The Bramson logarithmic delay in the cane toads equations

Abstract

We study a non-local reaction-diffusion-mutation equation modelling the spreading of a cane toads population structured by a phenotypical trait responsible for the spatial diffusion rate. When the trait space is bounded, the cane toads equation admits travelling wave solutions as shown in an earlier work of the first author and V. Calvez. Here, we prove a Bramson type spreading result: the lag between the position of solutions with localized initial data and that of the travelling waves grows as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 3 slash left-parenthesis 2 lamda Superscript asterisk Baseline right-parenthesis right-parenthesis log t"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> <mml:mi>log</mml:mi> <mml:mo>⁡</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">(3/(2\lambda ^*))\log t</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result relies on a present-time Harnack inequality which allows one to compare solutions of the cane toads equation to those of a Fisher-KPP type equation that is local in the trait variable.