Built on Simplicity: A Mathematical Analysis of Hydrozoa and Bryozoa 

Aizaz Ahsan Wahla, Ocean Burland, Dante Daniel Garcia Perez & Jessica Michalski 

Abstract 

As superorganisms, hydrozoan and bryozoan colonies are inherently complex. Their modular nature gives rise to hierarchical structures and morphologies that may appear chaotic at first, yet the physical structures making up these colonies are governed by comparatively simple mathematical models and principles. Allometric studies of both taxa show that depending on its function, hydrozoan medusae exhibit different growth patterns that suit their reproductive mode, either sexual or asexual. Bryozoa also exhibit allometric growth patterns that show varying metabolic rates across developmental stages, resulting in larger offspring being more metabolically efficient. These organisms also exhibit scaling relationships in their dimensions that fit their function. Hydrozoans in particular show isometric growth about bell height and allometric scaling about bell diameter, which suits functional adaptations like jet propulsion or rowing. Although bryozoans form large, branching colonies, the patterns underlying these bioconstructions are governed by simple geometry and rules modeled as continuous or discontinuous branching. Lastly, bryozoan assemblages exhibit competitive hierarchies that help stabilize their ecological communities when subjected to harsh environmental disturbances like ice scour. Mathematical models and coexistence theory show that hierarchy, where some competitors dominate others, counterintuitively helps mitigate instability and promotes coexistence. 

Introduction 

Hydrozoans and bryozoans are both primarily marine, colonial superorganisms characterized by their complex modular architecture. Their sessile or motile structures are built on modular units called zooids and exhibit ordered development to carry out various functions. Hydrozoans are colonial organisms of the Cnidaria phylum and often exist in different forms depending on their life cycle stages. They exist in the form of free-swimming larvae, benthic branching polyp colonies or pelagic medusae (Leclère et al., 2016). These medusae typically carry out reproduction either sexually or asexually by budding and propel themselves using jet propulsion or rowing motions (LaDouceur, 2021; Technau et al., 2015). Bryozoans, members of the phylum Bryozoa, similarly construct themselves into colonies by the fission and aggregation of zooids that organize themselves into complex structures, such as sheets or arborescent forms. Similarly, their zooids sustain colonies by specialization into niche functions (Schwaha, 2020).  

Allometry in Hydrozoa and Bryozoa 

Comparing allometry in sexual and vegetative hydrozoan medusae  

Many hydrozoan species are known for their ability to reproduce both sexually by egg and sperm and asexually via the polycytogenic process of budding, enabling them to live semi-sessile lives. This is the case for the Eleutheria dichotoma species. Yet, the sexually and vegetatively reproducing medusae belonging to this species have been found to significantly differ in their body morphologies, suggesting a differing allometric growth pattern. To better understand their differing trends, a study was conducted at the medusa stage of the E. dichotoma of both vegetative and sexual medusa (meaning that medusae exhibiting both sexual and asexual reproductive characteristics were discarded). They compared the dimensions of the diameter of the umbrella, tentacle knobs, and buds; areas of umbrella tentacles, and buds; and lengths of the tentacles for all findings (Table 1) (Schierwater, 1989). 

Table 1. MR is the mode of reproduction as either sexual or asexual, N is the number of samples of each type. Under knob diameter, the number of medusae tested was only 27 since 18 values were lost because of a computer error. Statistical parameters include the mean areas and diameters of the comparative body parts along with their standard deviation ±SD. The correlation coefficient r and the significance of the results P following a U test (a null hypothesis test) were reported for all  findings (Schierwater, 1989). 

Results showed that sexual medusae exhibit a slightly larger body size and biomass than vegetative medusae because of their bigger diameter of 185 to 550 micrometers compared to 211 to 437 micrometers, with a significance level of P = 0.049. However, the difference in bell diameter had a positive and statistically significant correlation between all measured dimensions of all medusae types. Moreover, the differences in size based on reproductive modes can be explained by the idea that sexual medusae need more area to store their embryos within the umbrella. The differences in biomass can rather be explained by how sexual E. dichotoma might require higher protein storage to accompany embryonic development (Schierwater, 1989). 

