Stability and bifurcation analysis of a reaction–diffusion equation with distributed delay

Abstract

Dynamics of a general reaction–diffusion equation with distributed delay are considered. The effects of the weak kernel and the strong kernel on the dynamics of the system are both investigated. By analyzing the characteristic equations in detail and taking the average delay as a bifurcation parameter, the stability of the constant equilibrium and the existence of Hopf bifurcations are obtained. The absolute stability and the conditional stability can be explicitly determined by the coefficients of the linearized system. For the case of the strong kernel, the average delay may induce the stability switches, but it is not able to occur for the case of the weak kernel. The algorithm for determining the direction and stability of the bifurcating periodic solutions is derived. Finally, the obtained theoretical results are applied to several single-species models, and the numerical simulations are illustrated to verify the theoretical results.

Appendices

Appendix 1

Proof of Lemma 2

Differentiating the two sides of (7) with respect to $\tau$ leads to

$$\begin{aligned} \frac{\mathrm{d}\lambda }{\mathrm{d}\tau }=\frac{2\tau \lambda ^2+\lambda \left( 2\tau \left( \frac{\mathrm{d}n^2}{l^2} -a\right) +2\right) +2\left( \frac{\mathrm{d}n^2}{l^2}-a-b\right) }{3\lambda ^2\tau ^3+2\lambda \tau ^3 \left( \frac{\mathrm{d}n^2}{l^2}-a+\frac{2}{\tau }\right) +2\tau ^2\left( \frac{\mathrm{d}n^2}{l^2}-a\right) +\tau }. \end{aligned}$$

For convenience, denote $\tau =\tau _{1n}$ or $\tau =\tau _{2n}$ by $\tau ^*$, by (8), and then we have

$$\begin{aligned} Re\left( \left. \frac{\mathrm{d}\lambda }{\mathrm{d}\tau }\right| _{\tau =\tau ^*}\right)&= Re\left( \frac{-2\tau ^*\omega _n^2+i\omega _n\left( 2\tau ^*\left( \frac{\mathrm{d}n^2}{l^2} -a\right) +2\right) +2\left( \frac{\mathrm{d}n^2}{l^2}-a-b\right) }{-3\omega _n^2{\tau ^*}^3+ 2i\omega _n{\tau ^*}^3\left( \frac{\mathrm{d}n^2}{l^2}-a+\frac{2}{\tau ^*}\right) +2{\tau ^*}^2 \left( \frac{\mathrm{d}n^2}{l^2}-a\right) +\tau ^*}\right) \\&= Re\left( \frac{-\tau ^*\omega _n^2+i\omega _n\left( \tau ^*\left( \frac{\mathrm{d}n^2}{l^2}-a \right) +1\right) +\frac{\mathrm{d}n^2}{l^2}-a-b}{{\tau ^*}^3\left( -\omega _n^2+i\omega _n \left( \frac{\mathrm{d}n^2}{l^2}-a+\frac{2}{\tau ^*}\right) \right) }\right) \\&= \frac{1}{\tilde{\varDelta }}\left( \omega _n^2\tau ^*-\left( \frac{\mathrm{d}n^2}{l^2}-a-b \right) +\left( \tau ^*\left( \frac{\mathrm{d}n^2}{l^2}-a\right) +1\right) \left( \frac{\mathrm{d}n^2}{l^2} -a+\frac{2}{\tau ^*}\right) \right) \\&= \frac{1}{\tau \tilde{\varDelta }}\left( 4\tau ^*(\frac{\mathrm{d}n^2}{l^2}-a)+b\tau ^* +(\frac{\mathrm{d}n^2}{l^2}-a)^2{\tau ^*}^2+3\right) \\&= \frac{1}{\tau ^*\tilde{\varDelta }}\left( 1-\left( \frac{\mathrm{d}n^2}{l^2}-a\right) ^2(\tau ^*)^2 \right) ,~~\text {by}~~(8)_n. \end{aligned}$$

where $\tilde{\varDelta }={\tau ^*}^3\left( \omega _n^2+\left( \frac{\mathrm{d}n^2}{l^2}-a +\frac{2}{\tau ^*}\right) ^2\right)$. Since $\tau _{1n}\tau _{2n}=\frac{1}{\left( \frac{\mathrm{d}n^2}{l^2}-a\right) ^2}$, and $\tau _{1n}<\tau _{2n}$, we have

$$\begin{aligned} \tau _{1n}<\frac{1}{|\frac{\mathrm{d}n^2}{l^2}-a|},~\tau _{2n}>\frac{1}{|\frac{\mathrm{d}n^2}{l^2}-a|}. \end{aligned}$$

