Joon Hyuk Kang
This paper explains the uniqueness of positive steady state of general Lotka-Volterra competition model of two species of animals in the same environment.
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An age-structured predator-prey system with diffusion and Holling-Tanner-type nonlinearities is considered. Regarding the intensity of the fertility of the predator as bifurcation parameter, we prove that a branch of positive coexistence steady states bifurcates from the marginal steady state with no prey. A similar result is obtained when the fertility of the prey varies.
Reaction-diffusion systems with a Lotka-Volterra-type reaction term, also known as competition-diffusion systems, have been used to investigate the dynamics of the competition among $m$ ecological species for a limited resource necessary to their survival and growth. Notwithstanding their rather simple mathematical structure, such systems may display quite interesting behaviours. In particular, while for $m=2$ no coexistence of the two species is usually possible, if $m \ge 3$ we may observe coexistence of all or a subset of the species, sensitively depending on the parameter values. Such coex…
In this paper we consider a system of parabolic reaction-diffusion equations with strong competition and two related scalar reaction-diffusion equations. We are mainly concerned with the case of periodic coefficients and periodic solutions. We show that, for sufficiently large periods, these models have stationary, non-constant, fully non-trivial and stable solutions. We compare our results with already known results about the existence and non-existence of such solutions. Finally, we provide ecological interpretations for these results in terms of resistance against an invasion.
In this paper, the positive solutions of a diffusive competition model with saturation are mainly discussed. Under certain conditions, the stability and multiplicities of coexistence states are analyzed. And by using the topological degree theory in cones, it is proved that the problem has at least two positive solutions under certain conditions. Finally, investigating the bifurcation of coexistence states emanating from the semi-trivial solutions, some instability and multiplicity results of coexistence state are expressed.
Competition-exclusion and coexistence in a two-strain SIS epidemic model in patchy environments
本研究探讨了斑块环境中两株系 SIS 传染病模型的动力学，给出了无病平衡点全局稳定的条件，分析了株系的竞争排斥与共存，并证明了共存地方病平衡点的存在性。
In this paper, we give a sufficient condition for the existence, nonexistence and uniqueness of coexistence of positive solutions to a rather general type of elliptic competition system.
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Neumann boundary conditions in dumbbell-like domains. Under suitable non-degeneracy assumptions, we show that, as the competition rate grows indefinitely, the system reaches a state of coexistence of all the species in spatial segregation. Furthermore, the limit configuration is a local minimizer for the associated free energy.
In this paper, we consider a two species chemotaxis system of parabolic-parabolic-elliptic type with Lotka-Volterra type competition terms in heterogeneous media. We first find various conditions on the parameters which guarantee the global existence and boundedness of classical solutions with nonnegative initial functions. Next, we find further conditions on the parameters which establish the persistence of the two species. Then, under the same set of conditions for the persistence of two species, we prove the existence of coexistence states. Finally we prove the extinction phenomena in the s…
Morse index of steady-states to the SKT model with Dirichlet boundary conditions
本文研究带Dirichlet边界条件的SKT模型稳态的Morse指标，构造了维数等于Morse指标的局部不稳定流形。
In this paper, we gives a complete classification of the global dynamics of two- species Lotka-Volterra competition models with nonlocal dispersals: where K, P represent nonlocal operators, under the assumptions that the nonlo- cal operators are symmetric, the models admit two semi-trivial steady states and 0<bc<1. In particular, when both semi-trivial steady states are locally stable, it is proved that there exist infinitely many steady states and the solution with non- negative and nontrivial initial data converges to some steady state. Furthermore, we generalize these results to the case th…
In this paper, the global dynamics of two-species Lotka-Volterra competition models with nonlocal dispersals is studied. Under the assumption that dispersal kernels are symmetric, we prove that except for very special situations, local stability of semi-trivial steady states implies global stability, while when both semi-trivial steady states are locally unstable, the positive steady state exists and is globally stable. Moreover, our results cover the case that competition coefficients are location-dependent and dispersal strategies are mixture of local and nonlocal dispersals.
In this paper, we study a diffusive Lotka-Volterra competition model under homogeneous Dirichlet boundary conditions. We shall discuss the effects of dispersal rate and spatial heterogeneity on population dynamics. More precisely, we establish the main results about the global asymptotic stability of semitrivial as well as coexistence steady states.
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed by balls connected with thin corridors, we show the occurrence of pattern formation (coexistence and spatial segregation of all the species), as the competition grows indefinitely. As a result we prove the existence and uniqueness of solutions for a remarkable system of differential inequalities involved in segregation phenomena and optimal partition problems.
Diffusion contrast induced dynamics in a time-periodic competition-diffusion system with equal total resources
本文针对总资源相等的周期竞争扩散系统，结合渐近分析与主特征值理论，揭示时间周期与空间异质性的相互作用可颠覆经典的“慢扩散者总是占优”原理，获得了丰富的生态动力学结果。
This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, w…
This paper studies the asymptotic behavior of coexistence steady states of the Shigesada-Kawasaki-Teramoto model as both cross-diffusion coefficients tend to infinity at the same rate. In the case when either one of two cross-diffusion coefficients tends to infinity, Lou and Ni (1999) derived a couple of limiting systems, which characterize the asymptotic behavior of coexistence steady states. Recently, a formal observation by Kan-on (2020) implied the existence of a limiting system including the nonstationary problem as both cross-diffusion coefficients tend to infinity at the same rate. This…
Steady-State and Dynamical Behavior of a PDE Model of Multilevel Selection with Pairwise Group-Level Competition
研究了一个考虑两两组群冲突的多层级选择PDE模型，刻画了测度值解的适定性，并证明群体间选择较弱时种群收敛至全背叛者均衡，同时给出了合作长期存活的条件及稳态收益上限。
The existence and instability of blowing-up steady states for the Shigesada-Kawasaki-Teramoto competition model with cross-diffusion
针对带交叉扩散的SKT双物种竞争模型，本文采用变换结合Lyapunov-Schmidt约化方法，证明了其影子系统及大交叉扩散系数下原模型的爆破正稳态的存在性、渐近结构与谱不稳定性。
Control of a Lotka-Volterra System with Weak Competition
本文研究弱竞争下Lotka-Volterra系统的边界约束可控性，给出有限时间达到共存状态及渐近达到单物种或灭绝状态的条件，并构造不可控场景下的障碍解，揭示竞争、扩散和区域大小对物种动态的影响。
In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to account for stable inhomogeneous steady states exhibiting spatial segregation, which describe a situation of coexistence of two competing species. We provide a deeper understanding on the conditions required on both the cross-diffusion and the reaction coefficients for non-homogeneous steady states to exist, by combining a detailed linearized analysis with advanced numerical bifurcation methods via the continuation software pde2path. We report some numerical experiments suggesting that, when cross-diffusion is taken into …
Existence and Asymptotic Stability of Traveling Waves of Two-species Lotka-Volterra Competition-Diffusion Systems via Geometric Singular Perturbations
本文通过几何奇异摄动理论，证明了两物种Lotka-Volterra竞争-扩散系统中连接共存态与平凡态的行波存在性，并分析了其单调性与渐近稳定性。
This paper investigates a reaction-advection-diffusion system modeling interspecific competition between two species over bounded domains. The kinetic terms are assumed to satisfy the Beddington-DeAngelis functional responses. We consider the situation that first species disperse by a combination of random walk and directed movement along the population density of the second species which disperse randomly within the habitat. For multi-dimensional bounded domains, we prove the global existence and boundedness of time-dependent solutions. For one-dimensional finite domains, we study the effect …
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