UNIFORM BOUNDEDNESS AND STABILITY OF SOLUTIONS TO A CUBIC PREDATOR-PREY SYSTEM WITH CROSS-DIFFUSION

Using the energy estimate and Gagliardo-Nirenberg-type inequalities,the existence and uniform boundedness of the global solutions to a strongly coupled reaction-diffusion system are proved. This system is a generalization of the two-species Lotka-Volterra predator-prey model with self and cross-diffusion. Suffcient condition for the global asymptotic stability of the positive equilibrium point of the model is given by constructing Lyapunov function.     

竞争-竞争-互惠交错扩散模型的整体解

应用能量估计方法和bootstrap技巧证明了竞争-竞争-互惠交错扩散模型在空间维数小于6时古典解的整体存在性.当反应函数的系数满足一定条件时,应用Lyapunov函数获得了该模型解的收敛性.     

具Ⅱ型Holling功能性反应强耦合椭圆系统的共存问题

本文研究带齐次Dirichlet边界条件的强耦合椭圆系统,首先证明了当食饵和捕食者的扩散率足够大,或者出生率足够小时,系统不存在共存现象,并给出半平凡解存在的充分条件．然后利用Schauder不动点定理,得到强耦合的椭圆问题至少有一个正解存在的充分条件．该条件说明只要捕食者的内部竞争强,物种的交叉扩散相对弱,或者捕获率足够小,物种的交叉扩散相对弱,强耦合系统就至少有一个正解存在．     

Qualitative analysis for a cross-diffusion model of forest

The existence of a bounded global attractor for a cross-diffusion model of forest with homogeneous Dirichlet boundary condition is proved under some condition on the parameters Project supported by the National Natural Science Foundation of China (Grant No. 19671005).     

三种群食物链交错扩散模型的整体

本文应用能量估计方法和Gagliardo-Nirenberg型不等式证明了一类强耦合反应扩散系统整体解的存在性和一致有界性，该系统是带自扩散和交错扩散项的三种群Lotka-Volterra食物链模型．通过构造Lyapunov函数给出了该模型正平衡点全局渐近稳定的充分条件．     

Global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems on domains of arbitrary dimensions

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in -dimensional domains (). Global existence of classical solutions under certain restrictions on the coefficients is established.

     

Analysis of Maxwell–Stefan systems for heat conducting fluid mixtures

The global-in-time existence of bounded weak solutions to the Maxwell–Stefan–Fourier equations in Fick–Onsager form is proved. The model consists of the mass balance equations for the partial mass densities and the energy balance equation for the total energy. The diffusion and heat fluxes depend linearly on the gradients of the thermo-chemical potentials and the gradient of the temperature and include the Soret and Dufour effects. The cross-diffusion system exhibits an entropy structure, which originates from the thermodynamic modeling. The lack of positive definiteness of the diffusion matrix is compensated by the fact that the total mass density is constant in time. The entropy estimate yields the a.e. positivity of the partial mass densities and temperature. Also diffusion matrices are considered that degenerate for vanishing partial mass densities.     

The spatial patterns through diffusion-driven instability in a predator-prey model

Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.     

Nonconstant steady states and pattern formations of generalized 1D cross-diffusion systems with prey-taxis

Cross-diffusion effects and tactic interactions are the processes that preys move away from the highest density of predators preferentially, or vice versa. It is renowned that these effects have played significant roles in ecology and biology, which are also essential to the maintenance of diversity of species. To simulate the stability of systems and illustrate their spatial distributions, we consider positive nonconstant steady states of a generalized cross-diffusion model with prey-taxis and general functional responses in one dimension. By applying linear stability theory, we analyze the stability of the interior equilibrium and show that even in the case of negative cross-diffusion rate, which appeared in many models, the corresponding cross-diffusion model has opportunity to achieve its stability. Meanwhile, in addition to the cross-diffusion effect, tactic interactions can also destabilize the homogeneity of predator–prey systems if the tactic interaction coefficient is negative. Otherwise, taxis effects can stabilize the homogeneity.     

Double-diffusive magnetoconvection

This review deals principally with the interaction between double-diffusive convection and an externally imposed vertical magnetic field in a Boussinesq fluid. Both linear and nonlinear (two and three-dimensional) theories have been discussed. Double-diffusive magnetoconvection is shown to exhibit a rich variety of dynamical behaviour unimaginable in a single component system and serves as a guide to the behaviour of all triple-diffusive systems. Finally, the effects of cross-diffusion, rotation and chemical reaction on double-diffusive magnetoconvection and pattern selection have been briefly touched upon. The author felicitates Prof. D S Kothari on his eightieth birthday and dedicates this paper to him on this occasion.