Waves analysis and spatiotemporal pattern formation of an ecosystem model

Existence and non-existence of spatial patterns in a ratio-dependent predator–prey model

“…Stationary and non-stationary patterns can be classified as hot spot pattern, cold spot pattern, labyrinthine pattern, stripe pattern, target pattern, spiral pattern, tip-splitting pattern, interacting spiral pattern and chaotic pattern Cantrell and Cosner, 2003;Malchow et al, 2008;Shoji et al, 2003;Shoji and Iwasa, 2005;Upadhyay et al, 2010). These wide varieties of patterns are already reported by several researchers based upon their works on reaction-diffusion models of predator-prey interactions (Alonso et al, 2002;Banerjee and Banerjee, 2012;Banerjee and Petrovskii, 2011;Baurmann et al, 2004Baurmann et al, , 2007Camara, 2011;Fasani and Rinaldi, 2011;Petrovskii et al, 2004;Petrovskii and Malchow, 1999;Sherratt et al, 1997;Wang et al, 2007). Interestingly one can find only spot pattern for parameter values lying within the Turing domain only and rest of the patterns appears for parameter values within the Turing-Hopf domain, outside the Turing domain and also as a result of specific choices of initial condition (Malchow et al, 2008;Medvinsky et al, 2002).…”

“…The reaction-diffusion models of population interaction with appropriate initial and boundary conditions are capable to produce spatial patterns due to Turing instability (Murray, 2002;Malchow et al, 2008;Okubo and Levin, 2001). Apart from the stationary Turing pattern formation, the nonstationary and spatiotemporal chaotic patterns are also capable to explain the patchy distribution of the species Banerjee and Petrovskii, 2011;Baurmann et al, 2004Baurmann et al, , 2007Camara, 2011;Morozov et al, 2004;Petrovskii et al, 2004;Petrovskii and Malchow, 1999;Sherratt et al, 1997Sherratt et al, , 1995Tian, 2010;Tian andUpadhyay et al, 2012).…”

Shearing in flow environment promotes evolution of social behavior in microbial populations

“…Diffusion can cause an instability that leads to the formation of intriguing patterns Turing ( 1990 ), which among other fields, have been investigated in ecological context Tian et al ( 2011 ); Camara ( 2011 ); Baurmann et al ( 2007 ). These so called Turing patterns typically form when an inhibiting agent has a diffusion length greater than that of an activating agent.…”

Mathematical analysis and numerical simulation of pattern formation under cross-diffusion

“…[13][14][15][16][17]). Recently, the pattern formation mechanisms for (1.1) without cross-diffusion have been investigated by [18][19][20][21][22][23][24]. Shi et al [25] showed that cross-diffusion can destabilize or stabilize a uniform equilibrium in a reaction-diffusion system.…”