We formulate and analyze a simple dynamical systems model for climate–vegetation interaction. The planet we consider consists of a large ocean and a land surface on which vegetation can grow. The temperature affects vegetation growth on land and the amount of sea ice on the ocean. Conversely, vegetation and sea ice change the albedo of the planet, which in turn changes its energy balance and hence the temperature evolution. Our highly idealized, conceptual model is governed by two nonlinear, coupled ordinary differential equations, one for global temperature, the other for vegetation cover. The model exhibits either bistability between a vegetated and a desert state or oscillatory behavior. The oscillations arise through a Hopf bifurcation off the vegetated state, when the death rate of vegetation is low enough. These oscillations are anharmonic and exhibit a sawtooth shape that is characteristic of relaxation oscillations, as well as suggestive of the sharp deglaciations of the Quaternary.

Our model's behavior can be compared, on the one hand, with the bistability of even simpler, Daisyworld-style climate–vegetation models. On the other hand, it can be integrated into the hierarchy of models trying to simulate and explain oscillatory behavior in the climate system. Rigorous mathematical results are obtained that link the nature of the feedbacks with the nature and the stability of the solutions. The relevance of model results to climate variability on various timescales is discussed.

Climate has an important effect on vegetation. Plant growth is
affected by temperature, carbon dioxide (CO

This vegetation–albedo feedback appears to be important in
semi-arid regions

The uptake of CO

Although vegetation plays an essential role in the climate
system, it has only been rather recently included as an active
player in climate models. The hierarchy of climate models

The simplest climate models are conceptual models, which are
usually governed by a small number of ODEs. These models do not
claim to be realistic in the sense of making precise
quantitative predictions, but allow one to study basic
underlying mechanisms. They are also useful for exploring
qualitative changes in the climate system's behavior, commonly
known in dynamical systems theory as bifurcations

Studying conceptual models can also provide guidance in
interpreting results from larger, more detailed models

In the latter idealized climate-oscillator models, vegetation is not usually included. There are, however, some simple models that explore the interaction between climate and vegetation, and we will briefly review some noteworthy ones in the next section.

A pioneering model dealing with vegetation and climate was
Daisyworld

Beyond the possibility of multiple steady states, it is of
interest to examine in the simplest conceptual models the
possibility of internal, self-sustained oscillations.
Oscillatory behavior has been observed in Daisyworld-like
models, for example when an explicit temperature equation is
added

So far, though, simple, 0-D climate–vegetation models seem not
to have included an ocean component. Earth's oceans constitute
about 70 % of the area of the planet, and are a very
important factor in determining its climate

The purpose of this paper is to explore the solution space of
such a model and to relate its behavior to other models and to
observations. In the next section, we formulate the model, and
the numerical results are presented in
Sect.

Schematic representation of the model planet's surface, including a
fraction

The climate system contains several subsystems, all working
together to produce highly nonlinear behavior through its many
feedback mechanisms. Some of the simplest and most important
feedback effects act via the planetary albedo: darker areas – like
the vegetated ones – absorb more solar energy and thus
warm the planet, while lighter areas – such as those covered by
snow and ice – tend to cool it. The ice–albedo feedback was
included in energy balance models (EBMs) of climate long
ago

The model's two governing equations are given below:

Temperature

The values

Fraction of sea ice cover (blue) and ocean albedo (red) as a
function of temperature

The albedo of the ocean will be taken as a function of global
temperature

The function

The form of

Definition and values of the model parameters.

We will not attempt to estimate the most realistic
values of these or other model parameters – since that
is not the role of such simple, conceptual models

Vegetation cover

Outgoing energy as a function of temperature for different parametrizations.

The parameter

We start, as usual, by looking at the fixed points of
the system (Eq.

