Transient dynamics and pattern formation: Reactivity is necessary for Turing instabilities. Numerical Solution of Partial Differential Equations. A nonstandard finite difference scheme for a Fisher PDE having nonlinear diffusion. Spatiotemporal complexity of plankton and fish dynamics. Medvinsky, A., Petrovskii, S., Tikhonova, I., Malchow, H., Li, B.-L., 2002. Stability and Complexity in Model Ecosystems. Dynamical stabilization of an unstable equilibrium in chemical and biological systems. Convergence and stability analysis of an explicit finite difference method for 2-dimensional reaction–diffusion equations. Predator functional responses: Discriminating between handling and digesting prey. Fully discrete stability and invariant rectangular regions for reaction–diffusion systems. Experimental Ecology of the Feeding Fishes. Partial differential equations in ecology: Spatial interactions and population dynamics. The functional response of predators to prey density and its role in mimicry and population regulation. Some characteristics of simple types of predation and parasitism. Stability and convergence of finite difference methods for systems of nonlinear reaction–diffusion equations. Finite-Difference Equations and Simulations. Circles and spirals: Population persistence in a spatially explicit predator–prey model. Gurney, W., Veitch, A., Cruickshank, I., McGeachin, G., 1998. Functional responses for zooplankton feeding on multiple resources: A review of assumptions and biological dynamics. Gentleman, W., Leising, A., Frost, B., Strom, S., Murray, J., 2003. Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response. Real World Appl., submitted for publication. Analysis of two generic spatially extended predator–prey models. 57: Monographs and Textbooks in Pure and Applied Mathematics. Deterministic Mathematical Models in Population Ecology. The global dynamics of discrete semilinear parabolic equations. North-Holland, Amsterdam.Įlliott, C., Stuart, A., 1993. 4: Studies in Mathematics and its Applications. The Finite Element Method for Elliptic Problems. The Mathematical Theory of Finite Element Methods. On a uniformly accurate finite difference approximation of a singularly perturbed reaction–diffusion problem using grid equidistribution. Implicit–explicit methods for time-dependent partial differential equations. Mutual interference between predators can give rise to Turing spatial patterns. Users can download, edit, and run the codes from, to investigate the key dynamical properties of spatially extended predator–prey interactions.Īlonso, D., Bartumeus, F., Catalan, J., 2002. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs M ATLAB. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. Furthermore, there are implementational advantages of the methods. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. Prey distributions are often clumped, and predators respond by looking for patches where prey is dense and then searching within patches (Kramer 2001).We present two finite-difference algorithms for studying the dynamics of spatially extended predator–prey interactions with the Holling type II functional response and logistic growth of the prey. \): The black-browed albatross regularly flies hundreds of kilometers across the nearly empty ocean to find patches of food.
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