Length Scale Analysis.

For investigating the coherence length scale, the model was run on a 300300 grid with a mixed

population. N = 300 was, however, extremely computationally intensive, so the error analysis was only performed for one competition type (absolute asymmetry). The period over which the error analysis was carried out is T = 50 years, starting after t0 = 50 years. A constant k

was tted to nEn(section 2.1.5) for four cases: a mixed population of annuals and perennials,

monocultures of annuals and perennials and a mixed population with reduced annual seed dispersal (gure 23).

The coherence length scale is around nc'125 for the mixed population (gure 23a). There is,

however, very little error above nc '100, so that the grid size of N = 96 used in this chapter

is acceptable, particularly in view of the great computation involved in this model.

The mixed population shows slight negative spatial coherence at small scales (0 < n < 10) as En <E

0

n. Since @@n(nEn) > 0, an aggregating tendency is exhibited as n increases over a

wide range of scales (0 < n < 80), resulting in positive spatial coherence at intermediate scales (10 < n < nc). Disaggregation necessarily occurs at larger scales (75 < n < nc), as the mass

tends towards a random distribution above the coherence length scale (_@@n(nEn) < 0).

The annual monoculture (gure 23b) has a slightly lower coherence length (nc ' 105) and

starts with slightly more negative coherence than the mixture (0 < n < 10), with a stronger aggregating force as n increases for 0 < n < 80. The perennial monoculture (gure 23c) is, in contrast, always shows signicant positive coherence below nc'140 (En>E

0

n) with a very

slight aggregating tendency as n rises at smaller scales (0 < n < 50) and disaggregation at larger scales (50 < n < nc). Thus there is a clear dierence between the annuals, which have

the dispersing eect of seed scattering, and the perennials, whose clumping mechanisms are always dominant.

50 100 150 0 0.2 0.4 0.6 0.8 (a) sub-grid size error 50 100 150 0 0.2 0.4 0.6 0.8 (b) sub-grid size error 50 100 150 0 0.2 0.4 0.6 0.8 (c) sub-grid size error 50 100 150 0 0.2 0.4 0.6 0.8 (d) sub-grid size error

Figure 23: Error analysis for the annual-perennial system: numerical error nEn for the model

(|) and theoretical error nE 0

n (). (a) Total mixed population (Ps = 0:7). (b) Annual

plants only. (c) Perennial plants only. (d) Total mixed population for reduced seed dispersal (Ps= 0:3).

The length scale analysis is not valid above half of the total grid size (n = 150), because of the toroidal boundary conditions, so the results are only shown up to this value. However, the error values for n > 150 do in fact support the results by giving a constant value of nEn.

A lower annual seed scattering distance leads to a weaker negative coherence and to a stronger aggregating tendency as n increases (

@@n(n En)

is greater in gure 23d than in gure 23a),

but the coherence length scale is the same (nc '125). Thus the error analysis identies this

change to a mechanism with greater aggregation.

By examination of the coherence length scale of the model, it is clear that a large lattice is needed: doubt is cast on the many lattice models which have used small grids of sizes as small a 55 (Auld & Coote, 1980 Karlson & Jackson, 1981 Bard, 1981). Each model naturally

requires individual analysis as the coherence length scale can be a low as 15 (section 7.6.1), but this is likely to correspond to near random patterns where a spatial model is probably not needed.

Plant Populations.

Chapter Summary

The plant coupled map lattice of chapters 3 and 4 is extended to incorporate an explicit resource base, through the introduction of a second lattice variable. Resource depletion and crowding suppress plant growth and both positive and negative feedback processes mediate the formation of size hierarchies.

Environmental or resource heterogeneity is introduced using an algorithm for generating dierent sizes of patches a necessary balance of computational speed and accuracy is identied. Patchy resources are shown to lead to suppressed stand yields and increased size variation.

The result of chapter 3, asserting that hierarchy development can be used to discriminate between symmetric and asymmetric competition, is seen to hold with a homogeneous or moderately heterogeneous resource distribution. The model is extended to multiple years, allowing the coherence length scale to be found, demonstrating that a moderately-sized lattice is sucient.

An overview of the ecology of seed sizes is presented, focusing on the opposing selective forces on large seeds, which are sometimes adapted to tolerate adverse conditions, but are also suited to the intense competition of rich habitats. In a ho- mogeneous environment, higher resource levels favour large-seeded species, which exclude the small-seeded species the opposite happens with lower resources. Co- existence of the two species is seen in intermediate environments. Small- and large-seeded species are seen to coexist in a heterogeneous environment with suf- ciently large high and low resource patches.

`And nature must obey necessity'- William Shakespeare - Julius Caesar IV, iii

5.1. Introduction.

Resources - soil nutrients, light and physical space - naturally play a central role in plant ecosystems. The eect of resource availability and heterogeneity on both plant and animal systems have been studied for many years, in natural and agricultural systems47and via patch,

reaction-diusion and lattice models48. Resource patterns have been shown to have fundamen-

tal implications for community structure, species diversity and evolution of physiological and behavioural characters. Spatial and temporal resource heterogeneities challenge the survival of plant species (Fowler, 1988) and consequently dierent ecological types of plants have evolved various mechanisms to overcome unreliability of nutrient supply. Examples are dispersal of propagules to overcome spatial uctuations,dormancyas a response to temporal heterogeneity andphysiological integrationin clonal species to exploit spatial patchiness (Hutchings & Slade, 1988 Caraco & Kelly, 1991 Fahrig et al., 1994).

In chapters 3 and 4 the CML models of plant populations adopt the common assumption that occupation of space is representative of the acquisition of nutrients and light as well as physical space. In this chapter an explicit resource distribution is added to the CML, which aects growth rates and is depleted as individual plants grow. Using a separate variable for a generalised representative nutrient allows dierent environments, both homogeneous and heterogeneous, to be applied to the CML plant populations. Thus the CML can be used to assess the impact of nutrient levels on the structure of plant monocultures during a growing season. An algorithm is added to the CML to deal with heterogeneous resource distributions, in the form of patches of high and low resources. The consequences of such heterogeneities

47Cohen, 1971 Hartnett & Bazzaz, 1983 1985 Ferrara & Quinn, 1987 Inouye & Tilman, 1988 Hocking &

Steer, 1989 Naeem, 1990 Grieve & Francois, 1992 Jackson & Caldwell, 1992 Kelly & Canham, 1992 Jackson & Caldwell, 1993 Robertson et al., 1993 Birch & Hutchings, 1994 Mian & Nafziger, 1994 Gross et al., 1995 Kadmon, 1995.

48Horn & McArthur, 1972 Comins & Blatt, 1974 Ludwig et al., 1979 Shigesada et al., 1979 Chesson, 1981

Elderkin, 1982 Pacala & Roughgarden, 1982 Pacala, 1987 Abrams, 1988 Armesto et al., 1991 Colasanti & Grime, 1993 Vail, 1993 Lavorel et al., 1994 Oborny, 1994a 1994b.

are investigated in monospecic stands for single growing seasons and for competing species over several years and the role of dispersal in providing spatiotemporal integration of resource supplies is examined.

The CML is applied to the specic issue of the ecology of seed sizes. After a detailed study of the growth over a single season of species with dierent sizes of seeds, the controversial subject of the adaptation of seed sizes to high or low resource habitats is investigated.