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Bility and traveling wavefronts for a convolution model of mistletoes and birds with nonlocal diffusion

  • Author Bio: E-mail address: monnelh@163.com; E-mail address: tianyl@scnu.edu.cn
Supported by the NSF of China (11171120) and the Natural Science Foundation of Guangdong Province (2016A030313426).
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  • A convolution model of mistletoes and birds with nonlocal diffusion is considered in this paper. We first consider the stability of the constant steady states of the model by linearized method, and then the existence of traveling solutions. The main aim of this article is to challenge the hardness lying in the construction of upper-lowers for wave profile system. With the help of an additional condition, we at last obtain a pair of upper-lower solutions. A constant $c_{*}>0$ is obtained such that traveling wavefronts exist for $c\geq c_{*}$. Amongst the construction, we take advantage of the relation between two components of principle eigenvector for the linearized system to control the two components of upper solution. The method seems novel. Some simulations and discussions are given to illustrate the applications of our main results and the effect of parameters on $c_{*}$. A comparison for $c_{*}$ is also given with two different kernel functions.

    Mathematics Subject Classification: 45K05, 47G20, 92D25.


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  • Figure 1.  The solution of system (5) with parameters in Table 1 and initial value condition: $u_{10}(t, x)=0.1$, $u_{20}(t, x)=0.1$, $t\in[-1,0]$

    Figure 2.  (1) $\lambda_{1}(\nu)=-d_{m}+\frac{\alpha e^{-d_{i}\tau}}{\omega}\bar{k}(\nu)e^{-\lambda_{1}(\nu)\tau}$;
    (2) $\lambda_{2}(\nu)=D\bar{J}(\nu)-1-D$

    Figure 3.  (1) $\mathbf{\Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+\frac{\alpha e^{-d_{i}\tau}}{\omega}\frac{\bar{k}(\nu)}{\nu}e^{-\nu\mathbf{\Phi}_{1}(\nu)\tau}$;

    Figure 4.  (1) $\lambda_1(\nu)=-d_{m}+ \frac{\alpha e^{-d_{i}\tau}}{\omega }\bar{k}(\nu)e^{-\lambda_1(\nu)\tau}$;
    (2) $\hat{\lambda}(\nu)=d\bar{k}(\nu)+D\bar{J}(\nu)-1-D$;
    (3) $\mathbf{ \Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+ \frac{\alpha e^{-d_{i}\tau}}{\omega }\frac{\bar{k}(\nu)}{\nu}e^{-\lambda_1(\nu)\tau}$

    Figure 5.  (1) $\mathbf{\Phi}_{1}(\nu)=-\frac{d_{m}}{\nu}+\frac{\alpha e^{-d_{i}\tau}}{\omega\nu}e^{-\nu\mathbf{\Phi}_{1}(\nu)\tau}$;
    (2) $\mathbf{\Phi}_{2}(\nu)=\frac{D\bar{J}(\nu)-1-D}{\nu}$

    Figure 6.  The traveling wave solution found with parameters in Table 1 and initial value condition: $u_{10}(t, x)=0.001$, $u_{20}(t, x)=0.001$, $t\in[-1,0]$

    Table 1.  Parameter values for simulations

    $k(y)$ $J(y)$ $\bar{k}(\nu)$ $\bar{J}(\nu)$ $d_{m}$ $\omega$ $\alpha$ $d$ $d_{i}$ $\tau$ $D$
    $\frac{1}{\sqrt{4\pi}}e^{-\frac{y^{2}}{4}}$ $\frac{1}{\sqrt{4\pi}}e^{-\frac{y^{2}}{4}}$ $e^{\nu^{2}}$ $e^{\nu^{2}}$ 0.1 1 0.7 0.3 0.3 1 0.5
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