TY - JOUR

T1 - Classification of the spreading behaviors of a two-species diffusion-competition system with free boundaries

AU - Du, Yihong

AU - Wu, Chang Hong

N1 - Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/2

Y1 - 2022/2

N2 - In this paper, we revisit the spreading behavior of two invasive species modelled by a diffusion-competition system with two free boundaries in a radially symmetric setting, where the reaction terms depict a weak-strong competition scenario. Our previous work (Du and Wu in Cal Var PDE 57:52, 2018) proves that from certain initial states, the two species develop into a “chase-and-run coexistence” state, namely the front of the weak species v propagates at a fast speed and that of the strong species u propagates at a slow speed, with their population masses largely segregated. Subsequent numerical simulations in Khan et al. (J Math Biol 83:23, 2021) suggest that for all possible initial states, only four different types of long-time dynamical behaviours can be observed: (1) chase-and-run coexistence, (2) vanishing of u with v spreading successfully, (3) vanishing of v with u spreading successfully, and (4) vanishing of both species. In this paper, we rigorously prove that, as the initial states vary, there are exactly five types of long-time dynamical behaviors: apart from the four mentioned above, there exists a fifth case, where both species spread successfully and their spreading fronts are kept within a finite distance to each other all the time. We conjecture that this new case can happen only when a parameter takes an exceptional value, which is why it has eluded the numerical observations of Khan et al. (J Math Biol 83:23, 2021).

AB - In this paper, we revisit the spreading behavior of two invasive species modelled by a diffusion-competition system with two free boundaries in a radially symmetric setting, where the reaction terms depict a weak-strong competition scenario. Our previous work (Du and Wu in Cal Var PDE 57:52, 2018) proves that from certain initial states, the two species develop into a “chase-and-run coexistence” state, namely the front of the weak species v propagates at a fast speed and that of the strong species u propagates at a slow speed, with their population masses largely segregated. Subsequent numerical simulations in Khan et al. (J Math Biol 83:23, 2021) suggest that for all possible initial states, only four different types of long-time dynamical behaviours can be observed: (1) chase-and-run coexistence, (2) vanishing of u with v spreading successfully, (3) vanishing of v with u spreading successfully, and (4) vanishing of both species. In this paper, we rigorously prove that, as the initial states vary, there are exactly five types of long-time dynamical behaviors: apart from the four mentioned above, there exists a fifth case, where both species spread successfully and their spreading fronts are kept within a finite distance to each other all the time. We conjecture that this new case can happen only when a parameter takes an exceptional value, which is why it has eluded the numerical observations of Khan et al. (J Math Biol 83:23, 2021).

UR - http://www.scopus.com/inward/record.url?scp=85124384864&partnerID=8YFLogxK

U2 - 10.1007/s00526-021-02170-8

DO - 10.1007/s00526-021-02170-8

M3 - Article

AN - SCOPUS:85124384864

SN - 0944-2669

VL - 61

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

IS - 2

M1 - 54

ER -