Stability in a diffusive food-chain model with Michaelis–Menten functional response

Instability induced by cross-diffusion in reaction–diffusion systems

“…the solution (u * 1 , u * 2 ) of the nonlinear system (3.1) loses its stability. We consider the instability of (3.4) using the method similarly as in [18,19]. Let 0 = µ 1 < µ 2 < · · · be the eigenvalues of the operator −∆ on a bounded domain Ω with the no-flux boundary condition, and the linearization of (3.5) is equivalent to …”

Dynamics of food-chain models with density-dependent diffusion and ratio-dependent reaction function

“…In recent years, attention has been given to one-prey two-predator systems with various types of reaction functions, including ratio-dependent functional response (cf. [2,7,8,[12][13][14]25]). However, most of the discussions are either for semilinear reaction diffusion systems or for ordinary differential systems with either the traditional Lotka-Volterra type of reaction functions or ratio-dependent reaction functions (cf.…”

“…The same system was later extended by the authors to Dirichlet boundary condition in ( [13]). A similar 3-equation semilinear system with Michaelis-Menton reaction function was considered in [14] using Lyapunov method and in [7] using semi-group theory. In this paper, we consider the quasilinear systems (1.1) to (1.4) where m i > 1 for some or all i, and the boundary condition is either Dirichlet type (α i = 0, β i = 1) or Neumann-Robin type (α i = 1, β i ≥ 0).…”

Permanence and stability of a diffusive predator–prey model with disease in the prey

“…Predator-prey interaction is one of the well known fundamental structures in population dynamics. Studies for various predator-prey models are significant in theory and applications (see [1][2][3][4][5][6]). In parallel, epidemic models have been studied extensively (see [7][8][9]) since the introduction of the classical SIR (susceptible-infective-removal) model by Kermack and Mckendrick in 1927. To learn how a disease affects the dynamical behaviors of predator-prey, there have been some eco-epidemiological models introduced and studied.…”