Thus, $\left. Re(\frac{\mathrm{d}\lambda }{\mathrm{d}\tau })\right| _{\tau =\tau _{1n}}>0,~\left. Re(\frac{\mathrm{d}\lambda }{\mathrm{d}\tau })\right| _{\tau =\tau _{2n}}<0$.

Appendix 2

We first transform System (1) into the following operator equation:

$$\begin{aligned} \frac{\hbox {d}u(t)}{\hbox {d}t}=Au_t+G(u_t,\alpha ), \end{aligned}$$

(17)

where $u_t=u(t+\theta ),~\theta \in (-\infty ,0],\,A$ and $G$ are defined by

$$\begin{aligned} A\phi (\theta )= {\left\{ \begin{array}{ll} \frac{\hbox {d}\phi (\theta )}{\hbox {d}\theta },&{}\theta \in (-\infty ,0),\\ \left( -\frac{\mathrm{d}n^2}{l^2}+a\right) \phi (0)&{}\\ \quad +\,b\int _{-\infty }^0f(-s)\phi (s)\hbox {d}s,&{}\theta =0. \end{array}\right. } \end{aligned}$$

$$\begin{aligned}&G(\phi ,\alpha )= {\left\{ \begin{array}{ll}0,&{}\theta \in (-\infty ,0),\\ F''_{11}\frac{\phi ^2(0)}{2}+F''_{12}\phi (0)\int _{-\infty }^0f(-s)\phi (s)\hbox {d}s +\frac{F''_{22}}{2}\left( \int _{-\infty }^0f(-s)\phi (s)\hbox {d}s\right) ^2\\ ~~+\frac{F'''_{111}}{6}\phi ^3(0)+\frac{F'''_{112}}{2}\phi ^2(0) \left( \int _{-\infty }^0f(-s)\phi (s)\hbox {d}s\right) \\ ~~+\frac{F'''_{122}}{2}\phi (0)\left( \int _{-\infty }^0f(-s)\phi (s)\hbox {d}s\right) ^2\\ ~~+\frac{F'''_{222}}{6}\left( \int _{-\infty }^0f(-s)\phi (s)\hbox {d}s\right) ^3+O(4), &{}\theta =0,\end{array}\right. } \end{aligned}$$

(18)

where $a=F'_1(k,k),~b=F'_2(k,k)$ are as previously shown, $F''_{11},F''_{12},F''_{22},F'''_{111},F'''_{112},etc$ denote the high-order partial derivations at $(k, k)$.

The adjoint operator $A^*$ of $A$ is defined as

$$\begin{aligned}&A^*(\psi (s))\\&\quad ={\left\{ \begin{array}{ll} -\frac{\hbox {d}\psi (s)}{\hbox {d}s},&{}s\in (0,+\infty ),\\ \left( -\frac{\mathrm{d}n^2}{l^2}+a\right) \psi (0)&{}\\ \quad +b\int _{-\infty }^0f(-s)\psi (-s)\hbox {d}s,&{}s=0. \end{array}\right. } \end{aligned}$$

Define the bilinear pairing

$$\begin{aligned} \ll \psi ,\phi \gg&= \bar{\psi }(0)\phi (0)-b\nonumber \\&\times \int _{-\infty }^0\int _0^\theta \bar{\psi }(\xi -\theta )f(-\theta )\phi (\xi )\hbox {d}\xi d\theta .\nonumber \\ \end{aligned}$$

(19)

It can be verified that $q(\theta )=e^{i\omega \theta },~\theta \in (-\infty ,0]$ is an eigenvector of $A$ corresponding to $i\omega$. $q^*(s)=re^{i\omega s},~s\in [0,+\infty )$ is an eigenvector of $A^*$ corresponding to $-i\omega$, where

$$\begin{aligned} \bar{r}=\left( 1-b\int _{-\infty }^0e^{i\omega \theta }f(-\theta )\theta \hbox {d}\theta \right) ^{-1}. \end{aligned}$$

