1.1 Impacts of ocean warming on marine fisheries

Fish species are a vital component of marine ecosystems and remain one of major sources of food and nutrition feeding hundreds of millions of people around the world. This is especially important for coastal undeveloped and developing countries, where fish provide inhabitants with as much as 50% of animal protein intake [1] and are a major economic income. According to annual reports from FAO, the global food fish consumption and average fish consumption per capita increased at an average rate of 3.1% from 1960s, and 1.5% per year, respectively [2]. Despite their critical role in maintaining the sustainability of marine ecosystems and global food security, fish ecosystems are all along under increasing anthropogenic pressure from pollution, habitat degradation and climate change that has been steadily warming the sea water. Marine fish ecosystems are prototypical complex systems that are highly dynamical and sensitive to external environmental factors including sea temperature (ST). Ocean warming is altering fish ecosystems with deep impacts on their ecology (behavior, biomass, range and abundance to name just a few) and environment (for instance on biogeochemical cycles) and hence with negative feedback on the climate itself. Data from the US National Oceanic and Atmospheric Administration (NOAA) show that ST anomaly increased from -0.02°C in 1974 to 0.79°C in 2016 which is the largest anomaly observed in last 100 years [3]. By comparing ST data from last two decades (1999–2019) to two decades before that (1979–1999), ocean warming had dramatically sped up in last two decades [3, 4]. “Fish are like Goldilocks: they don’t like their water too hot or too cold, but just right.”, said Malin L. Pinsky, a co-author of the study [5], where scientists found that some few species populations benefited from ocean warming, but more of them suffered. These results stress the fact that ocean warming is making fish species diversity and populations decline, yet threatening the sustainability of marine ecosystems and food supply for millions of people around the world. Therefore, it is imperative to unravel the complex relationships between marine ecosystems and anomalous changes of ST caused by climate change, so that stakeholders can design optimal ecosystem restoration and policy to mitigate the negative effects of climate change on these ecosystems.

Due to increasing ecological and societal concerns about the sustainability of marine ecosystems –where fishery are following a rapid trend of depletion [6]—coupled to environmental degradation of habitats under ocean warming and development pressure, assessing the systemic vulnerability (mode, extent, location and species) of these ecosystems is considerably important. Fishes are particularly critical considering both their multitrophic (positive and negative) feedback on species (from microbes to humans; examples are emerging viruses in the oceans [7, 8]) and the environment (in terms of biogeochemical cycles affecting for instance algal blooms and carbon sequestration [9, 10]) and their high potential as multiscale biosensors of marine ecosystem health and change. On the one hand, most studies conduct these research fields only by visualizing taxonomic and abundance time series and observing the variations of species biodiversity, distribution and abundance at temporal scale. To identify possible environmental factors responsible for the deterioration of marine fish ecosystem, linear and non-linear relationships between taxonomic and abundance indicators and climate change are studied by comparing the observations to oceanographic climatic data. For example, [11] found that increasingly warming ST altered the composition of marine fish catches, with an increase in the proportion of warmer water species catches at higher latitudes and a decrease of subtropical species catches at lower latitudes by plotting sea surface temperature in the past four decades and calculating mean temperature as an indicator to describe the preference temperature of fish species. [5] applied temperature-dependent population models to measure negative effects of ocean warming on the productivity of 235 populations of 124 species in 38 ecological regions by analyzing the data of marine fishery production from 1930 to 2010 and computed maximum sustainable yield over time. [12] plotted time-series abundance and temperature collected in the North Sea since 1960s and west of Scotland since 1980s, and estimated the trend of length at age and growth parameters including absolute growth rate over time and confirmed the fact that the growth and maturation of fish species correlated to ST, and the mean length of herring populations steadily declined across multiple age groups. [13] examined influences of ST on global biomass transfers from marine secondary production to fish stocks, and predicted that trophic transfer efficiency (TTE) and biomass residence time (BRT) would decrease until 2040 and remain relatively stable after 2040, also by plotting the past observations of the trend of TTE and BRT in marine food webs from 1950 to 2010. As examples, these studies were conducted only by considering species abundance, composition and richness at a globally temporal scale, and the resultant conclusions in time domain were important and valuable, but only provided a coarse view in showing negative influences of the rise of ST on species diversity and populations.

On the other hand, the significant increase of ST is attributed to the gradual accumulation of imperceptible anomalous fluctuations within a season or shorter periods of time, whereas our knowledge on seasonal variation in the composition and population of species communities related to ST is still poor. Scientists have realized the importance of deeply capturing the relationship between marine communities and seasonal changes of ocean warming at a finer scale (considering season or shorter time periods). Some studies have also presented apparent seasonal fluctuations of fish abundance and biodiversity in marine ecosystems by assessing and understanding the composition and seasonal changes of the fish assemblage and doing relevant statistical analyses [14–17]. The results suggested that these seasonal variations could derive from the migration and arrival of species and their own seasonal changes of behavior related to the seasonality of temperature patterns. Although these results provided more specifics for observing linear or non-linear relationships between fish ecosystems and ST, only showing the seasonal variations of fish biodiversity and populations considering sea water temperature by analyzing the time-series observations of species composition and abundance in a traditional ecological way is still inadequate. Marine fish ecosystems themselves, as a prototype of holistic system, should be treated as an integral whole in a system view so that we can study the resilience, stability and the state evolution of marine ecosystems. This scenario allows us to further understand the feedback-and-response mechanism and how the change of ST affects the fish community in terms of collective dynamics.

1.2 Stability, sustainability and management of marine fish ecosystems

Oceans are home to a wondrous array of plants and animals including estimated 20,000 fish species, supporting the “ecological bank” of the planet [18], that supports the global food-web (including humans) and maintaining the stability of global ecosystems from ocean to land and atmosphere via the control of biogeochemical cycles [10]. Although it is impossible to know the exact number of species in the ocean (scientists estimate that more than 80% of oceanic species have yet to be discovered [19]), the number of species which people already knew in the ocean is decreasing. It is urgent and important to keep marine fish ecosystems stable, sustainable and well-managed for the environment and human health. Conventional methods to understand fish community trends are typically analyzing species abundance and/or richness fluctuations in comparison to variations of ST individually. We argue that these approaches, especially when adopting linear models, are far from adequate to deeply understand how fluctuations of ST affects the collective dynamics of fish communities that is deeply rooted into species interactions. The latter are the cause of cascading changes in diversity and abundance. In fish communities, there is a large number of fish species including native and invasive species, preys and predators that interact each other non-linearly. In a dynamical system purview, these species take distinct roles for maintaining the marine ecosystem stable (and sustainable) and always in conjunction with the collective. Therefore, investigating fish ecosystems accounting for ecological-environmental asymmetries and their temporal change is fundamental for understanding feedback between species and environmental determinants, as well as for defining appropriate solutions to improve ecosystem resilience of to ocean warming. Conventional linear models largely fail to capture the non-linearity—arising from time-delayed effects and multiple interactions—to study the response of fish communities to fluctuations of ST. Non-linearity that proves useful to identify the most affected species by temperature change, and the ones mostly responsible for system stability. Even though some studies have investigated the stability of ecosystem using mathematical models [20–23], integrated methodologies that incorporate non-linear network-inferred interaction and predictive models for understanding marine fish ecosystems are still lacking.

