Francesco Accatino
email: francesco.accatino@mail.polimi.it

Carlo De Michele
email: carlo.demichele@polimi.it

Renata Vezzoli
email: renata.vezzoli@polimi.it

Davide Donzelli
email: davide.donzelli@mail.polimi.it

Robert J Scholes
email: bscholes@csir.co.za

Tree-grass co-existence in savanna: Interactions of rain and fire

Keywords: savanna, tree-grass co-existence, fire, rainfall, soil water

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INTRODUCTION

The term savanna describes ecosystems characterized by the co-dominance of trees and grasses. The proportions of tree and grass can vary greatly. Savannas occur in areas with annual rainfall from 300 to 1800 mm, and are commonly split into dry and moist forms (semi-arid and wet, nutrient-rich and nutrient-poor, fine-leafed and broad-leafed are all synonyms for this division). In dry savannas, the grass primary production is a strongly increasing function of annual rainfall, while in moist savannas the relation is weak or absent. The transition between dry and moist savannas can be located around 500 700 mm of annual rainfall (e.g., [START_REF] Scholes | Savanna[END_REF]Walker, 1993, Sankaran et al., 2005).

What is special about the savanna environment that allows trees and grasses to coexist, as opposed to the general pattern in other areas of the world where either one or the other functional type is dominant?

This has been referred to as the 'savanna question' [START_REF] Sarmiento | The Ecology of Neotropical Savannas[END_REF]. The question has attracted the interest of many scientists in the last forty years: [START_REF] Walter | Ecology of Tropical and Subtropical Vegetation[END_REF], [START_REF] Walker | Ecology of Tropical Savannas[END_REF], [START_REF] Scholes | An African Savanna: Synthesis of the Nylsvley Study[END_REF], [START_REF] Scholes | Tree-grass interaction in savannas[END_REF], [START_REF] Sankaran | Tree-grass coexistence in savannas revisited insights from an examination of assumptions and mechanisms invoked in existing models[END_REF], [START_REF] Sankaran | Determinants of woody cover in African savannas[END_REF], D 'Odorico et al. (2006), Scheiter and Higgins (2007), [START_REF] Lüttge | Physiological Ecology of Tropical Plants[END_REF], [START_REF] Hanan | Do fires in savannas consume woody biomass? A comment on approaches to modelling savanna dynamics[END_REF], and [START_REF] Leibold | Ecology: return of the niche[END_REF], among others.

According to rangeland ecology literature (e.g., [START_REF] Westoby | Opportunistic management for rangelands not at equilibrium[END_REF][START_REF] Briske | Vegetation dynamics on rangelands: a critique of current paradigms[END_REF], savannas can be viewed in light of range succession or state-and-transition models. In range succession models, savannas are a point in a continuous spectrum whose extremes are grassland and forest. Disturbances like fire or herbivores just modify the tree-grass ratio along this spectrum with reversible transformations. In state-and-transition models multiple stable states are possible, and the rangeland dynamics is explained through transitions (reversible or irreversible and abrupt) due to disturbances between steady states. The existence of multiple stable states has been pointed out by [START_REF] Dublin | Vegetation dynamics in the Serengeti-Mara ecosystem: the role of elephants, fire, and other factors. In: Serengeti II: dynamics, management, and conservation of an ecosystem[END_REF], [START_REF] Van De Koppel | Catastrophic vegetation shifts and soil degradation in terrestrial grazing systems[END_REF], [START_REF] Anderies | Grazing management, resilience and the dynamics of fire-driven rangeland system[END_REF], [START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF][START_REF] D'odorico | A probabilistic analysis of fire-induced tree-grass coexistence in savannas[END_REF], [START_REF] Hanan | Do fires in savannas consume woody biomass? A comment on approaches to modelling savanna dynamics[END_REF], [START_REF] Okin | Impact of feedbacks on Chihuahuan desert grasslands: transience and metastability[END_REF], [START_REF] Baudena | An idealized model for tree-grass coexistence in savannas: the role of life stage structure and fire disturbances[END_REF].

According to savannas ecology literature (e.g., [START_REF] Sankaran | Tree-grass coexistence in savannas revisited insights from an examination of assumptions and mechanisms invoked in existing models[END_REF], the 'savanna question' is addressed within paradigms of equilibrium and disequilibrium. The former interprets the savanna as a long term stable state due to internal factors, being disturbances modifiers of the tree-grass ratio. According to the latter, savanna is an unstable state and its existence is due to disturbances which maintain the tree-grass mixture, preventing the achievement of the complete dominance of trees or grasses.

Within the equilibrium paradigm, the tree-grass co-existence has been explained through competition based mechanisms (e.g., root niche separation and balanced competition), while demographic bottleneck mechanisms are related to the disequilibrium paradigm (see [START_REF] Sankaran | Tree-grass coexistence in savannas revisited insights from an examination of assumptions and mechanisms invoked in existing models[END_REF].

The archetypal competition-based model of tree-grass co-existence in savannas is based on the 'Walter hypothesis' [START_REF] Walter | Ecology of Tropical and Subtropical Vegetation[END_REF], which assumes soil water to be the limiting resource, with grasses as the superior competitor, but having roots restricted to the topsoil layer whereas trees roots both in the topsoil and subsoil. The niche separation avoids the tree-grass competition and allows the stable co-existence. The absence of niche separation in fact would lead to one species equilibrium. Models based on the 'Walter hypothesis' have been widely applied in literature [START_REF] Walker | Stability of semi-arid savanna grazing systems[END_REF][START_REF] Walker | Ecology of Tropical Savannas[END_REF], Eagleson and Segarra, 1985[START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF]. Some experiments and observations have supported 'Walter hypothesis' [START_REF] Knoop | Interactions of woody and herbaceous vegetation in a Southern African savanna[END_REF][START_REF] Sala | Resource partitioning between shrubs and grasses in the Patagonian steppe[END_REF][START_REF] Pelaez | Water relations between shrubs and grasses in semi-arid Argentina[END_REF] but many others have cast doubts on the existence of vertical rooting niche separation [START_REF] Scholes | An African Savanna: Synthesis of the Nylsvley Study[END_REF][START_REF] Belsky | Tree/grass ratios in East African savannas: a comparison of existing models[END_REF][START_REF] Le Roux | Spatial partitioning of the soil water resource between grass and shrub components in a West African humid savanna[END_REF][START_REF] Mordelet | Tree and grass rooting patterns in an African humid savanna[END_REF][START_REF] Smit | The influence of tree thinning on the soil water in a semi-arid savanna of Southern Africa[END_REF][START_REF] Hipondoka | Vertical distribution of grass and tree roots in arid ecosystems of Southern Africa: niche differentiation or competition?[END_REF].

In 'balanced competition' models, the superior competitor limits its own abundance and the inferior competitor can grow; in other terms the intra-specific competition of the superior competitor is stronger than the inter-specific competition [START_REF] Amarasekare | Competitive coexistence in spatially structured environments: a synthesis[END_REF]. For example, the water scarcity can limit the abundance of trees in savannas.

In the past two decades, the disequilibrium paradigm has gained favor over the equilibrium one (Higgins et al., 2000[START_REF] Jeltsch | Ecological buffering mechanisms in savannas: A unifying theory of long-term tree-grass coexistence[END_REF][START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF][START_REF] D'odorico | A probabilistic analysis of fire-induced tree-grass coexistence in savannas[END_REF][START_REF] Gardner | Tree-grass coexistence in a Brazilian cerrado: demographic consequences of environmental instability[END_REF][START_REF] Hanan | Do fires in savannas consume woody biomass? A comment on approaches to modelling savanna dynamics[END_REF]. The near-universal finding that tree cover increased when fires are excluded from savannas (Trapnell, 1959, Shackleton and[START_REF] Shackleton | Impact of fire frequency on woody community structure and soil nutrients in the Kruger National Park[END_REF] and that elephants (see the several studies reviewed in [START_REF] Kerley | Effects of elephants on ecosystems and biodiversity[END_REF], giraffe [START_REF] Pellew | The impacts of elephant, giraffe and fire upon the acacia tortilis woodlands of the Serengeti[END_REF] and other herbivores can substantially modify savanna structure, lend evidence to this school of thought.

Fire is an intrinsic characteristic of many savanna ecosystem allowing tree-grass coexistence: trees at the seedling stage can be easily attachable by flames, thus fire acts as bottleneck in trees demography preventing the canopy closure. Fire has been explicitly included in many savanna models [START_REF] Hochberg | The influence of tree biology and fire in the spatial structure of the West African savannah[END_REF][START_REF] Anderies | Grazing management, resilience and the dynamics of fire-driven rangeland system[END_REF][START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF][START_REF] D'odorico | A probabilistic analysis of fire-induced tree-grass coexistence in savannas[END_REF][START_REF] Beckage | Vegetation, fire and feedbacks: a disturbance-mediated model of savanna[END_REF] and has often been -5-described as dependent on the availability of fuel load, that in arid ecosystem is given by dead grass. [START_REF] Hanan | Do fires in savannas consume woody biomass? A comment on approaches to modelling savanna dynamics[END_REF] and [START_REF] Baudena | An idealized model for tree-grass coexistence in savannas: the role of life stage structure and fire disturbances[END_REF] have considered explicitly the demography of trees underlining the asymmetry of the fire effects between saplings and adult trees. This asymmetry can allow the survivorship of trees population at low densities thanks to mechanisms of 'storage effect' that compensate the loss of young trees due to demographic bottlenecks with the very low mortality of adult trees [START_REF] Warner | Coexistence mediated by recruitment fluctuation: a field guide to the storage effect[END_REF]. In addition, various studies focused on fire as cause of bistability in rangelands [START_REF] Anderies | Grazing management, resilience and the dynamics of fire-driven rangeland system[END_REF][START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF][START_REF] Okin | Impact of feedbacks on Chihuahuan desert grasslands: transience and metastability[END_REF].

