Philippa Kaur

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Historically, species distributions have been modeled independently from each other due to unavailability of multispecies datasets and computational restraints. However, ecological datasets that provide insights about collections of organisms have become prevalent over the last decade thanks to efforts like Long Term Ecological Research Network (LTER), National Ecological Observatory Network (NEON), and citizen science surveys1. In addition, technology has improved our ability to fit modern statistical models to these datasets that account for both species environmental preferences and interspecies dependence. These advancements have allowed for the development of joint species distribution models (JSDM)2,3,4 that can model dependence among species simultaneously with environmental drivers of occurrence and/or abundance.Species distributions are shaped by both interspecies dynamics and environmental preferences5,6,7,8. JSDMs integrate both sources of variability and adjust uncertainty to reflect that multiple confounded factors can contribute to similar patterns in species distributions. Some have proposed that JSDMs not only account for biotic interactions but also correct estimates of association between species distributions and environmental drivers3,9, while others claim JSDMs cannot disentangle the roles of interspecies dependence and environmental drivers5. We address why JSDMs can provide inference distinct from their concomitant independent SDMs, how certain parameterizations of a JSDM induce confounding between the environmental and random species effects, and when deconfounding these effects may be appealing for computation and interpretation.Because of the prevalence of occupancy data for biomonitoring in ecology, we focus our discussion of community confounding in JSDMs on occupancy models, although we also consider a JSDM for species density data in the simulation study. The individual species occupancy model was first formulated by MacKenzie et al.10 and has several joint species extensions4,11,12,13,14,15,16. We chose to investigate the impacts of community confounding on the probit model since it has been widely used in the analysis of occupancy data4,13,17. We also developed a joint species extension to the Royle-Nichols model18 and consider community confounding in that model.We use the probit and Royle-Nichols occupancy models to improve our understanding of montaine mammal communities in what follows. We show that including unstructured random species effects in either occupancy model induces confounding between the fixed environmental and random species effects. We demonstrate how to orthogonalize these effects in the model and compare the resulting inference compared to models where species are treated independently.Unlike previous approaches that have applied restricted regression techniques similar to ours, we use it in the context of well-known ecological models for species occupancy and intensity. While such approaches have been discussed in spatial statistics and environmental science, they have not been adopted in settings involving the multivariate analysis of community data. We draw parallels between restricted spatial regression and restricted JSDMs but also highlight where the methods differ in goals and outcomes. We find that the computational benefits conferred by performing restricted spatial regression also hold for some joint species distribution models.Royle-Nichols joint species distribution modelWe present a JSDM extension to the Royle-Nichols model18. The Royle-Nichols model accounts for heterogeneity in detection induced by the species’ latent intensity, a surrogate related to true species abundance. Abundance, density, and occupancy estimation often requires an explicit spatial region that is closed to emmigration and immigration. In our model, the unobservable intensity variable helps us explain heterogeneity in the frequencies we observe a species at different sites without making assumptions about population closure. In the “Model” section, we further discuss the distinctions between abundance and intensity in the Royle-Nichols model.The Royle-Nichols model utilizes occupancy survey data but provides inference distinct from the basic occupancy model10. In the Royle-Nichols model, we estimate individual detection probability for homogeneous members of the population, whereas in an occupancy model, we estimate probability of observing at least one member of the population given that the site is occupied. Furthermore, the Royle-Nichols model allows us to relate environmental covariates to the latent intensity associated with a species at a site, while in an occupancy model, environmental covariates are associated with the species latent probability of occupancy at a site. Species