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Traveling waves of a full parabolic attraction-repulsion chemotaxis system with logistic source

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  • In this paper, we study traveling wave solutions of the chemotaxis system

    where $ \tau>0,\chi_{i}> 0,\lambda_i> 0,\ \mu_i>0 $ ($ i = 1,2 $) and $ \ a>0,\ b> 0 $ are constants. Under some appropriate conditions on the parameters, we show that there exist two positive constant $ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) $ such that for every $ c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2)\leq c<c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2, \lambda_2) $, (0.1) has a traveling wave solution $ (u,v_1,v_2)(x,t) = (U,V_1,V_2)(x-ct) $ connecting $ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $ and $ (0,0,0) $ satisfying

    where $ \mu\in (0,\sqrt a) $ is such that $ c = c_\mu: = \mu+\frac{a}{\mu} $. Moreover,

    $ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{**}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = \infty $


    $ \lim\limits_{(\chi_1,\chi_2)\to (0^+,0^+))}c^{*}(\tau,\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2) = c_{\tilde{\mu}^*}, $

    where $ \tilde{\mu}^* = {\min\{\sqrt{a}, \sqrt{\frac{\lambda_1+\tau a}{(1-\tau)_{+}}},\sqrt{\frac{\lambda_2+\tau a}{(1-\tau)_{+}}}\}} $. We also show that (1) has no traveling wave solution connecting $ (\frac{a}{b},\frac{a\mu_1}{b\lambda_1},\frac{a\mu_2}{b\lambda_2}) $ and $ (0,0,0) $ with speed $ c<2\sqrt{a} $.

    Mathematics Subject Classification: 35B35, 35B40, 35K57, 35Q92, 92C17.


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