Simple (non-spatial) models of savanna dynamics, available in literature, focus the attention principally on one mechanism at a time. Competition mechanisms are considered by [START_REF] Walker | Stability of semi-arid savanna grazing systems[END_REF], [START_REF] Walker | Ecology of Tropical Savannas[END_REF], Eagleson and Segarra (1985), Fernandez-Illescas and Rodriguez-Iturbe (2004), [START_REF] Baudena | An idealized model for tree-grass coexistence in savannas: the role of life stage structure and fire disturbances[END_REF], while disturbancesdriven mechanisms are used by [START_REF] Casagrandi | A minimal model for forest fire regimes[END_REF]Rinaldi (1999), D'Odorico et al. (2006), [START_REF] Hanan | Do fires in savannas consume woody biomass? A comment on approaches to modelling savanna dynamics[END_REF], [START_REF] Beckage | Vegetation, fire and feedbacks: a disturbance-mediated model of savanna[END_REF]. [START_REF] Sankaran | Tree-grass coexistence in savannas revisited insights from an examination of assumptions and mechanisms invoked in existing models[END_REF] point out the necessity to take into account both disturbances and the competition for resources, simultaneously in order to capture their relative importance in shaping the different types of savannas.

Based on data from 854 sites in Africa, [START_REF] Sankaran | Determinants of woody cover in African savannas[END_REF] identify a distinction between savannas receiving less that a650 700 mm of annual rainfall (dry savannas) and those receiving more (moist savannas). The former are stable, and disturbances modify the woody cover, but are not necessary for tree-grass coexistence; the latter are unstable and maintained by disturbances, in this case they are essential for the maintenance of a savanna.

van [START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF] represent the savanna vegetation through a simple model of two equations (one for tree and one for grass) considering the joint role of water (through the root niche separation mechanism) and disturbances (fire and herbivores) that remove grasses and trees. In arid and semiarid ecosystems the dynamics of soil water is closely linked to the the dynamics of vegetation [START_REF] Scholes | Savanna[END_REF]Walker, 1993, Rodriguez-Iturbe and[START_REF] Rodriguez-Iturbe | Ecohydrology of water controlled ecosystems: plants and soil moisture dynamics[END_REF], thus in our approach, we will consider explicitly the soil water in the root zone as a state variable.

Here, we address the 'savanna question' including in an ecohydrologic model balanced competition and bottleneck mechanisms. We propose a space implicit model of tree-grass dynamics competiting for soil water and perturbated by fire. In the Materials and Methods section, the savanna is described through a set of three differential equations including the dynamics of trees, grasses, and soil water, fed by rainfall, and disturbed by fire. The variability of the model's parameters is investigated and the stability analysis is presented.

In the Results section, changes of vegetation structure along gradients of rainfall and fire frequency are predicted using the stability map in the rainfall-fire frequency space. The role of rainfall and fire frequency in maintaining dry and moist savannas is illustrated.

MATERIALS AND METHODS

Tree-grass dynamics

Let us consider the space implicit model introduced by [START_REF] Tilman | Competition and biodiversity in spatially structured habitats[END_REF] to represent the temporal dynamics of tree and grass

°°® G G G TG c T G G c dt dG T T T c dt dT G T G T T ) (1 = ) (1 = (1)
where T and G are the fractions of area, occupied respectively by tree and grass. T and G are dimensionless, and range in the closed interval [0, 1]: T = 0 means that trees are not present in the area, T = 1 means that the area is fully covered by trees, and similarly for grasses. In addition, the values of T and G must satisfy also the condition 0 ≤ T+G ≤ 1. c T and c G are the colonization rates, whereas G T and G G are the 'offtake' rates for trees and grasses respectively. c T , c G , G T , G G are all positive and carry dimension [1/t], where t is the time. In Eq. ( 1), trees are the superior competitor, while grasses are the inferior competitor.

Tree can displace grass (the term -c T TG ), and colonize places where trees are absent (the term (1-T)), while grass can colonize places where both trees and grass are absent (the term (1-G-T)). See [START_REF] Tilman | Competition and biodiversity in spatially structured habitats[END_REF] for further details.

Fire forcings and tree-grass dynamics

There is much evidence that fire influences the balance between tree and grass in savannas (e.g., [START_REF] Scholes | An African Savanna: Synthesis of the Nylsvley Study[END_REF]. The occurrence and spread of fires depends on the presence of sufficient mass of dry plant material to provide a near-continuous fuel load.

The fuel for savannas fires is mostly provided by dead grass: the fire may only burn following years of adequate rainfall and low herbivory, when sufficient grass biomass has accumulated [START_REF] Bond | Fire and plants[END_REF].

The fire acts differently on grass and tree. Fire occurs mainly in the dry season, when the perennial grasses are dormant. It consumes grass leaves as fuel, but does not generally kill the dormant grass buds at/or below the soil surface. The fire also consumes tree leaf litter, but if grass is completely absent it is very difficult for fire to propagate. The living tree biomass is seldom ignited, but dead wood and leaf litter may be, and the aboveground live tree biomass can be killed by exposure to fire. Small trees are the most vulnerable, since they may be entirely within the flame zone. Mature trees are little affected, since their bark protects the main stem and their buds (dormant in the fire season) are held above the flame zone.

To take into account explicitly the fire forcing in our model system, Eq. ( 1), the removal term for grass, -G G G, is split in two: -fG describing the consumption of grass by fires, and -G GO G representing the grass reduction due to the other causes such as mortality and herbivores, similarly to van [START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF]; while in the equation for trees the reduction at aboveground cover due to fire (-G F fGT) is determined by the amount of grass fuel load (fG), the tree abundance itself (T) and the trees vulnerability to fire G F , similarly to [START_REF] Van Langevelde | Effects of fire and herbivory on the stability of savanna ecosystems[END_REF].

The system presented in Eq. ( 1) becomes:

°°® G G G fG G TG c G T G c dt dG fGT T T T c dt dT GO T G F T T ) (1 = ) (1 = .
(2)

In Eq. ( 2), the coefficient f [1/t] can be viewed as proxied by the fire frequency, while the coefficient G F [ ] is a function of both the frequency and intensity of fire, where grass fuel load is a proxy for fire intensity. In Eq. ( 2) no differentiation is made between juvenile and adult trees. However, the value of the parameter G F can provide information about the mean age of trees. High values of G F represent sites where trees are young, and vulnerable to fire, while low values of G F represent sites where trees are adult and little affected by fire.

Rainfall forcing, soil water and tree-grass dynamics

Soil water is a direct link between the hydrologic cycle and the ecological system. In arid and semi-arid environments, the water present in the rooting zone is the primary determinant of vegetation composition, structure and function. Root zone water content is highly variable in time. In contrast to the 'Walter hypothesis' [START_REF] Walter | Ecology of Tropical and Subtropical Vegetation[END_REF], where grasses have access only to the topsoil layer, and trees extend their roots both in topsoil and subsoil layers for water uptake, here we assume that the depth at which trees and grasses extract water is the same.

We further assume that the water table is so deep that it does not affect the water dynamics

w 1 = z × n.
Let S be the degree of saturation of the profile available water capacity, defined as the water volume present in the root zone relative to the maximum volume of water that can be held in this zone (the pore space in the volume w). The dynamics of S is described by the following mass balance equation in the control volume w:

. ) (1 ) (1 = 1 SG ST G T S S w p dt dS G T W W H
(3)

S is dimensionless assuming values in the closed interval [0, 1], in particular S = 0 corresponds to completely dry soil, and S = 1 to completely saturated soil. The condition S = 1 rarely happens in semiarid and arid savannas, particularly if the balance is investigated at seasonal or annual time scales.

The term p/w 1 , [1/t], is the rainfall rate (over the unit area) p ≥ 0 [A/t] normalized by

w 1 [A].
Here we assume that the soil surface is more-or-less horizontal, and that all the rainfall infiltrates in the soil. The term (p/w 1 )S is the deep percolation, i.e., the amount of rainfall exceeding the root zone capacity percolating beyond the root zone depth.

Note that in literature, the deep percolation is represented through a non-linear function of S (e.g., Rodriguez-Iturbe and Porporato, 2004). However, since the deep percolation is not a prominent feature of the hydrology of semi-arid systems (often less than 1% of the budget drains to the water table, e.g., [START_REF] Scholes | An African Savanna: Synthesis of the Nylsvley Study[END_REF]

Model of savanna dynamics

The dynamics of trees and grass are linked to the dynamics of soil water. Thus in Eq. ( 2), the colonization rates, of tree, c T , and grass, c G , are not constant, but functions of the soil moisture in the root zone. For simplicity, we assume a linear dependence of c T and c G on S, i.e., c T = J T S and c G = J G S, where J T , and J G are the maximum colonization rates, carrying the dimension [1/t].

