ONLINE SUPPLEMENTARY MATERIAL
Spatial and temporal signatures of fragility and threshold proximity
in modelled semi-arid vegetation
R.M. Bailey
School of Geography and the Environment, University of Oxford, UK.
SIMULATING ENVIRONMENTAL STRESS
The sources of stress affecting semi-arid vegetation are varied and include moisture availability (predominantly controlled by rainfall), nutrient availability, grazing pressure, fire and human disturbance. Rather than attempting to simulate these influences explicitly, a prescribed generic parameter (α) is used, which lumps together these‘external’ factors, and simply affects the overall probability of cell (plant) survival. The relationships between the external factors and α are not addressed here. The same environmental stress forcing (α) is imposed simultaneously on each grid cell. The probability of cell death increases with α, although the response of each cell to αdepends on the level of neighbourhood competition/facilitation andthe age of the cell.
AGE DEPENDENCIES
An age-dependence in the competitive and facilitative effects of real plants is expected, due in part to changes with age in biomass, of both the above and below ground parts. Even though the precise forms of such age dependencies are unknown, inclusion of relevant approximate parameterizations in the present model was judged preferable to the otherwise unrealistic situation where young plants (e.g. seedlings) and older (mature) plants exert, and respond to competitive/facilitative effects in identical manners. The conclusions from the present work are not, however, sensitively dependent on the fine details of these parameterizations. Qualitatively, the model behaviour (the spatial patterning and temporal dynamicsdescribed in the main body) is robustand, more importantly, preserved if all age-dependencies are excluded from the model. The spatial patterning was however better simulated (compared to observations) when age-dependence was included, and the problem of extended longevity avoided. A more detailed description of the age dependencies is given below, followed by quantitative descriptions and plots of the relevant functions (Figure A1).
Age-stress
An age-related stress component is included in the model. Without this term, under favourable (low-stress) conditions the probability of death is vanishingly small and individual cells continue to age. This is clearly unrealistic and an age-dependent stress is therefore included, which is greater than zero for very young individuals (reflecting relatively high mortality rates in younger plants; e.g. Lauenroth and Adler, 2008), drops to zero for fully established/mature individual and then rises to an asymptotic value (M) , reaching 0.99M by 50 yrs (reflecting the increased mortality of relatively old plants as overall plant function inevitably deteriorates; e.g. Roach et al., 2009, Pico and Retana, 2008). The effect of this term is that relatively young and relatively old occupied cells have an amplified susceptibility to stress of all kinds (from competitionand from environmental stress) and in general live cells have a progressively smaller probability of surviving as they age past maturity (even in favourable conditions).
Strength of competition and facilitation
Compared to seedlings, more mature plants in general possess greater above-ground biomass than younger plants, and a size-/age-dependent amplification of facilitation effects to neighbouring plants is expected (e.g. Barbier et al., 2008). Reflecting this, modelled facilitation strength grows monotonically with age, from zero to a maximum at full maturity, and remains constant thereafter (implicitly, above-ground biomass does not change significantly with age beyond full maturity, here arbitrarily defined as 25 yrs). For similar reasons, younger plants (assumed to be physically smaller) are expected to exert less competitive pressure on the adjacent area (specifically competition for soil moisture and nutrients), due to less extensive root systems (e.g. Barbier et al., 2008). Modelled competition strength rises monotonically with age, until full maturity and then eventually reduces with age, reflecting reduced plant function at greater ages (Roach, 2009).
Sensitivity to competition and facilitation
Empirical data support the notion that younger plants benefit proportionately more from facilitation than older plants, and that the degree of facilitation increases with the size (and age) of the facilitating plant (Kellman and Kading, 1992; Pugnaire et al., 1996, Callaway, 1997).Plant size is not included in the present model and instead occupied grid-cell age is used as a proxy for size. An age-dependence is given to the sensitivity of each occupied cell to both the competition and the facilitation exerted by cells in its neighbourhood: younger individuals benefit more from facilitation than older cells but also suffer more from competition; at full maturity (25 yrs), the cell is at its most resilient and sensitivity to competition reaches a minimum; beyond this, sensitivity to competition rises again, reflecting the aging of the plant (Roach et al., 2009).
DEFINITIONS AND CALCULATIONS
The neighbourhood of a given cell is defined as the five square shells around it (each being one cell thick, so 8 cells in the first shell, 16 in the second, etc.). Each shell is assigned a co-efficient (ci), with positive values representing net facilitative effects (shells 1,2), and negative values representing net competitive effects (shells 4,5) (zero being neutral; shell 3). A total score for each shell is calculated by summing age-dependent contributions from the occupied cells. Summing these values over all relevant shells gives the total neighbourhood score, T, yielding the net effect of all competitive and facilitative contributions.T is calculated for each cell, in each time-step, and defined
Eq.A1
where ai,j is the value of jth cell in the ith shell (a=1 for an occupied (live) cell and a=0 for an unoccupied cell); ci=competition or facilitation coefficient of shell i (i=1...5)(summing the co-efficients for all cells in shells c1 to c5yields values 2,1,0,-0.5,-1); kc=competition strength (<0); kf=facilitation strength (>0); sc=sensitivity to competition (≥0); sf=sensitivity to facilitation (>0); S=stress due to age (z=cell age and z≥1 for occupied cells). In the case of kc, kf, sc, sfand S, the form of the normal distribution is used in the definition of the age-dependence, where p and w are the peak position and width parameter respectively, in z(p=25, w=8). For convenience, the function N(z) is introduced, followed by other definitions: ; ; ( when , when ); ; ; , when , and , when . All relevant functions are plotted in Figure A1.
The T score is then normalized to the range 0-1 (asymptotic at extreme positive and negative values of T) and the additional stress due to cell age is subtracted, giving :
Eq.A2
serves as a critical value against which a uniformally-distributed random value (h) is compared. At each time-step, h is recalculated for each cell. If h≤, a dead/empty cell becomes occupied (a:01) and occupied cells live on (a=1), increasing their age by 1 year; if h>, a live cell dies (a:10), an empty cell remains empty (a=0).
The balance of total competitive and facilitative co-efficients (the sum of negative and positive c values, -1.5 and +3 respectively) was chosen to provide the greatest diversity of patterns, and the best (qualitative) agreement with observed semi-arid vegetation patterns. Increasing the total facilitative weighting (sum of positive c values), or reducing the total competitive weighting (sum of negative c values), gradually extinguishes the labyrinth-type patterns. Conversely, increasing the competition, or reducing the facilitation weightings, results in persistent and (in terms of semi-arid vegetation) un-realistic overly-angular patterning over a wide range of population densities.
For each model run, the model is initialized with the starting population (P0) in year 1, located randomly on the grid, with each occupied cell assigned a (uniformly distributed) random age between 1 and 50 years. The model is then run to equilibrium (at theα value chosen for the beginning of the experiment), allowing both the population density and age distribution to stabilize before the experiment commences.
References
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