Noise-induced transitions and shifts in a climate–vegetation feedback model

Motivated by the extremely important role of the Earth’s vegetation dynamics in climate changes, we study the stochastic variability of a simple climate–vegetation system. In the case of deterministic dynamics, the system has one stable equilibrium and limit cycle or two stable equilibria corresponding to two opposite (cold and warm) climate–vegetation states. These states are divided by a separatrix going across a point of unstable equilibrium. Some possible stochastic scenarios caused by different externally induced natural and anthropogenic processes inherit properties of deterministic behaviour and drastically change the system dynamics. We demonstrate that the system transitions across its separatrix occur with increasing noise intensity. The climate–vegetation system therewith fluctuates, transits and localizes in the vicinity of its attractor. We show that this phenomenon occurs within some critical range of noise intensities. A noise-induced shift into the range of smaller global average temperatures corresponding to substantial oscillations of the Earth’s vegetation cover is revealed. Our analysis demonstrates that the climate–vegetation interactions essentially contribute to climate dynamics and should be taken into account in more precise and complex models of climate variability.


Introduction
Whether or not all physical mechanisms and their correlations were determined or described in detail, the fact that they relate the climate and land surface changes was observed and reported even many years ago. So, for instance, the notes written by Thomas Jefferson that the Virginia winters were getting noticeably warmer [1]  equations for the global average temperature T and the fraction of land A covered by vegetation. The temperature T changes with time t as a result of balance between the incoming and outgoing energy fluxes: where C T is the heat capacity, Q 0 is the incoming solar energy and functions α(T, A) and R 0 (T) determine dependencies of the Earth's albedo and the outgoing energy flux, respectively. Introducing the planet fraction p that is land (1 − p is the fraction of ocean), one can express the first of these functions as where α v and α g represent the albedo of vegetation and ground so that α v < α g as forests and savannas absorb more energy than bare ground.
The ocean albedo α o should depend on temperature T and, in particular, on temperatures T α, and T α,u below and above which the ocean is ice-covered and ice-free, respectively. This dependence can be represented as a ramp function of the form [35,36] and α max and α min are the albedos of ice-covered and ice-free ocean. The energy flux outgoing from the Earth takes into account the fact that increasing carbon dioxide decreases the outgoing radiation where B 0 and B 1 are the model constants and T opt is the optimal vegetation growth temperature. The evolution of vegetation cover A can be described by the following logistic law: where γ designates the vegetation death rate and determines the parabolic growth rate in a certain temperature interval with a maximum temperature T = T opt (here k represents the growth curve thickness). The model (2.1)-(2.5) determines the evolutionary behaviour of the global average temperature and the land fraction covered by vegetation. Below we detail its deterministic dynamics and describe its new evolutionary scenario under the influence of stochastic forcing.

Deterministic dynamics
A bifurcation analysis of the deterministic climate-vegetation model (2.1)-(2.5) was carried out in Rombouts & Ghil [34]. The important details of the phase portraits for this nonlinear feedback model are demonstrated in figure 1. Here, phase trajectories leading to 'cold' attractor (snowball) are plotted by blue colour, and others leading to 'warm' attractor are shown by red colour. In the first instance, the model has a stable limit cycle (depicted on the right-hand sides of figure 1a-d by the closed red lines) embracing an unstable equilibrium (shown by the open blue circles inside the limit cycles). An interesting feature of the deterministic system is its behaviour near the separatrix dividing basins of attraction of 'warm' and 'cold' attractors. As can be seen from figure 1, the trajectories starting near the separatrix move along it for some time before approaching the attractors.
Note that with further increase of the vegetation death rate γ , the system undergoes a supercritical Andronov-Hopf bifurcation: the limit cycle and unstable equilibrium merge, and a stable equilibrium appears (see red filled circles in figure [34]: Generally speaking, there are only two opposite stable states of the climate-vegetation system under consideration: the limit cycle or stable equilibrium on the right (red colour) and the stable equilibrium on the left (blue colour). The first of them describes small climate-vegetation oscillations near the global average temperature 300 K (warm climate) whereas the second one corresponds to the snowball Earth scenario (242 K) when no fraction of land is covered by vegetation. These opposite stable regimes of climate-vegetation system are divided by the unstable equilibrium so that the phase trajectories (blue and red lines) go very close together in its vicinity (this occurs near 280 K in figure 1a-e and 286 K in figure 1f ). In this zone, even small temperature perturbations can throw the system across its separatrix and crucially change the dynamics of the system. An important point is the essential influence of vegetation death rate on the deterministic dynamics. The possible stable climate-vegetation oscillations expressed by the stable limit cycle become narrower with increasing vegetation death rate γ and degenerate into just a stable point in the phase diagram.