Intuitively speaking, biodiversity increases ecosystem stability and sustainability, but climate change or other human-driven perturbations may alter this positive relationship [24]. Therefore, this relationship has been contentious. In fact, the concept of ecosystem stability is complex due to multifaceted understanding and types [25–27]. As for a fish ecosystem, different types of stability describe different features that might lead to different patterns. The stability of fish ecosystems should be explored from the perspectives of species interactions underpinning complex food-webs (when inferred from abundance [28]), the topology (stability) and transitions defining resilience of communities to different types (point-source and diffused) of environmental perturbations [29].

For this purpose, an Optimal Information Flow model (OIF) developed in [28] is used to investigate ecosystem changes over time and temperature and its via metrics of inferred species interactions networks. OIF macroscopically analyzes the evolution of the fish ecosystem over time and ST, and “microscopically” identify critical species who play a significant role in maintaining functional stability. The dynamical analyses over time, functional stability and salient species assessment is novel with respect to [28]. Analyses are done as a function of time and temperature jointly (e.g. temperature-dependent time variability) and separately (e.g. atemporal seasonal patterns and temperature-independent stability).

1.3 Optimal information flow model and multiscale ecosystem analysis

In this study, to overcome limitations of conventional time-domain ecological analyses and explicate how the change of ST fluctuations affect fish communities at multiple scales, we adopt a complex system perspective. Specifically, a fish community in Maizuru Bay managed by the Maizuru Fishery Research Station of Kyoto University [21] is used as a prototype of our proposed information- and network-theoretic models.

Part of the work in modeling fish ecosystems lies in detecting causal interactions between species, that is a fundamental this but challenging task in real-world ecosystems. For this purpose we previously developed the information-theoretic OIF model in which causality is perceived as information flow or cross-predictability of species abundance, that makes causality as a practical computational problem of collective information gain (or uncertainty reduction equivalently) leaving aside all ecological details underlying cross-abundance variability [28]. OIF is based on Transfer Entropy (TE) with time delays that provides an asymmetric approach to measure directed information flow between random variables (species) from time-series data [30, 31], and associates cause-effect relationships with directed information flow considering the extent to which one species improves the prediction of another species’ future states in the context of the history of another species [32]. Information flows may relate to different types of interactions, for instance, to the collective behavior of species motion (considering data of relative position and alignment between fish species) [33, 34], or dynamical fluctuations of abundance as in this study. The cause-effect relationship can be interpreted as predictability between species to infer species interactions (also defined as predictive causality in [28]) in the fish community and reconstruct networks for describing the marine fish ecosystem, allowing us to study the ecosystem on both temporal and temperature-dependent scales in a systemic way. OIF model is improved with respect to [35] by considering its extension over time to reconstruct dynamical information networks, the varied Markov order of random variables and a refined pattern-oriented criteria to select optimal time delay and threshold based on maximization of mutual information. Therefore, OIF overcomes the limited TE for the directed uncertainty reduction scheme, for the consideration of maximum information (entropy) network [36], and MI-based maximization criteria to define the optimal time delay and interaction threshold to accurately predict systems’ patterns (e.g. biodiversity). It is noteworthy to mention that the optimal threshold on interactions is not necessarily within the scale-free or maximum entropy range of inferred collective behavior. In this way, interaction processes are clearly linked to patterns which provide relevance to the inference problem. Additionally, OIF is dependent on the choice of appropriate time delay between variables for TE calculation.

Specifically, to study the effects of ST on the fish ecosystem in a dynamical system purview, the long-term abundance time-series data are categorized into five groups considering five temperature ranges (TR): ≤10°C, 10–15°C, 15–20°C, 20–25°C, ≥25°C. We detect mutual relationships between species and reconstruct networks for each TR to investigate how the fluctuations of ST affect the collective behavior of particular fish species and the whole fish ecosystem by conducting network-based analyses on species interaction networks. For each TR, causal interactions between species are inferred by OIF model (computed as TE considering optimal time delay and threshold). These OIF-inferred species interactions define the connectivity among species, forming OIF networks for all TRs. Structural and functional patterns are recognized for OIF networks. Network-based species-specific analyses are also conducted by analyzing how much a particular species interacts others, and identifying the most salient links and critical nodes in networks.

Through comparison of functional features of interaction networks (characterizing network topology or structure) as well as of critical species among five TRs, we draw conclusions on the influence of ST fluctuations on the fish ecosystem and its stability. In particular, temperature-dependent but time-independent networks are inferred and analyzed to trace the collective organization of seasonal patterns; this informs about baseline stability, and in particular criticality (as Bak’s optimality [37]) in relation to TR network topology. Temporal networks are also implemented to capture how these seasonal patterns change and trigger instability (as critical slowing down a’ la Scheffer [38]), diversity, and keystone species when considering long-term ocean warming. The study is novel in providing: (i) connections between abundance and interactions patterns with identification of the most sensitive indicators to predict changes; (ii) non-linear causal attribution of temperature on species and collective stability (including the effect of environment-biota synchronization); (iii) entropic inference of distinct species groups dynamically (competitive and cooperative species); (iv) and interaction-based detection of keystone salient species for ecosystem organization. Thus, because of the novel multiscale analyses (in terms of biota from species to collective, and of the environment from TR to time) linking criticality to critical transitions, the work is providing a potent tool and indicators to understand and track multitrophic stability of marine ecosystems; fish community is an application but any species can be tracked. This is extremely valuable to formulate accurate science-based fishery policy to maintain ecosystems stable and sustainable toward desire states and potentially improve ecosystem resilience to ocean warming.

2.1 Fish community abundance time-series and seasonal categorization

Long-term time-series data (in total 285 time points) of the fish community were collected by scientists at Maizuru Fisheries Research Station, Kyoto University [21]. They conducted underwater visual census approximately every two weeks along the coast of Maizuru Bay from 1 January 2002 to 2 April 2014 [39]. Data collection for the only species of jellyfish (A. aurita) was different: jellyfish were counted from the pier at the Maizuru research station. For jellyfish data, it was count data, but divided by a research area (about 150 m²), so data for this species only have values below a decimal point. Such high-frequency time series enable the detection of potentially causal interactions between species. The fluctuation dynamics of species was later corroborated by eDNA species sampling [40]. Maizuru Bay is a typical semi-enclosed water area with nearly 50 m of the shore and at a water depth of 0–10m, located in the west of Wakasa Bay, Japan. Precipitation is rather high from summer to winter that is the rainy season in this area. Sea-surface temperature ranged from 5.2 to 31.8°C, sea bottom temperature from 8.5 to 29.6°C, which were measured near the surface and at the depth of 10 m underwater during the diving, respectively [41].

In this dataset, only 14 dominant fish species whose total observation counts were higher than 1000 (considering non-normalized abundance [21]), and 1 jellyfish species were included because rare species were not observed during most of census period. Rare species would bring large numbers of zeros in the time series which may lead to difficulty in analyzing data. As proven in other similar studies, ignoring rare species (in average abundance and without extreme fluctuations) does not significantly change fish ecosystem results since abundance dynamics of rare species contained within other species is typically very small; thus, the inclusion (or exclusion) of rare species does not alter the inferred ecosystem interaction topology beyond just adding poorly interacting species as found in [21, 28]. Jellyfish, a non-fish species, was involved in the dataset since it was thought to have prominent influences on the dynamics of the fish community due to its large abundance. Note that time-series data were normalized to unit mean and variance before analysis [21, 42] (i.e., technically normalized to zero mean and unit variance); this is done to ensure all species have the same level of magnitude for comparison and to avoid constructing a distorted state space.

