Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of `sufficiently small parameters', our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar $et\ al$, we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.Comment: 20 pages, submitte

Mackey-Glass type delay differential equations near the boundary of absolute stability

For equations $x'(t) = -x(t) + \zeta f(x(t-h)), x \in \R, f'(0)= -1, \zeta > 0,$ with $C^3$-nonlinearity $f$ which has negative Schwarzian derivative and satisfies $xf(x) < 0$ for $x\not=0$, we prove convergence of all solutions to zero when both $\zeta -1 >0$ and $h(\zeta-1)^{1/8}$ are less than some constant (independent on $h,\zeta$). This result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the Mackey-Glass type delay differential equations.Comment: 16 pages, 1 figure, accepted for publication in the Journal of Mathematical Analysis and Application

Global continuation of monotone wavefronts

In this paper, we answer the question about the criteria of existence of monotone travelling fronts $u = \phi(\nu \cdot x+ct), \phi(-\infty) =0, \phi(+\infty) = \kappa,$ for the monostable (and, in general, non-quasi-monotone) delayed reaction-diffusion equations $u_t(t,x) - \Delta u(t,x) = f(u(t,x), u(t-h,x)).$ $C^{1,\gamma}$-smooth $f$ is supposed to satisfy $f(0,0) = f(\kappa,\kappa) =0$ together with other monostability restrictions. Our theory covers the two most important cases: Mackey-Glass type diffusive equations and KPP-Fisher type equations. The proofs are based on a variant of Hale-Lin functional-analytic approach to the heteroclinic solutions where Lyapunov-Schmidt reduction is realized in a `mobile' weighted space of $C^2$-smooth functions. This method requires a detailed analysis of a family of associated linear differential Fredholm operators: at this stage, the discrete Lyapunov functionals by Mallet-Paret and Sell are used in an essential way.Comment: 21 pages, 3 figures, submitte