Nine stage classes of tree, the i-th of which (in cell x at time t) has number of individuals w_x,i(t) (i = 1, 2, …, 9), are modeled, based on height.  We define a tenth vegetation class, w_x,10(t), denoting the grass biomass (kg) in cell x at time t. These classes may be represented by the column vector

wx(t) = (wx,1(t),…, wx,10(t))',

where ' denotes the transpose of a vector. 

(See Box 1 for a flow-chart depicting the model procedure.)

 

Rainfall

To account for rainfall seasonality (Solbrig et al. 1996) and the southern Africa 20-year wet-dry cycles (Tyson and Dyer 1978, Gertenbach 1980), we model the rainfall in year [t, t+2] as a sine-wave plus noise overlaid in the long term, normalized to take the value of 1:

 

where is the amplitude (relative to the long-term mean) of wet-dry cycles of period years, and z(t) is a stochastic variable accounting for interannual variation around these cycles. 

 

Wet season woody plant dynamics

The change in the woody population during a wet season starting from time t is given by:

,

where w_x,i(t) is the number of individuals in cell x of woody class i at the start of the wet season, h_x,i(t) is loss of individuals due to encroachment by growth and expansion of larger individuals, and g_x,i (= g_x,i(t, w_x, r)) is the transition rate from class i to class i+1 for that cell and season.  The seedling class (i = 1) is given by

,

where g_x,0 is the proportion of these which successfully recruit.  c_x(t) is the expected number of new seedlings emerging in cell x at time t:

,

where m is the fecundity of mature trees, and is a dispersal parameter representing the proportion of seedlings parented by individual trees from the four neighboring cells. 

The transition rate from class i to i+1 is given by:

gx,i(t) = min (x,i(t), x,i(t)),

0 i 8,

where _x,i(t) is the maximum proportion of class i that can grow to class i+1 without causing the cell x to overfill.  _x,i(t) represents the underlying growth rate adjusted for competition and rainfall:

cx,0(t) = r(t)cx(t),
cx,i(t) = r(t)gi fx,i(wx(t)),	1 i 8,

where _i is the underlying growth rate from class i to i+1 and _x,i(w_x(t)) is a "competition coefficient" (sensu Getz and Haight 1989).  We approximate competitive effects by aggregating on a per-area basis:

,

where a_x,i(t) represents the proportion of area (in cell x at time t) controlled by class i, i.e.,

ax,i(t) = iwx,i(t),

1 i 9,

and _i is the "resource area" (ha; after Kiker 1998) occupied by one individual of class i.  Automatic growth within sapling classes means that _x,i(t) = 1 for i = 2, 3, 4. 

The expansion limiting coefficients, _x,i involve projecting total possible recruitment and then reducing that recruitment, in order of trees, shrubs, saplings, seedlings, in case of overflow:

This allows us to calculate g_x,7(t) and then proceed through the stage-classes (dropping the x and t arguments for convenience):

We set _i = 1 for those height classes deemed not to expand laterally upon growth to the next class, i.e., i = 2, 3, 4, 6, 8.

We introduce h_i as a "crowding coefficient", representing the proportion of plants overcrowded by the individuals growing from class i to i+1:

 

Wet season grass dynamics

We update grass cover to account for changes in woody vegetation, elephant grazing, and rainfall:

where u_x,10(t) is the proportion of the grass in cell x grazed by elephants.  The grass biomass then increases by the productivity of the area a_x,10(t+1), adjusted for wet season senescence (Illius and O'Connor 2000) and for elephant grazing:

,

 

where ₁₀ is annual grass productivity (kg/ha), and  is wet-season persistence ("survival") of grass biomass.

 

Elephant grazing

Elephant use of cell x is in proportion to the relative amount of grass biomass present:

,

 

where l(t) is elephant density (numbers/hectare) at time t, and I_g is the elephant intake rate of grass (kg/elephant/wet-season). 