In this study, we aim to study the effects of ST change on ecosystem dynamics and stability considering in particular how slow ST increase affects seasonal species organization and how that implicates long-term changes. This is done because ecosystems have memory and ST gradual changes accumulate over time leading to irreversible changes in the short-term dynamics (manifesting gradually or suddenly [38]), potentially. Initially, the normalized time-series abundance were categorized into five subsets considering five mean temperature range (the average of surface and bottom water temperature was considered), i.e. ≤10°C, 10–15°C, 15–20°C, 20–25°C, ≥25°C, resulting in five shortened time series with 16, 93, 58, 62, 56 time points. This allowed us to disentangle how the fish community fluctuates seasonally with the change of ST by separately analyzing time series of these five subsets. This is done independently of time considering the whole period to assess potential seasonal stability via probability distribution or network topology equivalently [43]. The data categorization picks the abundance observations at time points when the mean ST is within specific TRs and rearranges the selected abundance observations by preserving the time sequence, leading to new time series corresponding to TRs. The subset of time series after categorization in consideration of five TRs are still regarded as sequential time series, called TR-dependent time series here, even though the between-TR abundance interdependence is not considered as in the continuous temperature-dependent analysis (see Section 2.5).

2.2 Probabilistic portrayal of the fish community: Power-laws and transitions to exponential

We estimate the probability distribution function of normalized fish abundance and of inferred interactions between species (both are indicated as y as a generic random variable) using power-law or exponential distribution functions [44, 45]. Theoretically, the power-law function that has been forced a-priori (leaving the exponential distribution as the alternative function in case of non-fitting via Maximum Likelihood criteria on Kullback-Leibler divergence) has two types: discrete and continuous, of which the continuous form [46] is given by: (2.1) where y_min ≥ 0 is an estimated lower bound or cutoff for which the power-law starts to hold. γ is the power-law scaling exponent underlying the criticality of the studied variable [37]. γ = ϵ or Φ if Y = X (abundance) or TE (interactions). The power-law exponent indicates how mean, variance and higher statistical moments are defined (finite or infinite depending on γ [47]). Specifically: for γ ≤ 2 the regime is critical with mean and all moments that are infinite (accurately, this is the only heavy-tail distribution that almost always has finite-size effects); for 2 < γ ≤ 3 the regime is supercritical with finite mean and all higher moments as infinite; and, for γ ≥ 3 mean and all moments are finite. The latter regime is the last power-law regime before the exponential one. Strictly, power-laws should be defined for two orders of magnitude at least but we relaxed this condition due to data sparsity in this case study.

Distributions of species variables (abundance and interactions) are visualized by computing discrete exceedance probability distribution (epdf) that is defined as P(Y ≥ y) = 1 − P(Y < y), where P(Y < y) is the cumulative distribution function (cdf) [46] derived from probability distribution function p(y) (pdf), and by plotting the epdf on a log-log scale to emphasize the non-linear power-law dynamics useful to estimate the scaling exponent. Exponents were estimated using the approach and package of [48]. We also introduce the cutoff Y_break to the random variable whose probability distribution explicitly presents multiple regimes. In our study these probability distribution regimes are power-law and occur with different exponents that are separately estimated. This double Pareto distribution was highlighted in different ecosystems and analytically formulated by [47]: (2.2) where Y_break is the break point isolating two power-law distribution regimes, c₁ and c₂ are two coefficients, F(y) is a homogeneity function [47], γ₁ and γ₂ are power-law exponents of the two regimes. This “broken power-law” function (or double-Pareto distribution function) is a piecewise function consisting of two or more components with different power-law scaling exponents separated by break points [47, 49, 50]. Specifically in this study, a break point was set to divide the epdf of species interactions for the 15–20°C TR. The break point Y_break estimated from the model was much higher than the median value of species interactions for the first power-law regime. If a power-law distribution of abundance or interaction was not feasible, then an exponential distribution was fitted to the data, i.e. P(Y ≥ y) ∼ e^−λy.

2.3 Species diversity and abundance characterization

To characterize the fish community macroecologically, α-diversity is introduced to describe the fluctuations of species diversity over time and temperature. α-diversity refers to the biodiversity within a particular scale (time period, area or TR) and is computed as the number of species in that scale. Given a set of unique species S = {S₁, S₂, …, S_n} whose normalized abundances X = {x₁, x₂, …, x_n} change over time, time-dependent α-diversity α(t) is defined as in [35]: (2.3) where x_i(t) is the normalized abundance of species i at time point t. For each time point, sea surface temperature and bottom temperature were recorded. We average these temperatures as mean ST and reorganized α-diversity dependent on the mean temperature yielding temperature-dependent α-diversity α(T).

Additionally, fish species are distinguished into fish stocks (FS) [51, 52], native and invasive species groups as shown in S1 Table. This categorization was based on the information retrieved from Fishbase (http://www.fishbase.org/). We separately calculate the total abundance of all species, defined as Ecological Productivity (EP) that is proportional to the total biomass [53–55], and of these species groups, FS, native and invasive dependent on time and temperature as: (2.4) where T can be time or temperature, x_i(T) is the normalized abundance of species i at time or temperature T, n′ is the number of species in a particular subgroup (for all species group EP, n′ is n that is 15s).

2.4 Information-theoretic patterns and network inference

2.5 Continuous time- and temperature-dependent dynamical network analysis

Beyond the TR-constrained “seasonal” analysis (independent of time), OIF [28] was used to extract dynamical networks considering time and temperature as continuous dependent variables to analyze the ecosystem in long-term and for smaller steps along the temperature gradient. By necessity this is done by truncating the whole time series in time and temperature units among which networks are reconstructed. Discrete TR networks (for the groups in Section 2.1) are essentially extractable from the continuous analysis.

The total number of time-series units (number of networks and community features) on which dynamical network inference is performed by OIF is , where ⌊•⌋ rounds G to the smaller integer. When reconstructing time-dependent dynamical networks, L is the total number of samples (L = 285, i.e. the whole time period), l is the length of each time-series unit (time unit hereafter), Δl is the chosen inter-observation gap (or time step). In this study, each time period is set as one year (the length of time-series unit l is 24 time points that guarantees to have enough points for reliable TE inference), time step Δl is 2 time points that correspond to one month, and then the whole time series is truncated into 131 time-series units in total where networks are inferred. The overlap (and yet dependence of species) of data/networks is nine months vs. 2 weeks of [21] and we believe this is a more proper choice considering that species do not alter dynamics in a matter of weeks and more importantly independently of previous historical trends (earlier than two weeks from the present). In this regard, our approach considers long-term dynamics (l = 1 year data) for estimating short-term features at a scale Δl = 1 month. This is also the scale at which stability and effective diversity were calculated. The non-linearity between species is considered via time-delayed co-predictability of pdfs (i.e. TE); yet, more deeply than [21] that linearizes the non-linear community variability by considering ratios of delayed species abundance every two weeks. The latter is the basis of CCM [21], typically estimating local feature from local dynamics only (and without pdfs), unless larger time delays are selected but those cannot be too large without compromising computational efficiency.

