一类具变系数交错扩散的登革热模型

该文在登革热的传播模型中引入较复杂的异质性交错扩散,用于描述人群和蚊群的相互扩散现象,并探讨交错扩散对模型动力学的影响,以及根据风险阈值对稳态共存解存在性进行分析.结果 表明,风险阈值不仅与交错扩散有关,而且直接影响着模型的动力学,如果风险阈值大于1,并伴随其它条件成立,则人群和蚊群携带的病毒会共存,不利于登革热的控制...     

一类具扩散的Ivlev型捕食模型

研究一类具有扩散的Ivlev型捕食模型.首先利用线性化和特征值法给出正平衡解在常微分系统和加入自扩散的弱耦合偏微分系统下的局部渐近稳定性,其次,讨论在加入交错扩散的强耦合系统下会引起平衡解的Turing不稳定,最后给出数值模拟验证理论结果.     

一类三次捕食者-食饵交错扩散系统解的整体存在性和稳定性

首先引进一类三次捕食者-食饵交错扩散系统,该系统是两种群Lotka-Volterra交错扩散系统的推广,现有的已知结果目前很少.本文应用能量估计方法,结合Shauder理论和bootstrap技巧讨论该系统古典整体解的存在唯一性,并在反应函数的系数满足一定条件时,通过构造Lyapunov函数证明系统正平衡点的全局渐近性.     

一类具有阶段结构的三次捕食者-食饵交错扩散系统的整体解

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,该系统是具有阶段结构的两种群Lotka-Volterra捕食者-食饵交错扩散模型的推广.通过构造Lyapunov函数给出了该系统正平衡点全局渐近稳定的充分条件.     

种群动力学中椭圆系统在零解处的分支正解

考虑齐次Dirichlet边界条件下具有交错扩散压力的广义Lotka-Volterra两种群竞争反应扩散稳态系统. 首先借助Lyapunov-Schmidt约化方法考虑了系统在零解处小分支正解的存在性, 然后借助标准的线性化方法研究了这些分支正解的稳定性.     

一类具交错扩散的互惠模型的共存解

研究了Dirichlet边界条件下具交错扩散的两种群互惠模型.采用上下解方法,结合Schauder不动点理论,给出了问题共存解存在的充分条件.进一步,利用单调迭代序列的方法构造出问题的共存解.结果表明,当交错扩散相对弱时,问题至少存在一共存解.     

非均匀Chemostat中非单食物链模型稳态正解的存在性

研究一类由反应扩散方程组描述的非均匀Chemostat中微生物之间既表现竞争关系又表现捕食被捕食关系的模型．用特征值理论确定了系统正稳态解存在的必要条件，用锥映射不动点指数方法给出了系统正稳态解存在的充分条件．     

一类具有交错扩散的捕食者-食饵模型的稳定性分析

本文考虑一类具有交错扩散的捕食者-食饵模型,详细分析系统正常数平衡解的稳定性和Turing不稳定性,得到一些有意义的结论,并利用Matlab软件对所获得的理论结果给出了适当的数值验证.     

被捕食者带有第三边值的捕食模型的正稳态解的存在性

该文研究了一类被捕食者带有第三边值的捕食模型. 首先获得了它存在正稳态解的充要条件是a>mb+d₁λ₁; 然后研究了它的正稳态解的局部稳定性和唯一性;最后讨论了充分大的扩散参数对它的正稳态解的存在性的影响.     

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．     

Stationary pattern of a diffusive prey–predator model with trophic intersections of three levels

The paper is concerned with a diffusive prey–predator model subject to the homogeneous Neumann boundary condition, which models the trophic intersections of three levels. We will prove that under certain assumptions, even though the unique positive constant steady state is globally asymptotically stable for the dynamics with diffusion, the non-constant positive steady state can exist due to the emergence of cross-diffusion. We demonstrate that the cross-diffusion can create stationary pattern. Moreover, we treat the cross-diffusion parameter as a bifurcation parameter and discuss the existence of non-constant positive solutions to the system with cross-diffusion.     

Turing patterns of a strongly coupled predator-prey system with diffusion effects

The main purpose of this work is to investigate the effects of cross-diffusion in a strongly coupled predator-prey system. By a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, it is shown that Turing instability of the reaction-diffusion system can disappear due to the presence of the cross-diffusion, which implies that the cross-diffusion induced stability can be regarded as the cross-stability of the corresponding reaction-diffusion system. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions. These results exhibit interesting and very different roles of the cross-diffusion in the formation and the disappearance of the Turing instability.     

