POSITIVE EQUILIBRIUM SOLUTIONS OF SEMILINEAR PARABOLIC EQUATIONS

The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions： ＂subsolution ≮ supersolution＇, the existence and stability/instability of equilibrium solutions are obtained.

L^2 ASYMPTOTIC FOR THE EQUILIBRIUM SOLUTION OF THE F-M EQUATIONS IN THE SPACE-PERIODIC CASE

The L2 exponetial asymptotical stability for the equilibrium solution of the F-M equations in the space-periodic case (n=2) is considered. Under some assumptions on the external force, it can be shown that the weak solution of F-M equations with initial and boundary conditions in space-periodic case approaches the stationary solution of the system exponetially when time t goes to infinite.

NONTRIVIAL EQUILIBRIUM SOLUTIONS FOR A SEMILINEAR REACTION-DIFFUSION SYSTEM

By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.

Existence of Equilibrium Solutions to a Size-Structured Predator-Prey System with Functional Response

In this paper, we consider a nonlinear size-structured population model with functional response, which describes the dynamics of a predator-prey system living in a common habitat. We present a kind of functional response for the prey being a plant or algae, and explain its biological meanings. When the vital rates depend both on the individual＇s size and on the total population or only depend on the former, we obtain the existence of equilibrium solutions.

Influence of Fluoride Addition on the Composition of Solutions in Equilibrium with Acid Soils

Atmospheric emissions of fluoride from an aiuminium smelter-alumina refinery located on the northern coast of Galicia, NW Spain, increase the content of fluorine in soils and vegetation in the vicinity of the complex. The effects of the addition of fluoride solutions on the chemical properties of soil samples from the area surrounding the complex were investigated in laboratory experiments. Addition of fluoride to soils resulted in increases in pH and concentrations of Fe, A1, and organic matter in the equilibrium solutions and decreases in concentrations of Ca, Mg, and K. No consistent effects were observed on the Cu, Mn, or Zn concentrations. Most of the A1 in solution was bound to organic matter. Within the fraction ＂labile aluminium＇, the concentration of A1-OH complexes decreased and the A1-F complexes increased, especially A1F3 and A1F4^-, which are less toxic than Al^3＋ and A1-OH species.

V-L EQUILIBRIUM OF WEAK ELECTROLYTE SOLUTION OF THE NH_3-CO2-H_2O,NH_3-H_2S-H_2O AND NH_3-SO_2-H_2O SYSTEM

In this paper we have systematically studied V-L equilibrium in ternary aqueous solutions containingvolatile electrolytes by introducing a ternary interaction term into Edwards generalized molecular thermody-namic model and optimizing several adjustable parameters.The program PARA9 with flexible functions ofdoing a series of calculations has been developed and carried out on a TQ-16 computer.It can be usedeither for directly calculating the V-L equilibrium or for optimizing the adjustable parameters.For the sys-toms（NH3-CO3-H2O3,NH3-H2S-H2O and NH3-SO2-H2O）satisfactory results have been obtained withrelative mean deviation of 5-10%.Besides,several sets of adjustable parameters and valuable information ofactivity coefficients,equilibrium concentrations of ions and molecules in solutions are obtained.

ON THE MULTIPLE SOLUTIONS TO A THREE-COEFFICIENT SPECTRAL MODEL OF WIND-DRIVING OCEAN CURRENT EQUATION

In this paper,a research on the problem of multiple solutions of the three-coefficient low-spectrum model for the quasi-geostrophic ocean current equation with forcing and dissipation terms is carried out.The state of the ocean current under wind conditions such as those of typhoon is discussed carefully and the rela- tions between the multiple solutions and the coefficients R and ε are analyzed.It is seen that in an approxi- mate triangular region with the Rossby-coefficient R less than 0.5,and the friction-coefficient ε less than 0.22, there exist three equifibrium solutions,among which two are stable and one is unstable.For the former,the coefficient A or B in the expansion is rather large,while for the latter,A or B is relatively small.They respectively imply how much the ocean energy is fed back from the wind stress and the solution with a large A is much more stable than that with a larger B.

Awareness as the Most Effective Measure to Mitigate the Spread of COVID-19 in Nigeria

A mathematical model consisting of a system of four nonlinear ordinary differential equations is constructed.Our aim is to study the dynamics of the spread of COVID-19 in Nigeria and to show the effectiveness of awareness and the need for relevant authorities to engage themselves more in enlightening people on the significance of the available control measures in mitigating the spread of the disease.Two equilibrium solutions;Disease free equilibrium and Endemic equilibrium solutions were calculated and their global stability analysis was carried out.Basic reproduction ratio(R0)was also obtained,in this research𝑅𝑅0=3.0784.Data obtained for Nigeria is used to conduct numerical simulations in order to support the analytic result and to show the significance of awareness in controlling the disease spread.From the simulation result,it was shown that to mitigate the spread of COVID-19 in Nigeria there is need for serious awareness programs to enlighten people on the available control measures;social distancing,self-isolation,use of personal protective equipment(such as face mask,hand globes,overall gown,etc.),regular hand washing using soap or sanitizer,avoiding having contact with person showing the symptoms and reporting any suspected case.