When reconstructing temperature-dependent networks considering temperature as a continuous variable vs. discrete TR analysis (Section 2.1), L is the maximum temperature rounded to the largest integer (in this dataset the maximum mean temperature is 30.7°C, thus L is 31), l is the upper limit of the first temperature range (for convenience we picked the range [5 − 10]°C and then l = 10°C) and Δl (set to 1°C) is the temperature step establishing the moving window (or the difference between networks over temperature, equivalently). Therefore, the whole time series is truncated into 22 temperature units (calculated as ) corresponding to different temperature ranges whose difference is 1°C (and overlap is 4°C) for providing small-scale estimates by considering continuous smooth transitions along the temperature scale. Note that the temperature unit size (or range), over which temperature-dependent networks are estimated, is set to l* = 5°C that is granular enough to describe ecosystem variations. This is in analogy to the predictability overlap concept used for the temporal network inference: species interactions do not change abruptly for every Δ-temperature and yet non-linear superposition of species along the temperature-scale should be considered. The length of time series for different temperature units (representing somehow temperature niches) is different due to the fact the number of species in each unit can vary; note that the temporal sequence of abundance for different temperature units is preserved. This is different than temperature analysis where the number of samples every time units is always the same because of sampling design; yet the time unit size corresponds to the lower limit l. By selecting temperature units equivalent to the defined TRs in Section 2.1 it is possible to derive the discrete seasonal networks. Lastly we also performed TR-constrained dynamical analysis over time by combining time-dependent data conditional to the discrete TR groups defined in Section 2.1.

After time and temperature-dependent reorganization (i.e. picking abundance data within all temperature ranges), OIF is used to infer species interaction (TE) for each time-series unit (time or temperature unit), resulting in temporal and temperature-dependent interaction matrices W_i,j(g) and dynamical networks. Here, g is the time point of each time-series unit for temporal networks, while the lower limit of the temperature range for temperature-dependent networks. We analyze dynamical networks by computing total interaction (TI), local dominant eigenvalues and estimated effective α-diversity of each dynamical network. TI is defined as the sum of all TE asymmetrical interactions in the interaction matrix W_i,j(g), i.e.: (2.9)

The estimated effective α-diversity α_e(g) is defined as the number of nodes (species) involved in a dynamical network [28] that is formulated as: (2.10) where (2.11)

Therefore, α_e(g) denotes the total number of nodes whose functional degrees (considering asymmetrical interactions [35]) are not zero in a dynamical network. We use absolute values of W in Eq 2.11 for generality, in case interactions are generated by a model that identifies positive and negative interactions such as in CCM [21]; note that in OIF any negative TE value does not have physical sense (or is mathematically plausible, unless a negative sign is assigned a posteriori to opposite interaction directions between i and j) [28] and should be considered as a numerical artifact to remove.

3.1 Time and temperature-dependent abundance analyses

A simple analysis for original data of fish species abundance starts by looking into temporal trajectories and seasonal fluctuations of sea water temperature (Fig 1A), taxonomic α-diversity (Fig 1B) and abundance (Fig 2). It is evident that sea surface and bottom temperature, α-diversity of the fish community fluctuated over time synchronously and seasonally. Approximately since 2007, seasonal fluctuations of sea surface and bottom temperature were slightly getting higher except for 2009 with the least fluctuation (Fig 1A), while the global trend of α-diversity was decreasing (Fig 1B). This result implies that the change of biodiversity loss in the fish community may relate to the increasing fluctuations of ST. Total abundances of EP, FS, native and invasive groups over time (Fig 2A and 2B) show a slight increase with fluctuations since 2007, that is opposite to the global trend of species diversity, with the exception of a spike in 2009 and a downward spiral in 2011. This finding implies that increasing ST makes some species more abundant in global view, even though it negatively affects species diversity, and the latter is very sensitive to the change of ST fluctuations via species interaction alteration.

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Fig 1. Seasonal fluctuations of ST and α-diversity and relationship with mean temperature.

A: The sea surface temperature (red line) and bottom temperature (blue line) from June 2002 to April 2014. B: Taxonomic α-diversity over time. C: α-diversity over mean temperature (the average of sea surface and bottom temperature). Blue points, light green points, green points, yellow points and red points represent values of α-diversity corresponding to different TRs: ≤10°C, 10–15°C, 15–20°C, 20–25°C, ≥25°C, respectively. The black curve in plot C is a second degree polynomial fitting for α-diversity over temperature; that establishes an empirical Pareto optimal frontier for the climate niche of fish diversity exerted by temperature. The inset is showing the potential ecosystem landscape across the whole period where the entropy is proportional to 1 − pdf(α); transition seasons shows the highest entropy followed by summer.

https://doi.org/10.1371/journal.pone.0246222.g001

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Fig 2. Total species abundance of EP, FS, native and invasive species over time and mean temperature.

Total species abundance of EP (yellow) and FS (blue) species over time and temperature (A and B). Total species abundance of native (yellow) and invasive (blue) species over time and temperature (C and D). Abundance of species 1 (green) and 2 (red) over mean temperature (C and D).

https://doi.org/10.1371/journal.pone.0246222.g002

These results can be more explicitly confirmed in Fig 2 where EP, FS, total abundance of native and invasive species and species 1 (Aurelia aurita) are observed to increase exponentially over the seasonal increase in ST [70, 71]. The abundance of species 2 (Engraulis japonicus) that is a cooperative species with large geographic range (S15 Fig vs. S16 and S17 Figs for competitive species), as an exception, decreases with the rise of ST both seasonally and over time. It should be carefully noted that patterns over temperature (Fig 2B and 2D) characterize the dynamics of species only across seasons and not over time; rather, changes in the exponential growth parameter over time are meaningful for tracking temporal changes. Therefore, Fig 2B and 2D characterize the baseline exponential response (X = X_minexp^kT where k is the growth factor in the legend and X_m in is the abundance for the lowest temperature) of abundance across seasons. Discordance between these patterns and patterns over time for increasing temperature (e.g. for native species and fish stock) are warning of changes in species related to other factors, such as changes in species interactions. Table 1 shows the observed ecosystem response to temperature over time considering abundance and other metrics discussed hereafter.

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Table 1. Observed changes in ecosystem community and population metrics.

Metrics are evaluated considering 2002 and 2014 observed values. Negative Δ % are for decrease in values over time. Δ/year % are the yearly changes over 12 years. Ecosystem stability is based on the real part of the dominant eigenvalue of the whole community (S12A Fig).

https://doi.org/10.1371/journal.pone.0246222.t001

We also reorganize α-diversity considering ST, yielding α-diversity against temperature shown in Fig 1C. It shows that α-diversity grows with the increasing temperature (over seasons), while the rate of the increase of α-diversity (α′(C)) gradually declines (see the black line fitting the α(T) points in the plot). The decreasing α′(T) implies that fish diversity does not continuously grow with the increase of temperature. Within lower TRs, the fish community is more sensitive to the change of temperature relative to higher TRs.

