title: The diffusive Lotka-Volterra predator-prey system with delay.
abstract: Semi-analytical solutions for the diffusive Lotka-Volterra predator-prey system with delay are considered in one and two-dimensional domains. The Galerkin method is applied, which approximates the spatial structure of both the predator and prey populations. This approach is used to obtain a lower-order, ordinary differential delay equation model for the system of governing delay partial differential equations. Steady-state and transient solutions and the region of parameter space, in which Hopf bifurcations occur, are all found. In some cases simple linear expressions are found as approximations, to describe steady-state solutions and the Hopf parameter regions. An asymptotic analysis for the periodic solution near the Hopf bifurcation point is performed for the one-dimensional domain. An excellent agreement is shown in comparisons between semi-analytical and numerical solutions of the governing equations.
authors.affiliation: Mathematics and Statistics Department, Faculty of Science, University of Taif, Taif, Saudi Arabia.
authors.forename: K S
authors.initials: KS
authors.lastname: Al Noufaey
authors.affiliation: School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, N.S.W. 2522, Australia. Electronic address: tim_marchant@uow.edu.au.
authors.forename: T R
authors.initials: TR
authors.lastname: Marchant
authors.affiliation: School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, N.S.W. 2522, Australia.
authors.forename: M P
authors.initials: MP
authors.lastname: Edwards
mesh_terms: D000818:Animals; D017753:Ecosystem; D020387:Food Chain; D055641:Mathematical Concepts; D008954:Models, Biological; D011156:Population Density; D011235:Predatory Behavior
keywords: Hopf bifurcations; Lotka–Volterra predator–prey model; Reaction–diffusion equations; Semi-analytical solutions