Population growth rates of reef sharks with and without fishing on the great barrier reef: robust estimation with multiple models

Making Predictions Using Poorly Identified Mathematical Models

Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub . Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.

Demographic Analysis of Shortfin Mako Shark (Isurus oxyrinchus) in the South Pacific Ocean

The shortfin mako shark (Isurus oxyrinchus) demonstrates low productivity and is thus relatively sensitive to fishing. Natural mortality (M) and fishing mortality (F) data are critical to determine their population dynamics. However, catch and fishing effort data are unavailable for this species in the South Pacific Ocean, making stock assessments difficult. Demographic quantitative methods aid in analyzing species with limited data availability. We used a two-sex stage-structured matrix population model to examine the demographic stock status of mako sharks. However, data-limited models to determine fishery management strategies have limitations. We performed Monte Carlo simulations to evaluate the effects of uncertainty on the estimated mako shark population growth rate. Under unfished conditions, the simulations demonstrated that the mako sharks showed a higher finite population growth rate in the 2-year reproductive cycle compared to the 3-year reproductive cycle. Protecting immature mako sharks led to a higher population growth rate than protecting mature mako sharks. According to the sex-specific data, protecting immature male and female sharks led to a higher population growth rate than protecting mature male and female sharks. In conclusion, sex-specific management measures can facilitate the sustainable mako shark conservation and management.

Genomic insights into the historical and contemporary demographics of the grey reef shark

Analyses of genetic diversity can shed light on both the origins of biodiversity hotspots, as well as the conservation status of species that are impacted by human activities. With these objectives, we assembled a genomic dataset of 14,935 single nucleotide polymorphisms from 513 grey reef sharks (Carcharhinus amblyrhynchos) sampled across 17 locations in the tropical Indo-Pacific. We analysed geographic variation in genetic diversity, estimated ancient and contemporary effective population size (N_e) across sampling locations (using coalescent and linkage disequilibrium methods) and modelled the history of gene flow between the Coral Triangle and the Coral Sea. Genetic diversity decreased with distance away from the Coral Triangle and north-western Australia, implying that C. amblyrhynchos may have originated in this region. Increases in N_e were detected across almost all sampling locations 40,000-90,000 generations ago (approximately 0.6-1.5 mya, given an estimated generation time of 16.4 years), suggesting a range expansion around this time. More recent, secondary increases in N_e were inferred for the Misool and North Great Barrier Reef sampling locations, but joint modelling did not clarify whether these were due to population growth, migration, or both. Despite the greater genetic diversity and ancient N_e observed at sites around Australia and the Coral Triangle, remote reefs around north-western New Caledonia had the highest contemporary N_e, demonstrating the importance of using multiple population size assessment methods. This study provides insight into both the past and present demographics of C. amblyrhynchos and contributes to our understanding of evolution in marine biodiversity hotspots.

Parameter identifiability and model selection for sigmoid population growth models

Sigmoid growth models, such as the logistic, Gompertz and Richards' models, are widely used to study population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and parameter estimation are critical if these models are to be used to make practical inferences. However, the question of parameter identifiability - whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates - is often overlooked. We use a profile-likelihood approach to explore practical parameter identifiability using data describing the re-growth of hard coral. With this approach, we explore the relationship between parameter identifiability and model misspecification, finding that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards' models encounter practical non-identifiability issues. This analysis of parameter identifiability and model selection is important because different growth models are in biological modelling without necessarily considering whether parameters are identifiable. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and potentially misleading mechanistic interpretations. For example, using the Gompertz model, the estimate of the time scale of coral re-growth is 625 days when we estimate the initial density from the data, whereas it is 1429 days using a more standard approach where variability in the initial density is ignored. While tools developed here focus on three standard sigmoid growth models only, our theoretical developments are applicable to any sigmoid growth model and any appropriate data set. MATLAB implementations of all software are available on GitHub.

Evaluation of Biological Reference Points for Conservation and Management of the Bigeye Thresher Shark, Alopias superciliosus, in the Northwest Pacific

Full stock assessment of sharks is usually hindered by a lack of long time-series catch and effort data. In these circumstances, demographic and per-recruit analyses may provide alternate approaches to describe population status because these methods can be applied to estimate biological reference points (BRPs) for shark stocks. However, the appropriate level of BRPs for sharks is difficult to determine, given the expected low reproductive rates. To determine which BRPs are most appropriate for the CITES-listed species—bigeye thresher shark, Alopias superciliosus, a stochastic demographic model with Monte Carlo simulations and per-recruit models were used to estimate BRPs in this study. The results indicated that conventional fishing mortality-based BRPs (FBRPs) derived from per-recruit models may result in a clear population decline. Our analyses also demonstrated that the bigeye thresher population in the Northwest Pacific will stabilize only if demographic-based FBRP is implemented. The FBRP estimated based on the stochastic demographic model was 0.079–0.139 y−1, which was equivalent to SPR = 50–70%. The findings strongly suggested that more conservative threshold FBRPs should be implemented to ensure sustainable utilization of the bigeye thresher stock. The present study provides new and strategically important information on the population dynamics of the bigeye thresher in the Northwest Pacific, which can be used to help fishery managers to adopt more efficient management measures for this stock. It is also suggested that this approach can be applied to other shark species with limited catch and effort data.