Effect of cross-diffusion on the stationary problem of a diffusive competition model with a protection zone

This paper is concerned with the positive stationary problem of a Lotka–Volterra cross-diffusive competition model with a protection zone for the weak competitor. The detailed structure of positive stationary solutions for small birth rates and large cross-diffusion is shown. The structure is quite different from that without cross-diffusion, from which we can see that large cross-diffusion has a beneficial effect for the existence of positive stationary solutions. The effect of the spatial heterogeneity caused by protection zones is also examined and is shown to change the shape of the bifurcation curve. Thus the environmental heterogeneity, together with large cross-diffusion, can produce much more complicated stationary patterns. Finally, the asymptotic behavior of positive stationary solutions for any birth rate as the cross-diffusion coefficient tends to infinity is given, and moreover, the structure of positive solutions of the limiting system is analyzed. The result of asymptotic behavior also reveals different phenomena from that of the homogeneous case without protection zones.     

Stationary patterns of a ratio-dependent prey-predator model with cross-diffusion

This paper is concerned with a ratio-dependent predator-prey system with diffusion and cross-diffusion in a bounded domain with no flux boundary condition. We show that under certain hypotheses, the cross-diffusion can create non-constant positive steady states even though the corresponding model without cross-diffusion fails.     

Turing pattern formation in a three species model with generalist predator and cross-diffusion

In a natural ecosystem, specialist predators feed almost exclusively on one species of prey. But generalist predators feed on many types of species. Consequently, their dynamics is not coupled to the dynamics of a specific prey population. However, the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species. This is described by cross-diffusion in a generalist predator–prey model. First, the positive equilibrium solution is globally asymptotically stable for the ODE system and for the reaction–diffusion system without cross-diffusion, respectively, hence it does not belong to the classical Turing instability scheme. But it becomes linearly unstable only when cross-diffusion also plays a role. This implies that cross–diffusion can lead to the occurrence and disappearance of the instability. Our results exhibit some interesting combining effects of cross-diffusion, predations and intra-species interactions. Furthermore, we consider the existence and non-existence results concerning non-constant positive steady states (patterns) of the system. We demonstrate that cross-diffusion can create non-constant positive steady-state solutions.     

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.     

Stationary patterns created by cross-diffusion for the competitor-competitor-mutualist model

In this paper we consider a competitor-competitor-mutualist model with cross-diffusion. We prove some existence and non-existence results concerning non-constant positive steady-states (patterns). In particular, we demonstrate that the cross-diffusion can create patterns when the corresponding model without cross-diffusion fails.     

Cross-diffusion effects on a morphochemical model for electrodeposition

We analyze the effects of cross-diffusion on pattern formation in a PDE reaction-diffusion system introduced in Bozzini et al. 2013 to describe metal growth in an electrodeposition process. For this morphochemical model - which refers to the physico-chemical problem of coupling of growth morphology and surface chemistry - we have found that negative cross-diffusion in the morphological elements as well as positive cross-diffusion in the surface chemistry produce larger Turing parameter spaces and favor a tendency to stripeness that is not found in the case without cross-diffusion. The impact of cross-diffusion on pattern selection has been also discussed by the means of a stripeness index. Our theoretical findings are validated by an extensive gallery of numerical simulations that allow to better clarify the role of cross-diffusion both on Turing parameter spaces and on pattern selection. Experimental evidence of cross-diffusion in electrodeposition as well as a physico-chemical discussion of the expected impact of cross diffusion-controlled pattern formation in alloy electrodeposition processes complete the study.     

Effects of the self- and cross-diffusion on positive steady states for a generalized predator–prey system

This paper deals with a generalized predator–prey system with cross-diffusion and homogeneous Neumann boundary condition, where the cross-diffusion is included in such a way that the prey runs away from the predator. We first give a priori estimate for positive steady states to the system. Then we obtain the non-existence result of non-constant positive steady states. Finally, we investigate the stability of constant equilibrium point and the existence of non-constant positive steady states. It is shown that the system admits a non-constant positive steady state provided that either of the self-diffusions is large or the cross-diffusion is small.     

Stability and cross-diffusion-driven instability in a diffusive predator–prey system with hunting cooperation functional response

This paper presents a qualitative study of a diffusive predator–prey system with the hunting cooperation functional response. For the system without diffusion, the existence, stability and Hopf bifurcation of the positive equilibrium are explicitly determined. It is shown that the hunting cooperation affects not only the existence of the positive equilibrium but also the stability. For the diffusive system, the stability and cross-diffusion driven Turing instability are investigated according to the relationship of the self-diffusion and the cross-diffusion coefficients. Stability and cross-diffusion instability regions are theoretically determined in the plane of the cross-diffusion coefficients. The technique of multiple time scale is employed to deduce the amplitude equation of Turing bifurcation and then pattern dynamics driven by the cross-diffusion is also investigated by the corresponding amplitude equation.