# Integrate-and-fire models of insolation-driven entrainment of broadcast spawning in corals

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## Abstract

The circa-annual cycle of gametogenesis produces mature gametes at the spawning “season” for successful mass spawning of broadcast corals. We develop a bioenergetic integrate-and-fire model that reveals how annual insolation rhythms can entrain the gametogenetic cycles in tropical hermatypic corals to the appropriate spawning season, since photosynthate is their primary source of energy. In the presence of short-term fluctuations in the energy input, a feedback regulatory mechanism is likely required to achieve coherence of spawning times to within one lunar cycle, in order for subsequent signals such as lunar and diurnal light cycles to unambiguously determine the “correct” night of spawning. The feedback mechanism can also provide robustness against population heterogeneity that may arise due to genetic and environmental effects. We solve the integrate-and-fire bioenergetic model numerically using the Fokker–Planck equation and use analytical tools such as rotation number to study entrainment.

## Keywords

Bioenergetic integrate-and-fire model Broadcast spawning Entrainment Insolation Coral reproduction Biological rhythms Computational biology## Introduction

A large number of sessile and sedentary marine invertebrates (Giese 1959), including corals, gastropods, ascidians, bivalves (Babcock et al. 1992), to name a few, reproduce by broadcast spawning, i.e., by releasing their gametes—spermatocytes and oocytes—into the water column, where gametes from different individuals fertilize to produce larvae. Synchronous release of these gametes vitally determines and maximizes reproductive success and maintenance of genetic diversity in the population (Crean and Marshall 2008). One instance of extremely tight synchrony in spawning is observed in a subset of corals known as broadcast spawning corals.

Broadcast spawning corals reproduce in synchronized mass spawning events (Babcock et al. 1986) that most typically occur annually close to the period of peak insolation in a window of a few hours after sunset during the waning moon phase close to the night of the full moon. It is postulated that these corals have perfected the control of initiation of gamete development (gametogenesis) for spawning, since gametes have to be at the optimal stage of development for successful fertilization after release. These corals appear to have a hierarchical perception of time to facilitate synchronized spawning: *circa-annual* (of the order of months), *circa-lunar* (of the order of days), and *circa-diel* (of the order of hours) scales of timekeeping that we now outline based on the probable underlying coral physiology (similar observations have been made in Babcock et al. (1986) and van Woesik et al. (2006)).

The gametogenetic cycle spans typically 3–9 months, with oogenesis having a longer development time than spermatogenesis (Wallace 1985). Therefore, oogenesis and spermatogenesis are appropriately initiated to ensure that both gametes are concurrently mature at the time of spawning. Gametogenesis thus occurs in the *circa-annual* scale to produce gametes of sufficient maturity at the time of spawning. Solar insolation and sea surface temperature (SST) have been proposed as possible exogenous environmental inputs for scheduling reproductive processes in the circa-annual scale (Baird et al. 2009). Although most spawns occur at SST between 28°C and 30°C (close to the peak SST in the respective region), SST is unable to explain the differences in spawning times observed between the east and west coast of Australia with similar SST patterns (Baird et al. 2009). On the other hand, changes in solar insolation have been empirically shown to be good predictors of spawning times of *Montastraea annularis* in the Caribbean by van Woesik et al. (2006). The signals that drive synchrony in circa-annual time scales are often referred to as “ultimate” triggers whereas the drivers in the circa-lunar and circa-diel scales are termed ‘proximal’ triggers.

Spawning in several coral species occurs sometime between the full moon and several days after the full moon, depending on the species and geographical location. Nevertheless, spawning is consistent from year to year at the same location for the same species (Table 1, Mendes and Woodley 2002). This points to the presence of a *circa-lunar* (or intralunar cycle) tracking of time by corals, although the exact mechanism relating the time of spawning to the lunar cycle remains unknown. The understanding of this causation by lunar irradiance is complicated by the presence of secondary effects of the lunar cycle such as tides and length of dark period, whose effects cannot be decoupled.

