Travelling wave solutions in a negative nonlinear diffusion–reaction model

We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.

Minimum wave speeds in monostable reaction–diffusion equations: sharp bounds by polynomial optimization

Many monostable reaction–diffusion equations admit one-dimensional travelling waves if and only if the wave speed is sufficiently high. The values of these minimum wave speeds are not known exactly, except in a few simple cases. We present methods for finding upper and lower bounds on minimum wave speed. They rely on constructing trapping boundaries for dynamical systems whose heteroclinic connections correspond to the travelling waves. Simple versions of this approach can be carried out analytically but often give overly conservative bounds on minimum wave speed. When the reaction–diffusion equations being studied have polynomial nonlinearities, our approach can be implemented computationally using polynomial optimization. For scalar reaction–diffusion equations, we present a general method and then apply it to examples from the literature where minimum wave speeds were unknown. The extension of our approach to multi-component reaction–diffusion systems is then illustrated using a cubic autocatalysis model from the literature. In all three examples and with many different parameter values, polynomial optimization computations give upper and lower bounds that are within 0.1% of each other and thus nearly sharp. Upper bounds are derived analytically as well for the scalar reaction-diffusion equations.

Response of Immunotherapy to Tumour-TICLs Interactions: A Travelling Wave Analysis

There are several cancers for which effective treatment has not yet been identified. Mathematical modelling can nevertheless point out to clinicians tumour invasion properties that should be targeted to mitigate these cancers. We present a travelling wave analysis of a tumour-immune interaction model with immunotherapy. We use the geometric treatment of an apt-phase space to establish the intersection between stable and unstable manifolds. We calculate the minimum wave speed and numerical simulations are performed to support the analytical results.

On travelling wavefronts of Nicholson's blowflies equation with diffusion

This paper is devoted to the study of Nicholson's blowflies equation with diffusion: a kind of time-delayed reaction diffusion. For any travelling wavefront with speed c > c* (c* is the minimum wave speed), we prove that the wavefront is time-asymptotically stable when the delay-time is sufficiently small, and the initial perturbation around the wavefront decays to zero exponentially in space as x → −∞, but it can be large in other locations. The result develops and improves the previous wave stability obtained by Mei et al. in 2004. The new approach developed in this paper is the comparison principle combined with the technical weighted-energy method. Numerical simulations are also carried out to confirm our theoretical results.