intensity and occupancy may be governed by different mechanisms, and inference from an intensity model can be distinct from that provided by an occupancy model19,20,21. Cingolani et al.20 proposed that, in plant communities, certain environmental filters preclude species from occupying a site and an additional set of filters may regulate if a species can flourish. Hence, certain covariates that were unimportant in an occupancy model may improve predictive power in an intensity model.Community confoundingSpecies distributions are shaped by environment as well as competition and mutualism within the community8,22,23. Community confounding occurs when species distributions are explained by a convolution of environmental and interspecies effects and can lead to inferential differences between a joint and single species distribution model as well as create difficulties for fitting JSDMs. Former studies have incorporated interspecies dependence into an occupancy model4,11,12,13,14,15,16, and others have addressed spatial confounding1,17,24,25, but none of these explicitly addressed community confounding. However, all Bayesian joint occupancy models naturally attenuate the effects of community confounding due to the prior on the regression coefficients. The prior, assuming it is proper, induces regularization on the regression coefficients26 that can lessen the inferential and computational impacts of confounding27. Furthermore, latent factor models like that described by Tobler et al.4 restrict the dimensionality of the random species effect which should also reduce confounding with the environmental effects.We address community confounding by formulating a version of our model that orthogonalizes the environmental effects and random species effects. Orthogonalizing the fixed and random effects is common practice in spatial statistics and often referred to as restricted spatial regression27,28,29,30,31. Restricted regression has been applied to spatial generalized linear mixed models (SGLMM) for observations (varvec{y},) which can be expressed as$$begin{aligned} varvec{y}&sim [varvec{y}|varvec{mu }, varvec{psi }], end{aligned}$$
(1)
$$begin{aligned} g(varvec{mu })&= varvec{X}varvec{beta } + varvec{eta }, end{aligned}$$
(2)
$$begin{aligned} varvec{eta }&sim mathcal {N}(varvec{0}, varvec{Sigma }), end{aligned}$$
(3)
where (g(cdot )) is a link function, (varvec{psi }) are additional parameters for the data model, and (varvec{Sigma }) is the covariance matrix of the spatial random effect. In the SGLMM, prior information facilitates the estimation of (varvec{eta },) which would not be estimable otherwise due to its shared column space with (varvec{beta })30. This is analogous to applying a ridge penalty to (varvec{eta },) which stabilizes the likelihood. Another method for fitting the confounded SGLMM is to specify a restricted version:$$begin{aligned} varvec{y}&sim [varvec{y}|varvec{mu }, varvec{psi }], end{aligned}$$
(4)
$$begin{aligned} g(varvec{mu })&= varvec{X}varvec{delta } + (varvec{I}-varvec{P}_{varvec{X}})varvec{eta }, end{aligned}$$
(5)
$$begin{aligned} varvec{eta }&sim mathcal {N}(varvec{0}, varvec{Sigma }), end{aligned}$$
(6)
where (varvec{P}_{varvec{X}}=varvec{X}(varvec{X}varvec{X})^{-1}varvec{X}’) is the projection matrix onto the column space of (varvec{X}.) In the unrestricted SGLMM, the regression coefficients (varvec{beta }) and random effect (varvec{eta }) in (1) compete to explain variability in the latent mean (varvec{mu }) in the direction of (varvec{X})27. In the restricted model, however, all variability in the direction of (varvec{X}) is explained solely by the regression coefficients (varvec{delta }) in (4)31, and (varvec{eta }) explains residual variation that is orthogonal to (varvec{X}). We refer to (varvec{beta }) as the conditional effects because they depend on (varvec{eta }), and (varvec{delta }) as the unconditional effects.Restricted regression, as specified in (4), was proposed by Reich et al.28. Reich et al.28 described a disease-mapping example in which the inclusion of a spatial random effect rendered one covariate effect unimportant that was important in the non-spatial model. Spatial maps indicated an association between the covariate and response, making inference from the spatial model appear untenable. Reich et al.28 proposed restricted spatial regression as a method for recovering the posterior expectations of the non-spatial model and shrinking the posterior variances which tend to be inflated for the unrestricted SGLMM.Several modifications of restricted spatial regression have been proposed30,32,33,34,35. All restricted spatial regression methods seek to provide posterior means (text {E}left( delta _j|varvec{y}right)) and marginal posterior variances (text {Var}left( delta _j|varvec{y}right)), (j=1,…,p) that satisfy the following two conditions36:

