TY - JOUR
T1 - Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator–prey model
AU - Bürger, Raimund
AU - Ruiz-Baier, Ricardo
AU - Tian, Canrong
PY - 2017/2/1
Y1 - 2017/2/1
N2 - Time delay is an essential ingredient of spatio-temporal predator–prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator–prey reaction–diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction–diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge–Kutta scheme.
AB - Time delay is an essential ingredient of spatio-temporal predator–prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator–prey reaction–diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction–diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge–Kutta scheme.
KW - Finite volume element discretization
KW - Limit cycle
KW - Pattern selection
KW - Spatio-temporal patterns
KW - Time delay
UR - http://www.scopus.com/inward/record.url?scp=85002125720&partnerID=8YFLogxK
U2 - 10.1016/j.matcom.2016.06.002
DO - 10.1016/j.matcom.2016.06.002
M3 - Article
AN - SCOPUS:85002125720
SN - 0378-4754
VL - 132
SP - 28
EP - 52
JO - Mathematics and Computers in Simulation
JF - Mathematics and Computers in Simulation
ER -