Time series of normalized abundance are categorized into five groups considering five TRs: ≤10°C, 10–15°C, 15–20°C, 20–25°C, ≥25°C. Epdf for these five TRs are visualized in Fig 3A. Epdf plots show that the distribution of abundance, and hence abundance uctuations, becomes more scale-free with the increase of temperature; note that this is a seasonal analysis that does not reveal any temporal change. Lower power-law scaling exponent for higher TRs means fatter tail of power-law distributions (with slower decay than exponential distribution, yet less resilient Scheffer sensu). This indicates that the distribution of species abundance is less even since some species become exceedingly more uctuating in abundance (with “black swan” extremes) that may or may not be persistent over time (see S1 and S2 Figs show the epdf of abundance for each species individually and competitive/cooperative groups). Power law implies less evenness (the similarity in abundance among species is lower despite common species are more even) but higher long-range organization and this is a beneficial properties of biodiversity, also in an energetic sense considering the lowest entropy of scale-free organization [35]. This is particularly evident for cooperative species (in terms of distribution rather than extreme values) as also show in [28] and Fig 8. These results suggest that for higher temperature, fish species present wider distribution in abundance compared to lower TRs. Fig 3B shows the abundance-based Taylor’s law, i.e. standard deviation against mean abundance for each species corresponding to each TR. The regression of log standard deviation vs. log mean abundance for all TRs gives scaling laws with slopes less than 1. Positive slopes address that the total variability of species abundance increases with the rise of mean species abundance. Slopes less than 1 suggest that the per capita variability of species abundance for all TRs decreases with the increasing mean species abundance [72]. Therein, slopes of scaling law for lower TRs (≤10°C, 10–15°C, 15–20°C) slightly increase with the increasing ST, and are obviously lower than those for higher TRs (20–25°C, ≥25°C). It indicates that the variability of species abundance is on average higher for higher TRs compared to lower TRs. This result confirms the finding from Fig 1C that the fish community in lower TRs (≤10°C, 10–15°C, 15–20°C) is more sensitive to the change of ST compared to higher TRs (20–25°C, ≥25°C). The increase of scaling exponents implies that the fish community experiences a significant change in species populations with impacts on the collective behavior (in terms of interactions) around 20°C.

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Fig 3. Zipf’s and Taylor’s law of species abundance for temperature groups.

A: Exceedance probability distribution functions (epdf) are fitted by power-law distributions. |−ϵ + 1| is the scaling exponent of the distribution. B: Taylor’s law as scaling law between standard deviation and mean species abundance for five TRs; ν/2 is the scaling exponent of Taylor’s law that is typically construed by using the variance [72], 〈x²〉 ∼ 〈x〉^ν; smaller ν means that fluctuations in abundance are more even and species are more regularly distributed; vice versa, species are more power-law distributed (as supported by Zip’s law) driven by stronger environmental effects [85] (in this case ST) determining portfolio effects with potential stabilizing effects [86] (this however neglects interaction topology). All exponents for five TRs are interpolated by black dashed lines in insets to emphasize abundance patterns transitions across TRs/seasons; transitions show a gradual second-order phase transition with higher Pareto-distributed fluctuations for higher temperature.

https://doi.org/10.1371/journal.pone.0246222.g003

3.2 Interaction inference and temperature-dependent network characterization

Different patterns in epdf and Taylor’s law of species abundances shown in Fig 3 envision the discrepancy in species interactions and dynamics of ecosystem for different TRs. In this study, species interaction is quantified by TE that is an information-theoretic variable measuring the amount of directed information flow and evaluating connectivity between species. Networks for the whole period and five TRs (see Fig 4) are therefore inferred by TE-based OIF model, and graphically visualized with Gephi [73]. To refine network structure, links whose interactions are lower than 0.01 are discarded by setting a threshold to filter TE. Optimal Information Flow (OIF) networks present different structural and functional properties for particular TR fish groups. Network for the whole period shown in Fig 4A provides a static overview of the fish community without considering the dynamical change of temperature or seasonality. It outlines causal relationships between species in the fish community, but is inadequate to tackle the dynamics of fish ecosystems driven by the fluctuation of temperature, and identify ecological and dynamical responses. Therefore, to understand how temperature affects the fish community in detail, networks for five TRs (≤10°C, 10–15°C, 15–20°C, 20–25°C and ≥25°C) are reconstructed and listed in Fig 4 (from Fig 4B to 4F). Here we first define network size as an indicator that is proportional to the number of nodes and the total amount of interactions between species. By looking at the structure of networks (Fig 4B–4F) and taking total interactions over temperature (S12F Fig) into consideration, it is obviously observed that network size is the largest for the fish community within a higher TR. Nodes connected by links with warm colors are regarded as a cluster in which species are significantly affecting others or affected by others. Network of 15–20°C group shows a larger cluster with stronger interactions between species compared to other TRs. Yet, the distribution of species interactions in this TR presents more evenness considering the gradient of links’ color in networks. This finding can be obviously observed by comparing Fig 4B’–4F’ describing the phase mapping of interaction matrices. OIF-based interactions for 15–20°C group are distributed more randomly. This randomness reveals that OIF-inferred network for 15–20°C group seems to be less scale-free versus other networks, and stands on a critical point where the fish ecosystem is experiencing a phase transition from a stable state to a metastable state. When considering specific species within different TRs, except for the ≤10°C and 10–15°C groups in a globally stable state, species 6, 7, 8, 9 are always the nodes with warm color (high OTE) (see Fig 4A and 4C–4F) in networks. Yet, interactions between these species are higher than others considering the phase mapping of TE matrices shown in Fig 4.

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Fig 4. OIF-inferred species interaction networks and matrices.

The OIF model is used to infer the causal interactions between all pairs of species, yielding average interaction (TE) matrices for the whole time period and five TRs. TE values in interaction matrices are normalized to 1 and drawn in plots A’, B’, C’, D’, E’ and F’. After removing weak interactions by setting a threshold (TE = 0.01) to filter TEs, species interaction networks are reconstructed using Gephi (https://gephi.org/) and shown in plots A, B, C, D, E and F. The size and color of nodes is proportional to the Shannon Entropy of species abundance and the total outgoing transfer entropy (OTE) (the greater OTE, the warmer the color); the width and color of the link between species are proportional to species interactions computed as TE (the greater TE, the warmer/wider the link’s color width). Each network is plotted independently from the others so colors and sizes are relative to each TR range. The arrow of links defines the direction of species interaction (TE) that are bidirectional and asymmetrical. High/low values of TE define competitive/cooperative species dynamics that is associated to different network motifs (small-world and scale-free, respectively); increases in temperature over time leads to scale-free loss and increased high interactions with dominance of competitive species with summer-like dynamics.

https://doi.org/10.1371/journal.pone.0246222.g004

We also set a threshold as top 20% TE considering the Pareto principle (also known as the “80/20 rule”) which specifies that roughly 80 percent of the consequences come from 20 percent of the causes in many events [65]. TE threshold following this rule is generally higher than 0.01. It would further reduce network size, making inferred networks briefer relative to the networks shown in Fig 4. OIF networks (S6 Fig) for the whole time period and five TRs with the threshold of top 20% TE are reconstructed using the same regulation used in Fig 4. After the further reduction of network size, networks in S6 Fig ignore more functional details, but are clearer to identify the structural difference between networks compared to Fig 4. Furthermore, it is possible to statistically analyze the structural degree, in- and out-degree of nodes for each network. Statistical analysis for nodal degree is shown in S7 Fig. Structural degree, in- and out-degree get higher with the increase of temperature as a whole. This result implies that warmer conditions make the fish community more socially connected. Separately, pdf of structural degree (see S7A Fig) shows that structural degree of nodes increases with the increasing temperature, and favors bimodal distribution with different shapes for ≤10°C, 10–15°C, 15–20°C, 20–25°C groups, while roughly uniform distribution for ≥25°C group. For ≤10°C group, most structural degrees are distributed within the lower unimodal range from 1 to 3. With the increasing temperature, more and more values of structural degree are distributed in the higher unimodal range. However, for the highest TR (≥25°C) group, the pdf of structural degree presents more evenness within the range from 0 to 10 (uniform distribution) compared to other groups. This result implies that structural degree does not increase continuously with ST, and that the fish community would become less interacting when ST is too high. Analogous features can be observed in the distribution of in-degree (S7B Fig) and out-degree (S7C Fig). For each TR, OTE and average abundance are calculated for each species without considering any threshold for TE. Species’ OTE favors bimodal distribution with short range for ≤10°C and 10–15°C groups, and broad range for 15–20°C, 20–25°C and ≥25°C groups (see S8A Fig). This result suggests that overall effects of species on others grow with the increasing temperature. In addition, OTE is scaling with mean abundance (a recurrent pattern known as Kleiber’s law as found for other ecosystems [35]) and abundance standard deviation that is mass-independent (S11A and S11B Fig, respectively).

