Propagation of Delayed Lattice Differential Equations without Local Quasimonotonicity

This paper is concerned with the traveling wave solutions and asymptotic spreading of delayed lattice differential equations without quasimonotonicity. The spreading speed is obtained by constructing auxiliary equations and using the theory of lattice differential equations without time delay. The minimal wave speed of invasion traveling wave solutions is established by presenting the existence and nonexistence of traveling wave solutions

Asymptotic behaviour of general nonautonomous Nicholson equations with mixed monotonicities

A general nonautonomous Nicholson equation with multiple pairs of delays in {\it mixed monotone} nonlinear terms is studied. Sufficient conditions for permanence are given, with explicit lower and upper uniform bounds for all positive solutions. Imposing an additional condition on the size of some of the delays, and by using an adequate difference equation of the form $x_{n+1}=h(x_n)$, we show that all positive solutions are globally attractive. In the case of a periodic equation, a criterion for existence of a globally attractive positive solution is provided. The results here constitute a significant improvement of recent literature, in view of the generality of the equation under study and of sharper criteria obtained for situations covered in recent works. Several examples illustrate the results.Comment: 23 page

Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

We consider a bistable (0\textless{}\theta\textless{}1 being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $\theta$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large

Asymptotic behaviour for a class of non-monotone delay differential systems with applications

The paper concerns a class of $n$-dimensional non-autonomous delay differential equations obtained by adding a non-monotone delayed perturbation to a linear homogeneous cooperative system of ordinary differential equations. This family covers a wide set of models used in structured population dynamics. By exploiting the stability and the monotone character of the linear ODE, we establish sufficient conditions for both the extinction of all the populations and the permanence of the system. In the case of DDEs with autonomous coefficients (but possible time-varying delays), sharp results are obtained, even in the case of a reducible community matrix. As a sub-product, our results improve some criteria for autonomous systems published in recent literature. As an important illustration, the extinction, persistence and permanence of a non-autonomous Nicholson system with patch structure and multiple time-dependent delays are analysed.Comment: 26 pages, J Dyn Diff Equat (2017