Estimating allometric energy allocation between somatic and gonadic growth

Growth patterns of fish result from their energy allocation strategy, life stages and the environmental conditions in which they reside. Temperature, for example, can potentially drive size-dependent metabolic rates, then influence physiological functions such as food consumption leading to growth variation (Barneche & Allen, 2018; Madenjian et al., 2018). Previous studies suggest that common temperature-related reproductive traits various populations, faster growth, earlier maturation, shorter span smaller body size, are often associated with warmer temperature (Angilletta 2004; Atkinson, 1994; van Rijn 2017); see also nonlinear effects whereby decreased occurs when exceeding optimal (Matthias Variations both somatic gonadic rates hence shape different life-history characteristics amongst individuals, even within same species, underpin diversity marine communities. Changes may alter these prompt shifts abundance, composition geographical distribution a whole (Damuth, 1981; Oindo 2001). Thus, better quantification patterns, consequently, mechanisms, relation those driving factors, will provide new practical insights into conservation management fish. Mathematical models play pivotal role describing extent vary over lifetime, specifying flow determines The model is expressed form either mass or length; power transformation one another. Successful examples include variants von Bertalanffy–Pütter (Pütter, 1920; Bertalanffy, 1934) delineating monophasic curve time-independent parameters. However, offer little theoretical explanation regarding at post-maturation stage (Boukal 2014). Therefore, recent (Mollet 2010; Quince, Abrams, 2008; Shuter, 2008) have advanced diphasic (biphasic) models, accommodating pre- phases, between differs, thus, interrelated changes maturation (Honsey 2016; Lester 2004) be modelled. For chronological summary Wilson al. (2017). Notwithstanding promising feature considerable challenges emerge estimating parameters via numerical methods. takes differential analytically intractable; therein involve change point, estimators correlated each other. These mainly stem two aspects: nature type data available. parameter estimation task becomes particularly challenging it relies on fisheries (fish population data), most collected—which essentially one-time observation age size measurement individual. As opposed longitudinal contain repeated observations individual its do not vital information, consecutive measures time points maturity spawning. Yet, mathematical techniques heavily rely upon this lack information consequently hinders standard algorithms. To cope challenges, much previous research adopts predetermined value some (Mollet, Dieckmann Rijnsdorp, Mollet Mollet, Poos 2008). Doing so, however, introduces implicit assumptions limit interpretation; particular, suggested by entirely depends assumptions. emerging recognition tightly connected inseparable (Wilson 2017) heightens need understand investigate potential modelling approaches constructive manner. Such comprehension would crucial bases developing estimable interpretable advance biological insight underlying mechanism. date, seems received attention practice. Here, present study develops suite allometric scaling generalise framework studied West (2001) (2008). process identifies three typical (Sections 3.2.1–3.2.3, 3.2.1–3.2.3) arising based data. An effective procedure proposed reconcile well enhance interpretation. A simulation undertaken supports accurate estimates. illuminates fact relative reproduction specified time-delay factor model. Using subtropical cutlassfish Trichiurus japonicus collected northern southern coasts Taiwan (Wang Heino, 2018), highlights intraspecific strategies sites conditions. were monthly samplings trawl landing (Tsukuan Kengfang latitudes 22 24°N, hereafter referred T K respectively) 2013 2015; total 254 (site K) 150 T) individuals sampled study. stratified random sampling design was used consistent months, boat range mesh sizes sites; Wang (2017) Heino (2018) more details about -th , () weights measured; here average otolith annuli reported independent readers, recorded resulted 6-month time-scales . weight calculated difference weights, up 6 years age, relatively narrow range, partly due exploitation rapid history species. See detailed description measurements estimation. analysed triplet fish, Strictly speaking, point solely ; precise notation should therefore since only made specimen simplicity, denoted throughout manuscript simpler notation, is, unless stated otherwise. This aspect distinctive records. T. japonicus, benthopelagic distributed coastal waters western North Pacific (Munekiyo, 1990; Tzeng 2016). Most adult spawn neritic shelf water, larvae stay near-shore areas. Their juveniles subadults subsequently inhabit mid-shelf zones (Martins Haimovici, 2000). They restricted dispersal ability, large-scale cross-basin migration adults less likely (Tzeng display sites, slower during juvenile stage, older larger length north-eastern coast compared southwestern T). spatial differences histories considered an adaptive response local level genetic differentiation low (He 2014; 2016), likelihood unknown. (standard deviation) samples are, respectively, g mm site K, Equation (1a) states trivial sum constraint weights—that must equal weight, Equations (1b)–(1c) determine In (1b), first term, expresses rate available unknown constants second represents cost apportioned after maturity, defined (1c). analogous but another set coefficients, constants, together trajectory thus estimated observed later sections. instance, linearly increases if exhibits exponential increase mass-independent allocation, higher values shorten amount double qualitative interpretation similarly applied growth. quantities, purely cannot obtained because no individual, equivalent quantities once parameters—namely —are data, allocated quantified through (2) (3). Since challenging, had assume specific priori exponents Dieckmann, Poos, idea generalises (2001); (2008), (2008); (2010) simple instance dynamic budget