Allometry of hydrozoan medusae bell diameters 

Secondly, hydrozoan species ontogenetic shape change follows a specific scaling of mass to length. Indeed, to understand this, we must first understand the power relationship between length and mass in many pelagic species, expressed by the following equation (Hirst, 2012): 

(1)  

Or equivalently:

(2)  

In the equations above, a and b are constants, M is the mass, and L is the linear measurement of the body. The variable b is keen in assessing the power relationship between mass and length. If b is a value of 3, this means that all major body proportions will remain constant (meaning there is an isometric enlargement of all body parts) during growth. If b is less than 3, there will be a more rapid increase in length, and if b is greater than 3 there will be an increase of other axes compared to the length, such as thickness. Thus, in the case of hydrozoan medusae, studies on their length determined by both their bell diameter and height were analyzed. Results with correlation coefficients, r, of over 0.894 suggest that for hydrozoans assessed by bell height, their mean b value is 2.73 (6 0.37, 95% CI), which is not significantly different from 3. This confirms that they grow isometrically. However, hydrozoans assessed by bell diameter as the criterion for length have a b of 2.40 (6 0.19, 95% CI). This demonstrates that while bell diameter increases, the rest of the body increases at a proportionally lower rate (Fig. 1a). This can be explained by the fact that hydrozoans that are assessed by bell height have fineness ratios (the ratio of the length of a streamlined body to its maximum width or diameter) between 0.6 and 2.3 and use jet propulsion as their means of movement in viscous regimes (Fig. 1b) (Merriam-Webster Dictionnary, 2024) (Hirst, 2012).  

Fig. 1 Reported b values of body mass in relation to length for scyphozoans and hydrozoans. Minus sign indicated b significantly less than 3 (p ≤ 0.05), and ns not significantly different from 3 (p > 0.05). (b) Fineness ratios for each medusae grouping, where diamonds indicate means, circles individual data points. Horizontal lines join those groups that are significantly different with indicated p-values. (Hirst, 2012) 

Thus, the increase in mass with increase in linear dimension requires different pressures (Fig. 2). Yet, hydrozoans assessed by bell diameter have a more oblate form and locomote by rowing, meaning it requires their body mass to be allocated along the bell in a way such that they can generate vortices that increase locomotory and feeding efficiencies. 

Fig. 2 Change in mass (M) in relation to length (L) in pelagic taxa. All taxa are scaled relative to an initial length and mass of 1, with increases derived from mean b values for wet mass (except in the case of appendicularians, cladocerans, and chaetognaths, where dry mass. For nauplii and copepodites, overall average b values are (Hirst, 2012). 

Allometry of metabolic activation in polyps 

In terms of metabolic activation in hydrozoan species, the most obvious increase in metabolic scope (that is the amount that metabolic rate can increase relative to its resting value) is that metabolic rates will increase with muscle activity for movement in free-swimming hydrozoan medusae. However, metabolic activation is much less obvious in the case of sessile polyps, since they are immobile, and their only active muscular contractions are due to their myoepithelium as it drives gastrovascular fluid. To truly understand both the increase in metabolic scope, as well as the relation between size and metabolic rates, known as metabolic scaling, studies on the Hydractinia symbiolongicarpus feeding polyps were conducted. When it came to the polyp’s metabolic activation and increasing its rate, they found that there was an increase in metabolic rate by around 250% and an increase in oxygen uptake relative to the resting metabolic rate, when fed compared to unfed (Fig. 3). Moreover, in terms of metabolic scaling, they found that there was an isometric relationship between resting metabolic rate and size, and that large colonies do not exhibit a greater metabolic scope when compared to small sized animals (Almegbel et al., 2019). 

