Super-linear spreading in local and non-local cane toads equations

Abstract

In this paper, we show super-linear propagation in a nonlocal reaction-diffusion-mutation equation modeling the invasion of cane toads in Australia that has attracted attention recently from the mathematical point of view. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. In this paper, we are concerned with the case when the diffusivity can take unbounded values, and we prove that the population spreads as t 3/2 . We also get the sharp rate of spreading in a related local model.

Super-linear propagation for a general, local cane toads model

“…[8] predicted that the location of the front is of order (4/3)t 3/2 . This was then verified in the local model by Berestycki, Mouhot, and Raoul [6] and by Bouin, Henderson, and Ryzhik [11] using probabilistic and analytic techniques, respectively. It was also shown in [6] that in a windowed non-local model the propagation speed is the same, while [11] obtained weak bounds of order t 3/2 for the full non-local model.…”

“…This was then verified in the local model by Berestycki, Mouhot, and Raoul [6] and by Bouin, Henderson, and Ryzhik [11] using probabilistic and analytic techniques, respectively. It was also shown in [6] that in a windowed non-local model the propagation speed is the same, while [11] obtained weak bounds of order t 3/2 for the full non-local model. A model with a trade-off term, that is, a penalization for large of traits, has been proposed and studied by Bouin, Chan, Henderson, and Kim [9].…”

“…It is well-known that these two properties imply that D is a homogeneous polynomial. The case most often considered is, up to translation in θ, D(θ) = θ + θ for some θ > 0, whence D(θ) = θ, see [6,8,11]. A non-trivial example is D(θ) = θ(1 + log(θ + 1) + sin(θ)), (1.2) which, despite having arbitrarily large oscillations, nevertheless satisfies D(θ) = θ.…”

Accelerating invasions along an environmental gradient

“…When the trait space is unbounded, propagation of the level sets of order O(t 3/2 ) has been predicted in [13]. This was then proved rigorously in [12] with a local competition term, and in [14] for both local and nonlocal (in trait) competition, using probabilistic and analytic arguments respectively. Notice however that acceleration is not induced here by initial heavy tails, but by a phenotype-dependent term before the spatial diffusion.…”

“…Typical examples are "lighter heavy tails" (13), algebraic tails (14), and "very heavy tails" (15), that is…”