By (19) and $Aq(0)=i\omega q(0),~A^*q^*(0)=-i\omega q^*(0)$, we can compute that

$$\begin{aligned} \ll q^*,q\gg =1,~\ll q^*,\bar{q}\gg =0. \end{aligned}$$

And $\langle \cos {\frac{nx}{l}},\cos {\frac{nx}{l}}\rangle =1,~n=0$ and $\langle \cos {\frac{nx}{l}},\cos {\frac{nx}{l}}\rangle =\frac{1}{2},~n=1,2,\ldots .$

Then, the center subspace of linear equation of (17) with $\alpha =0$ is given by $P_{CN}\fancyscript{C}$, where

$$\begin{aligned} P_{CN}\fancyscript{C}=\left\{ \left( q(\theta )z+\bar{q}(\theta )\bar{z}\right) \cos {\frac{nx}{l}},~z\in C\right\} . \end{aligned}$$

Following the notation as Hassard et al. [40], the solution $u_t$ of Eq. (17) with $\alpha =0$ is as follows:

$$\begin{aligned} u_t=\left( q(\theta )z+\bar{q}(\theta )\bar{z}\right) \cos {\frac{nx}{l}}+W(\theta ), \end{aligned}$$

(20)

where

$$\begin{aligned} W(\theta )&= W(z,\bar{z},\theta )=W_{20}(\theta )\frac{z^2}{2}+W_{11}(\theta ) z\bar{z}\nonumber \\&\quad +\,W_{02}(\theta )\frac{\bar{z}^2}{2}+\ldots . \end{aligned}$$

(21)

On the center manifold $\fancyscript{C}_0$, the local coordinates $z$ in the direction of $q^*$ satisfy

$$\begin{aligned} \dot{z}&= i\omega z + \left\langle q^*,G\left( (q(0)z+\bar{q}(0)\bar{z})\cos {\frac{nx}{l}} \right. \right. \nonumber \\&\left. \left. +W(0),0\right) \right\rangle \nonumber \\&\triangleq i\omega z+g(z,\bar{z}), \end{aligned}$$

(22)

where

$$\begin{aligned} g(z,\bar{z})&= \overline{q^*}(0)\left\langle G\left( (q(0)z+\bar{q}(0)\bar{z})\cos {\frac{nx}{l}}\right. \right. \nonumber \\&\left. \left. +W(0),0\right) , \cos {\frac{nx}{l}}\right\rangle \nonumber \\&= \frac{\bar{r}}{l\pi }\int _0^{l\pi }G\left( (q(0)z+\bar{q}(0)\bar{z})\cos {\frac{nx}{l}}\right. \nonumber \\&\quad \left. +\,W(0),0\right) \cos {\frac{nx}{l}}\hbox {d}x\nonumber \\&= g_{20}\frac{z^2}{2}\!+\!g_{11}z\bar{z}\!+\!g_{02}\frac{\bar{z}^2}{2}\!+\!g_{21}\frac{z^2 \bar{z}}{2}\!+\ldots .\qquad \end{aligned}$$

(23)

Noticing that

$$\begin{aligned}&\int _{-\infty }^0e^{i\omega s}f(-s)\hbox {d}s = \frac{1}{(i\omega \tau ^*+1)^2},\\&\int _{-\infty }^0e^{-i\omega s}f(-s)\hbox {d}s = \frac{1}{(1-i\omega \tau ^*)^2}, \end{aligned}$$