This result implies that a planet without vegetation
will necessarily drop into a very cold
“snowball-Earth” state. Note that this result differs
from that of EBMs with no vegetation, in which two
stable fixed points co-exist for sufficiently high

The (

A quick check shows that, for

In the presence of vegetation,

To study the stability of these fixed points, we
consider the Jacobian matrix

For the no-vegetation fixed point

The model's phase plane with the two nullclines (dashed lines) and the three fixed points (open and filled dots); the two filled dots are stable, while the open one is unstable. The stable manifold of the saddle (open dot) is shown in black and the unstable one in magenta. The coordinates of the fixed points are (242.04, 0) for the unvegetated state, and (280.8, 0.898) and (297.9, 0.129) for the unstable and stable vegetated states, respectively.

Let us now look at the other two fixed points, shown in
Fig.

The leftmost of these two fixed points, where

The proof is given in Appendix A, and it does not
depend on any particular parameter values, only on
their relative signs; nor is the exact parametrization
of the growth rate

The stable and unstable manifolds of the saddle were
also computed numerically, and they are plotted in
Fig.

For the stability of the latter, rightmost fixed point,
Proposition 1 above says nothing, since

We can compute the trace of

We investigate next this loss of stability and the more complex model behavior to which it leads.

We study now in greater detail the stability of the
fixed point with non-zero vegetation (

The proof in the Appendix relies simply on rewriting the classic conditions on the Jacobian.

We know that

The proof of Proposition 1 in the Appendix implies, however, that this condition
is already necessary for a fixed point to

Bifurcation diagram, with the vegetation

Typical oscillatory model solution, shown here for

As shown in the caption of Fig.

As announced at the end of Sect.

We can also see in Fig.

We mentioned in Sect. 3.1 above that the
parameter change needs to make tr(

This product is a ratio of timescales, with

This analogy should not be pushed too far,
however. White daisy growth is favored by warmer
temperatures, whereas sea ice appears at low
temperatures. A comparison between our
Fig.

Another interesting difference between the two models
lies in the value of

Oscillatory behavior is also observed in the Daisyworld
variant studied by

This again shows the importance of timescales: only
when the effect of temperature on the growth rate of
the vegetation is substantially delayed does the system
exhibit internal oscillations. The discussion on timescales in Daisyworld is the subject of a recent paper
by

Figure

Such sawtooth-like behavior characterizes the ice
volume evolution during Quaternary glaciations; see,
for instance,

Given the importance of sea ice feedback in our model,
sea ice extent is plotted in Fig.

We described a simple dynamical systems model for climate–vegetation interaction. The model planet has a large ocean, which can be covered by sea ice, and a land area, which can be covered by vegetation. The system variables are temperature and vegetation cover, and the coupling between those is given by the growth rate of the latter and by the albedo, both of which are temperature-dependent.

The model is similar to Daisyworld and related models, the main difference being the inclusion of an ocean. Our model is also related to EBMs and, in this respect, the novelty lies in the inclusion of vegetation as a main variable.

The model exhibits two stable states, one with and one without vegetation. The vegetated state can lose its stability through a Hopf bifurcation and give rise to a limit cycle. This happens when the typical timescale for vegetation overturn becomes high enough. The influence of the model's sea ice in reinforcing the albedo feedback is essential to the presence of oscillatory behavior. The oscillations observed are anharmonic and have a sawtooth shape, reminiscent of deglaciations during the Quaternary. We have obtained analytical results on the role of the feedbacks in determining the system's behavior and compared the resulting oscillations with other models.

Although some parameter values in our highly idealized model are not entirely realistic, the results add to the evidence that vegetation, in combination with other feedback effects, can play an important role in affecting climate. The model studied herein is also interesting because it is one of the simplest ODE models for climate–vegetation interactions that exhibits oscillatory behavior. This intrinsic variability is a basic manifestation of how vegetation affects climate, and constitutes an example of how complex behavior arises in the Earth system even at the lowermost levels of the modeling hierarchy. Once demonstrated and understood at such a level, one can ask next whether similar behavior does persist in more detailed models, and whether it does reflect the actual behavior of the natural climate–vegetation system.

While the sea ice albedo feedback plays a crucial role
in our model's oscillatory behavior, the amplitude of
the oscillations in sea ice extent is quite small, from 0
to a mere 6 %.

