Complex patterns in a predator–prey model with self and cross-diffusion

Reprint of Domain decomposition multigrid methods for nonlinear reaction–diffusion problems

“…In particular, they have been used along the years to model chemical systems or population dynamics (see, e.g., [29,30,47]). A classical example is given by the Fisher equation, for which rðuÞ ¼ buð1 À uÞ and f 0, with b > 0, first proposed as a one-dimensional model for the propagation of a mutant gene (cf.…”

Pattern Formation in a Diffusive Ratio-Dependent Holling-Tanner Predator-Prey Model with Smith Growth

“…Segel and Jackson [4] first used reaction-diffusion system to explain pattern formation in ecological context based upon the seminal work by Turing [3]. Since then, a lot of studies have been devoted to spatiotemporal patterns which were produced by reactiondiffusion predator-prey, models with either a prey-dependent or a ratio-dependent predator functional response, for example, [1,2,[5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and references cited therein.…”

Self-Organized Patterns Induced by Neimark-Sacker, Flip and Turing Bifurcations in a Discrete Predator-Prey Model with Lesie-Gower Functional Response

“…A classic example is the discrete logistic model that exhibits period-doubling cascade and a route to chaos [21]. On the contrary, the continuous logistic model shows a simple "S" form curve, but never demonstrates the above dynamic complexity.…”

“…Different reaction-diffusion models focus on different functional responses that describe different predation relationships. There are a variety of responses [18], such as Lesie-Gower functional responses [19,20] Beddington-DeAngelis functional responses [14,[21][22][23][24] and so on. Among these functional responses, the Leslie-Gower formulation is based on the assumption that a reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food.…”