Ecological Archives E085-007-A1

Stephen H. Roxburgh, Katriona Shea, and J. Bastow Wilson. 2004. The intermediate- disturbance hypothesis: patch dynamics and mechanisms of species coexistence. Ecology 85:359–371.

Appendix A. Additional details on the simulation models.

(a) Between-patch model

Landscape attributes

The model is based on a landscape of 60 × 0 cells, with wrap-around edges to form a torus. Disturbances occur as patches, 25 cells × 25 cells in extent, and are located at random within the landscape. A disturbance event kills all species within a patch, and disturbances occur at regular intervals through time.

Species attributes

There are four parameters that define the biologies of the two species. These are defined below, and their values used in the simulations are given in Table A1.

(i) Adult survival probability (= longevity):

This is the probability that an individual occupying a cell at time t survives to time t + 1. For all simulations this was set to 0.95 for both species, corresponding to an average life expectancy of approximately 20 time-steps.

(ii) Age to maturity:

Following establishment of an individual within a cell, that individual must persist in that cell for 10 time-steps before it is capable of reproduction.

(iii) Fecundity

After reaching maturity, each individual produces five seeds at every time-step until death.

(iv) Dispersal distance

This parameter defines the patch around a mature individual within which its seeds are dispersed. This value is manipulated to give the trade-off between competitive ability and dispersal ability. Two variations of the model were run, one including the trade-off, and one without.

In the simulation without a trade-off, the dispersal area for both species is a 21 × 21 block of cells centred on the cell containing the parent individual. The seeds that are dispersed are assigned independently and at random to cells within this area. Although more than one seed is allowed to disperse to a given cell, only one individual is allowed to reach maturity within that cell, with the remaining seeds being lost from the system. For the inferior competitor, seeds successfully establish only if they land in an empty cell. For the superior competitor, seeds can establish in empty cells, and can also establish in cells that already contain a mature individual of the inferior competitor, resulting in the competitive exclusion of the inferior competitor from that cell. Seeds of either species dispersing to cells already occupied by members of the same species, either as seeds or adults, are also lost from the system.

To introduce the appropriate trade-off, the inferior competitor is again permitted to disperse its seeds within a square area with dimensions 21 cells by 21 cells, centered on the target cell. However the dispersal area of the superior competitor is restricted to adjacent cells only, i.e., an area 3 cells by 3 cells centered on the target cell. This provides the required trade-off between competitive ability and dispersal ability.

Initial conditions

For each run of the model the landscape was initialized with a mixture of the two species, randomly assigned so that 2% of cells were initially occupied by the inferior competitor, 2% by the superior competitor, and 96% were empty.

Determining coexistence

Coexistence was determined using a regression technique. After initialization, the simulation is run until one species is excluded, or until both species coexist for 20,000 time-steps. After this time the abundances of both species are retained at each subsequent time-step, and a linear regression of the abundance of each species vs. time is sequentially calculated as time progresses. The slopes of these two regressions are saved. As the simulation progresses, if these slopes converge towards zero, this is taken as evidence for coexistence. To detect this convergence, coexistence is accepted if the regression slopes for both species fall within +/- 1 × 10 -7, or if 2 × 106 time-steps are achieved without competitive exclusion.

The order of the algorithm

At the start of each time-step

(i)  The age of each individual in each cell is incremented by one, to keep track of which individuals have reached reproductive maturity.

(ii) For those cells containing reproductively mature individuals, reproduction and dispersal takes place.

(iii) Following dispersal, competition is then calculated according to the rules described above.

(iv) Natural mortality is then calculated probabilistically.

(v) If appropriate, a disturbance is then applied.

(b) Within-patch model

The within-patch model is based on the between-patch model, with the following differences. First, a belowground seed bank is added to each cell. However, unlike aboveground where only one individual of each species is permitted at a given time, the seed bank can contain seeds from both species. Second, disturbances are global. This means that when a disturbance does occur, all aboveground individuals are killed. Third, there is no dispersal-distance parameter, as dispersal of seeds for both species is assumed independent and random across the landscape.

