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We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

• The normal form formula of double-Hopf bifurcation is generated for general differential equations with the nonlocality. • The formula is applicable to functional differential equations and partial differential equations with or without the nonlocality. • The algorithm is favourable to reveal complex dynamics, e.g. periodic, quasi-periodic and multi-periodic oscillations. • Stable spatially nonhomogeneous periodic and quasi-periodic solutions are found in a nonlocal predator-prey model. This paper is concerned with the calculation of the normal form on the center manifold up to the third-order term at the double-Hopf singularity for general partial functional differential equations with the nonlocal effect. We derive explicit formulas of the normal form, that can be applied for both functional differential equations and partial differential equations with or without nonlocal effects in a bounded spatial domain. This provides an effective tool to establish existences of multi-periodic and quasi-periodic oscillations of a double-Hopf singularity for such equations. As an example, the Holling-Tanner predator-prey model with the nonlocal intraspecific competition of prey is considered. It turns out that double-Hopf bifurcation will occur because of the nonlocal interaction. Many spatio-temporal dynamics are found, including stable spatially homogeneous or nonhomogeneous periodic solutions, and stable spatially nonhomogeneous quasi-periodic solutions. [ABSTRACT FROM AUTHOR]