3.3 Interaction spectrum, phase transition and system stability

The probability distribution of species interactions is characterized by calculating discrete exceedance probability, then fitted using power-law function (see Fig 5). Among five TRs, the greatest interaction occurs in the highest TR (≥25°C) group, indicating that species interactions overall increase with ST. The scaling exponent of the power-law distribution (computed as 1 − φ) is lower for ≤10°C, 10–15°C, ≥25°C groups and the first regime (black fitting) of 15–20°C group compared to 20–25°C group and the second regime (green fitting) of 15–20°C group. The power-law model fits the epdf of TE better for ≤10°C and 10–15°C groups relative to other TRs. The better fitting demonstrates that the distribution of species interactions presents more scale-free patterns, implying a globally stable state for these two groups. With the rise of temperature, the fish community begins to get more interacting gradually and two regimes therefore appear in the distribution of species interactions for 15–20°C group. The distribution of species interactions for 15–20°C group is divided into two sections by a break point (y_min = 0.584 that is higher than the median value of TE, 0.263) estimated from power-law fitting; yet, the scaling exponents of the two epdf regimes are very different (-0.357 and -4.468, respectively) identifying a critical and subcritical regime, respectively [49, 50]. The critical regime is more stable (optimality sensu) due to higher scale-free organization vs. the subcritical that is less optimal but more dynamically stable or resilient (a la’ Scheffer). The two different regimes for the 15–20°C range suggest that the fish community stands on a critical point where the fish ecosystem is experiencing a phase transition from global stability to metastability, and that the collective dynamics of species is experiencing a significant change driven by ST. The change of collective behavior is also reflected by the fluctuations of species populations and richness. The power-law of species interactions for 20–25°C group shows the highest scaling exponent, and the distribution of species interactions tend to be the most exponential among five TRs. The power-law model was forced for the tail of the distribution in order to provide a phase-space plot as a function of power-law exponents; however, the maximum likelihood solution is an exponential distribution for TR 20–25°C and above 25°C. Species interactions continue to increase in magnitude as temperature rises, and a large number of weak interactions emerge simultaneously, leading to a distribution pattern where most interactions are clustered from low to intermediate range forming a small-world network topology. This distribution pattern exhibits a less scale-free topology compared to other TRs. Therefore, fish ecosystems within 15–20°C and 20–25°C groups are metastable (also see Fig 4D, 4E, 4D’ and 4E’). With the continuous increase of ST (≥25°C), the power-law scaling of the species interaction distribution shows a significant decrease that is opposite to ST. The power-law fitting model fits the epdf of species interactions better for ≥25°C compared to the critical group of 15–20°C and exponential group of 20–25°C. This result indicates that several major species interactions continue to rise, while minor interactions disappear, leading to a flatter distribution and more scale-free pattern relative to 15–20°C and 20–25°C groups. The fish ecosystem returns to another relatively stable state after perturbations caused by ST. This finding confirms the fact that the fish community would not get more interacting continuously with the increase of ST, yet the activity of fish species would decrease when temperature is too high. The trend of scaling exponents (Fig 5F) shows that the fish ecosystem is subjected to a phase transition from global stability to metastability in 15–20°C group, and finally returns to local stability (globally unstable Bak sensu, but resilient Scheffer sensu) with the increase of temperature. A fold-like bifurcation occurs considering the transition of species interaction at 15–20°C. This result reveals that, during the phase transition from global stability to local stability, the fish ecosystem stands at a critical point within 15–20°C where the fish community experiences a significant change in collective dynamics. These changes result in different functional features for the fish ecosystem (and potential environmental transitions, e.g. in biogeochemical cycling) within different TRs as graphically shown by functional interaction networks (see Fig 4).

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Fig 5. Exceedance probability distribution function of species interactions and phase-space dependent on surface temperature.

A, B, C, D and E: Epdfs of OIF-based interactions (after filtering with a threshold of 0.01) between all pairs of species for five TRs: ≤10°C, 10–15°C, 15–20°C, 20–25°C, ≥25°C. The fit is with a power-law function. In plot F, all power-law exponents are connected by dashed lines. Scaling exponents for different TRs are displayed in a phase-space where x-coordinates are middle values of TRs (note that 7.5°C and 27.5°C are selected as middle temperature values for TRs lower than 10°C, and higher than 25°C, respectively); points are empirically connected by a dashed line from low to high temperature to emphasize phase-transition of interactions. Note that for the 15–20°C group, epdf of species interactions presents two power-law regimes that are separately fitted by a power-law functions. The double-Pareto distribution reflects the presence of two strange-attractors and a critical unstable transition at 15–20°C. Critical regimes are defined according to power-law criteria defining finiteness of statistical moments (see [35]).

https://doi.org/10.1371/journal.pone.0246222.g005

In this work, the dominant eigenvalue of species interaction (TE) matrix is studied as a representation of network structure and to analyze the stability of the fish ecosystem. According to the above analysis of species interaction spectrum, OIF-inferred networks representing the fish community present different structural patterns for different TRs. As a part of the complex marine ecosystem, these structural differences may relate to numerous relevant biotic and abiotic factors. To prove ST to be an important factor affecting the structure and function of networks, we infer dynamical networks over time for each TR, then calculate the information flow from ST to the real part of the dominant eigenvalues of dynamical TE matrices underlying network stability (see Table 2). Table 1 reports various ecosystem metrics of species populations and community considering time variability of the environment. The information ow from ST to network structure increases from ≤10°C to 20–25°C group, then slightly declines. The lower information ows for ≤10°C and 10–15°C groups manifest less effect of ST on these TRs, implying more stable states with higher resilience to the fluctuation of ST compared to other groups. In contrast, 15–20°C and 20–25°C groups show high information ows from ST to network structure, implying strong effect of ST on the fish community. This corresponds to the relatively unstable state (metastability) where these groups undergo a significant change in system state and dynamics. The slight decrease in information ow from 15–20°C and 20–25°C groups to ≥25°C group means the reduction in the effect of ST on (the improvement of system stability for) ≥25°C group. These results extraordinarily conform to the findings from the probabilistic analysis of species interactions for TRs. Therefore, the change of ST is considered as an important factor that drives structural changes and state shifts of the fish ecosystem.

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Table 2. Predicted causality between temperature and ecosystem stability for different temperature groups.

