.

Two new types of conserved quantities deduced by Noether–Mei symmetry of nonholonomic mechanical system are studied. The definition and criterion of Noether–Mei symmetry for the system are given. A coordination function is introduced, and the conditions under which the Noether–Mei symmetry leads to the two types of conserved quantities and the forms of the two types of conserved quantities are obtained. An illustrative example is given. The coordination function can be selected according to the demand for finding the gauge function, and the choice of the coordination function has multiformity, so more conserved quantities deduced from Noether–Mei symmetry of mechanical system can be obtained.

The Mei symmetry and conserved quantity of general discrete holonomic system are investigated in this paper. The requirement for an invariant formalism of discrete motion equations is defined to be Mei symmetry. The criterion when a conserved quantity may be obtained from Mei symmetry is also deduced. An example is discussed for applications of the results.

Based on our previous paper (Commun. Theor. Phys. 39 (2003) 417) we derive the convolution theorem of fractional Fourier transformation in the context of quantum mechanics, which seems a convenient and neat way. Generalization of this method to the complex fractional Fourier transformation case is also possible.

Our primary purpose of this work is to explicitly construct the general multipartite Einstein–Podolsky–Rosen (EPR) entangled state in multi-mode Fock space for a system with different masses of particles, which makes up a new quantum mechanical representation owing to completeness relation and orthogonal property. Its entanglement can be seen more clearly by analyzing its standard Schmidt decomposition. In addition, some applications of the multipartite entanglement are proposed including deriving the generalized Wigner operator and squeezing operator.

We propose a scheme for approximately and conditionally teleporting an unknown atomic-entangled state in dissipative cavity QED. It is the further development of the scheme of [Phys. Rev. A 69 (2004) 064302], where the cavity mode decay has not been considered and the state teleportated is an unknown atomic state. In this paper, we investigate the influence of the decay on the approximate and conditional teleportation of the unknown atomic-entangled state, which is different from that teleportated in [Phys. Rev. A 69 (2004) 064302] and then give the fidelity of the teleportation, which depends on the cavity mode decay. The scheme may be generalized to not only the teleportation of the cavity-mode-entangled-state by means of a single atom but also the teleportation of the unknown trapped-ion-entangled-state in a linear ion trap.

We propose two schemes for preparing four-photon cluster state through cross-Kerr nonlinearity. Two coherent fields interact when they enter a nonlinear Kerr medium. If the interaction time is chosen appropriately in each Kerr medium, four-photon cluster state can be generated based on the results of two homodyne detectors in the first scheme. These schemes only use Kerr medium and homodyne measurements on coherent light fields, which can be efficiently made in quantum optical laboratories. In addition, weak cross-Kerr nonlinearity is sufficient. All of the properties make these schemes feasible in experiments.

We propose two schemes for quantum secure direct communication (QSDC) and deterministic secure quantum communication (DSQC) over collective dephasing noisy channel. In our schemes, four special two-qubit states are used as the quantum channel. Since these states are unchanged through the collective dephasing noisy channel, the effect of the channel noise can be perfectly overcome. Simultaneously, the security against some usual attacks can be ensured by utilizing the various checking procedures. Furthermore, these two schemes are feasible with present-day technique.

A controlled teleportation scheme is presented. In this scheme, quantum information of a single-qubit state or an entangled two-qubits state is transmitted from a sender (Alice) to a receiver (Charlie) via a four-particle cluster state under the control of the supervisor (Bob). The feature of this scheme is that the teleportation between the sender and the receiver depends on the control of the supervisor.

Recently Jiang et al. [Chin. Phys. Lett. 24 (2007) 1144] gave a scheme for probabilistic controlled teleportation of a triplet W state from the sender Alice to the distant receiver Bob. The m controlled qubits are shared by m(s₁, s₂, ..., s_m) spatially-separated supervisors. Based on transformation operator, we can extend to teleporting an arbitrary three-qubit state. The relation between the transformation operators and the Bob's unitary transformation is also obtained.

