Importance of allelopathy as peudo-mixotrophy for the dynamics and diversity of phytoplankton

Understanding the dynamics and diversity of marine phytoplankton is essential for predicting oceanic primary production, oxygen generation and carbon sequestration. Several top-down and bottom-up factors lead to complex phytoplankton dynamics. Complexities further arise 
from inter-species interactions within phytoplankton communities. Consequently, some of the basic questions on phytoplankton diversity, identified long ago, still puzzle the ecologists: for example, what regulates the diversity in simple systems where species compete for limiting 
resources? In this context, allelopathic interaction among phytoplankton species has been identified as a potential driver of their dynamics and regulator of their diversity. This chapter deals with the importance of allelopathy in regulating the outcome of nutrient competition 
among phytoplankton species, through analysis of a resource-competition model. It demonstrates that, through the mechanism of pseudo-mixotrophy - proposed earlier by the author - allelopathy provides essential growth advantage to weaker competitors, and stabilizes resource competition, which ensures the coexistence of two phytoplankton on a single nutrient. 
In simple nutrient-phytoplankton interactions where higher-trophic influences are negligible, this mechanism theoretically promotes phytoplankton diversity, and can potentially support high diversity in natural phytoplankton communities.


Introduction
Phytoplankton are responsible for oceanic primary production and oxygen generation; and essential for regulating the global carbon cycle [1]. The dynamics and diversity of phytoplankton are constrained by several top-down and bottom-up effects. Complexities further arise from inter-species interactions within phytoplankton communities. Resources available for the growth of phytoplankton (e.g., light and dissolved nutrients) are often limited. But, despite the presence of limited variety of resources, phytoplankton are capable of maintaining an extreme level of species diversity [1][2][3]. This diversity is paradoxical to the theory of competitive exclusion [3], which suggests that in the steady state the number coexisting species cannot exceed the number of limiting resources [4,5]. The mechanisms proposed to explain phytoplankton diversity include environmental fluctuations, periodic fluctuations, spatial heterogeneity, deterministic chaos, life cycles, grazing, and chemical interactions (detailed in [6]). But, when the top-down effects and external factors are negligible, it is difficult to explain the 'building block' of the extreme diversity of phytoplankton, i.e., the stable coexistence of two phytoplankton on a single limiting resource.
There is a growing body of evidence, both theoretical and experimental, suggesting that allelopathic interactions among phytoplankton species have a major role in shaping phytoplankton-zooplankton dynamics and regulating phytoplankton diversity [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Some of these studies [20,21] suggested that 'toxin-allelopathy' can prevent competitive exclusion in Lotkta-Volterra interactions. Further, the allelopathic effect can potentially mediate resource competition in a chemostat. Focusing on simple resource-competition models, Roy [22] proposed that two phytoplankton can stably coexist on a single resource in a homogeneous media without any external factors when allelopathy acts as 'pseudo-mixotrophy'. This chapter elucidates how this mechanism ('if you cannot beat them or eat them, just kill them by chemical weapons' [22]) determines the outcome of resource competition between two phytoplankton, and how it potentially contributes to maintaining phytoplankton diversity in natural waters.

Mixotrophy and allelopathy
Mixotrophy is known to influence species interactions within a food web [24]. Mixotrophic algae that can combine phototrophy and phagotrophy are an important component of phytoplankton communities (e.g., [25]). Mixotrophy can be an effective strategy for securing essential carbon required for the survival of algae in adverse conditions, such as, low radiation, unfavourable temperature, salinity or pH [26,27]. Studies further suggested that certain algae (e.g., Prymnesium) can simultaneously be toxin producer and mixotrophic to 'kill and eat' [28]. However, not many species are known to follow this dual strategy that combines allelopathy and mixotrophy. But, several species are known to be allelopathic as they produce toxic or allelopathic chemicals (e.g., [13]). Studies suggested that the dynamics of phytoplankton with competitors and grazers are modulated by the presence of toxic species (e.g., [21,[29][30][31]). Allelopathy of toxin producers affects the growth and competitive ability of sensitive species. Allelopathy alone can potentially overturn the outcome of interspecific competition by providing 'additional' competitive and growth advantages to the allelopathic species [20,22]. Roy [22] proposed that theoretically the effect of allelopathy can be viewed as pseudo-mixotrophy for the survival or coexistence of phytoplankton in nutrient competition. In the rest of the chapter, this mechanism will be discussed.

