Environmental Engineering Reference
In-Depth Information
Chapter 7
Partial Differential Equations
Michael Sieber and Horst Malchow
Abstract Spatially homogeneous processes of change are the subject of the pre-
ceding chapter. Partial differential equations are one method to model the interplay
of these processes with spatial phenomena such as movement of individuals and/or
a heterogeneous environment. Random motion of organisms might be described as
diffusion, and directed motion as advection. The latter can be composed of loco-
motion and motion of the surrounding medium. The focus of this chapter is on
classical systems of no more than two interacting and diffusing populations. The
potential of such systems to exhibit spatiotemporal pattern formation is studied.
7.1
Introduction
Mathematical models for spatially homogeneous processes, with their potential for
multiple steady states and complicated temporal dynamics, are the cornerstone of
ecological modelling (see Chap. 6). Yet only by taking into account the spatial
dimension of species growth, interaction, locomotion and transport is the full diver-
sity of population dynamics realized. The possible spatiotemporal dynamics include
stationary and dynamic patchy patterns, regular and irregular oscillations, propagat-
ing fronts, target patterns and spiral waves amongst others. Historically, possibly the
best-known examples for spatiotemporal patterns come from physics and physical
chemistry, cf. the B´nard convection cells (B´nard 1900) and the waves in the
Belousov-Zhabotinskii reaction (Belousov 1959). Similar patterns have been found
in biological and, in particular, population-dynamical systems, such as the bio-
convection of up-swimming microorganisms (Hill and Pedley 2005; Wager 1911),
travelling waves in cyclic populations (Sherratt and Smith 2008), the wavy dynamics
of amoeba (Gerisch 1968) and striped vegetation patterns (White 1971), cf. Fig.
7.1
.
Partial differential equations (PDEs) are one way to incorporate the spatial dimension