We propose a scheme for generating multi-ion graph states using many trapped ions in thermal motion. Our generation scheme is insensitive to external state since the interaction between ions and laser fields does not involve the external degree of freedom. The scheme can be well realized within the current experimental technique.

Using a quantum channel consisting of a GHZ state exposed to noisy environment, we investigate how to remotely prepare an entangled state and a qubit state, respectively. By solving the master equation in the Lindblad form, the influence of the various types of noises on the GHZ state is first discussed. Then we use the fidelity to describe how close the remotely prepared state and the initial state are. Our results show that the fidelity is a function of the decoherence rates and the angles of the initial state. It is found that for each of the two RSP schemes, the influence of the noise acting simultaneously in x, y, and z directions on the average fidelity is the strongest while the influence of the noise acting in x or z direction on the average fidelity is relatively weaker.

We present a very simple scheme for generating four-qubit cluster states with one step using parity measurement based on dipole-induced transparency in a cavity-waveguide system. The scheme only uses the photon detectors to check the parity of the spatially separated dipole, which are the same (even parity) or different (odd parity) through measuring the light fields in the waveguide. The initial entangled states remain after nondetective identification and they can be used for successive tasks.

We classify biqutrit and triqutrit pure states under stochastic local operations and classical communication. By investigating the right singular vector spaces of the coefficient matrices of the states, we obtain explicitly two equivalent classes of biqutrit states and twelve equivalent classes of triqutrit states respectively.

For two-component disk-shaped Bose–Einstein condensates with repulsive atom-atom interaction, the small amplitude, finite and long wavelength nonlinear waves can be described by a Kadomtsev–Petviashvili-I equation at the lowest order from the original coupled Gross–Pitaevskii equations. One- and two-soliton solutions of the Kadomtsev–Petviashvili-I equation are given, therefore, the wave functions of both atomic gases are obtained as well. The instability of a soliton under higher-order long wavelength disturbance has been investigated. It is found that the instability depends on the angle between two directions of both soliton and disturbance.

We investigate the generation of entanglement of coherent excitonic states in coupled quantum dots placed in a cavity by meaning of the state preparation fidelity [Nature (London) 404 (2002) 256; Phys. Rev. A 65 (2002) 012107; J. Uffink, Phys. Rev. Lett. 88 (2002) 230406.] The effect of the number of excitons and the coherent intensity |α| of the cavity field on the entanglement is also studied.

This paper has been addressed to the well-known problem of energy in gravitational waves. We have investigated the energy of cylindrical gravitational waves in the context of General Relativity and teleparallel theory of gravity. For this purpose, the prescriptions of Einstein, Landau–Lifshitz, Bergmann–Thomson, and Möller are used in both the theories. It is shown that these energy-momentum complexes do not provide equivalent results in the two theories. However, these turn out to be constant for all the prescriptions except Möller in both the theories at large distances.

The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicative non-Gaussian noise and additive Gaussian white noise is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker-Planck equation is obtained by applying the unified coloured noise approximation, the small time delay approximation and the Novikov Theorem. The functional analysis and simplification are employed to obtain the approximate expressions of MFPT. The effects of non-Gaussian parameter (measures deviation from Gaussian character) r, the delay time τ, the noise correlation time τ₀, the intensities D and α of noise on the MFPT are discussed. It is found that the escape time could be reduced by increasing the delay time τ, the noise correlation time τ₀, or by reducing the intensities D and α. As far as we know, this is the first time to consider the effect of delay time on the mean first-passage time in the stochastic dynamical system.

By virtue of the invariant eigen-operator method we search for the invariant eigen-operators for some Hamiltonians describing nonlinear processes in particle physics. In this way the energy-gap of the Hamiltonians can be naturally obtained. The characteristic polynomial theory has been fully employed in our derivation.

Anti-synchronization between different hyperchaotic systems is presented using Lorenz and Liu systems. When the parameters of two systems are known, one can use active synchronization. When the parameters are unknown or uncertain, the adaptive synchronization is applied. The simulation results verify the effectiveness of the proposed two schemes for anti-synchronization between different hyperchaotic systems.