 

Dry season woody plant dynamics

In our model, woody plant mortality, elephant browsing and fire are limited to the dry season (see Box 1):

 

where s_i is the mean survival rate of woody class i, and F_i is a concatenation of the functions, f_surv,i, f_ele,i and f_fire,i, that incorporate the effects of mortality, elephant browsing and fire respectively: i.e.,

.

 

Woody plant survival

We assume that survival of subadult woody plants depends on the rainfall for that year, but that mature trees only experience increased mortality in droughts lasting more than two years (Scholes 1985):

,

 

where  is additional mortality experienced by mature trees in drought conditions, weighted for drought severity using the function (t): 

,

where describes the steepness of the drought response (cf. density response in Getz 1996), and k is the value of the three-year mean which causes 50% of the drought-related mortality.

 

Elephant browsing

Using vector notation, the effect of elephant browsing on the woody vegetation is:

,

 

where f_ele is the column vector (f_ele,1, f_ele,2, …, f_ele,9)′, E is an elephant-effect matrix mapping the rates of death and stunting resulting from elephant browsing, I is the (9´9) identity matrix and U_x(t) is a 9´9 diagonal matrix with the elements of u_x(t), an elephant-use vector for cell x, on the diagonal.  The vector u_x(t) is composed of the elements u_x,i(t), representing the proportion of each stage-class used.  We assume that elephants allocate their foraging effort between cells in proportion to the cells "perceived attractiveness", and within cells in proportion to each stage class's (preference-weighted) availability (Croze 1974; see also Lewis 1987):

,

,

 

where n is the number of cells in the grid (n = 100), v_i is the "preference-value" for stage-class i (v₁ = 0 as we assume seedlings are not browsed), I_1-7 is the elephant intake rate of sub-adult woody plants, (plants per elephant per dry season), and I_8-9 is the elephant intake rate of adult trees (plants per elephant per dry season).  If w_x,i(t) = 0 then we also set u_x,i(t) = 0. 

The elephant-effect matrix E adjusts the woody vegetation for the impact of elephant browsing, incorporating both mortality and sub-lethal effects:

where  is the proportion of those in class i killed by elephants, and _ik is the proportion of class i which are reduced in height to class k following browsing. 

 

Fire

The occurrence of fire in each cell is recorded by the binary variable b_x(t) and the condition for a cell burning (b_x(t) = 1) is

bx(t) = 1	if	Rx <wx,10(t),
bx(t) = 0	if	Rx ³ wx,10(t),

 

where R_x is a uniform random variable drawn on [0, 1] for each cell x, and is a constant which scales biomass to a fire probability.  This linear relationship between grass biomass and fire probability is a close approximation to the logistic regression formula produced by van Wilgen et al. (2000) from Kruger National Park fire history data.  The spread of fire is modeled by repeating this procedure for every non-burning cell with a burning neighbor, for each burning neighbor, until no additional cells burn.  We assume that fire intensity is linearly related to grass biomass (Trollope 1984, Higgins et al. 2000):

,

 

where _x(t) is the fire intensity relative to that yielded by a grass biomass of w_max.

The woody vegetation is adjusted for fire effects as follows:

,

where _i is the proportion of top-killed plants in stage-class i that resprout following fire.  The proportion of woody plants in class i experiencing topkill is given by _x,i(t):

where w* is the grass fuel required to kill 100% of saplings, and  and  are the topkill rates of shrubs and mature trees, respectively, under maximum intensity fires. 

 

Dry season grass dynamics

We assume that grass burns entirely in fires, and otherwise senesces (depending on rainfall):

,

where  is the dry-season persistence ("survival") of grass biomass.

 

LITERATURE CITED

Baxter, P. W. J.  2003.  Modeling the impact of the African elephant, Loxodonta africana, on woody vegetation in semi-arid savannas.  Dissertation.  University of California, Berkeley, California, USA.

Croze, H.  1974.  The Seronera bull problem.  II. The trees.  East African Wildlife Journal 12:29–47.