Finally, most observed coral spawnings have been after dark or twilight, and the spawning event is completed in a window of approximately 1 h. The times of spawning within the day have also been remarkably consistent at a given location for a particular species. At the *circa-diel* level, the hour of spawning has been hypothesized to be “triggered” by diurnal light cycles and time of spawning has been successfully manipulated by altering the light conditions (Knowlton et al. 1997; Brady et al. 2009). There is also some evidence of synchrony between coral colonies being possibly driven by chemical exchanges of hormones through diffusion in the water column (Atkinson and Atkinson 1992).

Although considerable research has been directed toward understanding the factors that contribute to the robust synchrony in spawning, relatively little is known about the pathways involved in the “triggers”, a problem that is exacerbated by the incomplete knowledge of coral physiology. In this paper, we take an analytical approach to elucidating mechanisms that permit *entrainment* to environmental cycles with minimal assumptions regarding coral physiology. We address the questions: Can a single environmental driver such as insolation with annual periodicity in conjunction with simple bioenergetic mechanisms robustly (in the presence of noise) entrain a coral to an circa-annual cycle to limit the possible spawning to about 1 month, whereupon the drivers acting on the finer circa-lunar and circa-diel time scales can trigger spawning at the ‘correct’ time? If this is possible, what are the evolutionarily selected, biologically consistent mechanisms that can achieve this and how narrow a window can spawning be restricted to?

Tropical hermatypic corals engage in a symbiotic relationship with *Symbiodinium* (micro-algae) that live inside the coral tissue. There is sufficient evidence (Muscatine et al. 1984; Edmunds and Davies 1986) to conclude that hermatypic corals derive most of their energy (up to 80%) from the *Symbiodinium*, especially in shallow reef corals in well-lit tropical waters (as mentioned earlier, empirical evidence of this correlation was presented in van Woesik et al. (2006)). *Symbiodinium* produce energy-rich compounds (photosynthate) from photosynthesis and to a first approximation, the photosynthate produced by the *Symbiodinium* is proportional to the amount of insolation incident on the coral. This photosynthate fufills the high energy demands of reproduction (Rinkevich 1989) allowing us to use insolation as a key driver determining the corals’ capability to spawn. This hypothesis is also reinforced by the observation that spawning in many species occurs during periods of peak insolation (Penland et al. 2004).

Although our results are equally applicable to other broadcast spawning organisms in response to environmental forcing, we present our findings using the entrainment of coral spawning to exogenous insolation as an example. In this work, we do not consider the evolution of synchrony from interactions between coral individuals, as suggested in Atkinson and Atkinson (1992). We also defer to a separate treatment the effect of other exogenous signals, such as lunar phase, lunar irradiance, and diurnal light cycles on the spawning time at the circa-lunar and circa-diel scales.

We use a bioenergetic model of the coral, in which some part of the energy derived from photosynthesis by the endosymbiotic *Symbiodinium* is allocated to the reproductive reserves. We use the accumulated reserves as an index of the reproductive state of the coral. When sufficient reserves have been accumulated, spawning can occur. In this regard, we develop a simple “integrate-and-fire” (I&F) model for corals that is mathematically similar to models for neurons (Knight 1972). We regard this model as an extreme simplification of the dynamic energy budget theory-based models, such as the one recently proposed for coral symbiosis by Muller et al. (2009). Although a great deal is known about coral physiology, the interactions between the host, the symbiont, and the environment are highly complex, making more detailed mechanistic models uninstructive. Finally, we find these low dimension I&F models extremely attractive for ultimately studying populations of interacting coral individuals.

The *entrainment* mechanisms we study, where individuals achieve coherence with an external periodic driver, is complementary to the mechanisms of *synchrony* where coherence is achieved through interaction (or coupling) between individuals; henceforth, we use the terms “entrainment” and “synchrony” to refer specifically to these two complementary mechanisms. A combination of entrainment and synchrony is used in many biological systems to produce robust rhythms, such as the mammalian circadian system (To et al. 2007).

Our bioenergetic entrainment model of synchronized reproduction shares many similarities with the globally coupled map models of synchronized “masting” or seeding in trees (Satake and Iwasa 2000, 2002). Analogous to our model, the map of masting proposed by Satake and Iwasa (2000, 2002) also tracks the reproductive reserves of individuals supplied by energy from photosynthesis, and masting occurs when the reserves reach a threshold. Using a globally coupled map, Satake and Iwasa (2000, 2002) capture the presence or absence of masting in a year and the correlation in the number of seeds produced in a year by different masting individuals. Thus, this model of the number of seeds produced by an individual tree from year-to-year (discrete-time) coupled by pollen exchange explores a mechanism of “synchrony” rather than a mechanism of “entrainment”.