Allelopathy mediating competition for a single nutrient
To demonstrate the allelopathic effect on nutrient competition, a standard resource-competition model (presented in Table 1) is considered, which is a generalised version of the model analysed by [22].
Eq. (2): Non-allelopathic species  Yield coefficient of species 1 dimensionless 1.0 η 2 Yield coefficient of species 2 dimensionless 1.0 Table 2. Functions and parameters with their meanings used in the nutrient competition model with allelopathic effect. The quantities N, P 1 and P2 are the concentrations of the nutrient, non-allelopathic species and allelopathic species, respectively.
The nutrient (with concentration N) uptakes by the non-allelopathic species (with concentration P 1 ) and allelopathic species (with concentration P 2 ) are described by the functions f 1 (N) and f 2 (N). In particular, these functions can take the standard Michaelis-Menten forms ( Table 3). The parameters of the model are described in Table (2). Allelopathy of species 2 imposes a higher mortality to the non-allelopathic species, which can be described by an 'additional' mortality term in the form of a phenomenological function φ(P 1 , P 2 ). This function may be a high-order interspecific product of P 1 and P 2 (see , Table 3) -a particular case of which was considered in [22]. In the absence of allelopathy, the model takes the form of a standard resource-competition model, which predicts the persistence of one of the two species depending on the lowest minimum nutrient requirements (i.e., depending on minimum R * [5]). So, if φ(P 1 , P 2 ) = 0, and if the non-allelopathic species has a lower minimum nutrient requirement, it will win over the allelopathic species in nutrient competition. However, if φ(P 1 , P 2 ) � = 0, allelopathy provides advantage to species 2 by imposing a higher mortality to species 1.

Coexistence of two phytoplankton on single nutrient
As mentioned in the previous section, the loss rate of species 1 due to allelopathy of species 2 can be described by a high-order product of P 1 and P 2 : φ(P 1 , A particular case was analysed in [22], where the exponents were taken as β 1 = 1 and β 2 = 2. For φ(P 1 , P 2 ) = γ P β 1 1 P β2 2 , it can be derived (following the analysis of [22]) that there exist a critical value γ c for the allelopathy parameter γ, such that, if γ < γ c , no coexisting steady state is possible. However, if γ > γ c , two alternative steady states are possible, Importance of Allelopathy as Peudo-Mixotrophy for the Dynamics and Diversity of Phytoplankton 3 http://dx.doi.org/10.5772/59055 and depending on the initial conditions the system will settle to one of the two steady states (see, Fig. 1). Therefore, for γ > γ c one can find suitable initial concentrations of P 1 and P 2 for which stable coexistence two phytoplankton on a single nutrient is possible: Fig. 1-(b) shows that in this case the ratio of P 2 to P 1 is stabilised to a non-zero value. Ratio of P2 to P 1 corresponding to the stable and unstable dynamics presented by (a), respectively. Condition for no recycling was used with other parameters and functions fixed at their default values/forms as in Table (2).

