Traveling waves for the diffusive Nicholson's blowflies equation

Abstract

We consider traveling wave front solutions for the diffusive Nicholson's blowflies equation on the real line. The existence of such solutions is proved using the technique developed by J. Wu and X. Zou (J. Dyn. Differ. Equations 13 (3) (2001)). Some numerical simulation using the iteration formula of Wu and Zou [7] is also provided.

Introduction

Consider the reaction–diffusion equation with a discrete time delay

$$\text{∂}\text{N(t,x)}\text{∂}\text{t}\text{=}\text{∂}{}^2\text{N(t,x)}\text{∂}\text{x}{}^2\text{−δN(t,x)+pN(t−τ,x)}\text{e}{}^{\mathrm{-aN(t-\tau ,x)}}\text{,}$$

where $\text{x∈}\text{R}$ and t⩾0. Such an equation had been studied in [4], [5], [8]. For the case when there is no spatial dependence, the corresponding delay equation was referred to as Nicholsons's blowflies equation, cf. [1] after the experiments of Nicholson [2], [3] had been extensively studied. Eq. (1) can be derived based on first principles by making use of the spatial (this leads to the diffusion term) and age structures (this leads to the discrete delay τ) of the population. The general theory of reaction–diffusion equations with delays can be found in [6].

Assume p/δ>1. Then there are two equilibria: N₀=0 and N_e=1/aln(p/δ). A traveling wave front is a solution of (1) of the form u(t,x)=φ(x+ct), where c>0, φ(t) is monotone increasing and it satisfies

$$\text{cφ}{}^{\mathrm{\prime }}\text{(t)=φ″(t)−δφ(t)+pφ(t−cτ)}\text{e}{}^{\mathrm{-a\phi (t-c\tau )}}\text{,}\text{φ(−∞)=N}{}_0\text{,}\text{φ(+∞)=N}{}_{\mathrm{e}}\text{.}$$

The main result of this paper is:

Theorem

If 1<(p/δ)⩽e, then there exists c^*>0 such that for every c>c^* there exists a traveling wave front for (1) with speed c.

In Section 2, we will prove this theorem by applying [7, Theorem 3.6], which is for delayed reaction–diffusion systems. Since we are considering a scalar equation here, for the convenience of reference and for simplicity, we only need a scalar version of this theorem, which is stated below.

Consider the delayed reaction–diffusion equation

$$\text{∂}\text{u(t,x)}\text{∂}\text{t}\text{=D}\text{∂}{}^2\text{u(t,x)}\text{∂}\text{x}{}^2\text{+f(u}{}_{\mathrm{t}}\text{(x)),}$$

where $\text{t⩾0,}\text{x∈}\text{R}\text{,}\text{u∈}\text{R}$ with D>0, and $\text{f:}\text{C([−τ,0],}\text{R}\text{)}\text{→}\text{R}$ is continuous and u_t(x) is an element in $\text{C([−τ,0],}\text{R}\text{)}$ parameterized by $\text{x∈}\text{R}$ and given by

$$\text{u}{}_{\mathrm{t}}\text{(x)(s)=u(t+s,x),}\text{s∈[−τ,0],}\text{t⩾0,}\text{x∈}\text{R}\text{.}$$

Looking for traveling wave solutions of the form u(t,x)=φ(x+ct) leads to a second-order functional differential equation

$$\text{Dφ″(t)−cφ}{}^{\mathrm{\prime }}\text{(t)+f}{}_{\mathrm{c}}\text{(φ}{}_{\mathrm{t}}\text{)=0,}\text{t∈}\text{R}\text{,}$$

where $\text{f}{}_{\mathrm{c}}\text{:}\text{X}{}_{\mathrm{c}}\text{=C([−cτ,0],}\text{R}\text{)}\text{→}\text{R}$ is defined by

$$\text{f}{}_{\mathrm{c}}\text{(ψ)=f(ψ}{}^{\mathrm{c}}\text{),}\text{ψ}{}^{\mathrm{c}}\text{(s)=ψ(cs),}\text{s∈[−τ,0].}$$

We assume

• (A1) There exists K>0 such that $\text{f}{}_{\mathrm{c}}\text{(}\text{0}\text{̂}\text{)=f}{}_{\mathrm{c}}\text{(}\text{K}\text{̂}\text{)=0}$ and $\text{f}{}_{\mathrm{c}}\text{(}\text{u}\text{̂}\text{)≠0}$ for u∈(0,K), where û denotes the constant function taking the value u on [−cτ,0].

• (A2) (Quasi-monotonicity). There exists β⩾0 such that

  $$\text{f}{}_{\mathrm{c}}\text{(φ)−f}{}_{\mathrm{c}}\text{(ψ)+β[φ(0)−ψ(0)]⩾0}$$

  for φ,ψ∈X_c with $\text{0⩽ψ(s)⩽φ(s)⩽K,}\text{s∈[−cτ,0]}$.

If for some c>0, (4) has a monotone solution φ satisfying $\text{lim}{}_{\mathrm{<mtext>t</mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\to </mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>-\infty </mtext>}}\text{φ(t)=0}$ and $\text{lim}{}_{\mathrm{<mtext>t</mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\to </mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\infty </mtext>}}\text{φ(t)=K}$, then u(t,x)=φ(x+ct) is called a traveling wave front of (3) with speed c.

Next we define the profile set for traveling wave fronts of (3) by

$$\text{Γ=φ∈C(}\text{R}\text{;}\text{R}{}^{\mathrm{n}}\text{),}$$

• (i) φ is nondecreasing in $\text{R}$,

• (ii) $\text{lim}{}_{\mathrm{<mtext>t</mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\to -</mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\infty </mtext>}}\text{φ(t)=0,}\text{lim}{}_{\mathrm{<mtext>t</mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\to </mtext><mspace\; sp="0.12"\; width="2px"\; linebreak="nobreak"\; is="true"></mspace><mtext>\infty </mtext>}}\text{φ(t)=K.}$

A function $\text{φ∈C(}\text{R}\text{,}\text{R}\text{)}$ is called an upper (resp., lower) solution of (4) if it is differentiable almost everywhere (a.e.) and satisfies

$$\text{cφ}{}^{\mathrm{\prime }}\text{⩾Dφ″(t)+f}{}_{\mathrm{c}}\text{(φ}{}_{\mathrm{t}}\text{),}\text{a.e.}\text{in}\text{R}\text{,}$$

resp.,

$$\text{cφ}{}^{\mathrm{\prime }}\text{⩽Dφ″(t)+f}{}_{\mathrm{c}}\text{(φ}{}_{\mathrm{t}}\text{),}\text{a.e.}\text{in}\text{R}\text{.}$$

Now we are in the position to state a scalar version of [7, Theorem 3.6].

Theorem A

Assume that (A1)–(A2) hold. Suppose that (4) has an upper solution $\text{φ}$ in Γ and a lower solution $\text{φ}$ (which is not necessarily in Γ) with $\text{0⩽}\text{φ}\text{(t)⩽}\text{φ}\text{(t)⩽K}$ and $\text{φ}\text{(t)≢0}$ in $\text{R}$, then (3) has a traveling wave front.