In contrast, we are interested in the coherence in spawning times within a year produced by a seasonal energy input with fluctuations through an entrainment mechanism without interaction between individuals (uncoupled) and without regard for the number of gametes released in a spawn. Moreover, individuals in a population receive the same mean insolation input with fluctuations uncorrelated between individuals. Satake and Iwasa (2000) further show that, under the conditions imposed on our model, individual trees in their model only exhibit chaotic masting.

## Bioenergetic entrainment model

*N*noninteracting coral individuals. Each coral individual receives photosynthate produced by the

*Symbiodinium*from the incident solar insolation at an average rate

*S*

_{ t }. The average incident insolation and, hence, the average photosynthate production rate are assumed to be the periodic entrainment driver with a period of 1 year, i.e.,

*S*

_{t + 1}=

*S*

_{ t }. The maximum insolation at a given geographical location is dependent on the latitude and time of the year. The maximum insolation profile has two peaks between the tropics (on the days when the sun is exactly overhead), while it has only one peak outside the tropics. Nevertheless, for simplicity, we consider a sinusoid with seasonal variation

*α*and a period of 1 year as the mean energy input (idealized insolation in energy equivalents):

*η*

_{ t }in the energy input with a coefficient of variation (CV) with respect to the mean energy input

*S*

_{ t }.

*t*is characterized by the quantity of its reproductive reserves,

*X*

_{ t }. A fraction

*λ*of the photosynthate received by each coral is allocated to reproduction (see Fig. 1). This allocation fraction

*λ*might dependent on the quantity of reproductive reserves

*X*

_{ t }and time

*t*, i.e.,

*λ*=

*λ*(

*X*

_{ t },

*t*), and 0 ≤

*λ*≤ 1. Each possible choice of allocation

*λ*can be considered one “strategy” to achieve entrainment. We further assume that all corals are identical and employ the same strategy. Since our premise is that the coral has no intrinsic timekeeper and thus has no “sense” of time, we restrict our study to strategies where the allocation only depends on the quantity of reserves and the mean energy input rate, i.e.,

*λ*=

*λ*(

*X*

_{ t },

*S*

_{ t }).

The energy allocated to reproduction might not lead to an equal increase in the reserve level due to maintenance costs.^{1} Here we consider a maintenance cost *f*(*X* _{ t },*t*) that is linearly dependent on the quantity of reserves *X* _{ t } and can vary with the amount of solar insolation. Thus, the maintenance cost is of the form *f*(*X* _{ t },*t*) = *γm*(*t*) *x*, where *γ* is the maintenance rate constant and *m*(*t*) is the time dependence of the maintenance cost.

*X*

_{ t }reaches the normalized threshold of 1 and its reproductive reserves are expelled as gametes (

*X*

_{ t }is reset to 0).

*X*

_{ t }is the total energy invested in the reproductive apparatus, such as gonads and gametes, by the coral; we represent all quantities in our model in energy equivalents.

*N*(

*x*,

*t*) is the number of coral individuals with reproductive reserves

*X*

_{ t }=

*x*at time

*t*; since the corals are identical and noninteracting,

*N*(

*x*,

*t*) and its dynamics are sufficient to represent the spawning behavior of the population. At any given time, since there is no coral birth or death, the total number of corals in the system is constant, \(\int_0^1 N(\xi,t) d\xi = N\). With a sufficiently large coral population, we can work equivalently with the fraction of individuals \(P(x,t) = \frac{N(x,t)}{N}\) with reserves

*X*

_{ t }=

*x*at time

*t*. The dynamics of the fraction of individuals at each state at time

*t*is characterized by the following partial differential equation (PDE) (the development of this Fokker–Planck PDE from a stochastic energy allocation model for the coral is presented in the Appendix A):

*T*

_{0}is the interval between successive spawning events for an individual coral under constant energy input

*α*= 0, no fluctuations

*η*

_{ t }= 0, and no maintenance costs

*γ*= 0. The spawning event followed by the reset of the reserve to 0 is imposed on Eq. 1 with boundary conditions

*t*, which is

*λ*and the maintenance

*f*(see the Appendix B for details), the fraction of individuals spawning at time

*t*reaches a periodic steady state of unit period after the transients die out. This steady-state

*spawning pattern*is designated

*F*

^{ ∞ }(

*t*). This spawning pattern represents the spawning behavior of the entire population and can be used to determine the coherence in spawning among individuals in the population. We numerically compute these spawning patterns for different biologically consistent allocation strategies and different maintenance cost functions using the computational approach detailed in the “Appendix B”.