Thus the coupled dynamics of tree, grass, and soil moisture in the presence of fire are described by a system of three differential equations:

°°°°°°® G J J G G J W W H fG G STG G T SG dt dG fGT T T ST dt dT SG ST G T S S w p dt dS GO T G F T T G T ) (1 = ) (1 = ) (1 ) (1 = 1 .
(4) Equation ( 4) is a minimal ecohydrologic model of savanna, characterized by 10 parameters: Their stochastic behavior can be considered through the statistical dynamical approach outlined by De [START_REF] De Michele | A minimal model of soil watervegetation interactions forced by stochastic rainfall in water-limited ecosystems[END_REF]. The analytical results obtained here do not depend on rainfall and fire stochasticity, so our mathematical treatment is kept as simple as possible.

z, n (w 1 = z × n), H, W T , W G , J T , J G , G T , G GO , G F
The model proposed in Eq. ( 4) is an extension of the model proposed by

Values of the parameters

The values of the parameters in Eq. ( 4) are determined as follows. The depth of the root zone z is in the range 0.1 1 m, while the porosity n in the range 0.05 0.55, [START_REF] Scholes | An African Savanna: Synthesis of the Nylsvley Study[END_REF]. The colonization and loss rates are obtained by consideration of the time scales necessary to attain the steady state. For the loss rates we consider one plant functional group at a time, in absence of fire and soil water. Under these conditions, the vegetation declines to zero exponentially, i.e. v exp( Gt). Making hypotheses about the time needed for each functional type to go close to extinction starting from its maximum value, we can estimate the mortality rate G. Savanna trees have a lifespan in the order of |10 100 years, consequently G T | 0.03 0.3 y -1 (e.g., a value of 0.0255 is used in [START_REF] Bampfylde | Modelling rain forest diversity: the role of competition[END_REF]. Grass tillers, on the other hand, live for 1 3 years, consequently G GO | 1 3 y -1 . For the colonization rates, once we have fixed a value for the loss rates, we estimate the time interval necessary to the vegetation to attain its steady state, with unlimited resources (S = 1) and in the absence of competitors. We assume for trees that this takes 5 100 years, consequently J T | 0.15 2.5 y -1 ; and for grass, 20 180 days and consequently J G | 20 200 y -1 . The parameter H is determined assuming complete bare soil condition (i.e., T = 0, G = 0) and absence of rainfall ( p = 0). In these conditions S o 0 with an exponential function S(t) = exp( Ht), thus using a desiccation time (i.e., the time to go from S = 1 to S = 0) of 35 50 days [START_REF] Wythers | Bare-soil evaporation under semi arid field condition[END_REF], H | 20 30 y -1 . The values of the uptake parameters, W G and W T , are determined fitting Eq. ( 4), at the steady state to the upper bound of the woody cover data collected by [START_REF] Sankaran | Characteristics of African savanna biomes for determining woody cover[END_REF], having fixed 1) gives a comparison among the experimental data collected by [START_REF] Sankaran | Characteristics of African savanna biomes for determining woody cover[END_REF], the maximum tree cover obtained using a 99th quantile piece-wise linear regression by [START_REF] Sankaran | Determinants of woody cover in African savannas[END_REF], and the tree coverage calculated using Eq. ( 4) at the steady state. A range of 1 50 y -1 is obtained for W G and W T , we will use W T = 30 y -1 , W G = 10 y -1 . The parameter G F varies from 0.02 if trees are very fire resistant to 0.6 if trees are easily killed by fire, in according to [START_REF] Casagrandi | A minimal model for forest fire regimes[END_REF]. We consider the annual rainfall in the range p | 0 1800 mm y -1 . According to [START_REF] Belsky | Influences of trees on savanna productivity: tests of shade, nutrients and treegrass competition[END_REF], in Africa, the fire occurrence in moist savannas is one every 1 2 years, while in dry savanna it is one every 3 to 10 years. For Brazilian cerrados and the wetter regions of Africa (the most frequently burnt ecosystems in the world) [START_REF] Bond | Fire as a global 'herbivore': the ecology and evolution of flammable ecosystems[END_REF] report values of the fire frequency that reaches a maximum of 2 y -1 . Here, we consider f | 0 1.2 y -1 . Table 1 reports the range for the parameters and environmental forcings involved in Eq. ( 4).

z = 1 m, n = 0.345, H = 20 y -1 , J T = 2 y -1 , J G = 180 y -1 , G T = 0.04 y -1 , G GO = 2.8 y -1 and assuming f = 0. Figure (
range succession models philosophy. In this case, savannas can be considered as broad ecotones between grasslands and closed woodlands as noticed in [START_REF] Walter | Ecology of Tropical and Subtropical Vegetation[END_REF]. This type of savanna is commonly called dry savanna, [START_REF] Sankaran | Determinants of woody cover in African savannas[END_REF].

Vegetation changes along a rainfall gradient with frequent fires

The transect C-D for high fire frequency (f = 1 y -1 ) intersects regions of bistability: this implies that the vegetation model exhibits an hysteresis cycle (i.e., abrupt changes) in terms of both tree and grass. 5, 6, 7 could be classified as fire-dependent ecosystems, according to [START_REF] Bond | What controls South African vegetation -climate or fire?[END_REF].

The alternation of stable states along the rainfall gradient and the presence of abrupt transitions are coherent with the disequilibrium paradigm and the state-and-transitions models philosophy. Fire-dependent savannas exist because of fires causing bottlenecks in tree demography preventing the canopy closure. Moreover, the persistence of grass in firedependent ecosystem is strengthened by the positive feedback between grasses and fire: more grasses means more fuel load, more fire implies less trees, and consequently more space available to grasses and, thence, an increase in fire fuel load; in case of grasses extinction, trees grow undisturbed due to the absence of grass and, thence, of fire.

The grassland in region 6 can be denoted as false grassland according to [START_REF] Acocks | Veld types of South Africa[END_REF] and [START_REF] Bond | What controls South African vegetation -climate or fire?[END_REF], because in absence of fire the vegetation should be more wooded.

For high values of annual rainfall, p ≥ 1100 mm y -1 , the co-existence of tree and grass is permitted only by the high levels of fire disturbance. This type of savannas, disturbancedriven, is often called moist savanna.

Vegetation changes along gradients of fire frequency

Here a fire gradient, 0.2 ≤ f ≤ 1 y -1 at a value of annual rainfall p = 650 mm y -1 is considered. 4), starting from point P of savanna equilibrium, and increasing the fire disturbance in the range 0.2 ≤ f ≤ 0.8 y -1 , savannas persist. A further increase of the fire disturbance, f > 0.8 y -1 , leads to a transition from savanna to forest in panel (a), a counter-intuitive result. In practice, fires so frequent cannot be sustained at this rainfall level, especially in presence of trees. In panel (b), with 0.8 ≤ f ≤ 1 y -1 , the system comes into regions 6 and 7 of bistability. Tolerance to the fire, especially in the juvenile life stages, has been noted by [START_REF] Bond | What controls South African vegetation -climate or fire?[END_REF] as a key requirement for savanna trees. In panel (b), the system can stay in savanna (when the system is in region 7) and then pass into grassland (when the system is in region 6). This is in agreement with the school of thought according to which: fire might be, in the long term, the only management tool to maintain healthy grass populations (e.g., Zimmermann et al., 2009 and reference therein). If the increase of fire is not gradual, abrupt transitions from savanna to grassland or from savanna to forest are feasible.

CONCLUSIONS

We present a simple ecohydrologic model of savanna vegetation, described through a set of three differential equations. The space implicit model includes the dynamics of tree, grass, and soil water as forced by rainfall and fire. This analytical (as opposed to numerical simulation) model is unusual in that it combines both competition-and disturbance-based mechanisms, and shows that they occupy a continuum rather than being alternative

  in the root zone. Let z [A] indicate the root zone depth, n is the porosity (fractional pore volume [ ]), w [A 3 ] the control volume which is assumed having unit area and depth z, w = 1 × z, and w p [A 3 ] the pore space in the volume w, w p = 1 × z × n=1 × w 1 , indicating with

  references therein), we have made the linear assumption to keep the model tractable as first approximation. The term HS(1 T G) is the evaporation from bare soil. It depends on the fraction of bare soil, (1 T G), and the available soil moisture S. The term W T ST is the water uptake by tree and W G SG the water uptake by grass. In Eq. (3), the parameters H, W T , W G , normalized by w 1 , are: maximum evaporation rate from the bare soil, water uptake rate by trees, and water uptake rate by grass respectively. The parameters H, W T , W G are all positive, and carry dimension [1/t].

  . The annual rainfall, p, and the fire frequency, f, are environmental forcings treated here as constants. In reality, they are stochastic variables as assumed by Fernandez-Illescas and Rodriguez-Iturbe (2004) in the case of rainfall forcing and by D'Odorico et al. (2006) and Hanan et al. (2008) in the case of fire disturbances.

De

  [START_REF] De Michele | A minimal model of soil watervegetation interactions forced by stochastic rainfall in water-limited ecosystems[END_REF], and[START_REF] Vezzoli | Dryland ecosystems: The coupled stochastic dynamics of soil water and vegetation and the role of rainfall seasonality[END_REF] for the coupled dynamics of soil moisture and one plant functional group in arid and semiarid ecosystems. The groups of parameters represented by the ratios O T = G T /J T and O T = (G GO +f)/ J G are of particular interest in the next.

