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Alex Genin edited this page Jun 19, 2015 · 12 revisions

Forest Gap Model

Description

This model can use either an explicit height for trees, in which case states can be anywhere in a range [Smin..Smax] (Solé et al., 1995), or use only two states, vegetated (gap) and empty (non-gap) (Kubo et al., 1996). Here we focus on the version that uses only two states: gap (+) and non-gap (0). Without spatial spreading of disturbance (all cells are independent), a cell follows the following rules, using the birth b and death d probabilities.

       b
    ------>   
(0)         (+)
    <------   
       d

where b and d are fixed transition probabilities.

However, gap expansion occurs in nature as trees having empty (non-vegetated) surroundings are more likely to fall due to disturbance (e.g. wind blows). Let p(0) be the proportion of neighbouring sites that are gaps. We can implement this expansion effect by modifying the death rate into d + ẟ p(0). Since 0 <= p(0) <= 1, ẟ represents the maximal added death rate due to gap expansion (i.e. the spatial component intensity).

In their simulations, the authors (Kubo et al. 1996) use a 100x100 torus-type lattice (with random initial covers?).

Parametrization

The authors consider two cases: one in which the recovery of trees is proportional to the global density of vegetated sites, and one where the recovery is proportional to the local density of vegetation. We use only the first case as it the only one producing bistability.

The birth rate b is replaced with ⍺⍴+ where ⍴+ represents the global density of non-gap sites and ⍺ is a positive constant. This can produce alternative stable states over a range of ẟ values within 0.15-0.2 (⍺ is fixed to 0.20 and d to 0.01).

The state transition probabilities thus become:

      ⍺⍴+
    ------>   
(0)         (+)
    <------   
    d + ẟp₀

Example result

Behavior as implemented (compare with Kubo 1996, fig 3.b)

Compare with Kubo 1996, fig 3.b

References

Kizaki, S. and Katori, M. (1999) Analysis of canopy-gap structures of forests by Ising–Gibbs states – Equilibrium and scaling property of real forests. J. Phys. Soc. Jpn 68, 2553–2560

Katori, M. (1998) Forest dynamics with canopy gap expansion and stochastic Ising model. Fractals 6, 81–86

Kubo, T. et al. (1996) Forest spatial dynamics with gap expansion: total gap area and gap size distribution. J. Theor. Biol. 180, 229–246

Solé, R.V. and Manrubia, S.C. (1995) Self-similarity in rain forests: evidence for a critical state. Phys. Rev. E. 51, 6250–6253

Sole ́, R.V. and Manrubia, S.C. (1995) Are rainforests self-organized in a critical state? J. Theor. Biol. 173, 31–40