Critical conditions for coexistence
The critical level of allelopathy γ c is a crucial quantity, which can be computed from the parameters of the model. Corresponding to γ c , there exists an unique coexisting steady state N * , P c 1 , P c 2 , where the magnitudes of P c 1 and P c 1 depend on the model parameters. Extending the analysis of [22] , the magnitudes of these quantities can be derived explicitly for all possible forms of the function φ(P 1 , P 2 ) ( Table 3). When γ > 0, the critical conditions for the existence of the unique steady state can alternatively be derived with respect to N 0 -the input nutrient concentration. The allelopathy parameter γ would depend on the inherent biological properties of the allelopathic species, and hence its magnitude cannot normally be altered using experimental conditions. However, the parameter N 0 associated with the experimental conditions can very well be controlled. Rearranging the expressions of γ c (Table 3), one can derive the corresponding threshold magnitudes of the input nutrient concentration, say, N c 0 , so that, for N 0 > N c 0 alternative steady states are possible leading to the stable coexistence of two phytoplankton. The explicit expressions of N c 0 for different forms of φ(P 1 , P 2 ) are 10.5772/59055 φ(P 1 , P 2 ) P c 1 , P c 2 γ c for a given N 0 N c 0 for a given γ γ P 2 Does not exist -- Table 3. Parametric conditions for stable coexistence when allelopathy acts as pseudo-mixotrophy in nutrient competition models. The allelopathic effect is denoted by φ (P 1 , P2), the critical steady state by (N * , P c 1 , P c 2 ), the critical level of allelopathy by γ c , and the threshold level of N0 by N c 0 . The quantities c 1 , c2, c3 and A used in the table are defined as:  Table (2).
presented in Table (3). The results in Table (3) can be used to address how the critical values γ c , N c 0 , P c 1 and P c 2 may change due to uncertainties in describing the allelopathic effect by a phenomenological function. Considering the general form φ(P 1 , P 2 ) = γ P β 1 1 P β2 2 , the magnitudes of γ c , N c 0 , P c 1 and P c 2 are computed for a range of values of the exponents β 1 and β 2 (Fig. 2). If the model parameters are fixed, γ c or N c 0 would be minimum when β 1 is the lowest and β 2 is the highest (Fig. 2-a, b). The unique steady states of P 1 and P 2 depend on both β 1 and β 1 : for a given β 1 , P c 1 decreases but P c 2 increases with β 2 (Fig. 2-c, d).

Allelopathy as pseudo-mixotrophy
The function of allelopathy in mediating the coexistence can be understood from the Figs.
(3) & (4). Under nutrient-limiting conditions, allelopathy of the weaker competitor helps increase the availability of nutrient the by killing the stronger competitors: an illustration of this process based on the model of [22] is given in Fig. (3-a). In the simplest scenario,  Table (2).
when recycling of nutrient is 'turned off' in the model, and no killing by allelopathy takes place, the level of available nutrient decreases and stabilises to a low value where the non-allelopathic species alone survives eventually ( Fig. 3-a, b). However, the extra (higher) mortality of species 1 (P 1 ) due to killing by allelopathy of species 2 (P 2 ) leads to elevation of the nutrient concentration (and further prevents it from decreasing gradually) (Fig. (3-a); the nutrient concentration eventually stabilises to a level where both species stably coexist ( Fig. 3-a, b). The ratio of per-capita nutrient uptake by P 2 to that of P 1 decreases to a low value when killing by allelopathy does not take place ( Fig. 3-b, Fig. 4-a); however, this ratio stabilises to a considerably higher value when allelopathy kills stronger competitors ( Fig. 3-b,  Fig. 4-b). When nutrient recycling is incorporated, killing by allelopathy increases the dead cells (Fig. 4-c,d), and the recycling process releases a portion of the nutrient quota of the dead competitors available for uptake ( Fig. 4-d). The recycling process coupled with killing by allelopathy thus generates an extra amount of nutrient ( Fig. 4-d) available for uptake by the species. Therefore, by imposing higher mortality to stronger competitor, allelopathy provides clear advantage to the weaker competitor. This mode of action of allelopathy that provides growth advantage to the allelopathic species, not through direct predation but through killing of the competitors (e.g., Fig. 4-b, Fig. 4-d), was termed as pseudo-mixotrophy [22]. In this process killing by allelopathy provides a positive feed-back by increasing of the growth limiting resource that reduces the competition pressure (e.g., Fig. 4). Clearly, this feedback loop provides crucial benefit to the growth rate of the allelopathic algae, and modulates the dynamics of the resource competition within a common trophic level.

