Spatiotemporal dynamics of two generic predator–prey models

Abstract

We present the analysis of two reaction-diffusion systems modelling predator-prey interactions, where the predator displays the Holling type II functional response, and in the absence of predators, the prey growth is logistic. The local analysis is based on the application of qualitative theory for ordinary differential equations and dynamical systems, while the global well-posedness depends on invariant sets and differential inequalities. The key result is an L(∞)-stability estimate, which depends on a polynomial growth condition for the kinetics. The existence of an a priori L(p)-estimate, uniform in time, for all p ≥ 1, implies L(∞)-uniform bounds, given any nonnegative L(∞)-initial data. The applicability of the L(∞)-estimate to general reaction-diffusion systems is discussed, and how the continuous results can be mimicked in the discrete case, leading to stability estimates for a Galerkin finite-element method with piecewise linear continuous basis functions. In order to verify the biological wave phenomena of solutions, numerical results are presented in two-space dimensions, which have interesting ecological implications as they demonstrate that solutions can be 'trapped' in an invariant region of phase space.

Fourier spectral method for higher order space fractional reaction–diffusion equations

“…In the work of Murray [19] and other researchers [18,30,31,32], studies have shown that diffusion contributes in no small measure to stability in a system, it increases the concentrations of lower regions, and decreases concentrations of the higher regions. As a result, pattern formation in the form of nonhomogeneous steady-state and periodic solutions is eminent.…”

“…There will be no diffusion-driven instability. Proof Known that the point (u * , v * ) = (1, 0) is the homogeneous solution for system (31). The matrix J for the Gray-Scott reaction-diffusion system (31) is…”

Global existence for semilinear reaction–diffusion systems on evolving domains

“…Remark (Further applications) The analysis can be applied to a large number of problems unrelated to the theory of pattern formation. Garvie and Trenchea [2009] provide an example applicable to ecology and the aforementioned paper of Morgan [1989] contains further examples as well as the numerous citations of said paper that use the approach on various problems.…”

Numerical simulations of two components ecological models with sinc collocation method

“…Jiang et al in [14] consider a predator-prey model with Beddington-DeAngelis functional response subject to the homogeneous Neumann boundary condition. Moreover, Wang et al [15] investigated the spatial pattern formation of a predator-prey system with prey-dependent functional response of Ivlev type and reaction-diffusion whereas the analysis of predator-prey systems showing the Holling type II functional response is examined in Garvie and Trenchea [16]. Moreover, we can point to other efficient numerical methods for solving the reaction-diffusion equations arising in biology such as [17,18,19,20,21].…”