Geometric singular perturbation of a nonlocal partially degenerate model for <i>Aedes aegypti</i>

Abstract

&lt;p style='text-indent:20px;'&gt;This paper is devoted to investigate the existence of traveling wave solutions for a partially degenerate &lt;i&gt;Aedes aegypti&lt;/i&gt; model with nonlocal effects. By taking specific kernel forms and time scale transformation, we transform the nonlocal model into a singularly perturbed system with small parameters. A locally invariant manifold for wave profile system is obtained with the aid of the geometric singular perturbation theory, and then the existence of traveling wave solutions is proved provided that the basic reproduction number &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \mathcal{R}_0&amp;gt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; through utilizing the Fredholm orthogonal method. Furthermore, we study the asymptotic behaviors of traveling wave solution with the help of asymptotic theory. The methods used in this work can help us overcome the difficulty that the solution map associated with the system is not compact. Numerically, we perform simulations to demonstrate the theoretical results.&lt;/p&gt;