Invasion waves for a diffusive predator–prey model with two preys and one predator

In this paper, we study the existence of invasion waves of a diffusive predator–prey model with two preys and one predator. The existence of traveling semi-fronts connecting invasion-free equilibrium with wave speed [Formula: see text] is obtained by Schauder’s fixed-point theorem, where [Formula: see text] is the minimal wave speed. The boundedness of such waves is shown by rescaling method and such waves are proved to connect coexistence equilibrium by LaSalle’s invariance principle. The existence of traveling front with wave speed [Formula: see text] is got by rescaling method and limit arguments. The non-existence of traveling fronts with speed [Formula: see text] is shown by Laplace transform.

Entire solutions for a delayed lattice competitive system

In this paper, we investigate the existence of entire solutions for a delayed lattice competitive system. Here the entire solutions are the solutions that exist for all $(n,t)\in \mathbb{Z}\times \mathbb{R}$ ( n , t ) ∈ Z × R . In order to prove the existence, we firstly embed the delayed lattice system into the corresponding larger system, of which the traveling front solutions are identical to those of the delayed lattice system. Then based on the comparison theorem and the sup–sub solutions method, we construct entire solutions which behave as two opposite traveling front solutions moving towards each other from both sides of x-axis and then annihilating. Moreover, our conclusions extend the invading way, which the superior species invade the inferior ones from both sides of x-axis and then the inferior ones extinct, into the lattice and delay case.

Symmetry breaking in the embryonic skin triggers a directional and sequential front of competence during plumage patterning

ABSTRACTThe development of an organism involves the formation of patterns from initially homogeneous surfaces in a reproducible manner. Simulations of various theoretical models recapitulate final states of natural patterns1-4 yet drawing testable hypotheses from those often remains difficult4,5. Consequently, little is known on pattern-forming events. Here, we extend modeling to reproduce not only the final plumage pattern of birds, but also the observed natural variation in its dynamics of emergence in five species. We built a unified model intrinsically generating the directionality, sequence, and duration of patterning, and used in vivo experiments to test its parameter-based predictions. We showed that while patterning duration is controlled by overall cell proliferation, its directional and sequential progression result from a pre-pattern: an initial break in surface symmetry launches a traveling front of increased cell density that defines domains with self-organizing capacity. These results show that universal mechanisms combining pre-patterning and self-organization govern the timely emergence of the plumage pattern in birds.

Global Stability of Traveling Waves for a More General Nonlocal Reaction-Diffusion Equation

The purpose of this paper is to investigate the global stability of traveling front solutions with noncritical and critical speeds for a more general nonlocal reaction-diffusion equation with or without delay. Our analysis relies on the technical weighted energy method and Fourier transform. Moreover, we can get the rates of convergence and the effect of time-delay on the decay rates of the solutions. Furthermore, according to the stability results, the uniqueness of the traveling front solutions can be proved. Our results generalize and improve the existing results.