Decline of coastal apex shark populations over the past half century

Overexploitation of large apex marine predators is widespread in the world's oceans, yet the timing and extent of declines are poorly understood. Here we reconstruct a unique fisheries-independent dataset from a shark control programme spanning 1760 km of the Australian coastline over the past 55 years. We report substantial declines (74-92%) of catch per unit effort of hammerhead (Sphyrnidae), whaler (Carcharhinidae), tiger shark (Galeocerdo cuvier) and white sharks (Carcharodon carcharias). Following onset of the program in the 1960s, catch rates in new installations in subsequent decades occurred at a substantially lower rate, indicating regional depletion of shark populations over the past half a century. Concurrent declines in body size and the probability of encountering mature individuals suggests that apex shark populations are more vulnerable to exploitation than previously thought. Ongoing declines and lack of recovery of vulnerable and protected shark species are a cause for concern.

Refining mortality estimates in shark demographic analyses: a Bayesian inverse matrix approach

Leslie matrix models are an important analysis tool in conservation biology that are applied to a diversity of taxa. The standard approach estimates the finite rate of population growth (λ) from a set of vital rates. In some instances, an estimate of λ is available, but the vital rates are poorly understood and can be solved for using an inverse matrix approach. However, these approaches are rarely attempted due to prerequisites of information on the structure of age or stage classes. This study addressed this issue by using a combination of Monte Carlo simulations and the sample-importance-resampling (SIR) algorithm to solve the inverse matrix problem without data on population structure. This approach was applied to the grey reef shark (Carcharhinus amblyrhynchos) from the Great Barrier Reef (GBR) in Australia to determine the demography of this population. Additionally, these outputs were applied to another heavily fished population from Papua New Guinea (PNG) that requires estimates of λ for fisheries management. The SIR analysis determined that natural mortality (M) and total mortality (Z) based on indirect methods have previously been overestimated for C. amblyrhynchos, leading to an underestimated λ. Updated distributions of Z and λ were produced for the GBR population and corrected obvious error in the demographic parameters for the PNG population. This approach provides opportunity for the inverse matrix approach to be applied more broadly to situations where information on population structure is lacking.

Scene through the eyes of an apex predator: a comparative analysis of the shark visual system

The eyes of apex predators, such as the shark, have fascinated comparative visual neuroscientists for hundreds of years with respect to how they perceive the dark depths of their ocean realm or the visual scene in search of prey. As the earliest representatives of the first stage in the evolution of jawed vertebrates, sharks have an important role to play in our understanding of the evolution of the vertebrate eye, including that of humans. This comprehensive review covers the structure and function of all the major ocular components in sharks and how they are adapted to a range of underwater light environments. A comparative approach is used to identify: species-specific diversity in the perception of clear optical images; photoreception for various visual behaviours; the trade-off between image resolution and sensitivity; and visual processing under a range of levels of illumination. The application of this knowledge is also discussed with respect to the conservation of this important group of cartilaginous fishes.

Impact of biology knowledge on the conservation and management of large pelagic sharks

Population growth rate, which depends on several biological parameters, is valuable information for the conservation and management of pelagic sharks, such as blue and shortfin mako sharks. However, reported biological parameters for estimating the population growth rates of these sharks differ by sex and display large variability. To estimate the appropriate population growth rate and clarify relationships between growth rate and relevant biological parameters, we developed a two-sex age-structured matrix population model and estimated the population growth rate using combinations of biological parameters. We addressed elasticity analysis and clarified the population growth rate sensitivity. For the blue shark, the estimated median population growth rate was 0.384 with a range of minimum and maximum values of 0.195–0.533, whereas those values of the shortfin mako shark were 0.102 and 0.007–0.318, respectively. The maturity age of male sharks had the largest impact for blue sharks, whereas that of female sharks had the largest impact for shortfin mako sharks. Hypotheses for the survival process of sharks also had a large impact on the population growth rate estimation. Both shark maturity age and survival rate were based on ageing validation data, indicating the importance of validating the quality of these data for the conservation and management of large pelagic sharks.