Figures

  Figures (3c, d) give the values of T and G along the transect C-D. Starting from point E in

  Figure (4) gives the bifurcation diagram for two different values of the fire sensitivity parameter G F : 0.05 and 0.35. The first value is representative of fire resistant trees, panel (a), while the second value represents fire-vulnerable trees, panel (b). The stability map in Fig. (4) can refer to plant communities with different fire vulnerabilities of trees, but also to different life stages of trees at a single site, e.g., panel (a) for adult trees, and panel (b) for juvenile trees. In both panels of Fig. (

Figure 3 Figure 4

 34 Figure 3 Changes in vegetation stable states over rainfall gradients in low and high

  Figure 1

Table 2

 2 Steady states of the model in Eq. (4).

	Steady state

Acknowledgements

The research was co-funded by Comune di Milano through BIO-DESCESA project (BIOdiversity and DEsertification SCEnarios in South Africa under climate change projections) and by the bilateral science agreement between Italy and South Africa.

-24-Eagleson, P.S., Segarra, R.I. (1985). Water-limited equilibrium of savanna vegetation system. Water Resour. Res., 21(10), 1483Res., 21(10), -1493. .

Steady states

Equation (4) admits five steady state solutions, i.e., dS/dt = dT/dt = dG/dt = 0, but only four satisfy the conditions 0 ≤ T ≤ 1, 0 ≤ G ≤ 1, and 0 ≤ T+G ≤ 1. Of these, the first solution is characterized by the absence of vegetation, which we will refer to as unvegetated i.e., T = 0, G = 0. The second solution has grass only, i.e., (desert) grassland, T = 0, G > 0. Solution three has trees only i.e., forest, T > 0, G = 0. The fourth solution has a coexistence of tree and grass, i.e., savanna, T > 0, G > 0. Table 2 reports the analytical expressions for the four stable states. For simplicity the equilibrium of tree-grass coexistence is given in the implicit form.

Stability analysis

Conditions for the spreading of a single functional type

The conditions for the spreading of a single functional type are determined by considering one functional type at a time. The type can have a positive stable equilibrium if its colonization rate is greater than its mortality rate. In Eq. ( 4) this is satisfied by the condition S > O T for trees, and to S > O G for grass, where S is the soil moisture at the equilibrium.

Note that if O G > 1, then the conditions for the grass spreading can never be met because S takes values in the closed interval [0, 1]. Similarly, if O T > 1 then the condition for the tree spreading can never be satisfied.

Stability map

A stability analysis of the system described in Eq. ( 4) is performed in the space of parameters p f, i.e., the environmental space defined by rainfall and fire.

The other parameters are assumed to be: z = 1 m, n = 0.345, H = 20 y -1 , W T = 30 y -1 ,

Within the ecologically-feasible domain, i.e., positive values of the environmental forcings and state variables, the system has 7 different regions of stability and bistability, see Fig. (2). Region 1 is unvegetated: the rainfall, p ≤ 100 mm y -1 , is not sufficient to support either grasses or trees. In region 2, the stable equilibrium is (desert) grassland: p is sufficient for the existence of grasses, but not trees. In region 3, rainfall allows the occurrence of both trees and grasses, and the stable equilibrium is thus savanna. For high rainfall values (p ≥ 1100 mm y -1 ), within region 3, the equilibrium values of tree canopy cover are T ≥ 0.8, in other words, although grass is present, this vegetation is a forest, closed woodland or thicker, [START_REF] Scholes | Savanna[END_REF]. In Fig.

(2) the region 3 has been divided into two subregions (3a) and (3b) by the isoline T = 0.8 (grey line), corresponding to the cover fraction corresponding to the canopy closure, above which virtually no grass would be present because of shading.

In region 4, the stable equilibrium is forest, as well as in region 5. In region 4 grass cannot grow because the mortality due to fire is too high (i.e., very high fire frequencies can only be tolerated if the rainfall is also high). In region 5, the fire is not sufficient to prevent the closure of the tree canopy -a condition that is widely encountered. In both regions 4 and 5, the direct effect of fire on grass is greater than the indirect effect on trees. Regions 6 and 7 are characterized by bistability. In region 6, the stable equilibria are grassland and forest, while in region 7 are savanna and forest.

The bistability between savanna and forest has been found also by da Silveira Lobo [START_REF] Da Silveira Lobo | Savanna-forest hysteresis in the tropics[END_REF]. The equilibrium state of the system in the bistability regions depends on the initial condition and on the history of disturbances [START_REF] Solbrig | Biodiversity and tropical savanna properties: a global view[END_REF].

The bistability in regions 6 and 7 is associated to the high fire frequency. Depending on initial condition, two scenarios are possible: 1) fire completely removes grasses leaving only trees (forest), 2) fire is enough to completely remove trees (region 6), or to limit their abundance (region 7), but some grass is left.

Note that the boundaries of the regions are parameter-dependent.

RESULTS

In this Section we discuss the stability map in detail showing the changes in vegetation that it predicts along gradients of rainfall and fire. Moving up a rainfall gradient,

Vegetation changes along a rainfall gradient with infrequent fires

The transect 50 ≤ p ≤ 1800 mm y -1 at f = 0.2 y -1 is characterized by a continuous change of vegetation, without abrupt transitions, from unvegetated, to (desert) grassland, to savanna, to forest. This sequence has been described as the sub-tropical succession according to [START_REF] Shmida | Plant growth-form strategies and vegetation types in arid environments[END_REF]. In Fig. (3) note that the boundary between grassland and savanna is located at different values of annual rainfall (p) depending on the fire frequencies (f). If f o 0 y -1 the region of grassland between unvegetated and savanna is small. In Fig.

(2), the transect A-B is representative of fire conditions with 0 ≤ f ≤ 0.6 y -1 . In absence of fire disturbance (f = 0 y -1 ), the same vegetation sequence is observed with the only difference being the tree coverage at a given rainfall, see Fig.s (3a,b). This means that the fire influences the tree-grass ratio in region 3, and the grass coverage in region 2, while the rainfall availability determines the type of vegetation. Regions 2 and 3 could be designated climate-dependent ecosystems, following the nomenclature suggested by [START_REF] Bond | What controls South African vegetation -climate or fire?[END_REF].

For values of annual rainfall in the range 100 ≤ p ≤ 600 mm y -1 the co-existence of tree and grass is permitted by the limited amount of water resource, while the fire influences the tree-grass ratio only. The water scarcity limits the density of the superior competitor, allowing the co-existence through mechanisms of balanced competition.

Along the rainfall gradient the savanna equilibrium varies continuously without abrupt or irreversible transition. This behavior is coherent with the equilibrium paradigm and the descriptions of savanna dynamics.

The stability map in the parameters space of rainfall and fire synthesizes the conditions for the different types of vegetation: unvegetated, (desert) grassland, savanna, and forest. It helps to 1) clarify that the co-existence of tree and grass is due to the simultaneous influence of rainfall and fire, and 2) distinguish the two main types of savanna. For 100 ≤ p ≤ 600 mm y -1 the dry savanna co-existence is permitted by the balanced competition for limited rainfall and fire influences only the tree-grass ratio. The system would still be a savanna, even in the absence of fire. For rainfall above 1100 mm y -1 the moist savanna co-existence can only occur in the presence of a high level of fire disturbance, because the ecosystem would be a forest in the absence of fire. In the intermediate range, 600 ≤ p ≤ 1100 mm y -1 , savanna is the result of the co-occurence of water limitation and fire. The stability map shows how dry savannas are stable equilibria, while moist savannas are a bi-stable condition with forest. The stability map allows also to predict the vegetation structure changes that occur along gradients of rainfall and fire frequency, and to clarify the distinction between climate-dependent ecosystems and firedependent ecosystems.

TABLES

Table 1 Range for parameters and environmental forcings of the model in Eq. ( 4).

Symbol

Unit Range (2005). The continuous line is the tree coverage calculated using Eq. ( 4) at the steady state, with the parameter values set at z = 1 m, n = 0.345, H = 20 y -1 , W T = 30 y -1 , W G = 10 y -1 ,


Ricard V Solé

Joan Saldaña

Jose M Montoya

Douglas H Erwin

Simple model of recovery dynamics after mass extinction

Biotic recoveries following mass extinctions are characterized by a complex set of dynamics, including the rebuilding of whole ecologies from low-diversity assemblages of survivors and opportunistic species. Three broad classes of diversity dynamics during recovery have been suggested: an immediate linear response, a logistic recovery, and a simple positive feedback pattern of species interaction.

Here we present a simple model of recovery which generates these three scenarios via differences in the extent of species interactions, thus capturing the dynamical logic of the recovery pattern. The model results indicate that the lag time to biotic recovery increases significantly as biotic interactions become more important in the recovery process.

I. INTRODUCTION

Until recently paleontologists have focused on understanding the dynamics of the extinction process and assumed that with the amelioration of the causes of the extinction recovery would be straightforward: diversification of new species and migration of surviving species would ensure rapid repopulation of niches vacated by the extinction. Little attention was paid to the actual dynamics of recovery. Logistic models derived from ecology were commonly seen as the appropriate theoretical framework (e.g. [START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity I. Analysis of marine orders[END_REF][START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity II. Early Phanerozoic families and multiple equilibria[END_REF][START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity III. Post-Paleozoic families and mass extinction[END_REF], coupled with the dynamics of diversification into empty niches (Valentine 1980; [START_REF] Valentine | Diversity trends within a model taxonomic hierarchy and evolution[END_REF], where the rate of diversification was simply assumed to be diversity dependent. However the lag of some 4 million years between the end of the end-Permian mass extinction and the onset of recovery of most marine and terrestrial ecosystems (Erwin 1998a(Erwin , 1998b;;[START_REF] Erwin | Increasing returns, ecological feedback and the Early Triassic recovery[END_REF][START_REF] Sahney | Recovery from the most profound mass extinction of all times[END_REF]) and the presence of many 'Lazarus' taxa which disappear from the fossil record before the extinction but then reappear late in the recovery, suggests something of the complex ecological and evolutionary dynamics of postextinction intervals.

