Delayed PDE Modeling and Simulations (Part 1)

Numerical Simulations of a Nonlocal Delayed Reaction-Diffusion Model

To illustrate the formation and stability of wave solutions we turn our attention to a class of nonlocal delayed RD equation proposed by So et al (2001, Proc. R. Soc. Lond. A, 457, 1841–1853). In particular, the authors adopted Smith-Thieme’s approach to obtain the following model of single species population.

\( \frac{\partial w}{\partial t} = D_m\,\frac{\partial^2 w}{\partial x^2} - d_m w + \varepsilon \int_{-\infty}^{\infty} b\!\big(w(y,t-\tau)\big)\, f_{\alpha}(x-y)\,dy, \)

where  \( x \in \mathbb{R}, \) and \(0 \le \epsilon \le 1\)  and \( w(x,t)\) represents the total mature population \(u(x,a,t)\) at age \(a\), time \(t\) and position \(x\) that is given by

\(w(x,t)=\int_{0}^{\infty} u(x,a,t)\,da.\)

The kernel function is given by

\( f_{\alpha}(x)=\frac{1}{\sqrt{4\pi\alpha}}\,e^{-x^2/(4\alpha)} \quad \text{with} \quad \alpha=\tau D_I>0,\ \tau>0 \)

 

We study

1. Impacts of the maturation time delay on PDE solution
2. Formation of oscillatory traveling wave solutions

 

1. Impacts of the maturation time delay on PDE solution

We study the impacts of the maturation time delay \(\tau\) with respect to the PDE solution, which  (i) oscillates in the spatial domain due to the increased delay value; and  (ii) converges to the trivial solution (i.e., the population goes extinct).

 

PDE solution with birth function \(b(w)=pwe^{aw}\)

 



b1

 

 

PDE solution for τ=80

t80

 

 

 



b4dde



b4

 

 

PDE solution with birth function b(w) = p.w^2.exp(aw)


b4dde

More simulations

b4

 

 

 

2. Formation of oscillatory traveling wave solutions

Impacts of the diffusion rates: increased values of the immature-mature diffusion ratio (DI /Dm) results in oscillatory traveling wavefronts due to the loss of stability of the nontrivial constant steady state.

Example 1. Wavefront with the birth function b(w) = p.w.exp(aw) 

w3


The corresponding phase-plane

w3


Example 2. Wavefront with the birth function b(w) = p.w^2.exp(aw)


b4ft

The corresponding phase-plane


bf