Because of Proposition 1 and the shape of the growth curve, it is essential that
the fixed point for which the Hopf bifurcation occurs
have a temperature higher than

Such distinctions between local and global variables are hard to make in highly idealized, ODE models, and it is clear that there is a need to pursue this type of analysis in more detailed, spatially dependent models, either with several climate zones or with continuous dependence on latitude. Such an extension would allow the model, for example, to have vegetated taiga and tundra interact with sea ice at higher latitudes and desert vegetation interact with atmospheric and oceanic temperatures at lower latitudes.

Spatially dependent versions of Daisyworld have already
been studied and yielded interesting
results.

Earth system models of intermediate complexity provide
some evidence that vegetation plays a role in
Quaternary climate variations. In particular, the
albedo changes that accompany shifts in vegetation act
to amplify orbital forcing

The concept of a hierarchy of models – already fairly
well accepted in climate modeling

The conditions

In our model, the feedback that produces oscillations
is negative: increasing temperature lowers vegetation
cover, which in turn lowers energy absorption and
decreases temperature. Due to the large number of
interacting processes in nature, the effective sign of
the feedback is not known directly from observations,
and it might be different in different areas

One such process is the hydrological cycle.

In addition to the influence of vegetation on the hydrological cycle, there is a whole range of other feedback effects between vegetation and several components of the climate system. Of particular importance here is the biogeochemical feedback, as opposed to the biogeophysical effects, such as albedo change, that we have explored. Biogeochemical effects include the influence of vegetation on the carbon cycle, and therefore on the greenhouse effect and the radiation balance of the Earth.

A more detailed model than the one studied here could
include a simplified version of these
effects.

Another interesting model extension that comes to mind
is the inclusion of oceanic vegetation,
i.e., phytoplankton. Plankton interacts with climate in
different ways, through its possible effect on cloud
formation

The results of this paper have to be viewed in the broader perspective
of the hierarchy of climate models already mentioned repeatedly in
its preceding sections

Within this hierarchy, the role of the simple models, sometimes referred to
as “toy” models, is to provide insight and help understand the behavior of
the more complex models, as well as of the climate system itself. The role of
the intermediate models is to refine these insights and bridge the gap
between the toy models and the GCMs

Finally, GCMs allow an extensive comparison with the observations and can
thus help invalidate

The results of our toy model suggest that vegetation might play a larger role in climatic variability – whether in apparent jumps between two or more types of near-stationary states or in oscillatory behavior – than heretofore suspected. In particular, it might contribute to more-or-less regular, but not necessarily simply periodic variability.

It is clear, for instance, that the clouds' contribution to planetary albedo
is larger than that of vegetation

Recall also that, in the early days of energy balance models

While some Earth models of intermediate complexity do indeed show multiple
equilibria, these appear to be mostly of local relevance, for instance in the
Sahel.

Our paper is only trying to make a case for the possibility of vegetation
playing a more important role than contemplated heretofore and does not claim
in the least to have definitively proven that this is so. A similar argument
about local versus global effects has been made with respect to the oceans'
thermohaline circulation. Recall that the

There is no better way of concluding this broader assessment of our toy
model's results than by citing Karl Popper: “Science may be described as the
art of systematic oversimplification”

In this appendix, we prove the propositions from the main text. Recall the
definition

For a fixed point (

This proves Proposition 1, since, when

We will now use the same notation to prove Proposition 2, which we repeat here for the reader's convenience.

We substitute the values into

The statement about the stability of the focus follows from the fact that a
sufficient condition for the focus to be stable is that
tr(

It is a pleasure to thank K. Pakdaman for useful discussions on the presentation of this work. M. Crucifix and an anonymous referee have helped improve the paper by judicious and helpful comments and suggestions. J. Rombouts was supported through an Erasmus Mundus scholarship of the European Union, while M. Ghil benefited from partial support of the European Union's ENSEMBLES project, as well as from the US Department of Energy grant DE-SC0006694 and the National Science Foundation grant OCE-1243175. The paper is the result of research carried out during J. Rombouts's stay at the Ecole Normale Supérieure and the Ecole Polytechnique in Paris, as part of an Erasmus Mundus programme. Edited by: S. Vannitsem Reviewed by: M. Crucifix and another anonymous referee