Fig. 3 Oxygen uptake values (measured in L^-1 min^-1) in fed and unfed states for Hydractinia symbiolongicarpus colonies.  Each point represents the individual colony (n=27 with size range of 34-139 feeding polyps, least squared regression slope ± SE = 2.36 ± 0.2, intercept ± SE = 0.017 ± 0.011, r² = 0.85 (Almegbel et al., 2019). 

Allometric scaling of metabolism in bryozoans 

Metabolic scaling with body size is known to be one of the most ubiquitous and contentious relationships in biology. Yet, one consensus is that an increase in mass rarely results in a proportional increase in metabolism, and the scaling exponent is often less than 1. Indeed, this was the case during a study conducted on two species of bryozoans: Bugula neritina and Watersipora subtorquata. However, there has been little approach on the static metabolic scaling when comparing the metabolic rate to the body size of animals within the same species at the same developmental level.  Thus, through this same experiment, results showed that in both cases, developmental stages had a significant impact (p<0.05) on their respective metabolic rates. For B. neritina, the metabolic rate was highest in the larval stage of development and lowest in the settler stage of development. For W. subtorquata, metabolic rate increased through ontogeny. Moreover, this study determined the allometric scaling of metabolism and the benefits of increased offspring size. In other words, offspring that depend on their maternal resources will have different metabolic efficiencies when comparing larger offspring with smaller offspring (Fig. 4). What they found is that larger offspring have lower metabolic rates compared to smaller ones, and that bigger larvae survive better and reproduce more as a colony. In addition, larger offspring are much more metabolically efficient compared to smaller offspring regarding the consumption of maternal energy. Indeed, due to their larger size, they require less resources from their maternal sources compared to smaller offspring. Finally, because fecundity is inversely proportional to offspring size, mothers producing smaller offspring will lose 47% of their total investment to metabolic costs of development, while mothers producing larger offspring will lose only 22% (Fig. 4b) (Pettersen et al., 2015). 

Fig. 4 Schematic showing proportion of energy used by the smallest (6.7µg) and largest (24.5µg) B. neritina larvae observed in the study. (a) Hypothetical isometric relationship (scaling exponent, b = 1; as assumed by life-history theory) versus an allometric relationship between size and metabolic rate up until the independent phase based on scaling exponents obtained from these results (larval stage; b = 0.76, early stage; b = 0.76, late stage; b = 0.29). According to the power function, MR = aM^b, energy use is directly proportional to body mass in an isometric relationship (where b = 1), while in an allometric relationship, smaller offspring use relatively more energy per unit body mass than larger offspring (b < 1). (b) The relative amount of energy that is consumed in the dependent phase for mothers with identical total reproductive but where one mother produces the largest observed offspring and the other produces the smallest observed offspring. As a proportion of total supplied energy, the fewer, larger offspring in total will use less than half of the maternally supplied energy relative to the many smaller offspring (i.e. mother producing larger offspring: total energy (50 000 mJ)/per offspring energy (87.9 mJ) = offspring number (569) individual energy burned (19.3) = total metabolic cost (10 958 mJ)) (Pettersen et al., 2015). 

Branching patterns in Bryozoa 

Although the underlying pattern of growth is common between members of the phylum, bryozoan growth habits vary from encrusting sheets to erect, arborescent forms, giving rise to complex colonial structures (Fig. 5) (Hageman et al., 1998). In arborescent bryozoans, the branches originate from a single base, providing a tree-like appearance (Cheetham et al., 1980). Despite the inherent complexity of bryozoan colonies, the branching patterns are governed by simple geometry and can be modelled as such. Indeed, many mathematical models of bryozoan branching patterns are based on a small set of growth rules, or parameters, that control simulated branching (Bell et al., 1986). Further, models describing bryozoan branching patterns are generally based on the concept of regular growth, which refers to growth by repeated iterations of growth rules. In other words, the growth rules remain the same, or approximately the same, over successive bifurcations. In this section, the principles of modelling branch patterns will be introduced, and a branching model of adeoniform growth, a form of arborescent growth, will be further explored. 

