| Abstract: |
| In this talk, we first study the propagation dynamics of the following system
\begin{equation}\label{1}
\begin{cases}
\frac{\partial S(x,t)}{\partial t} =d \frac{\partial^2S(x,t)}{\partial x^2}+f(N(x,t))-\mu S(x,t)-\beta S(x,t)I(x,t), \
\frac{\partial I(x,t)}{\partial t} =d \frac{\partial^2I(x,t)}{\partial x^2}+\beta S(x,t)I(x,t)-(\mu +\gamma )I(x,t)+\delta R(x,t), \
\frac{\partial R(x,t)}{\partial t} =d \frac{\partial^2R(x,t)}{\partial x^2}+\gamma I(x,t)-(\mu +\delta )R(x,t),
\end{cases}
\end{equation}
in which $x\in\mathbb{R}, t>0, $ $N(x,t)=S(x,t)+I(x,t)+R(x,t)$, and all parameters are positive. We investigate the spreading speeds and traveling waves if $N$ has a wave form. Then the idea is utilized to study the propagation dynamics of the following delayed equation
\[
\frac{\partial u(x,t)}{\partial t}=\frac{\partial ^{2}u(x,t)}{\partial x^{2}}+u(x,t)F(x+ct,u(x,t),(J*u)(x,t)),\quad x\in\mathbb{R},t>0.
\] |
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