Non-local competition slows down front acceleration during dispersal evolution

Abstract

We investigate super-linear spreading in a reaction-diffusion model analogous to the Fisher-KPP equation, but in which the population is heterogeneous with respect to the dispersal ability of individuals and the saturation factor is non-local with respect to one variable. It was previously shown that the population expands as O(t 3/2 ). We identify a constant α * , and show that, in a weak sense, the front is located at α * t 3/2 . Surprisingly, α * is smaller than the prefactor predicted by the linear problem (that is, without saturation) and analogous problem with local saturation. This hindering phenomenon is the consequence of a subtle interplay between the non-local saturation and the non-trivial dynamics of some particular curves that carry the mass to the front. A careful analysis of these trajectories allows us to characterize the value α * . The article is complemented with numerical simulations that illustrate some behavior of the model that is beyond our analysis.Résumé. -Nous examinons le phénomène de propagation surlinéaire pour un modèle de réaction-diffusion analogue à l'équation de Fisher-KPP, mais pour lequel la population est hétérogène vis-à-vis du taux de dispersion de chaque individu, et de plus, le terme de saturation est non-local par rapport à la variable de dispersion. Il avait été démontré que la population s'étend comme un O(t 3/2 ). Ici, nous identifions une constante α * telle que le front d'expansion est localisé autour de α * t 3/2 , dans un sens faible. Curieusement, la constante α * est strictement inférieure au préfacteur obtenu à partir du problème linéarisé (en omettant la saturation), ce dernier coïncidant par ailleurs avec celui obtenu à partir du problème avec saturation locale. Ce phénomène de ralentissement est la conséquence d'une interaction subtile entre la saturation non-locale et la dynamique non-triviale de certaines trajectoires qui amènent la masse au front d'invasion. Une analyse très précise de ces courbes particulières nous permet de caractériser algébriquement la valeur de α * . En complément de ce travail, des simulations numériques viennent illustrer le comportement attendu des solutions, au-delà des résultats analytiques. Strategy of proofThis section is intended to sketch the main ideas underlying the proof. As a byproduct, we explain, in rough terms, the reason for slower propagation than linearly determined.

Adaptation of a quantitative trait to a changing environment: new analytical insights on the asexual and infinitesimal sexual models

“…To understand this relationship, the approximation of the phenotype distribution appeared necessary. This approach is indeed robust, as shown by several studies following the same methodology in spatial structured population models: discrete patches ([Mirrahimi, 2017] in asexual model and [Dekens, 2020] in the infinitesimal sexual model); dispersal evolution ([Perthame and Souganidis, 2016, Lam and Lou, 2017, Lam, 2017, W Hao, 2021, Calvez et al, 2018 in the asexual case and [Dekens and Lavigne, 2021] in the infinitesimal sexual case). Moreover, this methodology is expected to be efficient to investigate other structured population models.…”

Adaptation of a quantitative trait to a changing environment: New analytical insights on the asexual and infinitesimal sexual models

“…To understand this relationship, the approximation of the phenotype distribution appeared necessary. This approach is indeed robust, as shown by several studies following the same methodology in spatial structured population models: discrete patches ( (Mirrahimi, 2017) with an asexual model and (Dekens, 2020) with the infinitesimal sexual model); dispersal evolution ( (Perthame and Souganidis, 2016;Lam and Lou, 2017;Lam, 2017;W Hao, 2021;Calvez et al, 2022a;Lam et al, 2022) in the asexual case and (Dekens and Lavigne, 2021) in the infinitesimal sexual case). Moreover, this methodology is expected to be efficient to investigate other structured population models.…”

A Liouville-Type Result for Non-cooperative Fisher–KPP Systems and Nonlocal Equations in Cylinders

“…where u(t, x, θ) is a population density structured with respect to a phenotypic trait θ ∈ [θ, θ] ⊂ [0, +∞]. This eco-evolutionary model has also attracted attention recently (e.g., [2,4,6,[8][9][10][11]14,27,31]), especially due to an acceleration phenomenon when d(θ) = θ and θ = +∞ but also because, just like the nonlocal KPP equation, it does not satisfy the comparison principle and requires new techniques.…”