Patterned solutions of a homogenous diffusive predator-prey system of Holling type-III

Nonexistence of nonconstant positive steady states of a diffusive predator-prey model

“…and the existence of spatially inhomogeneous periodic orbits and nonconstant steady state solutions were obtained by the bifurcation method and Leray-Schauder degree theory. The Hopf bifurcation of model (1.3) was also investigated in [26,32]. We remark that there are also many results on the pattern formations for other diffusive predator-prey models and chemical reaction-diffusion models, see [1,13,17,18,20,21,23,28,29,30,31,34] and references therein.…”

Complexity Analysis of a Modified Predator-Prey System with Beddington–DeAngelis Functional Response and Allee-Like Effect on Predator

“…en, there is a chance of bifurcation around this higherorder singular point. Here we choose m 2 as a bifurcation parameter and select eigenvector v � 0 1 corresponding to the zero eigenvalue for matrix (30). e eigenvector corresponding to the zero eigenvalue for the transpose of matrix…”

“…Here D 1 and D 2 are two positive diffusive constants; Δ is the Laplacian operator; u � u(t, x) and v � v(t, x) are the densities of prey and predator, respectively; Ω is an one-dimensional bounded domain with smooth boundary z(Ω); the symbol z ] is the outer flux, and no flux boundary condition is imposed; thus, the system is closed [29]; the admissible initial functions u 0 (x) and v 0 (x) are all continuous functions on Ω; to describe an environment surrounded by dispersal barriers, we take zero flux at z(Ω) [28]. e method of calculating the first Lyapunov coefficient l 1 in Section 5.3 is a reference for deducing of Hopf bifurcation direction in reaction-diffusion systems [29,30]. ough the dynamical behavior of predator-prey systems in single species or multispecies has been researched by many previous literatures, we still need further study in biomathematics, especially the phytoplankton and zooplankton systems.…”