Z. Angew. Math. Phys. (2018) 69:10
2017 Springer International Publishing AG,
part of Springer Nature
published online December 15, 2017
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
Patterns in a nonlocal time-delayed reaction–diﬀusion equation
Abstract. In this paper, the existence, stability, and multiplicity of nontrivial (spatially homogeneous or nonhomogeneous)
steady-state solution and periodic solutions for a reaction–diﬀusion model with nonlocal delay eﬀect and Dirichlet/Neumann
boundary condition are investigated by using Lyapunov–Schmidt reduction. Moreover, we illustrate our general results by
applications to population models with one-dimensional spatial domain.
Mathematics Subject Classiﬁcation. Primary 34K18, 35B32, 35B35; Secondary 35K57, 35Q92, 92D40.
Keywords. Reaction–diﬀusion, Nonlocal delay eﬀect, Hopf bifurcation, Stability.
Recently, there has been a growing interest in the dynamical behavior of spatial nonlocal and time-
delayed population systems. Many researchers used the theory of monotone semiﬂows, the comparison
arguments, and the ﬂuctuation method to study spreading speeds, traveling waves, and the global stability
(see, for example, [25,26,31,32,34,35,38,39,46–48,50,53]). However, researches on these problems become
relatively rare for nonmonotone delayed reaction–diﬀusion systems (see [13,19,51]). The main reason is
that it is diﬃcult to establish an appropriate expression for solutions to study the solution semiﬂow under
The objective of the present paper is to investigate the dynamics of spatially homogeneous/
nonhomogeneous steady-state solutions of the following time-delayed reaction–diﬀusion equation with
u(x, t)=Δu(x, t)+f(u(x, t)) +
ϑ(α, x, y)S(u(y, t− τ))dy, x ∈ Ω,t>0,
Bu(x, t)=0,x∈ ∂Ω,t>0,
where α>0, τ ≥ 0, Δ denotes the Laplacian operator on R
, f ∈ C
(R, R) satisﬁes f(0) = 0, S ∈
(R, R) satisﬁes S(0) = 0 and S
(0) = 0, Ω is a connected bounded open domain in R
(n ≥ 1) with a
smooth boundary ∂Ω.
As far as boundary conditions are concerned, we shall study both the homogeneous Dirichlet problem
(i.e., the homogeneous boundary operator B is the identity Id) and the homogeneous Neumann problem
(i.e., B =
is the diﬀerentiation in the direction of the outward normal n to ∂Ω). The former models
a habitat surrounded by a completely hostile external environment, and the latter models a closed habi-
tat with boundaries through which the individuals in the population cannot pass. The kernel function
ϑ(α, x, y) accounts for the probability that an individual born at location y can survive the immature
period [0,τ] and has moved to location x when becoming mature (τ time units after birth). Therefore, the
This work was partially supported by the National Natural Science Foundation of P.R. China (Grant No. 11671123).