Gertenbach, W. P. D.  1980.  Rainfall patterns in the Kruger National Park.  Koedoe 23:35–43.

Getz, W. M.  1996.  A hypothesis regarding the abruptness of density dependence and the growth rate of populations.  Ecology 77:2014–2026.

Getz, W. M., and R. G. Haight.  1989.  Population harvesting: demographic models of fish, forest, and animal resources.  Princeton University Press, Princeton, New Jersey, USA.

Higgins, S. I., W. J. Bond, and W. S. W. Trollope.  2000.  Fire, resprouting and variability: a recipe for grass-tree coexistence in savanna.  Journal of Ecology 88:213–229.

Illius, A. W., and T. G. O'Connor.  2000.  Resource heterogeneity and ungulate population dynamics.  Oikos 89:283–294.

Kiker, G. A.  1998.  Development and comparison of savanna ecosystem models to explore the concept of carrying capacity.  Dissertation. Cornell University, Ithaca, New York, USA.

Lewis, D. M.  1987.  Fruiting patterns, seed germination, and distribution of Sclerocarya caffra in an elephant-inhabited woodland.  Biotropica 19:50–56.

Scholes, R. J.  1985.  Drought related grass, tree and herbivore mortality in a southern African savanna.  Pages 350–353 in J. C. Tothill and J. J. Mott, editors.  Ecology and management of the world's savannas.  Commonwealth Agricultural Bureaux, Canberra, Australia.

Solbrig, O. T., E. Medina, and J. F. Silva.  1996.  Determinants of tropical savannas.  Pages 31–41 in O. T. Solbrig, E. Medina, and J. F. Silva, editors. Biodiversity and Savanna Ecosystem Processes.  Springer-Verlag, Berlin, Germany.

Trollope, W. S. W.  1984.  Fire behaviour.  Pages 199–218 in P. de V. Booysen and N. M. Tainton, editors.  Ecological effects of fire in South African ecosystems.  Springer-Verlag, Berlin, Germany.

Tyson, P. D., and T. G. J. Dyer.  1978.  The predicted above-normal rainfall of the seventies and the likelihood of droughts in the eighties in South Africa.  South African Journal of Science 74:372–377.

van Wilgen, B. W., H. C. Biggs, S. P. O’Regan, and N. Marè.  2000.  A fire history of the savanna ecosystems in the Kruger National Park, South Africa, between 1941 and 1996.  South African Journal of Science 96:167–178.

TABLE A1. Parameters and variables used (in alphabetical order). For parameters, roman type indicates default values and italics indicate initial conditions.  Asterisks in the "Values" column are used to indicate functions, calculated in the course of each simulation.  Parameters values were derived from an extensive review of the African savanna literature (Baxter 2003).