and combining (18), (20)–(23), we have,

$$\begin{aligned} g_{20}&= \frac{\bar{r}}{l\pi }\int _0^{l\pi }\left( F''_{11}+\frac{2F''_{12}}{(1 +i\omega \tau ^*)^2}\right. \\&\quad \left. +\,\frac{F''_{22}}{(1+i\omega \tau ^*)^4}\right) \cos ^3{\frac{nx}{l}}\hbox {d}x\\&= {\left\{ \begin{array}{ll} 0,\quad n=1,2,\ldots ,&{}\\ \bar{r}\left( F''_{11}+\frac{2F''_{12}}{(1+i\omega \tau ^*)^2}+\frac{F''_{22}}{(1 +i\omega \tau ^*)^4}\right) ,&{}n=0. \end{array}\right. }\\ g_{11}&= \frac{\bar{r}}{l\pi }\int _0^{l\pi }\left( F''_{11}+\frac{F''_{12}(2-2 \omega ^2{\tau ^*}^2)+F''_{22}}{(1+{\tau ^*}^2\omega ^2)^2}\right) \\&\times \cos ^3{\frac{nx}{l}}\hbox {d}x\\&= {\left\{ \begin{array}{ll} 0,\quad n=1,2,\ldots ,&{}\\ \bar{r}\left( F''_{11}+\frac{F''_{12}(2-2\omega ^2{\tau ^*}^2)+F''_{22}}{(1 +{\tau ^*}^2\omega ^2)^2}\right) ,&{}n=0.\end{array}\right. }\\ g_{02}&= {\left\{ \begin{array}{ll} 0,\quad n=1,2,\ldots ,&{}\\ \bar{r}\left( F''_{11}+\frac{2F''_{12}}{(1-i\omega \tau ^*)^2}+\frac{F''_{22}}{(1-i\omega \tau ^*)^4}\right) ,&{}n=0. \end{array}\right. }\\ g_{21}&= \frac{\bar{r}}{l\pi }\int _0^{l\pi }\left( F''_{11}(2W_{11}(0)+W_{20}(0)\right) \\&\quad +\,F''_{12}\left( 2\int _{-\infty }^0W_{11}(s)f(-s)\hbox {d}s\right. \\ \end{aligned}$$

$$\begin{aligned}&\quad \left. +\,\int _{-\infty }^0W_{20}(s)f(-s)\hbox {d}s+\frac{W_{20}(0)}{(1-i\omega \tau ^*)^2}\right. \\&\quad \left. +\,\frac{2W_{11}(0)}{(1+i\omega \tau ^*)^2}\right) \\&\quad +\,F''_{22}\left( \frac{\int _{-\infty }^0f(-s)W_{20}(s)\hbox {d}s}{(1-i\omega \tau ^*)^2}\right. \\&\quad \left. +\,\frac{2\int _{-\infty }^0 W_{11}(s)f(-s)\hbox {d}s}{(1+i\omega \tau ^*)^2}\right) \cos ^2{\frac{nx}{l}}\hbox {d}x\\&\quad +\,\frac{\bar{r}}{l\pi }\int _0^{l\pi }(F'''_{111}+F'''_{112}\left( \frac{1}{(1 -i\omega \tau ^*)^2}\right. \\&\quad \left. +\,\frac{2}{(1+i\omega \tau ^*)^2}\right) \\&\quad +\,F'''_{122}\left( \frac{2}{(1-i\omega \tau ^*)^2}+\frac{1}{(1+i\omega \tau ^*)^2}\right) \\&\quad +\,F'''_{222}\frac{1}{(1-i\omega \tau ^*)^2(1+i\omega \tau ^*)^4})\\&\times \cos ^4{\frac{nx}{l}}\hbox {d}x,~~n=0,1,2\ldots . \end{aligned}$$

Since $W_{11}(\theta )$ and $W_{20}(\theta )$ are included in $g_{21}$, we will compute them.

By Wu [18], $W(z,\bar{z})$ satisfies

$$\begin{aligned} \dot{W}=AW+H(z,\bar{z}), \end{aligned}$$

(24)

where

$$\begin{aligned}&AW=AW_{20}(\theta )\frac{z^2}{2}+AW_{11}(\theta )z\bar{z}\nonumber \\&\quad \qquad \qquad +\,AW_{02}(\theta ) \frac{\bar{z}^2}{2}+\ldots ,\nonumber \\&H(z,\bar{z})=-2Re\left( g(z,\bar{z})q(\theta )\right) \cos {\frac{nx}{l}}\nonumber \\&\quad \qquad \qquad +\,G\left( (zq(\theta )+\bar{z}\bar{q}(\theta ))\cos {\frac{nx}{l}}+W(\theta ),0\right) \nonumber \\&\quad \qquad \quad \triangleq H_{20}\frac{z^2}{2}+H_{11}z\bar{z}+H_{02}\frac{\bar{z}^2}{2}+\ldots . \end{aligned}$$