As paleontologists have begun to explore these complex dynamics over the past decade most have followed a largely empirical approach, documenting local and regional recovery patterns, exploring the structure of recovery communities and investigating the correlation to changes in stable isotopes (e.g. d'Hondt et al 1998; [START_REF] Payne | Large perturbations of the carbon cycle during recovery from the end-Permian extinction[END_REF]. As valuable as such data is, increased understanding of the dynamics of post-extinction diversification also requires an appropriate theoretical frame-work, both for understanding and interpreting existing data and for guiding new research. Recent efforts in this direction include the explicit modeling of recovery dynamics using trophic network models (Solé et al., 2002a;[START_REF] Roopnarine | Extinction cascades and catastrophe in ancient food webs[END_REF]; [START_REF] Roopnarine | Trophic network models explain instability of Early Triassic terrestrial communities[END_REF]; see also [START_REF] Solé | Selforganization in complex ecosystems[END_REF]Yedid et al., 2009). These models consider some sort of trophic interaction pattern between species belonging to different levels. The number of such levels, their species composition as well as the number and strength of links evolve with speciation and extinction.

Here we consider three different recovery scenarios: First, immediate biotic recovery following a mass extinction, modeled as a linear response. This appears similar to at least two events in the fossil record, the end-Ordovician mass extinction about 439 million years ago (Ma) and possibly the end-Triassic mass extinction 199 Ma. This may also apply to some smaller biotic crises, and roughly corresponds to the standard description of so called neutral communities (e.g. Hubbell 2001; [START_REF] Alonso | The merits of neutral theory[END_REF], where the intensity of species interactions do not affect the recovery process. Second, a logistic expansion, which is the traditional expectation of paleontologists (e.g. [START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity III. Post-Paleozoic families and mass extinction[END_REF]. It is not clear that there are good empirical examples of this pattern. Third, we also present a simple model of diversification driven by biotic interactions between pairs of species. This model is a first effort at examining the relevance of positive feedback processes in evolutionary diversifications and provides well-defined predictions of the time lags to recovery that should be expected. In each of the models we consider the equilibrium case where diversity returns to the pre-extinction level. In other words, the maximal diversity in the system is pre-determined. In reality of course, the most interesting cases are those where post-extinction diversity increases.

Identifying the pattern of recovery might help to establish the relevance of biotic interactions in rebuilding functional ecosystems at evolutionary scales. In a related ecological context, recent attempts to explain current patterns of biodiversity at local and regional scales (e.g. Hubbell 2001) assumed ecosystems as neutral entities, where interactions among species are not important. This perspective has recently being challenged [START_REF] Wooton | Field parameterization and experimental test of the neutral theory of biodiversity[END_REF]; seealso Pueyo 2006 and Pueyo et al), showing that neutral models predicts poorly field experimental results, indicating and essential role of species interactions. In this context, more general models can be defined by expanding the neutral approach and considering different levels of heterogeneous interactions (Solé et al., 2002b).

The results obtained from our model suggest that the greater the interactions among pairs of existing species in the generation of new species, the longer the delay in recovery. The long delay together with a rapid rise of diversity could result from two different processes: (a) increasing numbers of ecological interactions might provide the context for new opportunities to speciate, but building new links is a slow, costly process, strongly dependent on external and internal constraints. Alternatively, (b) as the number of species and thus interaction grows, the potential combinations eventually may allow a rapid increase in species. Our approach is the simplest one able to capture these two ingredients.

II. MATHEMATICAL MODEL

The simplest model able to cope with biotic recovery would be a logistic one [START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity III. Post-Paleozoic families and mass extinction[END_REF]. Here, if S indicates the number of species and μ is speciation rate, we have:

dS dt = μS 1 - S M (1)
where M is the maximum number of allowed species.

In classical population dynamics (Case, 1999) this maximum is the so called carrying capacity (usually indicated as K) and gives the maximal number of individuals, whereas μ would be the per capita growth rate. Here speciation is defined as the difference μ = oe between origination (o) and extinction (e) rates. Specifically, as described by [START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity III. Post-Paleozoic families and mass extinction[END_REF] these are per-taxon rates and are expected to be stochastic parameters. In a deterministic context (to be followed here) these parameters are considered average values roughly constant over (geologic) time. This model however is unable to explain the spectra of recovery patterns observed in the fossil record even if we widely vary the model parameters. The reason is that the underlying assumptions involve a system in which ecological interactions are solely based on the available resources, with no explicit interaction among species. The situation is schematically drawn in figure 1(a), where species are indicated as gray balls and they monotonously fill the available space.

One way of generalizing the previous approach is to consider a dependence of speciation dynamics on species interactions. The underlying assumption, schematically drawn in figure 1(b) is that post-extinction biotic recovery (and indeed ecosystem rebuilding in general) is not described by means of a neutral process. Instead, it deals with the full spectrum of biotic interactions displayed by species belonging to different trophic levels and having different ecological interactions, from mutualism to parasitism. In such a context, the opportunities available to a given species to speciate are dependent upon the biotic structure of the current community. Actually, some general patterns exhibited by complex ecosystems over evolutionary time scales, such as community rebuilding after a major extinction event are closely related to some phenomena occuring at short, ecological time scales, such as ecological succession (Solé et al., 2002;[START_REF] Solé | Selforganization in complex ecosystems[END_REF].

We can make such approach explicit by considering three basic scenarios of diversity increase. The logic of these scenarios is sketched in figure 2a-c. We will reduce our description to a single macroscopic variable S (number of species) and a single free parameter μ representing speciation rate. The three diagrams indicate how these two ingredients influence each other. Here we do not indicate explicitely the negative feedback associated with limited resources. As will be shown below, these three plots can be directly translated into explicit equa- Here S indicates the number of species and μ speciation rates. In (a) a constant, diversity-independent rate is at work. In (b) speciation rate is species-dependent: the more species present the more species are generated. The third case (c) involves a positive feedback where the presence of species also enhances speciation rates.

tions for diversity dynamics. In order to further define our three models, let us consider a more microscopic description. If "0" denotes an empty niche (allowed to exist provided that resources are available) and S i corresponds to a given species i, it is possible to describe the three previous possibilities in terms of three simple transitions among states. These are:

0 μ -→ S i (2) 0 + S i μ -→ S i + S i (3) 0 + S i + S j μ -→ S i + S i + S j (4)
where each of these transitions occurs at a rate μ. The first case would correspond to figure 2a, a situation in which empty niches are simply refilled by invading species and thus μ effectively represents immigration rate. The second (figure 2b) implies the presence of empty niche and an available species (here indicated as S i ) in order to speciate and obtain a new species (here indicated as S i ). Finally, if species S i requires species S j to generate a new species (since S i is, say, a parasite of species S j ) then the two of them are necessary to effectively obtain an speciation event (figure 2c). In order to take into account the three situations, the previous equation ( 1) can be generalized as:

dS dt = μΦ β (S) 1 - S M (5)
where the function Φ β (S) includes the possibility of considering the key role of existing species to rebuild the ecosystem through speciation. One of the simplest choices is a power functional form, i. e.

Φ β (S) = S β (6)
with β ≥ 0. This choice is consistent with the previous representation of the three scenarios in terms of reactions. By considering these three case studies, the model naturally produces the three main types of recovery pattern. These correspond to:

A. Linear model, β = 0

In this case we have a system in which the increase in species numbers would follow a linear equation:

dS dt = μ 1 - S M (7)
where available resources (such available space and nutrients) allow species diversity to linearly increase through time until the maximum is reached. Using hereafter (without loss of generality) M = 1, it is not difficult to solve this equation, which gives an asymptotic increase:

S(t) = 1 -(1 -S 0 )e -μt (8)
This corresponds to a situation in which species coming from an external (biogeographic) pool invade the area of interest. Examples would include re-forestation after a devastating forest fire or re-population following a regionally extensive volcanic eruption. The resulting pattern (figure 3a) shows a monotonous increase on diversity takes place promtly reaching pre-extinction levels. In this figure (right column) we also show the corresponding spindle diagram, which provides a complementary picture of the diversification process, showing how relative diversity expands or shrinks through (geological) time. This model belongs to the class of models used in island biogeography, where islands free of species are colonized by the mainland pool. Such colonization process is typically assumed to be a linear function of the species abundances. Similarly, the extinction rate is also considered a constant process.

B. Density-dependent, β = 1

This corresponds to the logistic case [START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity III. Post-Paleozoic families and mass extinction[END_REF]. Species essentially interact because of their limiting resources (i.e. they compete), and no further biotic interactions are involved. The solution to [START_REF] Case | An illustrated guide to theoretical ecology[END_REF] with M = 1 now reads:

S(t) = 1 + 1 -S 0 S 0 e -μt -1 (9)
again asymptotically reaching the carrying capacity. The two basic trends present in logistic growth (initial exponential increase followed by saturation) are clearly observed in figure 3(b).