Fig. 5 Various growth habits on modern bryozoans, all specimens in the South Australian Museum (SAM) collection, Adelaide. 1) Adeona sp., mature, × 1, SAM L743; 2) Cribricellina rufa, × 2, SAM L744; 3) Cigclisula verticalis, × 0.5, SAM L745; 4) Flustra denticulata, × 1, SAM L746; 5) Caberea grandis, × 2, SAM L742 (Hageman et al., 1998). 

Overview of branching growth models 

Branching models can generally be characterized in a couple of ways, depending on the organisms of interest and the growth rules being used. For instance, branching patterns may be referred to as ‘blind’ or ‘sighted’ patterns in relation to branch initiation. In ‘blind’ patterns, branching is uniquely controlled by the growth rules, without input from the organism or the environment. The model described by Cheetham & Hayek (Cheetham & Hayek, 1983) for adeoniform growth is an example of a ‘blind’ branching pattern and will be further discussed. In contrast, ‘sighted’ patterns can be influenced by factors in the immediate ‘neighbourhood’ of the module (zooid in the case of bryozoans). For example, the fossil encrusting bryozoan Stomatopora has been modelled using a ‘sighted’ pattern, where the proximity of neighbouring zooids disrupts the branch growth (Bell et al., 1986).  Alternatively, growth models can be classified as continuous, where every branched path is divisible into the same number of increments, or discontinuous, where a main path has more increments than other paths (Fig. 6). For the purposes of modelling arborescent bryozoans, including adeoniform bryozoans, a continuous growth model is considered the most accurate (Cheetham et al., 1980).  

Fig. 6 A schematic showing the branching structures formed by A) continuous growth and B) discontinuous growth. For convenience, the first bifurcation is taken as the origin (t = 0). Note in A) that each path from the origin has five increments of lengths g_a, g_b, or a combination of the two, whereas in B), the inner right path (main path) has five increments, and a lateral path of 3-4 increments. Note also that g_a = g_b = g in B). [Adapted from (Cheetham et al., 1980)]. 

To understand continuous growth branching models, there are some parameters that must be defined. First, are the growth increments, g_a and g_b, which will bifurcate after a and b intervals of growth, respectively. The quantities a and b also determine the series (infinite sum) representing the increase in the number of growing tips. In fact, the number of growing tips, G_t, at any time t is given by: 

(3)  

Where a and b are positive integers, relatively prime (the only positive divisor of them both is 1), and such that a is greater than or equal to b. If this is the case, then the series, G, by which the number of growing tips increases, behaves as a power series, which is a series of the form: 

(4)  

Here, represents some constant or function of n. When a = 2 and b = 1, then G follows the Fibonacci series ( from Eqn. 3) while G follows the geometric series when a = b = 1 (Cheetham et al., 1980). Although the adeoniform growth model described by Cheetham & Hayek (Cheetham & Hayek, 1983) does not directly use the series defined in Eqn. 4, it is helpful for visualizing the general process of branching in arborescent bryozoans. 

Adeoniform branching growth model 

Adeoniform growth is a type of arborescent growth characterized by rigidly erect, branching colonies with calcified bases. The initially flattened branches are composed of two back-to-back layers of zooids (Fig. 7) (Cheetham & Hayek, 1983). 

Fig. 7 Early growth of an idealized adeoniform colony. A) Growth from the ancestrula (founding zooid), which is initially similar to sheet-like encrusting growth, with subsequent zooids growing erect. It has been suggested that adeoniform growth closely resembles the earliest known erect cheilostome bryozoans. B) Two layers of zooids with basal walls in contact with the median plane of the branch extend distally by budding at the growing tip. C) Multiplication and lateral separation of zooid rows produce the first bifurcation, which is shown in a single plane, but is 3-dimensional. [Adapted from (Cheetham & Hayek, 1983)]. 