Stochastic dynamics
To study the effect of external stochastic forcing on the climate-vegetation dynamics, let us consider the following nonlinear system: where ξ (t) is a standard Gaussian white noise with parameters ξ (t) = 0, ξ (t)ξ (τ ) = δ(t − τ ), and ε designates the noise intensity (hereafter its dimension is omitted for the sake of simplicity). The stochastic system (4.1) follows from equations (2.1) and (2.5) after the replacement of deterministic growth rate β(T) by its corresponding stochastic function β(T) + εξ (t). This means that we study how diverse fluctuations induced by different natural and anthropogenic processes are able to change the vegetation growth rate and, thus, the evolutionary scenario of the climate-vegetation system overall. Figure 3 shows that the additional stochastic disturbances lead to drastic changes in the system dynamics. Note that for numerical simulations of random trajectories, we have used the Euler-Maruyama scheme with time step 0.01 year. While the noise intensity ε is small enough (blue and green colour), in the time interval 0 ≤ t ≤ 5000 yr, the global temperature T slightly fluctuates near its mean value that is close to 300 K. In figure 3a, for γ = 0.02, one can see the noisy oscillations (green) near the deterministic small-amplitude periodic trajectory (blue). In figure 3b,c, for γ = 0.1 and γ = 0.035, there are small-amplitude stochastic oscillations around the deterministic stable equilibria. In the presence of such small noises, the vegetation land cover A fluctuates near either the deterministic periodic trajectory (figure 3a) or the stable equilibria (figure 3b,c). Note that the amplitude of these stochastic oscillations grows with increasing noise intensity ε.
When the noise intensity becomes large enough, the global temperature demonstrates a sharp stepwise decrease to the Earth's snowball state (242 K). What is more, this stepwise transition becomes steeper with increasing vegetation death rate (compare the red curves in figure 3). The level of vegetation cover A in the course of this transition tends to zero, as is demonstrated in figure 3a(ii)-c(ii).
Strictly speaking, there is a range of noise intensity leading the climate-vegetation system to its breakdown state of the snowball Earth. This range essentially depends on the vegetation death rate γ . It is illustrated in figure 4, where we demonstrate different random states of the system under consideration. So, for instance, the transitions occur in the interval 0.15 ε 0.2 for γ = 0.01 (figure 4a), and 0.35 ε 0.55 for γ = 0.1 (figure 4b). If the noise intensity exceeds this interval, the climatevegetation system inevitably tends to the Earth's snowball state. As this takes place, the smaller γ , the smaller noise intensity ε that determines this breakdown transition (the deterministic attractor is a limit cycle for figure 4a whereas it is a stable equilibrium for figure 4b).
Some details of this transformation of the stochastic dynamics of system (4.1) with increasing noise are shown in figure 5 for γ = 0.35. Here, the stochastic phase trajectories and probability density functions for different noise intensities ε are shown. In the case of small ε, the stochastic system is localized in the temperature range of approximately 290 T 300 K. As this takes place, a noise-induced shift to lower

Conclusion
In summary, we consider a simple climate-vegetation model consisting of a land surface covered by some vegetation and ocean. Temperature changes induce the growth of vegetation and alter the Earth's surface albedo, which in turn modifies the energy balance and, thus, induces temperature changes. At first, the nonlinear feedback model coupling the Earth's climate and vegetation is studied in the idealized case of deterministic dynamics. This analysis details the main features of deterministic dynamics and, in particular, it demonstrates the system attractors. Further we show that the evolutionary behaviour of the climate-vegetation system drastically depends on externally induced stochastic forcing which, in turn, can be caused by different anthropogenic and natural processes influencing the nonlinear dynamics.
Let us summarize below the main features of the initial deterministic climate-vegetation model [34]. This model possesses two stable states which are divided by a separatrix going across the unstable equilibrium. These stable states are either the limit cycle and equilibrium or two equilibria depending upon the value of the vegetation death rate. The system attractor of snowball Earth therewith is always the stable equilibrium. Another stable state describing the warm climate is the limit cycle embracing the unstable equilibrium or the stable equilibrium appearing as a result of their merging. An important point is that the oscillation periods of the global average temperature and the land fraction covered by vegetation decrease with increasing vegetation death rate.
Note that deterministic phase trajectories going to two opposite climate-vegetation states can be located very close to each other. As a result, even small external stochastic forcing can throw the system across the separatrix, and two opposite stable climate-vegetation states (having the warm and cold climate) become stochastically interconnected.
What is more, this transition between system attractors becomes steeper when the vegetation death rate increases. In addition, increasing this rate gives the greater noise intensities leading to the snowball state.
An important outcome of our stochastic analysis lies in the fact that the climate-vegetation system undergoes a noise-induced shift into the range of smaller global average temperatures where it fluctuates with a broad range of possible land fractions covered by vegetation. This shift can be responsible for substantial climate changes. Therefore, the coupled climate-vegetation model should be taken into account in conjunction with more precise and complete models of climate variability incorporating diverse interactions between the ocean, ice and atmosphere [24][25][26][37][38][39][40].
Data accessibility. This article has no additional data. Authors' contributions. All three authors equally contributed to the work, setting up the model and the relative simulations, and writing the manuscript. All authors gave final approval for publication.
Competing interests. We declare we have no competing interests. Funding. This work was supported by the Ministry of Education and Science of the Russian Federation (project number 1.9527.2017/8.9).