DCDS-B
Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations
Qihuai Liu Dingbian Qian Zhiguo Wang
Discrete & Continuous Dynamical Systems - B 2012, 17(5): 1537-1550 doi: 10.3934/dcdsb.2012.17.1537
In this paper, we prove the existence of positive quasi-periodic solutions for the Lotka-Volterra competition systems with quasi-periodic coefficients by KAM technique. The result shows that, in most case, quasi-periodic solutions exist for sufficiently small quasi-periodic perturbations of the autonomous Lotka-Volterra systems. Moreover, these quasi-periodic solutions will tend to an equilibrium of the autonomous Lotka-Volterra systems.
keywords: KAM iteration quasi-periodic solution small denominator. Lotka-Volterra competition system
DCDS
A new method for the boundedness of semilinear Duffing equations at resonance
Zhiguo Wang Yiqian Wang Daxiong Piao
Discrete & Continuous Dynamical Systems - A 2016, 36(7): 3961-3991 doi: 10.3934/dcds.2016.36.3961
We introduce a new method for the boundedness problem of semilinear Duffing equations at resonance. In particular, it can be used to study a class of semilinear equations at resonance without the polynomial-like growth condition. As an application, we prove the boundedness of all the solutions for the equation $\ddot{x}+n^2x+g(x)+\psi(x)=p(t)$ under the Lazer-Leach condition on $g$ and $p$, where $n\in \mathbb{N^+}$, $p(t)$ and $\psi(x)$ are periodic and $g(x)$ is bounded.
keywords: periodic nonlinearity Moser's theorem. boundedness Hamiltonian system at resonance
DCDS-B
Spatial propagation for a parabolic system with multiple species competing for single resource
Zhiguo Wang Hua Nie Jianhua Wu
Discrete & Continuous Dynamical Systems - B 2017, 22(11): 1-30 doi: 10.3934/dcdsb.2018237

A model of $m$ species competing for a single growth-limiting resource is considered. We aim to use the dynamics of such a problem to describe the invasion and spread of $m$ species which are introduced localized in space $\mathbb{R}^N$. The existence, uniqueness and uniform boundedness of the Cauchy problem are investigated by semigroup theory and local $L^p$-estimates. The asymptotic speed of spread is achieved by uniform persistence ideas. The existence of traveling wave is obtained by upper-lower solutions and sliding techniques. Our result shows that the asymptotic speed of spread for $m$ species is characterized by the minimum wave speed of the positive traveling wave solutions associated with this system.

keywords: Competition model spreading speed traveling wave reaction-diffusion system Cauchy problem

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