Growth and life history variability of the grey reef shark (Carcharhinus amblyrhynchos) across its range

For broadly distributed, often overexploited species such as elasmobranchs (sharks and rays), conservation management would benefit from understanding how life history traits change in response to local environmental and ecological factors. However, fishing obfuscates this objective by causing complex and often mixed effects on the life histories of target species. Disentangling the many drivers of life history variability requires knowledge of elasmobranch populations in the absence of fishing, which is rarely available. Here, we describe the growth, maximum size, sex ratios, size at maturity, and offer a direct estimate of survival of an unfished population of grey reef sharks (Carcharhinus amblyrhynchos) using data from an eight year tag-recapture study. We then synthesized published information on the life history of C. amblyrhynchos from across its geographic range, and for the first time, we attempted to disentangle the contribution of fishing from geographic variation in an elasmobranch species. For Palmyra's unfished C. amblyrhynchos population, the von Bertalanffy growth function (VBGF) growth coefficient k was 0.05 and asymptotic length L∞ was 163.3 cm total length (TL). Maximum size was 175.5 cm TL from a female shark, length at maturity was estimated at 116.7-123.2 cm TL for male sharks, maximum lifespan estimated from VBGF parameters was 18.1 years for both sexes combined, and annual survival was 0.74 year-1. Consistent with findings from studies on other elasmobranch species, we found significant intraspecific variability in reported life history traits of C. amblyrhynchos. However, contrary to what others have reported, we did not find consistent patterns in life history variability as a function of biogeography or fishing. Ultimately, the substantial, but not yet predictable variability in life history traits observed for C. amblyrhynchos across its geographic range suggests that regional management may be necessary to set sustainable harvest targets and to recover this and other shark species globally.

Reef sharks: recent advances in ecological understanding to inform conservation

Sharks are increasingly being recognized as important members of coral-reef communities, but their overall conservation status remains uncertain. Nine of the 29 reef-shark species are designated as data deficient in the IUCN Red List, and three-fourths of reef sharks had unknown population trends at the time of their assessment. Fortunately, reef-shark research is on the rise. This new body of research demonstrates reef sharks' high site restriction, fidelity and residency on coral reefs, their broad trophic roles connecting reef communities and their high population genetic structure, all information that should be useful for their management and conservation. Importantly, recent studies on the abundance and population trends of the three classic carcharhinid reef sharks (grey reef shark Carcharhinus amblyrhynchos, blacktip reef shark Carcharhinus melanopterus and whitetip reef shark Triaenodon obesus) may contribute to reassessments identifying them as more vulnerable than currently realized. Because over half of the research effort has focused on only these three reef sharks and the nurse shark Ginglymostoma cirratum in only a few locales, there remain large taxonomic and geographic gaps in reef-shark knowledge. As such, a large portion of reef-shark biodiversity remains uncharacterized despite needs for targeted research identified in their red list assessments. A research agenda for the future should integrate abundance, life history, trophic ecology, genetics, habitat use and movement studies, and expand the breadth of such research to understudied species and localities, in order to better understand the conservation requirements of these species and to motivate effective conservation solutions.

Connectivity in grey reef sharks (Carcharhinus amblyrhynchos) determined using empirical and simulated genetic data

Grey reef sharks (Carcharhinus amblyrhynchos) can be one of the numerically dominant high order predators on pristine coral reefs, yet their numbers have declined even in the highly regulated Australian Great Barrier Reef (GBR) Marine Park. Knowledge of both large scale and fine scale genetic connectivity of grey reef sharks is essential for their effective management, but no genetic data are yet available. We investigated grey reef shark genetic structure in the GBR across a 1200 km latitudinal gradient, comparing empirical data with models simulating different levels of migration. The empirical data did not reveal any genetic structuring along the entire latitudinal gradient sampled, suggesting regular widespread dispersal and gene flow of the species throughout most of the GBR. Our simulated datasets indicate that even with substantial migrations (up to 25% of individuals migrating between neighboring reefs) both large scale genetic structure and genotypic spatial autocorrelation at the reef scale were maintained. We suggest that present migration rates therefore exceed this level. These findings have important implications regarding the effectiveness of networks of spatially discontinuous Marine Protected Areas to protect reef sharks.

Odor tracking in sharks is reduced under future ocean acidification conditions

Recent studies show that ocean acidification impairs sensory functions and alters the behavior of teleost fishes. If sharks and other elasmobranchs are similarly affected, this could have significant consequences for marine ecosystems globally. Here, we show that projected future CO2 levels impair odor tracking behavior of the smooth dogfish (Mustelus canis). Adult M. canis were held for 5 days in a current-day control (405 ± 26 μatm) and mid (741 ± 22 μatm) or high CO2 (1064 ± 17 μatm) treatments consistent with the projections for the year 2100 on a 'business as usual' scenario. Both control and mid CO2 -treated individuals maintained normal odor tracking behavior, whereas high CO2 -treated sharks significantly avoided the odor cues indicative of food. Control sharks spent >60% of their time in the water stream containing the food stimulus, but this value fell below 15% in high CO2 -treated sharks. In addition, sharks treated under mid and high CO2 conditions reduced attack behavior compared to the control individuals. Our findings show that shark feeding could be affected by changes in seawater chemistry projected for the end of this century. Understanding the effects of ocean acidification on critical behaviors, such as prey tracking in large predators, can help determine the potential impacts of future ocean acidification on ecosystem function.