The potential effect of temperature variability over time on ecosystem stability is calculate as Transfer Entropy (TE) between temperature (within TR groups) and the real part of the dominant eigenvalue (see S12A Fig for the latter). The larger TE the larger the predictive causality.

https://doi.org/10.1371/journal.pone.0246222.t002

Eigenvalues of species interaction matrices for each TR are scattered together in a complex plane (Fig 6). This method displays the position of all eigenvalues in the form of an ellipse [22, 68]. There are 8 eigenvalues for ≤10°C group, 14 eigenvalues for 10–15°C group, 14 eigenvalues for 15–20°C group, 15 eigenvalues for 20–25°C group and 15 eigenvalues for ≥25°C group. Eigenvalues correlate with the stability and resilience of the fish ecosystem in both magnitude and variance. When only considering the eigenvalues with positive real parts that may result in instability, there are 2 eigenvalues with positive real parts (2.028 and 0.009) for ≤10°C group, 5 eigenvalues with positive real parts (2.262, 0.002, 0.002, 0.003 and 0.0003) for 10–15°C group, 1 eigenvalue with a positive real part (4.998) for 15–20°C group, 1 eigenvalue with a positive real part (4.934) for 20–25°C group and 2 eigenvalues with positive real parts (6.143 and 0.052) for ≥25°C group. It is observed that each TR has eigenvalues with positive real parts that may bring instability to the ecosystem, while positive real parts are greater for higher TRs (especially for 15–20°C, 20–25°C and ≥25°C groups). Sea temperature is quite likely the strong causal factor. This result confirms the finding that the fish ecosystem presents more instability for 15–20°C, 20–25°C and ≥25°C TRs.

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Fig 6. Eigenvalue spectrum of TE interaction matrices across temperature ranges and ecosystem relative stability over time.

Eigenvalue distribution of TE interaction matrices are scattered in a complex plane (five different colors correspond to five TRs) (A). Dynamical stability of the fish community is computed as the real part of the dominant eigenvalue of TE interaction matrices; this is show over time (B) for competitive (4–9 species) and cooperative species. The total interaction of dynamical networks over time (as sum of all TE interactions) (C) is mirroring the fluctuations of the eigenvalue relative stability. Note that fluctuations of the eigenvalue and interactions are increasing over time and manifesting critical slowing down (CSD) toward a suboptimal ecosystem state with small-world/random high interactions (Fig 5) with large fluctuations in abundance synchronized with temperature (Figs 1 and 3). Total interaction dynamics is dominated by cooperative species but extremes are regulated by competitive species that are much more synchronized with ST fluctuations. This is because cooperative species are more numerous and abundant than competitive species and the latter are synchronized with temperature extremes.

https://doi.org/10.1371/journal.pone.0246222.g006

3.4 Network-based ecological importance and critical species identification

Biological importance and critical species identification are performed by further analyzing structural features of OIF networks and combining interaction spectrum with abundance spectrum. To this end, we first measure the salience for all links in OIF networks [69]. Link salience computation is based on the effective proximity d_ij defined by the strength of mutual interactions: d_ij = 1/TE_ij. It is intuitively assumed that strongly (weakly) interacting nodes are close to (distant from) each other. Link salience matrices shown in Fig 7 illustrate that species 7 (Pseudolabrus sieboldi) is always the reference node of the most salient links with the exception of ≤10°C group where the reference node of the most salient links is species 15 (Rudarius ercodes) instead of species 7. According to the website of fishbase (http://www.fishbase.org/summary/Pseudolabrus-sieboldi.html), species 7 (Pseudolabrus sieboldi) is a native species in northwest pacific ocean and mainly distributed in Japan waters [74]. These results reveal that native species 7 has the most critical influence on other species in the fish community, and plays a vital role in maintaining structure and function of networks.

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Fig 7. Link salience matrices highlighting Pareto links.

Link importance is measured as link salience for all OIF networks corresponding to the whole period and five TRs, yielding 15 × 15 matrices. The values of link salience are normalized to 1, and drawn in plots A, B, C, D, E and F, respectively. Pareto links are between keystone species responsible for the stability of the ecosystem.

https://doi.org/10.1371/journal.pone.0246222.g007

We also calculate Shannon entropy for species to measure the fluctuation and uncertainty of species abundance, and compute species’ OTE as another nodal (species) property to quantify how much a species totally influences others in OIF networks. Then, species rankings considering top 5 Shannon entropy and top 5 OTE are listed and shown in S9 Fig. On the one hand, the species ranking of Shannon entropy (left figures in S9 Fig) shows that species 2 or (and) 5 always have the greatest Shannon entropy that represents the highest uncertainty in abundance for these two species. This finding also can be observed in the original time series of abundance plotted in S3 Fig. Species 2 and 5 have the highest fluctuations and diverse values of abundance. On the other hand, the species ranking of OTE (right figures in S9 Fig) shows that for all TRs except the ≤10°C group, species 7 always has the greatest OTE, and species 5, 6, 7, 8 and 9 are often involved in top 5 OTE ranking especially for higher TRs. These species are dynamically competitive species a’ la [28] and with higher temperature fitness as well as smaller geographic range tendentially. For instance species 5 Trachurus japonicus (Horse mackerel) in S15 Fig is native of Japan while species 9 Halichoeres tenuispinnis (Chinese wrasse) despite being mostly distributed in Japan is invasive with origin in SE China. These finding imply that these competitive species have significant effects on other species in the fish community, of which species 7 is the greatest. This result is in accordance with the results of link salience matrices shown in Fig 7. In a word, through measuring link salience and OTE, species 5, 6, 7, 8, 9 can be identified as the critical species in the fish community. Additionally, we also calculate TE, MI, species abundance vs. ST in semi-log scale, Pearson correlation coefficient (indicated as cc) and causality coefficient ρ of CCM [63] (Table 3) between species abundance and ST to identify which species are most affected by ST. The identification is done by comparing the aforementioned linear and non-linear methods that were also used to compare inferred species interactions (see S5 Fig, and [28]). The slope of the abundance-temperature relationship in semi-log scale is the parameter of the exponential function of species abundance growth dependent on ST (see S4 Fig). In Table 3, TE from ST to species abundance for species 5, 6 and 7 is higher than other species. MI, Pearson correlation coefficient and ρ are overall higher for species 5, 6, 7, 8 and 9. Therefore, species 5, 6, 7, 8 and 9 are considered as the species that are most affected by ST; this is not necessarily a negative impact of temperature but rather a signature of importance of temperature in constituting the ecological niche of these species. It is interesting that these species are also recognized as critical species in the fish community by measuring link salience (dependent on TE) and OTE ranking of importance for predicting collective dynamics. These competitive species (dynamically speaking as in [28]) are highly concordant with each other and more synchronized with temperature, yet more likely affected by temperature fluctuations directly. We also find that TE values from species abundance to ST are very small. This result is expected since fish species do not affect ocean temperature. By looking into S4 Fig, the abundance of species 5, 6, and 7 exponentially grows with ST more obviously than other competitive species. Competitive species have the strongest and positive exponential fit with seasonal temperature versus cooperative species whose response to seasonal temperature is less exponential and more frequently negative (implying a decay); the exception is for species 15 that is dynamically the most similar to competitive species. Competitive species (dynamically speaking as in [28]) are highly concordant with each other and more synchronized with temperature, yet more likely affected by temperature fluctuations directly. The exponential fit is not by chance a by-product of the synchronized exponential fluctuations of ST and fish abundance, while cooperative species fluctuations are more power-law distributed (S2 Fig) (although in terms of range than critical regime). This signifies that cooperative species are affected also by other factors such as fishing beyond linkages with competitive species-ST.