The dynamical behavior in the cortical brain network of macaque is studied by modelling each cortical area with a subnetwork of interacting excitable neurons. We find that the avalanche of our model on different levels exhibits power-law. Furthermore the power-law exponent of the distribution and the average avalanche size are affected by the topology of the network.

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physics systems. Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry. Furthermore, the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

A bilinear Bäcklund transformation is presented for the three coupled higher-order nonlinear Schrödinger equations with the inclusion of the group velocity dispersion, third-order dispersion and Kerr-law nonlinearity, which can describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses in wavelength-division multiplexed system. Starting from the Bäcklund transformation, the analytical soliton solution is obtained from a trivial solution. Simultaneously, the N-soliton-like solution in double Wronskian form is constructed, and the corresponding proof is also given via the Wronskian technique. The results obtained from this paper might be valuable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communication systems.

Using the form factors from light-cone sum rules, we study the branching ratios and forward-backward asymmetries (FBAs) of the exclusive decays B⁺_u → π⁺ℓ⁺ℓ⁻ and B⁺_u → ρ⁺ℓ⁺ℓ⁻ (ℓ = e,μ) in the standard model (SM) and the top quark two-Higgs-doublet model (T2HDM). From the numerical results, we find that the new physics contributions cannot provide very large enhancement to the branching ratios and the theoretical predictions are in good agreement with the SM ones. The T2HDM effects on FBAs of these decays are small. Precision measurements of the dilepton invariant mass distributions, especially in the lower dilepton mass region, and the FBAs in the decays B⁺_u → π⁺ (ρ⁺)ℓ⁺ ℓ⁻ will greatly help in discriminating among the SM and the new physics models.

Generalized Yang–Mills theory has a covariant derivative which contains both vector and pseudoscalar gauge bosons. Based on this theory, we construct a U(4) strong interaction model. By using this U(4) generalized Yang–Mills model, we obtain that mesons can be realized as the colorless pseudoscalar gauge bosons. We also obtain a gauge potential solution which can be used to explain the asymptotic behavior and color confinement.

The ground state properties of superheavy nuclei are systematically calculated by the macroscopic-microscopic (MM) model with the Nilsson potential. The calculations well produced the ground state binding energies, α-decay energies, and half lives of superheavy nuclei. The calculated results are systematically compared with available experimental data. The calculated results are also compared with theoretical results from other MM models and from relativistic mean-field model. The calculations and comparisons show that the MM model is reliable in superheavy region and that the MM model results are not very sensitive to the choice of microscopic single-particle potential.

The microscopic mechanism of nine experimentally observed bands in ¹⁷⁸W is investigated using the particle-number conserving method of the cranked shell model with monopole and quadrupole paring interactions. The experimental results, including the moments of inertia and angular momentum alignments of nine bands in ¹⁷⁸W, are reproduced well by the particle-number conserving calculations, in which no free parameter is involved. Calculations demonstrate that occurrence of sharp backbending comes mainly from the contribution of high-j intruder orbitals νi13/2 or πh11/2 and their interference effect with orbitals near the Fermi surface. The ω variation of the occupation probability of each cranked orbital and the contribution to moment of inertia from each cranked orbital are analyzed.

The effect of deuteron breakup in d-nucleus reaction is treated with the continuum discretized coupled channels (CDCC) approach, and the effects on the total reaction cross sections and elastic scattering angular distributions are studied by comparing the calculations of CDCC and spherical optical model with our global deuteron optical potential [Phys. Rev. C 73 (2006) 054605] below 200 MeV, for target nuclei ranging from ¹²C to ²⁰⁸Pb. The contributions from the closed channels to the total reaction and breakup cross sections, and angular distributions of elastic scattering are also seriously discussed.

