Majid Bani-Yaghoub, Ph.D.Associate Professor &
Associate Director
Office: Manheim 205 A
|
Numerical Simulations of a Nonlocal Delayed Reaction-Diffusion Model
1. Convergence of PDE solution to stationary wavefront
|
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.
where x is real 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
The kernel function is given by
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)
The corresponding phase-plane
Example 2. Wavefront with the birth function b(w) = p.w^2.exp(aw)
The corresponding phase-plane
where x is real 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
The kernel function is given by
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)
 |
The corresponding phase-plane
Example 2. Wavefront with the birth function b(w) = p.w^2.exp(aw)
The corresponding phase-plane