Stability Of Traveling Waves In Nicholson’s Model With Crossing-monostability

As a class of important reaction-diffusion equation, Nicholson’s blowflies model is derived from the experimental research about Australian lucilia cuprina, and has been extensively studied by many scholars. In the study of reaction-diffusion equations, traveling wave solution is an important topic, which can well describe the propagation with finite speed and the oscillations. Therefore, in this paper we study the stability of traveling waves of the following Nicholson’s model with nonlocal delay: where b(u)=pue-au. When the nonlinear term is monostable (1<p/r2≤e), the stability of traveling wave solutions can be obtained using the weighted energy method combined with the comparison principle. But when the nonlinear term is crossing-monostable (p/r>e), the comparison principle is no longer applicable. So we’ll consider the stability of the traveling waves by means of the weighted energy method combining continuation method. In this paper under the assumption of e<p/r<e9/5, we first state the local existence of solutions of the Cauchy problem for the equation, and establish a priori estimate with the weighted energy method. Then based on the local existence and the priori estimate of solutions, we apply the continuation method to prove the exponential stability of the traveling waves with large speed under the so-called small initial perturbation (i.e. the initial perturba-tion around the traveling wave is suitable small in a weighted norm).