Cheetham & Hayek modelled adeoniform growth using a 3-dimensional mathematical growth model, summarized in Table 2. Growth is simulated by three components quantified by nine parameters (Cheetham & Hayek, 1983).  

Table 2. Components of adeoniform branch growth and the mathematical quantities used to simulate them. [Adapted from (Cheetham & Hayek, 1983)]. 

The first component is the elongation and multiplication, or splitting, of branch axes (the center lines of the branch planes; Fig. 7). This component is the most complex aspect of the model and provides the 3-dimensional form of the colony (Cheetham & Hayek, 1983). The magnitudes of g_a, g_b, a, and b determine overall branch length, the lengths of fragments between bifurcations, and bifurcation frequency. The bifurcation angle, β, and twisting angle, τ, provide the spatial orientation of the branches. The bifurcation angle is the angle between a pair of branch axes lying in the plane of the bifurcation, with the axis of the mother branch also lying in this plane (Fig. 8A). A daughter branch of the bifurcation becomes the mother branch for the subsequent bifurcation, generating another plane of bifurcation that is twisted relative to the first about the common branch axis in the planes. In other words, the new plane of bifurcation is twisted about the branch axis of the daughter branch in the first bifurcation, which becomes the mother branch in the second bifurcation. The twisting angle is measured in a plane perpendicular to both planes of bifurcation (Fig. 9) (Cheetham & Hayek, 1983). 

In adeoniform growth, the median planes of the daughter branches diverge symmetrically from the median plane of the mother branch (Fig. 8). This symmetric divergence results in the twisting angle, τ, between subsequent planes of bifurcation (Fig. 9). The daughter-branch axes diverge at an angle of θ in the plane of the mother branch (X-Z plane; Fig. 8B). Meanwhile, the angle of divergence, φ, between the projections of the daughter-branch axes onto the plane normal to the median plane of the mother branch (Y-Z plane; Fig. 8C), results from the divergence of the daughter-branch planes. If the daughter branches exist on the same plane (Fig. 7C), then, φ equals zero, as does τ. As such, the bifurcation angle, β, will be equal to θ. Thus, the values of the angles φ and τ determine the 3-dimensionality of the colony (Cheetham & Hayek, 1983).  

Fig. 8 A representation of the geometry of adeoniform bifurcation. A) The plane of bifurcation includes the axes of the mother and both daughter branches, as well as the bifurcation angle, β. B) The median plane of the mother branch (X-Z plane) includes the angle of divergence, θ, between the projected axes of the daughter branches. C) The plane normal to the median plane of the mother branch (Y-Z plane) includes the angle of divergence, φ, between projected axes of the daughter branches (Cheetham & Hayek, 1983). 

Fig. 9 A representation of the geometry of subsequent adeoniform bifurcations. Each plane contains the axis of the mother branch and both daughter branches. The right daughter branch in the first bifurcation is the mother branch in the second bifurcation, thus the second plane of bifurcation is rotated about that branch axis. The twisting angle, τ, is perpendicular to both planes. [Adapted from (Cheetham & Hayek, 1983)]. 

The second component of the model is the establishment of the branch thickness at the growing tips and increasing thickness with elongation. Branch thickness is defined as the distance across the two layers of zooids, measured normal to the plane of the branch. Increases in branch thickness at the growing tip beyond the initial thickness are linearly proportional to branch elongation. Finally, the establishment of branch width is the final component of the model, which can be considered as a constant for the purposes of growth modelling for most adeoniform species (Cheetham & Hayek, 1983).  