The sigma meson production in p + ¹²C and p + ⁴⁰Ca reactions at the incident energy E_p = 1.5 GeV is investigated within the Quantum Molecular Dynamics model. The simulation results indicate a distinctive A dependence of the sigma production, that is, the increase of A is followed by an increase of the production cross sections. We find that the σ meson production in proton-induced reactions is strongly medium-dependent, and the produced σ mesons decaying in a denser medium experience a stronger mass shift towards lower masses. This mass shift is an experimentally accessible observable in the final state pion pairs, which do not suffer from reabsorption by the surrounding nucleons. It is pointed out that the ratio of measured sigma cross sections as a function of the sigma invariant-mass from various reactions is a good probe to explore the existence of the σ meson in a dense nuclear environment.

The spin distribution of the evaporation residue cross section of nuclei ¹⁹⁴Pb, ²⁰⁰Pb, ²⁰⁶Pb, and ²⁰⁰Os are calculated via a Langevin equation coupled with a statistical decay model. It is shown that with increasing the neutron-to-proton ratio (N/Z) of the system, the sensitivity of the spin distribution to the nuclear dissipation is decreased significantly. Moreover, for ²⁰⁰Os this spin distribution is no longer sensitive to the nuclear dissipation. These results suggest that to obtain a more accurate pre-saddle viscosity coefficient through the measurement of the evaporation residue spin distribution, it is best to yield those compound systems with low N/Z.

In this work, ionization potentials and quantum effects of 1s²np²P Rydberg states of lithium are calculated based on the calibrated quantum defect function. Energy levels and quantum defects for 1s²np²P bound states and their adjacent continuum states are calculated with the R-matrix theory, and then the quantum defect function of the 1s²np (n ≥ 7) channel is obtained, which varies smoothly with the energy based on the quantum defect theory. The accurate quantum defect of the 1s²7p²P state derived from the experimental data is used to calibrate the original quantum defect function. The new function is used to calculate ionization potentials and quantum effects of 1s²np²P (n ≥ 7) Rydberg states. Present calculations are in agreement with recent experimental data in whole.

Recently Yang, Chu, and Han [Phys. Rev. A 70 (2004) 022329] presented a new type of multipartite entangled states for implementing efficient many-party controlled teleportation of multiqubit quantum information. Here we propose a simple scheme for preparing such a type of multi-atom entangled states in cavity quantum electrodynamics (QED). The scheme involves atom-cavity interaction with large detuning, and is immune to the cavity decay and the thermal field states. Some practical analyses show its availability with the present technology.

Due to interaction with the vacuum of the radiation field, a K-type atomic system with near-degenerate excited and ground levels, which is driven by two strong coherent fields and two weak probe fields, has additional coherence terms — the vacuum-induced coherence (VIC) terms. In this paper, we find that, if the interference is optimized, the two-photon absorption properties of this atom system can be significantly modified and electromagnetically-induced transparency (EIT) is dependent on this interference. Furthermore, we find that in all the cases the coherence can suppress or enhance the partial two-photon transparency, while the complete transparency window is still strictly preserved, which means that it cannot be affected by the VIC. Another important result is the finding of the crucial role played by the relative phase between the probe and coupling fields: the relative height of absorption peaks can be modulated by the relative phase. The physical interpretation of the phenomena has been given.

Collisions of spatial solitons occurring in the nonlinear Schröinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect (and also diminished for the linear defocusing effect). In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.

An optimal current lattice model with backward-looking effect is proposed to describe the motion of traffic flow on a single lane highway. The behavior of the new model is investigated analytically and numerically. The stability, neutral stability, and instability conditions of the uniform flow are obtained by the use of linear stability theory. The stability of the uniform flow is strengthened effectively by the introduction of the backward-looking effect. The numerical simulations are carried out to verify the validity of the new model. The outcomes of the simulation are corresponding to the linearly analytical results. The analytical and numerical results show that the performance of the new model is better than that of the previous models.

By making use of the U(1) gauge potential decomposition theory and the ϕ-mapping topological current theory, we investigate the Schrödinger–Chern–Simons model in the thin-film superconductor system and obtain an exact Bogomolny self-dual equation with a topological term. It is revealed that there exist self-dual vortices in the system. We study the inner topological structure of the self-dual vortices and show that their topological charges are topologically quantized and labeled by Hopf indices and Brouwer degrees. Furthermore, the vortices are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points of the vector field .