### Entrainment, mode-locks, coherence width, quiescence, and spawn number

Spawning is defined to be entrained to the seasonal insolation cycle if, at steady state, there are a fixed number of spawning events in a given number of seasonal cycles (Gurney et al. 1992). Specifically, the coral is entrained to an *r*:*q* *mode-lock* with positive integers *q* and *r* if there are exactly *q* spawning events in a span of *r* seasonal cycles, which in our model is equivalent to *q* spawning events in *r* years; intuitively, the coral is entrained to a harmonic of the insolation cycle. Then *q* and *r* characterize entrainment for a particular choice of coral model parameters. However, in general, corals only spawn once annually at a fixed time of year. Although there is some evidence of biennial spawning and asynchronous spawning (Mangubhai and Harrison 2008), there is no consensus on such phenomena among coral biologists, and hence, we do not consider these scenarios here. Therefore, we only consider mode-locks of the form *r*:1. In other words, the coral spawns once every *r* years and is entrained to the *r*th harmonic of the driving insolation rhythm. If the corals were to lock on to an *r*:*q* mode, there would be *q* spawning events in *r* years and the time within a year when spawning occurs would not be fixed.

*F*

^{ ∞ }(

*t*) in Eq. 9 to describe its entrainment properties. The

*coherence width*

*W*(measured in days) measures how confined or focused spawning is to a certain time of the year (within one cycle of driving insolation rhythms):

However, the coherence width does not specify whether no firing occurred during significant parts of the driving cycle. We want to capture the lack of spawning outside the spawning window observed in field experiments with corals. We term periods of negligible firing in the spawning pattern *F* ^{ ∞ }(*t*) as *quiescence* periods and look for allocation strategies that can produce significant quiescence periods in the spawning pattern.

When a coral achieves an *r*:1 mode-lock, spawning occurs in 1 year and then skips the next *r* − 1 years before occurring again. Such a skipping phenomenon is not captured by the previous measures. The fraction of the population spawning at time *t* is composed of individuals that have already spawned varying number of times; we shall refer to the number of times an individual coral has spawned including a spawn in the current cycle as its *spawn number* *p*. Although we are not interested in the absolute spawn number of an individual, the spread of contributing spawn numbers of individuals in a certain cycle is a measure of if and how often coral individuals “skip” spawning cycles; a wide spread implies that corals often skip spawning cycles. We discuss the relevance of this skipping in the “Effect of parameter changes on spawning behavior”.

### Entrainment in noiseless coral I&F models

The simplest bioenergetic model for a coral individual uses purely time-dependent allocation strategies and maintenance cost functions, i.e., *f*(*X* _{ t },*t*) = *f*(*t*) and *λ*(*X* _{ t },*S* _{ t }) = *λ*(*S* _{ t }). We show in “Appendix C” that perfect matching of the parameters of the model and the driving seasonal signal (its period and strength) is necessary for achieving the required entrainment. Therefore, entrainment in such scenarios lacks robustness, and the addition of the noise perturbs the system away from these optimal parameter values; we corroborate this fact with simulations in the “Simple reserve accumulator” section. As such, state-independent scenarios in the stochastic model are incompatible with observed spawning synchrony in corals.

Intuitively, a generic state-independent system does *not* involve any feedback or decisions based on the state of system in order to compensate for the effects of noise and the lack of robustness is understandable. State dependence of the internal dynamics or allocation strategy introduces some feedback within the system and has the potential of providing robust entrainment to the forcing signal.