Species characteristics

The three parameters adult longevity, age to maturity, and fecundity are identical for each species, and have the same parameter values as in the between-patch model (Table A1). To link the seed bank with the aboveground dynamics three additional life-history parameters are added.

(i) Proportion of seeds incorporated into the seed bank

For a seed dispersed aboveground, there is a 0.2 probability that it will germinate in that cell immediately, and a 0.8 probability that it is incorporated into the seed bank for that cell. The number of seeds of each species accumulating in the seed bank for a cell is not recorded, hence each species is recorded as being either present or absent in a cell's seed bank.

(ii) Seed germination probability

This is the probability that the seeds of a given species in the seed bank of a cell will germinate. For both species it is set at 0.4. The underlying assumption is that, although a number of seeds may have accumulated in a cell over time, all of them germinate together. This is analogous to the dispersal of many seeds aboveground to the same cell, where eventually only one of the seeds successfully establishes. The successful establishment of seeds from belowground is based on the same rules as those defining success if the seeds had dispersed from aboveground, e.g., seeds of the superior competitor germinating from belowground are capable of competitively excluding an inferior competitor occupying the aboveground portion of the cell.

(iii) Seed survival probability (= seed longevity)

This is the probability of the seeds in the seed bank surviving to the next time-step, given that they did not germinate. Because the number of seeds accumulating in the seed bank is not recorded, when seed mortality occurs for each species in a cell, the seed bank for that cell is cleared of that species (although it may still contain the other species). It is this parameter which is manipulated to yield a trade-off between competitive ability and longevity of seed in the soil. In the version of the model without a trade-off, the seed survival probabilities of both species are set equal to 0.8. In the version incorporating the trade-off the seed survival probability of the inferior competitor is set to 0.8, and for the superior competitor it is set to 0.1.

The order of the algorithm

At the start of each time-step

(i)  Germination of plants from belowground occurs, including the competitive displacement of aboveground inferior competitors by germinating superior competitors.

(ii) The age of each individual in the aboveground portion of the cell is incremented by one, to keep track of which individuals have reached reproductive maturity.

(iii) For those mature aboveground plants, reproduction and dispersal take place according to the rules described above, i.e., with a probability that some seeds enter the seed bank.

(iv) Following dispersal, competition is then calculated according to the rules described above.

(v) The mortalities, both below and aboveground, are then calculated probabilistically.

(vi) If appropriate, a disturbance is applied.


(c) RYL (reciprocal-yield law) model

The parameters used for the RYL simulations are given in Table A2.

(d) Tables

Table A1. The parameter values used in the simulations for the within- and between-patch models.



Between-patch model

Within-patch model

 

With trade-off

Without trade-off

With trade-off

Without trade-off

Life-history parameters

Superior

Inferior

Superior

Inferior

Superior

Inferior

Superior

Inferior

Adults
 
 
 
 
 
 
 
 

Fecundity

5

5

5

5

5

5

5

5

Adult survival probability

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

Age to maturity

10

10

10

10

10

10

10

10

Dispersal area

21 × 21

3 × 3

21 × 21

21 × 21

-

-

-

-

Seed bank
 
 
 
 
 
 
 
 

Germination probability

-

-

-

-

0.4

0.4

0.4

0.4

Seed survival probability

-

-

-

-

0.1

0.8

0.8

0.8

Probability of a seed being incorporated into seed bank

-

-

-

-

0.8

0.8

0.8

0.8


Table A2. The parameter values used in the simulations for the RYL model.

 

Identical germination strategies

Contrasting germination strategies

Model parameters

Superior

Inferior

Superior

Inferior

Germination fraction, Gi

0.85

0.85

0.85

0.15

Seed survivorship, Si

0.15

0.15

0.15

0.85

Constant, Ci

0

0

0

0

Competition coefficients (i = superior competitor, j = inferior competitor)
 
 

aij

0.95

0.95

aji

1.05

1.05

aii

1.00

1.00

ajj

1.00

1.00

Disturbance parameter, K

0 in disturbance years; 1 otherwise



[Back to E085-007]