Explicit Continuous Space Reaction-Diusion Models.

`space has a key eect in controlling the dynamics of the populations that diuse in it'- Bas- compte & Solee 1994

While certain ecological systems are inherently patchy, others, especially marine systems, exist in relatively homogeneous environments or display continuous environmental gradients. Such situations motivate explicit treatment of spatial dimensions. Space as a continuum is mod- elled by partial dierential equations, which are often referred to as reaction-diusion (RD) systems (Levin & Segel, 1985 Holmes et al., 1994). In ecological terms, the reaction is the local population dynamic and the diusion is the regional or global dispersal of populations (gure 2b).

The key result of RD theory is the emergence of spatiotemporal patterns by the mechanism of dispersive or diusive instability. If a stable uniform equilibrium exists in the absence of diusion, then spatially inhomogeneous patterns may arise through instabilities driven by diusion13. The necessary conditions for the existence of non-uniform patterns have been

addressed in many studies. A class of systems admitting such heterogeneities involve activator- inhibitors (Tsonis et al., 1989), which contain appropriate feedback mechanisms, where the activators and inhibitors are, for example, chemicals or competing species. The original paper addressing this phenomenon concerns morphogenesis (Turing, 1952) and much further work

13Segel & Jackson, 1972 Evans, 1980 Cohen & Murray, 1981 Murray, 1981 McLaughlin & Roughgarden,

with spatial dierential equations in developmental biology has subsequently been done14.

Further applications in physiology include development of feather, shell and scale patterns (Murray et al., 1983 Murray & Oster, 1984 Meinhardt & Klingler, 1987 Cruywagen et al., 1994), animalcoat markings15, cell chemotaxis (Grindod et al., 1989 Murray, 1993a), modelling

of the nervous system (Swindale, 1980) and tumour growth (Chaplain, 1993 1994). Wound healing is also an interesting and highly applicable example of RD modelling. Understanding of the biochemical mechanisms by which wounds are dealt with by an animal has been increased and the optimal geometries for surgical incisions for minimal scarring have been explored with these approaches (Murray & Oster, 1984 Murray, 1993b).

Following on from the studies of spatiotemporal RD patterns in physiology have come ecolog- ical RD systems16, where applications have included predator-prey, plant-herbivore and host-

parasite systems17. As with patch models, continuum models have produced many results on

competitive communities and in particular the promotion of coexistence and stability by spatial extensiveness18, with results being sensitive to the size of spatial domains. Another example

from physiology illustrates this domain dependence well. Pattern formation in mammalian coats has been shown to reect the size of the animal. As the size increases a transition is seen between uniform colours and spots (Murray, 1993a). Similarly, postulated mechanochemical processes in the model explain the existence of animals with spotted bodies but striped tails, with the need for only a single mechanism acting in the dierent geometries (Murray, 1993c). A second phenomenon concerns the existence of spatial wave solutions. These are of two types: travelling waves and kinematic waves (Murray, 1989). Travelling waves result from

14Maginu, 1975 Murray, 1982 Murray & Oster, 1984 Murray, 1989, Maini, 1993b 1994. 15Bard, 1981 Murray, 1981 Murray & Oster, 1984 Murray, 1988 Savic, 1995.

16Levin & Segel, 1976 Rothe, 1976 Mimura & Murray, 1978 Shigesada et al., 1979 Evans, 1980 Mimura &

Kawasaki, 1980 Cohen & Murray, 1981 Kishimoto, 1981 1982 McLaughlin & Roughgarden, 1991.

17Segel & Jackson, 1972 Comins & Blatt, 1974 Levin & Segel, 1976 Rothe, 1976 Mimura & Murray, 1978

Ludwig et al., 1979 Kishimoto, 1982 McLaughlin & Roughgarden, 1991.

18Levin, 1974 Hastings, 1978b Ludwig et al., 1979 Shigesadaet al., 1979 Mimura & Kawasaki, 1980 Namba,

the transition between two stable equilibria of a non-spatial model, which occurs as diusion or convection is added kinematic waves arise from coupling of oscillators. A travelling wave moves through space without changing shape, whereas a kinematic wave deforms through space and time. While simple diusion leads to wave propagation at extremely slow speeds, which do not allow the level of information transfer observed in ecological or physiological systems, the addition of coupled local reaction dynamics increases wave speeds by many orders of magnitude, to realistic biological levels.

A wide range of wave solutions in spatial biological systems have been studied19: ecological

applications generally concern waves of invasion of species or waves of pursuit and evasion in predator-prey systems, while epidemiology considers waves of disease or pathogens. The stan- dard example of an oscillating system with spatial wave patterns is the Belousov-Zhabotinsky (BZ) reaction, a relatively straightforward chemical reaction which displays intricate spiral wave patterns (Zaikin & Zhabotinsky, 1970 Winfree, 1974 Hagan, 1982 Roux et al., 1981). Specic ecological examples include the spread of the grey squirrel (Okubo et al., 1989), the spread of rabies in foxes (Murray et al., 1986),waves of herbivores (Lewis, 1994). and the activities of colonies of social amoebae, such as the slime moldDictyostelium discoideium, that exhibit spiral waves via chemotactic mechanisms20.

While most models here involve the emergence of spatial patterns and other phenomena in homogeneous environments, a few studies have considered heterogeneous environments (Mc- Laughlin & Roughgarden, 1991 Shigesada et al., 1979) and environmental gradients (Comins & Blatt, 1974). In addition to its role in the production of patterns in uniform environments, diusion has been shown to be an amplier of pre-existing heterogeneities (McLaughlin & Roughgarden, 1991).

19Skellam, 1951 Maginu, 1975 Murray, 1975 Britton, 1982a Dunbar, 1983 Nagai & Mimura, 1983 Murray,

1989.

20Keller & Segel, 1970 Britton, 1982a Hagan, 1982 Monk & Othmer, 1989 Goldbeter, 1993 Martiel, 1993