On global behavior of the system of rational difference equations x_(n+1)=p+y_(n-k)/x_n,y_(n+1)=q+x_(n-k)/y_n

In this paper,we investigate the boundedness character,the global attractivity and the periodic nature of the system of rational difference equations:x=p+y/x,y=q+x/y,n=0,1,2…,where p>0,q>0,k∈{1,2,…} and the initial values xi,yi∈(0,∞),i=-k,-k+1,…,0. Some new results are obtained.

Bi-matrix Games with Fuzzy Strategies and Fuzzy Payoffs and Their Mathematical Programming Equivalents

A fuzzy bi-matrix game（FBG）,namely a two-person non-zero-sum game with fuzzy strategies and fuzzy payoffs is proposed.We have defined and analyzed the optimal strategies of this FBG,and shown that it can be transformed into a corresponding fuzzy mathematical programming issue,for which a ranking function approach can be applied.In addition,optimal strategies of FBG for both Player I and Player II can be gotten.

Study of Complexes Formed by Adrenaline with Nd^(3+),Gd^(3+) and Tm^(3+)

The solution reaction mechanisms,composition,donor atoms,properties of coordinate link and stability of the complexes on the systems of adrenaline with Na^(3+),Gd^(3+)and Tm^(3+)have been studied by pH-potentiometric titration and electronic spectroscopy.Adrenaline perturbs the f-f electronic transition of Nd^(3+)and Tm^(3+)so that the absorption spectra of their complexes are shifted toward longer wavelengths.The absorption spectra of the complexes formed by the three f-block element ions with adrenaline are shifted toward shorter wavelengths with the increase of PH in Δγ=350～650 nm

Optimal Strategy for Aircraft Pursuit-evasion Games via Self-play Iteration

In this paper,the pursuit-evasion game with state and control constraints is solved to achieve the Nash equilibrium of both the pursuer and the evader with an iterative self-play technique.Under the condition where the Hamiltonian formed by means of Pontryagin’s maximum principle has the unique solution,it can be proven that the iterative control law converges to the Nash equilibrium solution.However,the strong nonlinearity of the ordinary differential equations formulated by Pontryagin’s maximum principle makes the control policy difficult to figured out.Moreover the system dynamics employed in this manuscript contains a high dimensional state vector with constraints.In practical applications,such as the control of aircraft,the provided overload is limited.Therefore,in this paper,we consider the optimal strategy of pursuit-evasion games with constant constraint on the control,while some state vectors are restricted by the function of the input.To address the challenges,the optimal control problems are transformed into nonlinear programming problems through the direct collocation method.Finally,two numerical cases of the aircraft pursuit-evasion scenario are given to demonstrate the effectiveness of the presented method to obtain the optimal control of both the pursuer and the evader.

Distributed online mission planning for multi-player space pursuit and evasion

There are three important roles in evasion conflict: pursuer, target and defender. Pursuers’ mission is to access targets; targets’ mission is to escape from pursuers’ capture; defenders’ mission is to intercept pursuers who are potentially dangerous to targets. In this paper, a distributed online mission plan(DOMP) algorithm for pursuers is proposed based on fuzzy evaluation and Nash equilibrium. First, an integrated effectiveness evaluation model is given. Then, the details of collaborative mission planning which includes the co-optimization of task distributing, trajectory and corresponding maneuvering scheme are presented. Finally, the convergence and steadiness of DOMP are discussed with simulation results. Compared with centralized mission planning, DOMP is more robust and can greatly improve the effectiveness of pursuing. It can be applied to dynamic scenario due to its distributed architecture.

A Nonlinear Age-Structured Forest EvolutionSystem

In tnts paper,the Nonlinear Age-dependent forest evolution equation is discussed.The existence and uniqueness of the equation are have been proved.Also,we consider the equilibrium solution and nonequilibirium solution for the nonlinear forest systems with constant size,and give the necessary and sufficient conditions for the existence of equilibrium solution and nonequilibrium solution.

Vertex-Searching Method for Linear Interaction Programming

In order to solve linear interaction programming, the vertex-searching method is proposed in this paper. First, the existence of equilibrium is analyzed for the model of linear interaction programming. Then the conclusion is obtained in which the equi- librium is in the boundary of the restriction region of linear interaction programming. Also, a searching equilibrium solution is deduced from the conclusion.

Qualitative Analysis of a Strongly Coupled Prey-Predator Model

This paper investigates a strongly coupled reaction-diffusion model with Holling-II reaction function in a bounded domain with homogeneous Neumann boundary condition. The sufficient condition for the existence and non-existence of the non-constant positive solutions are obtained. Moreover, we prove that the nonlinear diffusion terms can create non-constant positive equilibrium solutions when the corresponding model without nonlinear diffusion term fails.

Stability of Control Model of Investment

Considers the investment system with time variousing parameter. Under some conditions, proves globally asymptotically stability of the equilibrium solution.

The Uniqueness of Market Economy in China

The purpose of this paper is to prove the equilibrium of general equilibrium of market economy in China. As a consequence,it is shown that, when the governmentplanning system gradually loses its function,the corresponding equilibrium point sequence of planned/market mixed economy converges to the unique equilibrium point of market economy. This demonstrates that the transition of China’s planned/market mixed economy to market economy will be stable.

Rent-seeking Game Analysis About Suppliers and Agencies in Government Procurement

In government procurement, government and suppliers are connected for their interests, government and agencies are connected for commissions. This paper focuses on these two kinds of relationship and use rent-seeking game model to analyze the behavior of the government.