C. Hyperbolic case, β = 2

This scenario considers the interaction among pairs of existing species. It assumes that speciation requires pairs of species to be interacting, not only through competition. This requirement strongly constrains the speed of speciation events, since interactions are necessary to evolve (co-evolve). A given parasite, for example, needs a host to reproduce. The presence of such a pair is also needed for the parasite to evolve new traits and eventually lead to a new species. A similar argument can be made with predator-prey or mutualistic interactions. The dynamics are now governed by: Here a pulse extinction occurs at time zero. The initial diversity after the pulse extinction is S0 = 0.01 and speciation rates are varied in a given range (2 ≥ μ ≥ 0). There is always a delay, even when large rates are used, although it becomes shortened in a nonlinear way. Here equation [START_REF] Erwin | The end and the beginning: recoveries from mass extinctions[END_REF] has been numerically integrated using Euler's method .

dS dt = μS 2 (1 -S) (10)
superficially resembling the logistic equation. However, the feedback implicit on the dependence of speciation on interactions among species leads to a very different dynamical behavior, characterized by a long lag after a rapid, strong perturbation is performed on the system [START_REF] Erwin | Understanding biotic recoveries: extinction, survival and preservation during the end-Permian mass extinction[END_REF] and is illustrated in figure 3(c), where two different speciation rates have been used. The long delay is clearly apparent in both cases. This situation corresponds to hyperbolic replicator dynamics, similar to the one displayed by models of hypercyclic organization (Szathmáry and Maynard [START_REF] Szatmáry | From replicators to reproducers: the first major transitions leading to life[END_REF]. In figure 4 we show the effects of continuously varying speciation rates on the time evolution of species numbers after the pulse extinction event. This hyperbolic growth model has been used to describe the dynamics of marine biodiversity through the Phanerozoic (Markov and Korotayev, 2007) and seems to better fit available data than the exponential one (see Benton, 1995). Such type of model has also been applied to the dynamics of world population and economics (Johansen and Sornette, 2001 and references cited).

In general, for any β ≥ 0 and if S 0 < 1, the solution S(t) ∈ (0, 1) for any t > 0. So, when comparing growth rates of the different solutions with the same initial condition S 0 , it follows

dS dt β<1 > dS dt β=1 > dS dt β>1 , (11)
where the derivatives are evaluated at the same moment.

In particular, it follows that the growth rate of the number of species (i.e. diversification rate) according to the model with β > 1 is very much lower than the the logistic case (β = 1) as long as S(t) 1, i.e., when the community is starting to define their biotic interactions. Therefore, β reflects the dependence of diversification rates on species interactions. In other words, how fast diversity will increase from a given initial condition (such as immediately after a major extinction) if β-dependent. Larger β values imply slower speeds of diversity growth (as indicated by the smaller derivatives) thus suggesting that perhaps the tempo and mode of recovery dynamics is strongly influenced by the ecological constraints associated to speciation events. If true, this should be observed in terms of different types and durations of recovery times, as shown in the next section.

III. DELAY TIMES TO RECOVERY

One of the fundamental questions relating recovery patterns involves the time lags to be expected until complete recovery is reached. As we noted before, the most interesting case occurs when total diversity is unconstrained in a non-equilibrium system, but here we continue to examine the restricted case where an equilibrium diversity is defined. Moreover, speciation rates must strongly influence the speed of recovery. This is illustrated in figure 4 for the β = 2 case using different speciation rates. As can be appreciated, a wide range of delay times is observable. In order to compare them, let us assume that the initial condition S 0 is the same for all models and satisfies S 0 1 (that corresponds to the condition after a mass extinction), and let us fix a value S * close to (but less than) the carrying capacity. Our goal here is to obtain well-defined estimates of recovery delay times T For β = 0, we have, from equation ( 7) and M = 1,

T (0) R = 1 μ S * S0 dS 1 -S (12)
which gives after integration to a delay time:

T (0) R = 1 μ log 1 -S 0 1 -S * (13)
Similarly, the recovery time for the logistic model (β = 1) can be easily obtained. By integrating equation ( 1) with M = 1, we obtain

μt = log 1 -S 0 1 -S(t) + log S(t) S 0 (14)
and so the recovery time for this case will be

T (1) R = 1 μ log 1 -S 0 1 -S * + log S * S 0 (15)
A similar calculation can be performed for β = 2, which gives a recovery time

T (2) R = 1 μ log 1 -S 0 1 -S * + log S * S 0 + 1 S 0 - 1 S * (16)
The recovery time for the general model with a given integer β ≥ 2 is given by:

T (β) R = T (1) R + 1 μ Γ(S 0 , S * , β) (17)
where Γ(S 0 , S * , β) is given by the sum:

Γ(S 0 , S * , β) = β-1 m=1 (S * ) m -S m 0 mS m 0 (S * ) m (18)
These previous expressions can be simplified under the previous assumptions on S 0 and S * , namely,

1 -S * ∼ O(S 0 ) ( 19
)
with S 0 1, that is, assuming a large extinction event and an almost full recovery of diversity. In this case, we have the following approximate estimations for delay times

T (i) R T (0) R ≈ - log S 0 μ , (20)
T (1) R ≈ - 2 log S 0 μ , (21)
T (2) R ≈ 1 μS 0 , ( 22
)
and, for the general model with β ≥ 2,

T (β) R ≈ 1 μ(β -1)S β-1 0 . ( 23
)
It is easy to compare the previous estimates and find that the following inequalities hold:

T (2) R T (1) R ∼ 2T (0) R (24)
in other words, recovery times rapidly increase as biotic interactions become more important within a community. Although lag times are comparable under scenarios not considering species interactions, the situation sharply changes at β = 2. This result confirms the expectation of a qualitatively different pattern when recovery relies upon ecosystem rebuilding, comprising both the presence of species and their interactions, The simplicity and generality of our analytic expressions allow to test our predictions with real data.

The relationship between delay times changes with both the size of the extinction event and the rate of speciation in the hyperbolic case (figure 5). The theory predicts a rapid decay in T

R inversely proportional to μ as well as a rapid increase with larger extinction events.

IV. DISCUSSION

Mass extinctions have received considerable attention from paleontologists, evolutionary biologists and others over the past several decades (see [START_REF] Jablonski | Mass extinctions and macroevolution[END_REF] and references therein). The subsequent process of biotic recovery has received much less attention, in part we think, because of the assumption that once the cause of the extinction was ameliorated the survivors would quickly respond. The growing number of studies of biotic recoveries [START_REF] Erwin | Understanding biotic recoveries: extinction, survival and preservation during the end-Permian mass extinction[END_REF](Erwin , 1998a[START_REF] Erwin | Lessons from the past: biotic recoveries from mass extinction[END_REF][START_REF] Benton | How to kill (almost) all life: the end-Permian extinction event[END_REF][START_REF] Lockwood | The K/T event and infaunality: morphological and ecological patterns of extinction and recovery in veneroid bivalves[END_REF][START_REF] Wagner | Abundance distributions imply elevated complexity of Post-Paleozoic Marine Ecosystems[END_REF][START_REF] Payne | Environmental and biological controls on the initiation and growth of a Middle Triassic (Anisian) reef complex on the Great Bank of Guizhou, Guizhou Province, China[END_REF][START_REF] Aberhan | Faunal evidence for reduced productivity and uncoordinated recovery in Southern Hemisphere Cretaceous-Paleogene boundary sections[END_REF] suggest more complex dynamics are occurring. Although paleontologists once assumed (often tacitly) that survivors simply diversified into the empty ecospace produced by the mass extinction (see Erwin 1993 for discussion) the biogeographic variability in recovery rates and processes (e.g. Jablonski 1998), as well as the recognition that some clades may survive the extinction only to succumb during the subsequent radiation, indicates that more complex processes may be operating. In addition, from theoretical grounds we would expect clades at different trophic levels (e.g. producers, herbivores, carnivores) and functional groups (e.g. grazers, detritivores) to respond at different rates.

In this paper we have taken a theoretical approach to the problem of recovery based on the simplest and most general approximation that can be taken. Without considering the particularities of the underlying ecological network, we have explored the consequences of including (or not) pairwise species interactions as part of the constraints involved in generating new species. Each of three recovery scenarios predicted by the model presented here may characterize different recovery periods. But they may also characterize and distinguish, within a particular period, the recovery dynamics of different taxa, trophic levels, or biogeographical areas. The rapid linear recovery of primary producers, for example, contrasts with observed delays in higher trophic levels (e.g. [START_REF] Hart | Recovery of the food chain after the Late Cenomanian extinction event[END_REF][START_REF] D'hondt | Organic carbon fluxes and ecological recovery from the Cretaceous-Tertiary mass extinction[END_REF]). Within taxa, or feeding groups, omnivore and detritivore species typically recover faster than strict hervibores and carnivores (Smith and Jeffrey 1998;[START_REF] Fara | Diversity of Callovian-Ypresian (middle Jurassic-Eocene) tetrapod families and selectivity of extinctions at the K/T boundary[END_REF]. Finally, biogeographic differences in recovery patterns (e.g. [START_REF] Jablonski | Background and mass extinction: The alternation of macroevolutionary regimes[END_REF][START_REF] Jablonski | Geographic variation in the molluscan recovery from the End-Cretaceous extinction[END_REF]) may also reflect differences in the relevance of biotic interactions for diversification among different areas. Therefore, identifying the pace of recovery informs on the processes responsible of diversification.