Geometric implications of this model 

The model indicates that the angles of bifurcation, β, and of twisting, τ, significantly affect the external and internal spatial relationships in simulated colonies, as they affect the branching pattern formed, particularly at later growth stages. The external spatial relationships deal with the shape of the outermost portions of the colony. These relationships are related to the height, radii, and lengths of branches relative to their distance from the substrate. In general, the height and radius demonstrate correlated growth but also appear at least partially in conflict. Indeed, variations in the angles of bifurcation (β) and twist (τ) seem to emphasize one parameter over the other, although both increased height and radius have advantages in terms of resource availability (Fig. 10) (Cheetham & Hayek, 1983). 

Fig. 10 Differences in the shape of simulated colonies at the final growth stage with the same relative growth rates and different values of β and τ. A) β = 50°, τ = 50°; B) β = 80°, τ = 50°; and C) β = 80°, τ = 20°. On the left are top views of the colonies, with front views of the colonies on the right. The X-, Y-, and Z-axes are relative to the median plane of the colony stem. Lost branches dipping below the substrate are show by dotted lines. [Adapted from (Cheetham & Hayek, 1983)]. 

Internally, within the constraints defined by the height and radius of the colony, branches multiply at an exponential rate, converging inwards at later growth stages. As such, branches become increasingly crowded, resulting interference between branches. Consequently, the limited space can place constraints of the growth of the colony, as well as the efficiency with which functions are accomplished. Convergence begins at the third bifurcation, following consecutive divergent bifurcations. Midway between the first and second growth stages of the fourth bifurcation, the branches crossover for the first time (Fig. 11A). Depending on the relative growth and the length of segments between bifurcations, the overlapping branches may form hexagonal patterns (Fig. 11B, C) (Cheetham & Hayek, 1983).  

Fig. 11 Convergence of branches in adeoniform growth beginning at the third bifurcation. Half-way between the first and second growth stages during the fourth bifurcation, branch axes cross for the first time. However, they do not intersect if τ is not zero.  A)  A representation of connected median planes of branches in an idealized adeoniform colony as the branch axes (show in solid black lines) first cross. Crossover distance, d_co, is shown by the dashed line between the crossing axes at the growing tips. B) and C) Stereopairs of branch axes of simulated colonies forming hexagonal patterns. [Adapted from (Cheetham & Hayek, 1983)]. 

Based on this model, it is evident that not all properties of the branching colony can be simultaneously optimized. Low angles of bifurcation (β) and twist (τ) emphasize height, reducing the loss of lower branches. However, this advantage comes at the expense of spacing (d_co), and the radius of the colony, resulting in interference between the branches. Alternatively, high angles of bifurcation (β) and twist (τ) emphasize the 3-dimensional nature of the colony and maximizing the crossover distance between branches at the first crossover. Yet, at later stages of colony growth, the height and radius of the colony are reduced, with the lower branches significantly shortened and the growing tips becoming more crowded. Ultimately, intermediate values of β and τ produce the most favourable long-term conditions, maximizing height, and radius of the colony, while reducing branch loss and overcrowding (Cheetham & Hayek, 1983). Thus, there is a trade-off made within the branching patterns of adeoniform bryozoans, which limits certain aspects of their growth, while maximizing other favourable outcomes. 

Competitive Hierarchies in Bryozoa 

Assemblages of shallow coastal bryozoans can be modelled using modern coexistence theory, a leading framework in community ecology to model intraspecific and interspecific competition over limited resources. The methods presented in this section are taken from that of Koch and colleagues in their work on analyzing competitive hierarchies within ecological communities (Koch et al., 2023).  

While originally developed to model the coexistence of two species, it now finds applications in modelling multiple species in competition and the stability of their populations across temporally and spatially variable environments (Barabás et al., 2018; Chesson, 2000). At its foundation, it proposes using the following quadratic approximation to model the growth rates of different species in an ecological community and to simulate competitive systems that include elements like residents and invaders of communities: 

(5)  

where n_j is the density of species j, t is time, S is the number of species and r_j is the per capita growth rate of species j. This is a function of E_j and C_j which are respectively a density-independent environmental parameter and a density-dependent interaction parameter. In other words, E_j considers the environmental effects that influence the population dynamics without being influenced by the abundance of the species, and C_j captures the effects of interactions between species that depend on population densities, such as competition or mutualism (Barabás et al., 2018).  