Symbol	Description	Units	Values
ax,i(t)	area controlled by woody vegetation classes in cell x at time t i = 1, 2 i 5, i = 6, 7, i = 8, 9.	ha	0.0012 0.1996 0.2075 0.0918
ax,10(t)	grass coverage in cell x at time t	ha	0.5000
bx	fire indicator for cell x	binary	0
cx	potential seedlings emerging in cell x	plants	0
E	elephant browsing effect matrix	–	*
F()	dry-season woody plant function comprised of fele, ffire, fmort.	–	*
fele()	dry-season elephant impact function for woody plants	–	*
ffire()	dry-season fire-impact function for woody plants	–	*
fsurv()	dry-season survival function for woody plants	–	*
gx,i	realized transition rate from class i to i+1 for cell x	–	*
hx,i	crowding coefficient resulting from growth to class i+1 in cell x	–	*
i	vegetation class index: woody classes (1 i 9), grass biomass (i = 10)	–	*
Ig	intake rate of grass by elephants, wet season only	kg/ele/season	4600
I1-7	elephant use of sub-adult woody plants, dry season only	plants/ele/season	5667
I8-9	elephant use of adult trees, dry season only	plants/ele/season	1000
k	value of mean relative rainfall resulting in 50% of the maximum drought-related tree mortality (mr)	–	0.8
l(t)	elephant density at time t	individuals/ha	0.0
m	fecundity of mature trees	seedlings/tree/yr	50
n	number of hectare cells in grid	–	100
Rx	uniform random variable drawn from U(0, 1)	–	*
r(t)	rainfall relative to the long-term mean (=0 for dry seasons)	–	1.0
si	survival of woody class i i = 1, 2 i 5, i = 6, 7, i = 8, 9.	–	0.950 0.990 0.994 0.995
s10D	dry season survival (non-senescence) of grass	–	0.2
s10W	wet season survival (non-senescence) of grass	–	0.8
t	time index	half-year	*
Ux	diagonal matrix with ux on the diagonal	–	*
ux	elephant woody plant use vector for cell x	–	*
ux,i	elephant browsing intensity on woody class i in cell x (dry seasons)	–	*
ux,10	elephant grazing intensity in cell x (wet seasons)	–	*
vi	elephant preference weighting for woody class i i = 1, 2 i 9.	–	0 1
wx,i(t)	number of woody plants of class i (1 i 9), in cell x, at time t i = 1, i = 2, i = 3, i = 4, i = 5, i = 6, i = 7, i = 8, i = 9.	individuals	1211.4 767.8 507.2 335.0 386.4 168.1 62.4 28.4 8.3
wx,10(t)	biomass of grass in cell x at time t	kg	1200
wmax	grass biomass yielding maximum intensity fire	kg	5000
w*	grass fuel required for a fire to kill 100% saplings	kg	2500
wx(t)	vector of woody structure in cell x at time t	individuals	*
x	cell index (1 x 100)	–	*
z(t)	rainfall random variable drawn from N(0, r2)	–	*
i	area controlled by one individual of class i i = 1, 2 i 5, i = 6, 7, i = 8, 9.	ha	1 × 10–6 1 × 10–4 9 × 10–4 25 × 10–4
x	relative fire intensity in cell x	–	*
i	expected transition rate (under mean, uncrowded conditions) from i to i+1 1 i 4, i = 5, i = 6, i = 7, i = 8.	–	1.000 0.353 0.300 0.300 0.150
10	net annual grass productivity under mean rainfall	kg/ha	2500
	proportion of seedlings dispersing to neighboring cells	–	0.05
	drought severity steepness control (shape parameter)	–	50
	amplitude of wet-dry rainfall cycles, relative to long-term mean	–	0.13
i	resprouting rate of woody class i following fire i = 1, 2 i 7, 8 i 9.	–	0.0 0.9 0.0
x,i	expansion-limiting coefficient for woody class i in cell x	–	*
6-7b	shrub topkill under maximum intensity fire	–	0.80
8-9b	mature tree mortality under maximum intensity fire	–	0.02
ie	mortality of class i plants browsed by elephants i = 1, 2 i 5, i = 6, 7, i = 8, 9.	–	1.00 0.25 0.25 0.80
8-9r	additional drought mortality for mature trees	–	0.05
	drought severity coefficient	–	*
x	proportion of elephant foraging allocated to cell x	–	*
'x,i	proportion of elephant foraging allocated to class i in cell x	–	*
ik	proportion of class i reduced to class k after elephant use 2 i 5 (k = 1), i = 6, 7 (k = 5), i = 8 (k = 5), i = 8 (k = 7).	–	0.5 0.5 0.1 0.1
r	standard deviation in annual rainfall relative to long-term mean	mm	0.30
i	topkill rate of woody class i due to fire	–	*
x,i	competition coefficient applied to growth rate of woody class i	–	*
x,i	net growth rate of woody class i given sufficient area for expansion	–	*
	scaling constant converting grass biomass to fire probability	kg–1	10–4
	period of wet-dry rainfall cycles	years	20