There is actually a simple connection between our model, which only includes a homogeneous set of species, and models considering multiple trophic levels. As an example, let us consider a two-trophic system composed by two species assemblages: producers and primary consumers. We thus consider two simultaneously evolving groups (figure 6a) which interact through coevolving ecological links. Indicating as S 1 and S 2 the number of species at each level, a macroscopic view of the ecologically-driven evolutionary dynamics can be obtained from the following set of equations:

dS 1 dt = μS 1 (M -S 1 ) -δS 1 S 2 ( 25
)
dS 2 dt = S 1 S 2 -eS 2 (26)
We can see that the previous equations, describing changes in the number of species through time, are not formally different from a standard Lotka-Volterra model of consumer-producer dynamics. However, the interpretation of the parameters must be based on a different criterion: μ is again speciation rate for producers, whereas δ and need to be interpreted as the consumer-driven species extinction rate and the consumer (externallydriven) extinction rate, respectively. An example of the dynamics displayed by this model is shown in figure 6. After a major extinction (reducing both assemblies to 5% of their pre-extinction equilibrium diversity) producers recover first, followed by consumers after a long delay (fig 6 b-c). The first group follows a rapid recovery characteristic of the simple model discussed here for β =1. Consistently with our prediction, consumers need a much larger time, because the speciation rate S 1 S 2 that appears in the right-hand side of equation ( 23) is the product of two species numbers, thus consistent with the β = 2 model discussed before. This observation helps interpret the meaning of our former model in light of the position of the group (e.g. trophic level) whose recovery pattern is analyzed within a given ecological network (e.g. food web). In this sense, long delays should be expected as we move up through the food chain (Solé et al., 2002). There are two ways to achieve greater understanding of recovery dynamics. Most paleontologists have followed a largely empirical approach of developing detailed analyses of the recovery patterns within local or regional areas, sometimes integrating the fossil data with geological or geochemical information. Here the hope is that through the accumulation of a wealth of data some empirical generalities might emerge. The alternative approach is to develop, test and refine models of the processes that may be involved in recovery, to serve as a guide to empirical exploration.

The number of available models of recovery is very limited, however. As mentioned, [START_REF] Sepkoski | A kinetic model of Phanerozoic taxonomic diversity III. Post-Paleozoic families and mass extinction[END_REF] invoked a simple pattern of logistic growth following the end of a biotic perturbation, although with different intrinsic rates of increase between his Evolutionary Faunas. The teserae models of [START_REF] Valentine | Determinants of diversity in higher taxonomic categories[END_REF] see also [START_REF] Valentine | Determinants of diversity in higher taxonomic categories[END_REF]Walker 1986, 1987) have had a more pervasive effect. Instantiations of earlier less-rigorously formulated ideas on filling empty ecospace, they explicitly considered how clades might diversify into open ecospace. These models invoked empty niches (not simply unutilized resources) as a critical component and did not examine how species interactions might facilitate diversification. These models, strongly influenced by the ecology of that time [START_REF] Macarthur | The limiting similarity, convergence, and divergence of coexisting species[END_REF]Levins 1967, Rosenzweig 1975) considered competition between species as the only interaction regulating diversity recoveries, which inextricably leads to logistic recovery patterns. However, other types of interactions, i.e. facilitation, mutualism or predation, played a crucial role in the rebuilding process and add complexity to the diversification process. Indeed, current ecological models of filling niche space assume species' niches that are partially determined by their interactions with other species within the community, both as resources and as consumers, and speciation occurs within this ecological context where network structure plays a key role ( We began developing a new generation of recovery models several years ago. Initially (Sole et al 2002) we evaluated a simple model with three trophic levels corresponding to primary producers, herbivores and carnivores and found that recovery would initiate at the lowest level of the food chain, and delays in recovery times increased as we move up in the food web. These results immediately suggested that empirical studies should investigate the relative timing of post-extinction recoveries at different trophic levels, using carbon isotope fluctuations as a proxy for primary productivity (e.g. d'Hondt et al 1998).

More useful, however, would be a suite of process-based models, testable with evidence from the fossil record, which would capture the range of possible post-extinction recovery dynamics [START_REF] Roopnarine | Trophic network models explain instability of Early Triassic terrestrial communities[END_REF]. This would allow paleontologists to plan field work with the intent of testing these alternative models and allow iterative refinement of such models. With this paper we have begun development of such a theoretical toolkit, with the first process-based model to capture species interactions as an essential component. The results make several specific predictions that can be further tested with data from different recovery events. In particular, if species interactions are a significant component of post-extinction diversification then the length of the delay before the onset of rapid recovery should scale with the speciation rate.

FIG. 1 :

 1 FIG.1: Recovery patterns. Recoveries might essentially consist of refilling available space (until some carrying capacity M is reached) as indicated in (a). But they also might require ecosystem rebuilding (b) relying in the reconstruction of lost interactions, functionality and other attributes.

FIG. 2 :

 2 FIG. 2: Three possible mechanistic scenarios for recovery.Here S indicates the number of species and μ speciation rates. In (a) a constant, diversity-independent rate is at work. In (b) speciation rate is species-dependent: the more species present the more species are generated. The third case (c) involves a positive feedback where the presence of species also enhances speciation rates.

FIG. 3 :

 3 FIG. 3: Recovery patterns under each of the three scenarios of the model, involving different types of ecological interactions: (a) β = 0, (b) β = 1 and (c) β = 2. Both the diversity over time (left) and the spindle diagrams (right) are shown. Here we have used μ = 1 (continuous lines) and μ = 0.5 (dashed lines, (c)). The spindle diagram for the third case corresponds to the later value.

FIG. 4 :

 4 FIG.4: Recovery pattern for the hyperbolic (β = 2) model. Here a pulse extinction occurs at time zero. The initial diversity after the pulse extinction is S0 = 0.01 and speciation rates are varied in a given range (2 ≥ μ ≥ 0). There is always a delay, even when large rates are used, although it becomes shortened in a nonlinear way. Here equation[START_REF] Erwin | The end and the beginning: recoveries from mass extinctions[END_REF] has been numerically integrated using Euler's method .

FIG. 5 :

 5 FIG. 5: Characteristic delay to recovery based on the approximate solution given by T (2) R (equation 19). The delay time quickly decays with increasing speciation or decreasing pulse size perturbation.

R

  for different β values.

FIG. 6 :

 6 FIG. 6: Patterns of recovery in a complex ecosystems involving two tropic levels (producers and consumers). In (a) a schematic diagram shows that the philogenetic branching associated to consumers (left tree) and producers (right tree) are linked through the presence of ecological interactions of different signs. Speciation within the producers trigger speciation in consumers (positive interaction) whereas increasing numbers of consumer species might eventually reduce the diversity of producers (negative arrow). A simple mathematical model (equations 22-23) illustrates the different nature of the recovery patterns for the two levels (see text). Parameters chosen here: M = 1.5, μ = 0.1, δ = 0.25, = 0.05, e = 0.06. Recovery patterns for the two trophic levels are different (bc).

  Williams & Martinez 2000, Drossel et al. 2004; Pascual and Dunne 2004; Montoya et al., 2006; Dunne et al. 2008).

Acknowledgments

The authors thank the members of the complex systems lab for useful discussions. This work has been supported by a grant FIS 2004-05442 (RVS), the McDonnell Foundation (RVS), the Santa Fe Institute (RVS and DHE) and the Thaw Cheritable Trust (DHE).


David J S Montagnes

Emily Christopher D Lowe

Clare Roberts

Mark Nicholas Breckels

Daniel E Boakes

Keith Davidson

Patrick J Keeling

Claudio Slamovits

Michael Steinke

Zhou Yang

Emily C Roberts

Phillip C Watts

An introduction to the special issue: Oxyrrhis marina, a model organism?

Keywords: review, dinoflagellate, experimental design Oxyrrhis marina, review, dinoflagellate, experimental design

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

 Nanny (1980), when discussing experiments with ciliates, referred to a group of select genera as "the chosen few", as these protists, notably ones like Paramecium, have long acted as successful experimental tools. This contention reflects Krogh's principle that "for such a large number of problems there will be some animal of choice, or a few such animals, on which it can be most conveniently studied." (see [START_REF] Krebs | The August Krogh principle: For many problems there is an animal on which it can be most conveniently studied[END_REF]. Clearly, many such organisms have been chosen for their ability to address questions related to a particular F o r P e e r R e v i e w 4 discipline, with an emphasis on a handful of taxa; e.g. these extend from the prokaryote Escherichia coli, to the fungus Sacchromyces cerevisae, to the invertebrates Drosophila melanogaster and Caenorhabditis elegans, and to the vertebrates Mus musculus and Rattus norvegicus, the last providing a strain of that gave rise to the vernacular "lab-" or "white-rat". Indeed, various model organisms have been adopted across countless biological fields, and like any model, they are chosen to amalgamate levels of simplicity and complexity needed to address specific questions within a particular discipline (e.g. [START_REF] Hedges | The origin and evolution of model species[END_REF][START_REF] Behringer | Emerging Model Organisms: v. 1: A Laboratory Manual[END_REF].

There can be no defined set of attributes to describes an ideal "lab rat" that provides a solution to all questions. However, there are general traits that we might consider as useful (Table 1). Furthermore, each field of research will inevitably impose unique criteria when selecting models, and organisms are thus often chosen for a specific need (see Bahls et al., 2003); although in some cases there appears to be some intellectual inertia, as a model originally used for one purpose is adopted for others, at times with serendipitous consequences (e.g. Drosophila melanogaster, [START_REF] Roberts | Drosophila melanogaster: the model organism[END_REF]. It is, therefore, not our intention to suggest that the heterotrophic dinoflagellate, O. marina, can act as a model in an absolute sense; instead, we address its general and specific versatility.

Oxyrrhis marina has been routinely used in experiments for over 100 years, and its application appears to be rising exponentially across a number of fields (Fig. 1a). Certainly, citations related to O. marina-related research are increasing rapidly (Fig. 1b), reflecting its widespread distribution, small size, utility in addressing ecological issues, and crucially its experimental tractability. Thus, we suggest that O. marina is an "emerging model organism".