Relevance to empirical and experimental studies
It is clear from the previous sections that allelopathy acting as pseudo-mixotrophy can theoretically stabilise nutrient competition of two phytoplankton on a single limiting nutrient. However, the applicability of this mechanism across natural phytoplankton is largely unexplored. An empirical or experimental evidence for pseudo-mixotrophy is still in demand. But, recent studies have shown promise that the role of allelopathy in maintaining biodiversity of natural phytopalnkton may be explored further. For example, chemical warfare has been shown to increases bio-diversity in microbial realm [32]; and [20] showed that allelopathy may be responsible for co-existence of the competing phytoplankton in the Bay of Bengal. The question of how much diversity of phytoplankton can be supported though allelopathy alone was addressed by [23], who derived a deterministic relationship between the abundance of the potential allelopathic species and the diversity of non-allelopathic phytoplankton (see, Fig. 5). The abundance-diversity relationship in Fig (5) shows a unimodal pathway through which the abundance of allelopathic phytoplankton regulates the diversity of non-allelopathic phytoplankton [23].

Concluding remarks
This chapter elucidates how phytoplankton allelopathy may function as pseudo-mixotrophy in determining the dynamics of nutrient-phytoplankton models, and how phytoplankton diversity is maintained in those systems. Firstly, the ecological conditions under which allelopathy functioning as pseudo-mixotrophy overturns the outcome of nutrient competition between two phytoplankton (e.g., [22]) is presented explicitly in terms of the model parameters. Secondly, the difficulties in mechanistically describing the allelopathic effect of a phytoplankton on its competitors is addressed by considering a phenomenological function, and the ecological conditions for the coexistence of phytoplankton species and stability of competition dynamics are derived. Thirdly, the competition dynamics is explored under the assumptions of 'no nutrient recycling' and 'continuous nutrient recycling'; and the effects of changing initial nutrient pool in culture media is explored. Therefore, a comprehensive set of constraints is derived under which allelopathy acts as pseudo-mixotrophy in nutrient-phytoplankton models. Finally, the evidences of allelopathic effects in determining the diversity of phytoplankton in natural systems are presented. In particular, how the increasing abundance of allelopathic species may regulate the diversity of phytoplankton (e.g., [23]), is discussed. The mechanism presented here would be useful for better understanding of the biodiversity and function of marine ecosystems. Allelopathy functions as 'pseudo-mixotrophy' in nutrient-phytoplankton models, which are often the basis of marine biogeochemical and ecosystem models. This mechanism has not been explored in relation to ocean biogeochemical models, which are generally used to predict phytoplankton species composition, and estimate the scale of oceanic carbon sink. Given the complexities in representing phytoplankton functional types in global biogeochemical models (e.g., [33]), it would be useful to understand how allelopathy or pseudo-mixotrophy of a phytoplankton type may affect the dynamics of the other types. The ecological conditions derived will be useful for investigating the role of 'pseudo-mixotrophy' in marine ecosystem models. The current challenges in monitoring, controlling and managing harmful algal blooms (HAB) (e.g., [34]), and predicting their consequences in aquatic ecosystems require better understanding of the dynamics of toxic or allelopathic species. Recent studies have also reported other roles of phytoplankton allelochemicals, e.g., defence against predators [35], and 'casual parasitism' that helps supplying organic nutrient to the mixotrophic donors by lysis of prey [36,37]. It will be worthwhile to further explore the mechanism presented here in relation to the succession of phytoplankton taxa that are known to form HABs. It is noteworthy that currently the mechanism has been explored using simple resource-competition models that can be tested in an experimental (chemostat) set up. Such an experiment will be helpful in formulating and parameterising resource-competition models including allelopathy, and for better understanding of the constraints of phytoplankton diversity.