(25)

Comparing the coefficient, we obtain,

$$\begin{aligned} H_{20}(\theta )&= {\left\{ \begin{array}{ll} -\left( g_{20}q(\theta )+\overline{g_{02}}\bar{q}(\theta )\right) \cos {\frac{nx}{l}},&{}\theta \in (-\infty ,0),\\ -(g_{20}+\overline{g_{02}})\cos {\frac{nx}{l}}+\left( F''_{11} +\frac{2F''_{12}}{(1+i\omega \tau ^*)^2}+\frac{F''_{22}}{(1+i\omega \tau ^*)^4} \right) \cos ^2{\frac{nx}{l}},&{}\theta =0. \end{array}\right. }\\ H_{11}(\theta )&= {\left\{ \begin{array}{ll} -\left( g_{11}q(\theta )+\overline{g_{11}}\bar{q}(\theta )\right) \cos {\frac{nx}{l}},&{}\theta \in (-\infty ,0),\\ -(g_{11}+\overline{g_{11}})\cos {\frac{nx}{l}}+\left( F''_{11} +\frac{F''_{12}(2-2\omega ^2{\tau ^*}^2)+F''_{22}}{(1+{\tau ^*}^2\omega ^2)^2} \right) \cos ^2{\frac{nx}{l}},&{}\theta =0. \end{array}\right. } \end{aligned}$$

According to (21), (22), (24), (25), we have

$$\begin{aligned} (2i\omega -A)W_{20}(\theta )=H_{20}(\theta ),~~-AW_{11}(\theta )=H_{11}(\theta ). \end{aligned}$$

(26)

By the definition of $A$ and (26), we can obtain

$$\begin{aligned} W_{20}(\theta )&= \left( \frac{g_{20}ie^{i\omega \theta }}{\omega } +\frac{\overline{g_{02}}ie^{-i\omega \theta }}{3\omega }\right) \cos {\frac{nx}{l}}\\&+E_1e^{2i\omega \theta },\\ W_{11}(\theta )&= \left( -\frac{ig_{11}e^{i\omega \theta }}{\omega } +\frac{\overline{g_{11}}ie^{-i\omega \theta }}{\omega }\right) \cos {\frac{nx}{l}}+E_2. \end{aligned}$$

Notice that, when $\theta =0$, by (26) and $Aq(0)=i\omega q(0)$ and $A^*q^*(0)=-i\omega q^*(0)$, we have

$$\begin{aligned}&b\int _{-\infty }^0f(-s)e^{i\omega s}ds=\frac{\mathrm{d}n^2}{l^2}-a +i\omega ,\\&b\int _{-\infty }^0f(-s)e^{-i\omega s}\hbox {d}s =\frac{\mathrm{d}n^2}{l^2}-a-i\omega . \end{aligned}$$

Hence,

$$\begin{aligned}&\!\!\!\left( 2i\omega +\frac{\mathrm{d}n^2}{l^2}-a\right) E_1-b\int _{-\infty }^0f(-s)e^{2i\omega s}\hbox {d}s E_1\\&\!\!\!\quad =\left( F''_{11}+\frac{2F''_{12}}{(1+i\omega \tau ^*)^2}+\frac{F''_{22}}{(1+i\omega \tau ^*)^4}\right) \cos ^2{\frac{nx}{l}}, \end{aligned}$$

which leads to

$$\begin{aligned} E_1=\frac{F''_{11}+\frac{2F''_{12}}{(1+i\omega \tau ^*)^2}+\frac{F''_{22}}{(1+i \omega \tau ^*)^4}}{2i\omega +\frac{\mathrm{d}n^2}{l^2}-a-\frac{b}{(1+2i\omega \tau ^*)^2}}\cos ^2{\frac{nx}{l}}. \end{aligned}$$

(27)

By the same way, we have

$$\begin{aligned} E_2=\frac{F''_{11}+\frac{F''_{12}(2-2\omega ^2{\tau ^*}^2)+F''_{22}}{(1 +{\tau ^*}^2\omega ^2)^2}}{\frac{\mathrm{d}n^2}{l^2}-a-b}\cos ^2{\frac{nx}{l}}. \end{aligned}$$

(28)