*r*:

*q*or quasiperiodic oscillations (Coombes and Bressloff 1999; Keener et al. 1981). We now present a metric to determine which of these two behaviors is exhibited by the model for a particular parameterization. In the absence of noise, the time of the

*n*th spawn

*t*

_{ n }completely determines the time of the (

*n*+ 1)th spawn

*t*

_{n + 1}. The transition map

*t*

_{n + 1}=

*T*(

*t*

_{ n }) that might be implicitly defined for some models provides a way of computing the sequence of spawning times. Starting from some initial time

*t*

_{0}, the

*n*th firing time

*t*

_{ n }can be computed by recursively applying the map: \(t_n = T^{(n)}(t_0) = T(T^{(n-1)}(t_0)),~T^{(1)}(t_0) = T(t_0)\). For a map

*T*(.), we define its rotation number

*ρ*as the average phase gained for every cycle of the input (Chpt. 11, Katok and Hasselblatt 1996):

*π*. Henceforth, we interchangeably use spawning time and spawning phase.

*ρ*is independent of the initial conditions

*t*

_{0}. Moreover, the coral I&F model achieves an

*r*:

*q*mode-lock if

*ρ*=

*q*/

*r*and exhibits quasiperiodic firing if

*ρ*is irrational (Chpt. 11, Katok and Hasselblatt 1996). These quasiperiodic oscillations are space filling or in other words, given enough time, spawning occurs at every time within a year. For example, in Fig. 2, the rotation number is shown for a range of maintenance rate constants

*γ*for the time-independent (

*m*(

*t*) = 1) and time-dependent (

*m*(

*t*) = 1 +

*α*−

*S*

_{ t }) models presented in the “Reserve accumulator with maintenance” section with natural spawning period

*T*

_{0}= 0.75. In addition to

*r*:1 mode-locks, some undesirable

*r*:

*q*mode-locks are also observed. This curve is different for other natural periods of the I&F model and the characterization of the entrainment behavior of the noiseless I&F model for different values of

*γ*and

*T*

_{0}is possible by computing Arnold tongue (the Arnold tongues for the time-independent maintenance model can be found in Fig. 3, Coombes and Bressloff (1999)).

The fundamental difference in the regions of locking with maintenance (state dependence) is that there exist intervals of parameter values that lead to the same mode-lock and are therefore robust. Thus, a population of heterogeneous individuals with different maintenance rates, but with locking parameters within the same interval, will similarly entrain to the seasonal driver. This robustness also makes entrainment in the presence of noise possible as perturbations to the system have a smaller effect on the dynamics. Moreover, in a heterogeneous population, individuals could have parameter values corresponding to different *r*:1 locks but still lead to synchronous spawning with individuals skipping different sets of spawns.

## Allocations to reproduction consistent with coral biology

In this section, multiple allocation strategies for an individual coral consistent with coral biology are discussed. Desirable allocation strategies are those that produce coral spawning patterns with coherence widths that are within the limits of the circa-lunar scale, i.e., *W* ≈ 30 days. Based on our hierarchical timekeeping hypothesis, once coherence to within the circa-lunar scale is achieved, processes acting on the circa-lunar and subsequently circa-diel scales can determine the precise time of spawning observed in the field.

Parameters in the I&F model with the nominal parameter values

Parameter | Description | Value |
---|---|---|

| Natural period of I&F model and length of reproductive cycle in years | 0.75 |

| Maintenance rate constant | 0.5 |

CV | Coefficient of variation of fluctuations in insolation as a fraction of | 0.1 |

| Magnitude of seasonal variation in mean insolation | 0.5 |

| Sets location of peak insolation at \(\phi+\dfrac{\pi}{2}\) | 0.5 |

### Simple reserve accumulator

The first class of models introduced in this section represents corals with no maintenance, i.e., *f*(*X* _{ t },*t*) ≡ 0. The bioenergetic implications of this model structure are that the energy allocated to reproduction entirely leads to augmentation of the reserves without any maintenance costs, i.e., incoming energy Δ*x* causes the reserves to increase by the same Δ*x*. We wish to verify the results of our analysis from the “Entrainment in noiseless coral I&F models” section that tight coherence in spawning is not achievable in these state-independent models and allocations.