Coexistence between two species in competition can be understood as dynamic forces at balance: a self-reinforcing feedback loop arises from the interaction between two species (a_ija_ji) which must be counteracted by at least equally strong self-damping feedback from intraspecific competition (a_iia_jj) (Fig. 12). Coexistence theory states that after a small perturbation, two species will return to their steady state if intraspecific competition is greater than interspecific competition (Chesson, 2000; Koch et al., 2023). 

Fig. 12 A 2-species system. Competition results in the positive feedback loop (a_ija_ji) where a_ij and a_ji are negative quantities. This feedback loop is counteracted by the same amount of negative feedback resulting from self-regulation (a_iia_jj) by intraspecific competition, thus leading to a system in balance. Asymmetry between the two species, represented by the thicker dark arrow (a_ji) decreases the product of a_ija_ji and consequently the amount of self-regulation (a_iia_jj) needed to ensure stability (Koch et al., 2023).  

Coexistence theory can be applied to model the stability of assemblages of shallow, coastal bryozoans from Arctic and Antarctic regions. Environmental factors (E_j) include ice scour and density-dependent factors (C_j) capture competition between bryozoan colonies growing into each other (Fig. 13) (Koch et al., 2023). 

Fig. 13 Examples of bryozoan assemblages. From left to right, two species and multiple colonies can be observed competing for resources and territories in the two first pictures. In the rightmost figure, an assemblage of various bryozoan species can be seen. [Adapted from (Koch et al., 2023)].  

Interaction between bryozoan species results in three situations: a win for species A over species B occurs when the former outgrows the latter by 5% in area, the loss of the latter, and a draw when both species cease growing along their boundaries or exhibit equal growth (Koch et al., 2023). Following the methods developed by Koch et al., the rate of biomass loss for a given species i resulting from its interactions with species j is calculated as the weighted sum of all the outcomes of contests between both. This is described as follows: 

(6)  

where W_ij, L_ij and D_ij are respectively the numbers of wins, losses and draws of species i over species j. The parameters p_w, p_L and p_D respectively describe fixed proportions of biomass loss for a given colony per time for wins, losses and draws (Koch et al., 2023). The dynamics of n species can be modelled by finding the rate of change of the population density of a species i using the following Lotka-Volterra type of differential equation: 

(7)  

where X_i represents the population density of species i, r_i is its intrinsic growth rate and c_ij is a constant describing competition intensity between species i and j. With the objective in mind of determining coexistence between species, an equilibrium state is achieved when the X_i population density remains constant and its rate of change equals zero due to the growth and loss components of the equation cancelling each other out. The dynamics leading to this outcome are represented by the partial derivatives of the system represented by the Jacobian matrix. These are called the interaction strengths between species i and j (Fig. 14) (Koch et al., 2023). 

Fig. 14 A 3-species system. From left to right, a matrix representing contact between species holds values representing the numbers of draws (D), wins (W) and total number of contacts (N). From this, an energy loss matrix bearing the energy loss rates of species can be found using Eqn. 6 by weighing the costs of each type of competitive outcome. The elements of the Jacobian matrix, the interaction strengths, are found by combining energy loss rates and abundance data. [Adapted from (Koch et al., 2023)]. 