Determining decision thresholds and evaluating indicators when conservation status is measured as a continuum

Categorization of the status of populations, species, and ecosystems underpins most conservation activities. Status is often based on how a system's current indicator value (e.g., change in abundance) relates to some threshold of conservation concern. Receiver operating characteristic (ROC) curves can be used to quantify the statistical reliability of indicators of conservation status and evaluate trade-offs between correct (true positive) and incorrect (false positive) classifications across a range of decision thresholds. However, ROC curves assume a discrete, binary relationship between an indicator and the conservation status it is meant to track, which is a simplification of the more realistic continuum of conservation status, and may limit the applicability of ROC curves in conservation science. We describe a modified ROC curve that treats conservation status as a continuum rather than a discrete state. We explored the influence of this continuum and typical sources of variation in abundance that can lead to classification errors (i.e., random variation and measurement error) on the true and false positive rates corresponding to varying decision thresholds and the reliability of change in abundance as an indicator of conservation status, respectively. We applied our modified ROC approach to an indicator of endangerment in Pacific salmon (Oncorhynchus nerka) (i.e., percent decline in geometric mean abundance) and an indicator of marine ecosystem structure and function (i.e., detritivore biomass). Failure to treat conservation status as a continuum when choosing thresholds for indicators resulted in the misidentification of trade-offs between true and false positive rates and the overestimation of an indicator's reliability. We argue for treating conservation status as a continuum when ROC curves are used to evaluate decision thresholds in indicators for the assessment of conservation status.

Quantifying shark distribution patterns and species-habitat associations: implications of marine park zoning

Quantifying shark distribution patterns and species-specific habitat associations in response to geographic and environmental drivers is critical to assessing risk of exposure to fishing, habitat degradation, and the effects of climate change. The present study examined shark distribution patterns, species-habitat associations, and marine reserve use with baited remote underwater video stations (BRUVS) along the entire Great Barrier Reef Marine Park (GBRMP) over a ten year period. Overall, 21 species of sharks from five families and two orders were recorded. Grey reef Carcharhinus amblyrhynchos, silvertip C. albimarginatus, tiger Galeocerdo cuvier, and sliteye Loxodon macrorhinus sharks were the most abundant species (>64% of shark abundances). Multivariate regression trees showed that hard coral cover produced the primary split separating shark assemblages. Four indicator species had consistently higher abundances and contributed to explaining most of the differences in shark assemblages: C. amblyrhynchos, C. albimarginatus, G. cuvier, and whitetip reef Triaenodon obesus sharks. Relative distance along the GBRMP had the greatest influence on shark occurrence and species richness, which increased at both ends of the sampling range (southern and northern sites) relative to intermediate latitudes. Hard coral cover and distance across the shelf were also important predictors of shark distribution. The relative abundance of sharks was significantly higher in non-fished sites, highlighting the conservation value and benefits of the GBRMP zoning. However, our results also showed that hard coral cover had a large effect on the abundance of reef-associated shark species, indicating that coral reef health may be important for the success of marine protected areas. Therefore, understanding shark distribution patterns, species-habitat associations, and the drivers responsible for those patterns is essential for developing sound management and conservation approaches.

Life-history correlates of extinction risk and recovery potential

Extinction risk is inversely associated with maximum per capita population growth rate (r(max)). However, this parameter is not known for most threatened species, underscoring the value in identifying correlates of r(max) that, in the absence of demographic data, would indirectly allow one to identify species and populations at elevated risk of extinction and their associated recovery potential. We undertook a comparative life-history analysis of 199 species from three taxonomic classes: Chondrichthyes (e.g., sharks; n = 82), Actinopterygii (teleost or bony fishes; n = 47), and Mammalia (n = 70, including 16 marine species). Median r(max) was highest for (and similar between) terrestrial mammals (0.71) and teleosts (0.43), significantly lower among chondrichthyans (0.26), and lower still in marine mammals (0.07). Age at maturity was the primary (and negative) correlate of r(max). In contrast, although body size was negatively correlated with r(max) in chondrichthyans and mammals, evidence of an association in teleosts was equivocal, and fecundity was not related to r(max) in fishes, despite recurring assertions to the contrary. Our analyses suggest that age at maturity can serve as a universal predictor of extinction risk in fishes and mammals when r(max) itself is unknown. Moreover, in contrast to what is generally expected, the recovery potential of teleost fishes does not differ from that of terrestrial mammals. Our findings are supportive of the application of extinction-risk criteria that are based on generation time and that are independent of taxonomic affinity.