#### Switched allocation

*c*controls the switching transition time between “off” (

*λ*= 0) and “on” (

*λ*= 1). The spawning pattern produced by this switched allocation is shown in Fig. 4a. Although this allocation produces spawning quiescence during a significant part of the year, the coherence width of spawning spans several months and is not desirable. Figure 4b, c is two possible representations of the distribution of spawning times for four successive firings of coral individuals as a probability density and a cumulative distribution, respectively. We designate an arbitrary spawn event as the reference and consider, among those individuals that participated in the reference spawn, the distribution of times when those individuals spawned for the first subsequent (p = 1) and second subsequent (p = 2) times, and so on. We observe that, within each spawn number, spawning is spread across about five insolation cycles.

The quiescence in the spawning pattern in Fig. 4a corresponds to the flat sections of the cumulative distribution of each spawn number in Fig. 4c. The cumulative distributions can be interpreted as the fraction of the individuals that spawn for the *p*th time before time *t* on the *x*-axis. Very narrow periods of relatively high spawning are revealed as sharply increasing cumulative distribution curves. Due to the clarity of the cumulative distributions over the probability distributions in Fig. 4b, henceforth we shall only show plots similar to Fig. 4c for the various models.

We strive to obtain allocations whose cumulative distributions have two highly desirable attributes (a) wide flat regions representing quiescence and (b) rapidly increasing curves connecting those flat regions representing the narrow spawning periods (in the extreme we want a step function). Note that the spawning pattern and the cumulative distribution are two different ways of visualizing the same data.

It appears that although the switched allocation introduces quiescence into the firing pattern, spawning at other times is not sufficiently coherent. The coherence width is 65.66 days, which encompasses two lunar cycles and, hence, insufficient to unambiguously determine the day and time of spawning using the circa-lunar and circa-diel scales.

#### Proportional allocation

Another variant of the previously described switching allocation is the following model that we term the proportional allocation (see Fig. 3): *λ*(*X* _{ t }, *S* _{ t }) = *S* _{ t }. The corals adopting this strategy allocate opportunistically to reproduction, i.e., they assign increased amounts of energy to reproduction if they have extra energy at hand and vice versa.

### Reserve accumulator with maintenance

We consider in this section systems with state- and possibly time-dependent maintenance costs to augmenting the reproductive reserves *X* _{ t }.

#### Proportional allocation with time-independent linear maintenance

In order to understand the effect of state-dependent maintenance, we consider a model with the simple proportional allocation strategy. The maintenance on the reserves is proportional to the current reserve level: *f*(*x*,*t*) = *γx*. Thus, it costs progressively more energy to raise the levels by the same amount as reserves near their full capacity. Such a maintenance would represent the biological situation where successive advanced stages of gamete production and maturation required more energy for the same relative progress through a stage.

#### Proportional allocation with time-dependent (seasonal) linear maintenance

This model retains the proportional allocation strategy as the previous model, but the maintenance costs are seasonally varying (time dependent). It has been hypothesized that SST affects the rate of gametogenetic development of corals. Assuming a perfect correlation between higher SST temperature and insolation (in reality, peak SST lags peak insolation), we assume the maintenance cost to commit to reproduction is higher at lower temperatures. Therefore, maintenance costs are negatively correlated with SST resulting in *f*(*x*,*t*) = *γ*(1 + *α* − *S* _{ t })*x*.

## Effect of parameter changes on spawning behavior

In the previous section, we presented the output of the bioenergetic models obtained from numerical computation of the model. Such analyses nevertheless do not provide a complete understanding of the dynamics of the model particularly with regard to other model parameterizations. We present here the performance of the stochastic models from the “Reserve accumulator with maintenanceReserve accumulator with maintenance” section for different maintenance rates and strength of the environmental fluctuations. We wish to investigate if the noiseless system dynamics provides insight into the stochastic performance.

*γ*distinct from the noiseless dynamics in Fig. 2, where specific intervals were obtained for different mode-locks. The coherence in spawning improves for increasing maintenance rates monotonically across different noise intensities.

We also observe in Fig. 8a, b that the performance is insensitive to the fluctuation intensities for small maintenance costs, but performance decreases with increasing noise at *γ* values close to 1. We believe that this is the direct result of the noiseless dynamics in Fig. 2, where the region of different *r*:1 mode-locks for both models is seen to lie between 0.8 and 1.2. In these mode-locking regions, noise introduces perturbations about the fixed point of the noiseless map, and hence, there is a “linear” relationship between the coherence width and noise intensity. At smaller *γ* values, mostly quasiperiodic oscillations are seen in the noiseless system (from the irrational rotation numbers), and the spread in the spawning pattern is a direct result of quasiperiodicity, which exists even in the absence of noise. In addition, the stochastic nature of the system causes different individuals to experience slightly different parameter values (for e.g., maintenance rate values *γ*), and this results in the lack of sharp boundaries between different dynamics as seen in the noiseless system.