In the following sections, we reflect on the broad appeal of O. marina to address a range of questions across a hierarchical scale, from genetic processes to aut-and synecological issues. Our review of the literature (see papers in this special issue) suggest that this protozoa (i.e. heterotrophic protist) appears to be joining the ranks of such unicellular "lab rats" as Paramecium, Tetrahymena, Amoeba, and Dictyostelium, not simply based on its historical application but on its appropriateness to answer specific questions, following the Krogh principle [START_REF] Krebs | The August Krogh principle: For many problems there is an animal on which it can be most conveniently studied[END_REF].

IS OXYRRHIS MARINA A USEFUL MODEL?

Although no one species can provide a model for all studies or questions, a convenient attribute is to fulfil the needs of many disciplines. This is not simply because of the practical benefits associated with the economy of size (e.g. availability, ease of routine maintenance, development of standard protocols) but more critically, that being able to answer many types of questions using a single species stimulates synergistic, interdisciplinary studies. Table 1 briefly explores the benefits and disadvantages of O. marina to answer questions and guides the reader to the papers in this special issue. In general though, by preparing this special issue we have recognised O. marina emerging as a model in three distinct disciplines: ecology, evolution, and biogeography.

An ecological model

Several of the papers in this special issue explore the utility of O. marina in aut-and synecological contexts. Although O. marina is rarely pelagic and is virtually never found in open ocean waters (Watts et al., this issue), it has been used as a proxy for a range of fundamental processes that relate to marine planktonic protozoa, which are recognised as a key component of pelagic food web dynamics (e.g. [START_REF] Sherr | Significance of predation by protists in aquatic microbial food webs[END_REF]. Mainly because this taxon is easy to obtain, identify, and rear in the laboratory (Lowe et al., et al., this issue), O. marina has been extensively characterised in terms of its growth, feeding, swimming behaviour, and response to chemical stimuli (Boakes et al., this issue; Breckels et al., this issue; Montagnes et al., this issue; Roberts et al., this issue). Furthermore, O. marina is consumed by a number of planktivores (Yang et al., this issue), and some data are available for predation rates upon it. In turn, O. marina-based responses have been embedded into various numeric models to assess a range of ecological processes (Davidson et al., this issue). Davidson et al. (this issue) provide a detailed consideration of the pros and cons of using O. marina as an ecological model, and we refer the reader to this work, but in general, there is a clear indication that, although it is far from perfect, O. marina is and will continue to be used as an ecological model organism.

An evolutionary model

Because of its phylogenetic position, O. marina constitutes an increasingly useful model to understand the origin and evolution of a range of unusual cellular and molecular features within the alveolates, particularly dinoflagellates (Slamovits and Keeling, this issue); this research has been facilitated by its ease of maintenance and near-global availability (Lowe et al., this issue-a,b; Watts et al., this issue). Oxyrrhis marina seems to have diverged very early in the branch leading to the dinoflagellate lineage, close to the time that dinoflagellates diverged from apicomplexans [START_REF] Saldarriaga | Multiple protein phylogenies show that Oxyrrhis marina and Perkinsus marinus are early branches of the dinoflagellate lineage[END_REF][START_REF] Slamovits | The highly reduced and fragmented mitochondrial genome of the early-branching dinoflagellate Oxyrrhis marina shares characteristics with both apicomplexan and dinoflagellate mitochondrial genomes[END_REF].

Consequently, it possesses morphological, cytological, and genetic features common with its near phylogenetic neighbours as well as exhibiting interesting taxon-specific characteristics (Slamovits and Keeling, this issue; Lowe et al., this issue-a). In addition the recognition of several genetically diverse lineages within the O. marina morphospecies represents a useful pool of genomic variation to elucidate processes that occurred during the evolution of the dinoflagellates and the development of derived ''Oxyrrhis'' characteristics [START_REF] Lowe | Patterns of genetic diversity in the marine heterotrophic flagellate Oxyrrhis marina (Alveolata: Dinophyceae)[END_REF]this issue-a). Advances in culturing strategies to allow axenic maintenance of O. marina (Lowe et al., this issue-b) combined with the increasing availability of low-cost highthroughput sequencing mean that O. marina is likely to continue to represent an important model for evolution and comparative genomics of alveolates in the coming decades.

A model for biogeography

A further benefit of its wide geographic range is that O. marina can be a useful model to examine hypotheses concerning factors that affect distribution and dispersal of protists (Watts et al., this issue). In addition, an increasing pool of intraspecific genetic diversity within O. marina has been uncovered, and there is some indication that this genetic variation is spatially structured [START_REF] Lowe | Patterns of genetic diversity in the marine heterotrophic flagellate Oxyrrhis marina (Alveolata: Dinophyceae)[END_REF]. Thus, although an area of research in its infancy, the pattern and distribution of O. marina clades appear to represent a useful model to examine both 1) the relative influences of evolutionary processes such as selection, dispersal, and genetic drift on the distribution of functional and neutral diversity and 2) the influences of historical contingency and anthropogenic transport in shaping patterns of biodiversity among populations of free-living marine protists. marina (e.g. [START_REF] Wootton | Biochemical prey recognition by planktonic protozoa[END_REF]; Keeling and Slamovits have uncovered exciting genetic patterns [START_REF] Slamovits | Plastid-derived genes in the nonphotosynthetic alveolate Oxyrrhis marina[END_REF]; Slamovits and Keeling, this issue) and together with Watts, Lowe, and Montagnes (Lowe et al., unpublished data) are producing a range of genomic resources (sequence data, EST libraries) that may have wide ecological and evolutionary applicability. Clearly, if O. marina continues to be studied at an escalating rate (Fig. 1), there will be a concomitant increase in both realised and potential synergies.

Synergism

A survey of 36 papers, since 1990, that mention O. marina in the title indicated ~55% were aut-or synecologically based, ~40% examined some aspect related to evolutionary or genetic biology, and ~5% were associated with distributional patterns; few of these ~5-10%, however, displayed any form of rigorous cross-disciplinary evaluation. There is thus scope to study many aspects of O. marina biology (Table 1), and there is clearly potential for crossdisciplinary studies. Consequently, it appears that this species may eventually become a robust model, following in the footsteps (pseudopodial streaks?) of such protists as Amoeba, Dictyostelium, and Paramecium. The remainder of this special issue sets the stage for such exploration and then reviews and explores means by which O. marina can and cannot act as a model to address questions. Table 1 Characteristics of a model species, partially obtained from Bahls et al. (2003), an indication of how well Oxyrrhis marina fits these, and a guide to relevant papers in this special issue.

A model species should be:

Level to which O. marina possesses this trait Paper in the special Positive Negative easier to study than the target it is modelling

• Generally this seems to be so relevant:

• ecologically

• economically

• evolutionarily

• genetically

• behaviourally inexpensive to obtain, maintain, and practical to use

• Can be collected all over the world, and cultures established with ease

• Poorly characterised strain variation; not all strains are easy to maintain in culture Lowe et al. (this issue-b) not rare or threatened

• Given the wide distribution and abundance, it appears not to be rare or threatened Watts et al. (this issue) easy to identify able to survive storage (e.g. cryopreservation)

• For many strains, cultures can be virtually ignored for weeks

• Has yet to be successfully cryopreserved Lowe et al. (this issue-b) fecund, with rapid generation times

• Up to ~ 2 divisions per day, which is reasonably rapid for protozoa of its size

• Life cycle poorly understood Montagnes et al. (this issue) able to produce clonal lines (to separate environmental from genetic effects)

• This has been well established Lowe et al. (this issue-b) stable in characteristics over many generations unemotive (e.g. not being a target for animal rights)

  Montagnes et al. 1. Introduction to the Oxyrrhis marina special issue 5

  Combining the potential of O. marina to act as an ecological, evolutionary, and biogeographic model offers opportunity for synergism. Here we present a few examples of such works, based on our own research: Lowe et al. (2005) have examined how O. marina ecophysiology and distribution may be related; Lowe et al. (2010) have explored the extent of biogeographical distributions of distinct O. marina clades; the Roberts-group are adapting Montagnes et al. 1. Introduction to the Oxyrrhis marina special issue 8 methods developed for metazoan phagocytic blood cells to investigate prey recognition by O.

  FIGURE LEGENDS

Fig. 2 .

 2 Fig. 2. Oxyrrhis marina. a. a bloom of O. marina in an intertidal pool on the Isle of Man

  this issue-a,b;

	Montagnes
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http://mc.manuscriptcentral.com/jplankt Journal of Plankton Research

Journal of Plankton Research

ACKNOWLEDGEMENTS

The Authors would like to thank Emma Wootton and Fiona Hobden for their constructive comments on this manuscript. We would also like to thank Susanne Menden-Deuer and two anonymous reviewers for their constructive suggestions. Finally, we wish to recognise the outstanding editorial efforts made by John Dolan and Roger Harris in the production of this special issue.

FUNDING

This work was, in part, supported by a UK NERC grant (NE/F005237/1) awarded to PCW, CDL, and DJSM; UK NERC grants NE/C519438/1 and NE/G010374/1 awarded to ECR and E. Dudley; SAMS/NERC Oceans 2025 programme supporting KD; grants from the Tula Foundation and the Canadian Institute for Advanced Research, awarded to CS; and a Project of International Cooperation and Exchanges from the National Natural Science Foundation of China (30970500) awarded to YZ.