Diffusive Spatial Movement with Memory

Diffusive spatial movement with memory and maturation delays

“…In this paper, we incorporate the maturation delay into the reaction-diffusion model with spatial memory, that was originally proposed in [23]. We show that the positive constant steady state is always unstable if the effect of memory-based diffusion is stronger than that of Fickian one, that is, |D 2 |u * > D 1 , no matter how the memory diffusion delay τ and the maturation delay σ vary.…”

“…where 0 = y ∈ X C and µ ∈ C. It follows from lemma 3.1 in [23] that (2.4) is equivalent to a family of transcendental equations with µ as eigenvalues:…”

“…This implies that for any (τ , σ) with τ > 0, (τ , σ) ∈ Θ n for large n. Then Θ = ∞ n=0 Θ n is an empty set, hence the constant equilibrium u = u * is linearly unstable with infinitely many pairs of complex roots with positive real parts. □ For σ = 0, it has been proved in [23] that (2.5) always have infinitely many complex roots, concentrated on the vertical line z ∈ C : Rez = ln |D2|u *…”

“…A recent review paper [8] emphasized the significance of integrating spatial memory into the modeling of animal movements, and the memory-based diffusion is extremely complicated and poorly understood. In [23], we first modeled the episodic-like spatial memory of animals and formulated the following mathematical model by considering a directed movement toward the negative gradient of density distribution function at past time: ∂u ∂t = D 1 ∆u + D 2 div(u∇u τ ) + g(u), x ∈ Ω, t > 0, ∂u ∂n (t, x) = 0,…”

“…Here u = u(t, x), u τ = u(t − τ , x); Ω is a bounded domain in R N ( N 1) with a smooth boundary ∂Ω; a homogeneous Neumann boundary condition is imposed so that there is no population movement across the boundary ∂Ω. It has been shown in [23] that the stability of a spatially homogeneous steady state fully depends on the relationship between the two diffusion coefficients but is independent of time delay. For (1.1), the reaction term, including both birth and death processes, is considered to occur instantaneously.…”