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Table 3. Indices measuring the relationship between ST and species abundance.

https://doi.org/10.1371/journal.pone.0246222.t003

Moreover, probability distribution functions of abundance for all species (see S2 Fig) show that species abundances of these species are more evenly distributed within relatively small ranges (the least range is [0,45) for species 7) compared to other species with wide distributions and extreme values. It demonstrates that trajectories of species abundance for species 6, 7, 8 and 9 are remarkably divergent from other species. Species with wider distributions and more extreme values present more uncertainty in abundance, that is, Shannon entropy of these species is higher than that of species 6, 7, 8, 9. Intuitively, TE-s from source variables (species) with low uncertainty (more deterministic information) in abundance (species 6, 7, 8, 9, for instance) to target variables with high uncertainty (see left figures in S9 Fig) are supposed to be high. Considering species 6, 7, 8 and 9 themselves, within the small range of abundance distribution, relatively speaking, these species also present transparent divergences (dissimilarity) in the distribution of species abundance. Therefore, causal interactions (transfer entropy) between these species are relatively high. These results address that the distribution of species abundance can be used as a proxy of interactions by comparing similarity and divergence. Take species 2 (Engraulis japonicus) as an extreme example, TE-s from species 2 to others are relatively low as expected since the original abundance data of species 2 have many zeros or values close to zero that lead to lower uncertainty despite the asynchrony and divergence.

Considering the whole time period, rather than solely estimate the simple continuous pdf of abundance for each species (S2 Fig), we probabilistically characterize distributions of abundance by computing epdf and using power-law fitting. The scaling exponent of power-law fitting is considered as another species property in terms of the distribution of abundance (see S1 Fig). Thereafter, abundance-interaction phase-space describing relationships between power-law scaling and species OTE (power-law exponents vs. OTE for all species, indeed) is shown in Fig 8.

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Fig 8. Abundance-interaction atemporal pattern and distribution of species interactions for competitive and cooperative species.

(A) Considering the whole time period for the 15 species, ϵ is the slope of the Zipf’s law (on pdf) of species abundance (see Fig 3) and OTE is the sum of all interactions of a species (that is how much one species influences the collective on average). Black continuous and dashed curves are the linear and non-linear (second-degree polynomial) fitting for the abundance-interaction pattern. Cooperative species, asynchronized with temperature, are characterized by more power-law distributed fluctuations in abundance (see S1, S3 and S4 Figs) manifested by the lower ϵ and wide scale-free distributed total interactions (Figs 4 and 5, and 8B) (with relatively low average) that are more stable over time (Fig 6); this underlines the weak-interaction induced stability of ecosystems. Increasing temperature leads to a predominance of small-world distributed high interactions of competitive species (whose abundance is synchronized with temperature and distributed more exponentially); this causes the loss (or a shift) of Pareto salient links (Fig 7) and the higher fitness of invasive species with “hybrid” dynamics. (B) Epdf of interactions of competitive and cooperative species. Interactions are estimated via the OIF model for the whole time period. The number of cooperative species is higher and their scale-free distribution wider than competitive species as well as much more stable due to the lower power-law exponent; this is when considering the extreme competitive interactions as power-law although the whole distribution is exponential (P(Y ≥ y) ∼ e^−λy). The rarity of competitive species is manifested by the narrower distribution with lower stability (Bak sensu). The legend shows in black the dominant dynamical regime for competitive and cooperative species (for the former only the exponential dynamics is substantiated); the inset is the species interaction network for the whole period (Fig 4A) with the competitive core highlighted.

https://doi.org/10.1371/journal.pone.0246222.g008

For a selected species, the higher the exponent of the power-law, the more abundance values are distributed within a narrower range (S1 and S2 Figs); additionally, the higher OTE, the more strongly the species interacts with others.

The phase-space mapping in Fig 8 shows a trend where the power-law exponent of species abundance is proportional to the species OTE with species 3 and 6 as outliers. This result suggests that species whose abundance has a wide distribution with a fat tail have relatively low total effects on other species (these are cooperative species, while species whose abundance is more evenly distributed within a narrow range have high total effects on other species (species 7, 8, and 9 but in general all competitive species). These findings, that highlight the duality between abundance and interaction distribution, are also extractable from the mapping of interactions for the whole time period shown in Fig 4A’. Previous research showed that species with more exponential distribution of abundance have a competitive role in the fish community dynamics [28]; this is beneficial for species abundance growth and regulation of other species. The former are acclimatized species where temperature affect them positively with higher ecological memory while the latter are affected by multiple environmental pressure such as temperature and fishing (temporally and likely spatially in the geographical area considered) and are likely more sensitive to short-term conditions. In S9 Fig, OTE vs. mean abundance shows that for most species, effects on other fishes are proportional to the abundance of species (S9A Fig) (this is known as Kleiber’s law [35]), while inversely proportional to the standard deviation of the abundance of species (S9B Fig).

3.5 Time- and temperature-dependent dynamical interactions and stability

Considering both dimensions of time and temperature, time- and temperature-dependent dynamical species interaction (TE) matrices are inferred from time series of each time and temperature unit, respectively. On the temporal scale, S12A Fig shows the real part of the dominant eigenvalue of TE matrix (blue line) and adjacency matrix (red line) underlying the structure of temporal networks. The real part of the dominant eigenvalue of TE matrix over time is lower than that of adjacency matrix, yet the former presents an obvious seasonal fluctuation. In the first half-year period (approx. from winter to early summer), it is observed that the real part of the dominant eigenvalue increases over time and reaches a spike during this period. Then it decreases in the second half-year period, while increases again at the end of year. The increasing trend or the high level of the real part of the dominant eigenvalue always appears within the time period with higher changes (fluctuations) of ST (from April to June or from October to December) in a year which probably corresponds to 15–20°C to 20–25°C TRs. These results suggest that the fluctuation of species interactions becomes higher and fluctuates more frequently during the time period with higher fluctuations of ST, and that the fish ecosystem tends to be metastable as the fish community becomes active. For adjacency matrix, the high level of the real part of the eigenvalue appears synchronously with that of TE matrix, while it is hard to identify the seasonality due to lower fluctuations. S12B Fig shows the magnitude of total interactions over time that is computed as the sum of TE values in interaction matrices of temporal networks. Total interaction over time presents clear seasonal fluctuations that highly synchronize with the seasonality observed in eigenvalues of TE matrix shown in S12A Fig (blue line). This finding verifies the result that species interactions in the fish community increase during the time period with high fluctuations of ST. S13 Fig shows how the dominant eigenvalue is the largest for intermediate and high TR as much as the functional network degree. Furthermore, species OTE is calculated for each temporal network, obtaining 256 OTE-s for each species. Pdfs of species OTE show that critical species 6, 7, 8, 9 have higher effects on other species in the fish community compared to others (see S14 Fig).

Considering temperature, S12E Fig shows how the real part of the dominant eigenvalue of TE matrix (blue line) and adjacency matrix (red line) underlying the structure of temperature-dependent networks changes with the increase of temperature, respectively. Both curves show that the real part of the dominant eigenvalue rises with the increasing temperature as a whole, while that of TE matrix has more informative fluctuations with higher frequency in contrast to the adjacency matrix. S12F Fig shows the magnitude of total interactions over temperature stemmed from TE matrices of temperature-dependent networks.

It is clearly observed that total interactions fluctuate analogously to the real part of the dominant eigenvalue of interactions (S12E Fig), as well as to the estimated effective α-diversity derived from dynamical top 20% TE interactions (S12H Fig). These results indicate that interactions between species in the fish community are more sensitive to fluctuations of ST compared to macroecological α-diversity that is a byproduct of interactions. Fluctuations of species interactions provide a potent indicator to monitor and predict ecosystem dynamics especially as a function of changes due to environmental stress. The effective α-diversity in temporal and temperature-dependent dimensions is shown in S12C, S12D, S12G and S12H Fig.