Supplementary material for the paper entiled

On the degradation of forest ecosystems by extreme climatic events: statistic model checking of a hybrid model

by Guillaume Cantin, Benoît Delahaye, Beatriz M. Funatsu.

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1/ Source code

FreeFem++ program

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2/ Animation of the reaction-diffusion system showing the formation of an ecotone

Figure 1 below shows an animation of the reaction-diffusion system defined by \[ \begin{cases} \begin{array}{lll} \dfrac{\partial u}{\partial t}(t,x) = \beta \delta w(t,x) - \gamma\big(v(t,x)\big) u(t,x) - f u(t,x), & t>0, &x \in \Omega, \\[1em] \dfrac{\partial v}{\partial t}(t,x) = f u(t,x) - h v(t,x), & t>0, &x \in \Omega, \\[1em] \dfrac{\partial w}{\partial t}(t,x) = d \Delta w(t,x) - \beta w(t,x) + \alpha v(t,x) & t>0, &x \in \Omega, \end{array} \end{cases} \] which corresponds to system (1) in our paper. The horizontal plane represents the domain $\Omega$ occupied by the forest. The vertical axis depicts the density of young trees $u(t, x)$. The green part of the surface corresponds to a high density of young trees, thus roughly models the forest. The yellow part of the surface corresponds to a very low density of young trees, thus corresponds to the non-wooded area of the domain $\Omega$. We observe that the spatio-temporal trajectory converges to a stationary state which is discontinuous in space. This discontinuous stationary spate can be interpreted as an ecotone (frontier between two ecosystems).

Fig. 1 - Animation showing the formation of an ecotone.

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3/ Simulation of the forest-climate model $(\mathcal{H})$

The previous reaction-diffusion system is incapable of reproducing the impacts of extreme climatic events on the dynamics of the forest. To remedy this lack, we couple the deterministic reaction-diffusion system with a discrete probabilistic process, so as to take into account the possible occurrences of localized extreme climatic events. In this probabilistic process, the times at which extreme events occur, as well as their geographical position, are determined by discrete random variables. The result of our approach is the hybrid model which is denoted $(\mathcal{H})$ is our paper (recall that $\mathcal{H}$ is equivalent to Equations (1) − (2) − (3) − (9) − (10) in the paper).

Figure 2 below shows a simulation of the hybrid model $(\mathcal{H})$, with the evolution for $t \geq 0$ of the density of young trees $u(x)$. Click on the Play/Pause button to activate the animation. The disk represents the domain $\Omega$ occupied by the forest. At distinct times of the simulation, a random process determines a subdomain $\omega \subset \Omega$ where an extreme event occurs. The extreme event causes a forest loss, which can be observed by the change of color: the yellow parts of the domain correspond to the impacts of the extreme events. After a certain time, we observe that the forest is globally impacted by the sequence of extreme events.