The only difference in performance between the time-independent and time-dependent models is observed at low maintenance rates, where the model with time-dependent maintenance (Fig. 8b) has a smaller coherence width across the entire range of noise intensities. On the other hand, the model with time-independent maintenance (Fig. 8b), while not significantly worse, has more noise intensity sensitive performance.

*T*

_{0}= 0.65 that regions of parameter space, where the noiseless system shows

*r*:1 mode-locking, are most sensitive to the choice of noise intensity as perturbations in the input cause “linear” changes in the output. The figure shows that the regions of parameter space that show 1:1 and 2:1 mode-locking are sensitive to the noise intensity whereas the coherence width for the spawning pattern is similar in regions with quasiperiodic oscillations (irrational rotation number).

## Discussion

We showed that simple allocation strategies can result in spawning once a year with coherence width within the resolution of the finer timing mechanisms in the circa-lunar and circa-diel scales and that energy considerations possibly play a direct role in the entrainment of coral spawning cycles. Allocation strategies that were independent of the current reproductive state of the coral require perfect matching of the model (and its parameters) to the forcing rhythm (insolation), i.e., the sets of parameters that lead to entrainment is a set of measure zero even in the noiseless scenario. On the addition of noise, the system is perturbed away from this optimal matching resulting in poor coherence and lack of quiescence in spawning. These factors make such purely time-dependent strategies extremely *non-robust* and, hence, unlikely as a class of mechanisms operating in biological situations, where populations tend to be heterogeneous due to genetic and environmental factors.

It is understandable that strategies that do not correct for the perturbations caused by noise based on the current state of the system, i.e., systems without feedback, would not be robust. We therefore explored state-dependent models that had a maintenance cost that depended linearly on the state of the system. This would be biologically associated with systems where the same percentage of progress through a stage costs progressively more in later stages. Entrainment of different modes could be observed over significant ranges of parameter space, even in the absence of noise. In the stochastic system, coherent spawning with coherence width of the order of a month and quiescence were observed. With significant regions of parameter space permitting the same *r*:1 mode-lock, these models accommodate heterogeneity in a population of individuals as is common in nature. This suggests that a strong regulatory mechanism exists for the management and commitment of energy toward reproduction, an observation made by Giese (1959, 1966) on a class of marine invertebrates.

While the noiseless system shows mode-locking of the expected mode over distinct subsets of parameter space, the stochastic system shows coherent spawning over a wide range of maintenance rate values in which the noiseless system with constant input achieved the spawning threshold. Moreover, the coherence width of spawning progressively decreased with increased maintenance rate. This seems to suggest that more synchronized spawning can be achieved by higher mode-locks where individuals skip cycles and spawn once every *r* cycles of the driver. In other words, corals can tighten their spawning coherence and improve their likelihood of reproductive success at the cost of expending more energy towards reproduction. Thus, there is a trade-off between synchrony (and spawning success) and the total energy investment per spawn.

The regions that showed *r*:1 mode-locks in the absence of noise, however, responded differently to changes in the noise intensity. Regions of noiseless entrainment showed linear dependence of the spawning coherence on noise intensity. These regions correspond to a fixed point on the spawning map about which the system can be linearized and thus explain this dependence of coherence width on the noise intensity. In this sense, these regions of noiseless entrainment are robust as they provide “good” performance when the driving energy conditions are “good” and vice versa. In regions of quasiperiodicity, the noiseless system displays wide variation in the spawning times from cycle to cycle and, therefore, has a spread of spawning times even in the absence of noise. Thus, no significant effect of noise is seen over the already present spread of spawning times.