The interaction strengths, that is the per-capita effect of the change in biomass of a species j on that of a species i, are of dimension [1/T]. Interspecific and intraspecific interaction strengths are respectively calculated as follows: 

(8)  

However, because the following is true of a system at equilibrium or a steady state: 

(9)  

The expression describing intraspecific interaction strengths (a_ii) simplifies to: 

(10)  

When analyzing a bryozoan assemblage of colonies, assuming that the system is near or at a steady state and that the energy loss rate for each species interaction (f_ij) equals that of the competition intensity as follows: 

(11)  

And taking the observed abundances B_i as the number of colonies of species i in an assemblage which can be equated to the equilibrium density X^*, an interaction strength, or an element a_ij of the Jacobian matrix, can thus be calculated as follows: 

(12)  

The stability of a system, which describes its ability to return to its original state after some disturbance shifts it away from equilibrium, can be described by taking the eigenvalues λ of the Jacobian matrix. The dominant eigenvalue λ_d serves as the indicator of the system’s stability and describes whether a perturbation increases according to a positive, real part Re(λ) or if the perturbation decays given a negative, real part Re(λ). However, this is not enough to measure stability of different systems in different time scales. A dimensionless metric, the critical level of self-regulation s^*, describes how far a system is from stability and is obtained by multiplying the diagonal of the Jacobian with a control parameter s such that the matrix sits at the threshold between stability and instability. Koch and colleagues used these methods to analyze 30 assemblages of shallow, coastal bryozoan assemblages and inferred that, contrary to the long-thought notion that competitive hierarchies are destabilising as opposed to transitive systems, where all competitors are equally strong and none can be pushed to competitive exclusion, hierarchy observed among bryozoans in competition appears to reduce instability (Koch et al., 2023).  

These assemblages exhibit strong competitive hierarchies, where certain stronger species dominate weaker ones, and by extension asymmetric energy losses in competition occur. Because intraspecific competition does not outweigh interspecific competition, these systems were found to be inherently unstable, and some species do indeed experience competitive exclusion. Despite this, asymmetries in the energy loss rates, caused by certain competitors dominating others as observed in the data analyzed by Koch and colleagues, keep feedback loops weaker and thus help reduce instability of a community. These findings complement the known environmental influences faced by these bryozoans and the niches they can occupy. Ice scour is an especially important influencing factor that periodically destroys colonies and is known to cause hierarchical patterns of competition among benthic communities. As a result, even though the community may be experiencing competitive exclusion, the richness of species is maintained by environmental disturbances that wipes out larger competitors and gives way to weaker, more opportunistic colonies. As they mature, these lose contests to stronger competitors that ensure hierarchical competition (Koch et al., 2023).  

In sum, bryozoan assemblages in the Arctic and Antarctic regions have evolved in relation to each other to coexist by reducing the strength of their feedback loops. These systems are subject to recurrent destructive perturbations in their environments, and to reduce the instability caused by the changes brought about by these, the effect of one species on the rate of loss of biomass of the other is minimized. Traditionally, hierarchy is thought to cause instability, but these colonial organisms leveraged hierarchy to ensure their mutual survival in a hostile environment (Koch et al., 2023).  

Conclusion

In brief, the development of hydrozoans and bryozoans adhere to fundamental, simple mathematical principles that govern their growth, reproduction and ecological interactions. These principles are used to guide the development of functional components and behaviors to optimize processes on which the life of these creatures depends. Hydrozoan medusae reproduce either sexually and are larger to accommodate embryonic development, or asexually and are optimized for budding. The success of bryozoan colonies depends on their larger offspring, where larger ones exhibit reduced metabolism as opposed to smaller offspring in order to efficiently utilize maternal resources. Space and resource access are important commodities among bryozoans, and their branching patterns, like sheets and arborescent structures, are governed by simple geometric rules that help optimize usage of these assets. Lastly, to survive to both competition and destructive events from their hostile environments, assemblages of bryozoans in polar regions evolved to coexist in competitive, hierarchical communities that reduce the effects of perturbations on their interspecific interactions and, by extension, their population densities.  Altogether, these findings shed light on the development of mathematical rules and systems to optimize processes that ensure the survival of these colonial superorganisms. 

References

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