theory (Kooijman, 1986), allowing all Unlike (1a)–(1c), single simultaneously estimated. practice, time, treated parameter. regarded (or expected) pattern caught, employed Typical threefold discussed following subsections. First, closed (1b) means fitting requires additional means. Second, spawning generally differ individuals; types missing Third, four five induces correlation lead convergence problems. subsections explore identify detail. (1b′-4) delineates function appears function, does explicit non-integrable involved. Note pre-maturity period, there exists analytical solution (Equation 1b′-4). situation technique, least-squares method (trajectory matching), easily extra alternative approach estimate without solving equation (see Section 4). offers circumvents data-oriented embedding probability algorithm require 4.2). contrast, (1a)–(1c) illuminate separately unique and, course, straightforward context dissected somatic- gonadic-related lengths; 7 further discussion. Standard equations follow approaches. One (Bates Watts, 1988; Seber Wild, 1989), known regression matching; other two-stage method, so-called gradient matching (Ramsay, 1996; Ramsay Hooker, 2017; Varah, 1982). former fits minimising squared errors almost exclusively studies. If unavailable, study, step required. That solve numerically steps iteratively until converge. long undesirable consequences (Ramsay 2007). inconvenient, Sections 3.2.1–3.2.3. latter implemented minimises gradients (derivatives) overcome addressed. consists steps: (a) expected processes smoothing specifically locally weighted loess (Cleveland, 1979; Cleveland Devlin, 1988); models. discuss technical introduced above. discussion (Section 7) how reconciles 3.2. overall results Table 1 summarises results, mean deviation 1,000 replicates bias simply reports empirical bias, estimates value. small inconsequential purposes. case exercise generalised linear (GLMs, McCullagh Nelder, 1989)—Equation (15), model, generated ( ), directly parameters, than Figure illustrates boxplots runs. red dots superposed correspond true generated. All median (the solid lines box) locate near dots), suggesting procedure. facilitate understanding strategy (Kengfang) (Tsukuan), Taiwan. 2 shows % bootstrap confidence intervals constructed bias-correction (Efron Tibshirani, 1994) samples, subset ranges 3 4 years. few 2), meaning extrapolation beyond care. reflected apparent comparing exponent, being ranges; exponent indicates stronger nonlinearity. nonlinearity muted T, owing fewer and/or site-dependent characteristics. Interestingly, similar, particularity accounting broader overlap interval. captures subtle plausible divergence afterwards sites. panels top rows illustrate curves total, observations. (Table 2) initial values: T)—these black green ones dashed segments indicate extrapolation. reference, commonly adopted close bottom reaches early period (2–3 years). It demonstrates sharper water temperature, mature K. 50% approximately year whereas K; details. instantaneous cumulative investments, represent taking invested towards plays effect 1b′-4), slows down. contrasts illustrating wither site. panel) investment stops increasing nearly around 6%–7% 2, right-bottom panel 2). On contrary, keeps lifetime. that, 3, ages. Roff (1983) suggests constant ratio contrast clearly variability age-dependent has developed depicts mechanisms assuming any priori. Although generalisation excellent flexible biologically modelling, faces field highlighted arise addressed unified disentangle issues. (two-stage least squares) missing. attractive appropriate situations non-closed obtained, 3.2.1). (nonlinear least-squares) convoluted unstable intensive computation Furthermore, given large under-represented identifying Honsey (2016) Wootton (2020). Another alleviating assigning fixed (2010), (2016). Unfortunately, conditional significant shortcoming limits interpretability mechanisms. demonstrated naturally accommodated probabilistic manner 8b) while gradient. expression (Equations 8a (1c) times Importantly, multiplicative term time-varying initiates smooth transition monotonic level. deal times, switches, minimisation carried out integral form. parameter, objective minimised non-differentiable respect situation, recommend profile method. Together avoids difficulties. maintaining crucial, interdependency (induced correlation) reflecting forms 7a 7b) ensures error structure underpins bases. Adopting inappropriate introducing unnecessary interpretability. appreciated focuses, done literature, (1b). dealt partially disentangled nonlinearity, expressing account constraint. neither (7a) (7b) shares fitted inseparability complexity successfully resolved issue collection measuring weights. focus length, rather mass. convert measures, cubic used: where product gravity, conversion space representing fraction cube occupy isometric assumed (Beverton Holt, 1954; Matthias Of split segments, separately. Rather, negates multiple explain been widely studied, convenience collection; effortless weight. share root terms complexity. reparameterised (1b′) (4) convenient eases worthy future investigation. addition, using approximated possibility easing Given could proxy remains negligible (1b′-4). According largest species 3). interplay availability leads case-specific tasks. comprehending investigating demonstrated. advances gain under conditions, enabling capture differences. More importantly, dependency indicating phenomena behaviour costs 2019; Pauli 2019). Clearly, interpreting our broaden perspectives raise questions maintenance diverse communities realm. authors declare competing interests. grateful associate editor anonymous reviewers comments suggestions greatly improved clarity paper. H.-Y.W. acknowledges support Ministry Science Technology, (MOST 108-2611-M-002-008). H.S. supported Japan Society Promotion (KAKENHI Grant Number: JP19K21569, JP21H03402). designed data; analysis modelling. equally contributed writing gave final approval publication agreed held accountable work performed therein. Data R code GitHub: https://github.com/hshimadzu/GrowthModelling deposited Shimadzu (2021).