A novel approach of characterizing single parametric model potential is proposed by equating total pair wise force to zero. Our well-established single parametric model potential is characterized using the proposed idea and compared the obtained parameter with parameters computed by previously used approaches. Thus characterized pseudopotential is then tested to compute total energy of alkali metals. The results establish the reliability of proposed idea of making total pair wise force to zero in determining the parameter of the pseudopotential.

We have calculated the electron energy of the ground and lower excited states for H⁺₂-like impurity states confined in finite spherical quantum dots in GaAs. Based on the characteristics of energy levels, we have proposed a scheme for realizing charge qubit composed by the the ground and the first excited states of this confined double donor system for the first time. In the proposed scheme the charge qubit is coded in terms of the located electronic states.

Taking into account the quantum size effect and the spin dependence of the electronic band structure, and including the spin dependence of the scattering from bulk impurities and two different sets of surface roughness, we present a theory on the electronic transport in magnetic film, in which the average autocorrelation function (ACF) for surface roughness is described by a Gaussion model. Our result shows that the conductivity is a sensitive function of surface roughness and exchange energy. It is also found that in the thin film limit and in the lower-order approximation of the surface scattering, the total conductivity is given by a sum of conductivities of all the subbands and the two spin channels, for each subband and each spin channel the scattering rates due to the impurities and two surfaces are additive.

Two kinds of topological soliton (skyrmion and magnetic vortex ring) in ferromagnets are studied. They have the common topological origin, a tensor H_αβ = · (∂_α × ∂_β), which describes the non-trivial distribution of local orientation of magnetization at large distances in space. The topological stability of skyrmion is protected by the winding number. Knot-like topological defect as magnetic vortex rings is also studied. On the assumption that magnetic vortex rings are geometric lines, we present their δ-function distribution in ferromagnetic materials. Furthermore, it is briefly shown that Hopf invariant is a proper topological invariant to describe the topology of magnetic vortex rings.

By means of an improved ligand-field theory, the "pure electronic" PS and the PS due to EPI of R line of MgO:V²⁺ have been calculated, respectively. The calculated results are in very good agreement with the experimental data. The behaviors of the pure electronic PS of R line of MgO:V²⁺ and the PS of its R line due to EPI are different. It is the combined effect of them that gives rise to the total PS of R line, which has satisfactorily explained the experimental results. The mixing-degree of of |t²₂(³T₁)e⁴T₂〉 and |t³₂²E) in the wavefunction of R level and its variation with pressure have been calculated and analyzed. The comparison between the feature of R-line PS of MgO:V²⁺ and that of MgO:Cr³⁺ has been made.

We propose a two-species infection model, in which an infected aggregate can gain one monomer from a healthy one due to infection when they meet together. Moreover, both the healthy and infected aggregates may lose one monomer because of self-death, but a healthy aggregate can spontaneously yield a new monomer. Consider a simple system in which the birth/death rates are directly proportional to the aggregate size, namely, the birth and death rates of the healthy aggregate of size k are J₁k and J₂k while the self-death rate of the infected aggregate of size k is J₃k. We then investigate the kinetics of such a system by means of rate equation approach. For the J₁ > J₂ case, the aggregate size distribution of either species approaches the generalized scaling form and the typical size of either species increases wavily at large times. For the J₁ = J₂ case, the size distribution of healthy aggregates approaches the generalized scaling form while that of infected aggregates satisfies the modified scaling form. For the J₁ < J₂ case, the size distribution of healthy aggregates satisfies the modified scaling form, but that of infected aggregates does not scale.

Understanding the cause of the synchronization of population evolution is an important issue for ecological improvement. Here we present a Lotka–Volterra-type model driven by two correlated environmental noises and show, via theoretical analysis and direct simulation, that noise correlation can induce a synchronization of the mutualists. The time series of mutual species exhibit a chaotic-like fluctuation, which is independent of the noise correlation, however, the chaotic fluctuation of mutual species ratio decreases with the noise correlation. A quantitative parameter defined for characterizing chaotic fluctuation provides a good approach to measure when the complete synchronization happens.