On the conjecture for the pushed wavefront to the diffusive Lotka–Volterra competition model. Journal of Mathematical Biology
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This paper concerns ecological invasion phenomenon of species based on the diffusive Lotka–Volterra competition model. We investigate the spreading speed (or the minimal wave speed of traveling waves) selection to the model and concentrate on the conjecture raised by Roques et al. (J Math Biol 71(2):465–489, 2015). By using an abstract implicit function theorem in a weighted functional space coupled with a perturbation technique, we not only prove this conjecture, but also show that the fast decay behavior of the first species is necessary and sufficient for the nonlinear speed selection of the whole system. This may lead to further significant results on the answer to the original Hosono’s conjecture, a problem that has been outstanding for more than twenty years.
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- الكلمات المفتاحية
Lotka–Volterra; Competition; Pulled and pushed waves; Speed selection
On a conjecture raised by Yuzo Hosono. J. Dynamics and Differential Equations
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In this paper, we study the speed selection mechanism for traveling wave solutions to a two-species Lotka–Volterra competition model. After transforming the partial differential equations into a cooperative system, the speed selection mechanism (linear vs. nonlinear) is investigated for the new system. Hosono conjectured that there is a critical value rc of the birth rate so that the speed selection mechanism changes only at this value. In the absence of diffusion for the second species, we obtain the speed selection mechanism and successfully prove a modified version of the Hosono’s conjecture. Estimation of the critical value is given and some new conditions for linear or nonlinear selection are established.
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- الكلمات المفتاحية
Lotka–Volterra; Traveling waves; Speed selection
Minimal-speed selection of traveling waves to the Lotka-Volterra competition model, Journal of Differential Equations.
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In this paper the minimal-speed determinacy of traveling wave fronts of a two-species competition model of diffusive Lotka–Volterra type is investigated. First, a cooperative system is obtained from the classical Lotka–Volterra competition model. Then, we apply the upper-lower solution technique on the cooperative system to study the traveling waves as well as its minimal-speed selection mechanisms: linear or nonlinear. New types of upper and lower solutions are established. Previous results for the linear speed selection are extended, and novel results on both linear and nonlinear selections are derived.
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- الكلمات المفتاحية
Lotka–Volterra; Traveling waves; Speed selection
Stability of traveling waves to the Lotka-Volterra competition model, Complexity
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In this paper, the stability of traveling wave solutions to the Lotka-Volterra diffusive model is investigated. First, we convert the model into a cooperative system by a special transformation. The local and the global stability of the traveling wavefronts are studied in a weighted functional space. For the global stability, comparison principle together with the squeezing technique is applied to derive the main results.
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Existence and stability of the steady state solution of a thin film on an inclined periodic solid substrate under gravity, Journal of Asymptotic Analysis
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In this paper, we investigate the dynamics of a liquid film flowing over a periodic wavy wall. This study is based on a long-wave model that is valid at near-critical Reynolds number. For the periodic wall surface, we prove the existence of a periodic steady-state solution to the model by the method of abstract contraction mapping in a particular functional space. Using the Floquet–Bloch theory and asymptotic method, we establish several analytic results on the stability of the periodic steady-state solution in a weighted functional space.
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Thin film flow, periodic solution, asymptotic analysis
Order Graph: A new representation of finite
groups
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There are many ways to associate a graph with a finite group.
Such an association yields many of the group properties. The link
established between a graph and a group is usually determined by the
definition of the adjacent vertices. In this research, the orders of the
elements will take a place in the graph creation. The order graph of
a finite group is the directed graph whose vertices are the elements of
the group order classes, and for two distinct vertices x and y there is
an arc from x to y if and only if x divides y. This paper will cover the
creation of the order graph in general and then concentrate on some
groups.
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- الكلمات المفتاحية
Element order - Order classes - Graph - Weighted graph
Periodic steady-state solutions of a liquid film model via a classical method. Canadian Math. Bulletin,
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In this paper, periodic steady-state of a liquid film flowing over a periodic uneven wall is investigated via a classical method. Specifically, we analyze a long-wave model that is valid at the near-critical Reynolds number. For the periodic wall surface, we construct an iteration scheme in terms of an integral form of the original steady-state problem. The uniform convergence of the scheme is proved so that we can derive the existence and the uniqueness as well as the asymptotic formula of the periodic solutions.
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- الكلمات المفتاحية
film flow; classical methods; asymptotic analysis