TY - JOUR
T1 - Stable patterns with jump-discontinuity for a phytoplankton–zooplankton system with both Allee and fear effect
AU - Zhang, Conghui
AU - Lu, Jin
AU - Liu, Maoxing
AU - Zhang, Hanzhi
N1 - Publisher Copyright:
© 2024 Elsevier B.V.
PY - 2025/2
Y1 - 2025/2
N2 - This paper is concerned with a phytoplankton–zooplankton system with both Allee and fear effect, in which zooplankton species diffuse but phytoplankton species do not diffuse. We show that this system may lead to a novel pattern formation phenomenon, i.e., far-from-the equilibrium patterns with jump discontinuity. Moreover, the L∞-stability of these discontinuous stationary solutions are demonstrated under appropriate conditions. In addition, we explore how diffusion, Allee and fear effect affect the system. Our results illustrate that (i) if both species diffuse, then the origin and the positive equilibrium are stable. Furthermore, no discontinuous stationary solutions exist; (ii) in the absence of Allee effect, the phenomenon of bistability disappears and only the positive equilibrium is stable. Besides, any discontinuous stationary solutions may be unstable; (iii) when excluding fear effects from the system, the density of zooplankton will be changed, more precisely, as fear costs increase, zooplankton population density declines.
AB - This paper is concerned with a phytoplankton–zooplankton system with both Allee and fear effect, in which zooplankton species diffuse but phytoplankton species do not diffuse. We show that this system may lead to a novel pattern formation phenomenon, i.e., far-from-the equilibrium patterns with jump discontinuity. Moreover, the L∞-stability of these discontinuous stationary solutions are demonstrated under appropriate conditions. In addition, we explore how diffusion, Allee and fear effect affect the system. Our results illustrate that (i) if both species diffuse, then the origin and the positive equilibrium are stable. Furthermore, no discontinuous stationary solutions exist; (ii) in the absence of Allee effect, the phenomenon of bistability disappears and only the positive equilibrium is stable. Besides, any discontinuous stationary solutions may be unstable; (iii) when excluding fear effects from the system, the density of zooplankton will be changed, more precisely, as fear costs increase, zooplankton population density declines.
KW - Allee and fear effect
KW - Phytoplankton–zooplankton system
KW - Stability
KW - Stationary solutions with jump discontinuity
UR - http://www.scopus.com/inward/record.url?scp=85211350724&partnerID=8YFLogxK
U2 - 10.1016/j.physd.2024.134481
DO - 10.1016/j.physd.2024.134481
M3 - Article
AN - SCOPUS:85211350724
SN - 0167-2789
VL - 472
JO - Physica D: Nonlinear Phenomena
JF - Physica D: Nonlinear Phenomena
M1 - 134481
ER -