Stochasticity also helps to genetically mix a population entrained to an *r*:1 mode-lock; in its absence, individuals that are together in an *r*:1 mode-lock always spawn together and never spawn concurrently with others that are in *r*:1 locks but in a different skipping pattern. Such a mechanism that favors greater genetic mixing might also be favored by evolution. Several hypotheses support synchronized spawning as the evolutionarily favored mechanism including higher likelihood of successful cross-fertilization, predator satiation, and genetic diversity. In the related work on masting, Yamauchi (1996) analyzed the evolutionary basis for synchronized seeding. Although similar analyses could be applied to study both the coherence and intermittency in spawning, such a study is beyond the scope and focus of this work.

We only considered maintenance rate values that resulted in spawning in the noiseless scenario. We know that a stochastic I&F system that does not reach the threshold exhibits stochastic resonance and the coherence width of spawning is minimum at an optimal noise intensity (Plesser and Tanaka 1997). Thus, tight spawning coherence in an I&F system can still be achieved outside the range of parameter values we consider. In this scenario, the organism accumulates energy to a subthreshold level and waits for a burst of fluctuations to move it across threshold. This is a possibility that we have not addressed and could have possible relevance in synchronizing spawns, but we are unaware of any experimental evidence pointing towards such a mechanism.

We are currently performing quantitative analyses to measure the level of an egg-specific protein vitellogenin in *Acropora digitifera* samples from the Republic of Palau as a function of time before spawning for use as a marker for the various stages of reproductive development, much like the reserve state that we model. These data can be fitted to the individual traces of state versus time shown in the “Allocations to reproduction consistent with coral biology” section and used for model discrimination and hypothesizing modified strategies. Another possibility would be to use diameter (size) of gametes (or gonadal index) (Giese 1959) as a measure of the stage of reproductive development as is often used by researchers studying gametogenetic cycles in corals and other spawning marine invertebrates. The limited data available on gamete diameter versus time (Fig. 5, Vargas-Ángel et al. 2006) appear to correlate well with trends seen in the traces of state shown here. Moreover, we have assumed here that coral individuals retain their residual gametes during the quiescence periods and continue development when favorable energetic conditions are reached. While coral fragments are known to reabsorb unreleased gametes after the spawning season, there is insufficient experimental evidence to support or contradict our assumption, although this merits further investigation.

We have posited here that insolation is the primary energetic driver in photosymbiotic corals. Insolation is known to be the primary driver of primary productivity in marine and aquatic ecosystems. Hence, reproduction in many broadcast spawning marine invertebrates typically follows this driver (Starr et al. 1990); clams that filter-feed on micro-algae typically begin gametogenesis in response to the blooms of these algae (Braley 1984), just as abalones and urchins that graze on kelp begin gametogenesis after these macro-algae resume their growth cycles. Other examples of synchronously spawning marine coral reef invertebrates can be found in Babcock et al. (1992). Variants of our model may thus be applicable to a larger class of spawning invertebrates than corals.

However, there are species-, system-, and location-specific phase delays in primary productivity that depend on local cycles of temperature, upwellings, and delivery of macro- and micronutrients. In the case of corals, it can be argued that the coral animal’s homeostasis buffers the endosymbionts from exogenenous ocean variability in nutrients that normally limit algal productivity and damps the influence of these secondary environmental factors that govern local primary productivity. This homeostasic buffering is not perfect and our models can explain the observed deviation of spawning dates of specific populations of corals (and other marine invertebrates) from strict calendar synchrony with other populations of the same species in the same hemisphere—a previously unresolved puzzle.

## Footnotes

- 1.
The term maintenance has many specific definitions especially in dynamic energy budget theory (Kooijman 2000). Here, we use it to represent energy costs to sustain a certain quantity of

*reproductive*reserves. - 2.
Brownian motion

*B*_{ t }is a stochastic process whose value is the sum of discrete jumps in value, each of which is uncorrelated with all the prior jumps (van Kampen 2007). The value of each jump is normally distributed with variance directly proportional to the time interval between jumps.

## Notes

### Acknowledgements

We would like to particularly thank Charlie Boch for enriching our understanding of coral biology especially through his extensive experimental work in the field. We also acknowledge numerous useful discussions with Erik Muller, Laure Pecquerie, Alison Sweeney, Pete Edmunds, and Aileen Morse. This work is supported by the Institute for Collaborative Biotechnologies through Grant W911NF-09-D-0001 from the US Army Research Office and National Science Foundation under Grant EF-0742521.

### **Open Access**

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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