The diffusive Lotka–Volterra predator–prey system with delay

• Highly accurate approximate solutions obtained for the diffusive delay Lotka–Volterra predator–prey system.
• Maps of the parameter space, in which Hopf bifurcations occur, are obtained for both one and two-dimensional domains.
• Asymptotic analysis near the Hopf point gives analytical expressions for the leading order limit cycle.
• Method for obtaining approximate solutions applicable to a wide range of population models, with biological and ecological applications.

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.