1.

(text {E}left( varvec{delta }|varvec{y}right) = text {E}left( varvec{beta }_{text {NS}}|varvec{y}right)) and,

2.

(text {Var}left( beta _{text {NS,}j}|varvec{y}right) le text {Var}left( delta _{j}|varvec{y}right) le text {Var}left( beta _{text {Spatial,}j}|varvec{y}right)) for (j=1,…,p),

where (varvec{beta }_{NS}) and (varvec{beta }_{Spatial}) are the regression coefficients corresponding to the non-spatial and unrestricted spatial models, respectively.The inferential impacts of spatial confounding on the regression coefficients has been debated. Hodges and Reich29 outlined five viewpoints on spatial confounding and restricted regression in the literature and refuted the two following views:

1.

Adding the random effect (varvec{eta }) corrects for bias in (varvec{beta }) resulting from missing covariates.

2.

Estimates of (varvec{beta }) in a SGLMM are shrunk by the random effect and hence conservative.

The random effect (varvec{eta }) can increase or decrease the magnitude of (varvec{beta }), and the change may be galvanized by mechanisms not related to missing covariates. Therefore, we cannot assume the regression coefficients in the SGLMM will exceed those of the restricted model, nor should we regard the estimates in either model as biased due to misspecification. Confounding in the SGLMM causes (text {Var}left( beta _j|varvec{y}right) ge text {Var}left( delta _j|varvec{y}right)), (j=1,…,p), because of the shared column space of the fixed and random effects. Thus, we refer to the conditional coefficients as conservative with regard to their credible intervals, not their posterior expectations.Reich et al.28 argued that restricted spatial regression should always be applied because the spatial random effect is generally added to improve predictions and/or correct the fixed effect variance estimate. While it may be inappropriate to orthogonalize a set of fixed effects in an ordinary linear model, orthogonalizing the fixed and random effect in a spatial model is permissible because the random effect is generally not of inferential interest. Paciorek37 provided the alternative perspective that, if confounding exists, it is inappropriate to attribute all contested variability in (varvec{y}) to the fixed effects. Hanks et al.31 discussed factors for deciding between the unrestricted and restricted SGLMM on a continuous spatial support. The restricted SGLMM leads to improved computational stability, but the unconditional effects are less conservative under model misspecification and more prone to type-S errors: The Bayesian analogue of Type I error. Fitting the unrestricted SGLMM when the fixed and random effects are truly orthogonal does not introduce bias, but it will increase the fixed effect variance. Given these considerations, Hanks et al.31 suggested a hybrid approach where the conditional effects, (varvec{beta }), are extracted from the restricted SGLMM. This is possible because the restricted SGLMM is a reparameterization of the unrestricted SGLMM. This hybrid approach leads to improved computational stability but yields the more conservative parameter estimates. We describe how to implement this hybrid approach for joint species distribution models in the “Community confounding” section.Restricted regression has also been applied in time series applications. Dominici et al.38 debiased estimates of fixed effects confounded by time using restricted smoothing splines. Without the temporal random effect, Dominici et al.38 asserted all temporal variation in the response would be wrongly attributed to temporally correlated fixed effects. Houseman et al.39 used restricted regression to ensure identifiability of a nonparametric temporal effect and highlighted certain covariate effects that were more evident in the restricted model (i.e., the unconditional effects’ magnitude was greater). Furthermore, restricted regression is implicit in restricted maximum likelihood estimation (REML). REML is often employed for debiasing the estimate of the variance of (varvec{y}) in linear regression and fitting linear mixed models that are not estimable in their unrestricted format40. Because REML is generally applied in the context of variance and covariance estimation, considerations regarding the effects of REML on inference for the fixed effects are lacking in the literature.In ecological science, JSDMs often include an unstructured random effect like (varvec{eta }) in (1) to account for interspecies dependence, and hence can also experience community confounding between (varvec{X}) and (varvec{eta }) analogous to spatial confounding. Unlike a spatial or temporal random effect, we consider random species effects to be inferentially important, rather than a tool solely for improving predictions or catch-all for missing covariates. An orthogonalization approach in a JSDM attributes contested variation between the fixed effects (environmental information) and random effect (community information) to the fixed effect.We describe how to orthogonalize the fixed and random species effects in a suite of JSDMs and present a method for detecting community confounding. In the simulation study, we test the efficacy of our method for detecting confounding, show that community confounding can lead to computational difficulties similar to those caused by spatial confounding31, and highlight that, for some models, restricted regression can improve model fitting. We also investigate the inferential implications of community confouding and restricted regression in JSDMs by comparing outputs from the SDM, unrestricted JSDM, and restricted JSDM of the Royle-Nichols and probit occupancy models fit to mammalian camera trap data. Lastly, we discuss other inferential and computational methods for confounded models and consider their appropriateness for joint species distribution modeling. More

Abiotic parametersThe temperature regime experienced by the embryos was purposefully natural and therefore varied between the three air exposure treatments. The subtidal treatment, where embryos were continuously submerged in water, remained around 9.5 °C for the duration of the experiment, while the intertidal treatments experienced dips in temperature during outside air exposure down to 2.5 °C and 0.8 °C, for the low and high intertidal respectively (Fig. 3A). Accumulated thermal units (ATU; days × temperature post collection until hatch) for each air exposure treatment were 79.6, 75.4 and 65.3 for subtidal, low intertidal and high intertidal, respectively. Despite differences in thermal regime, peak hatch was on the same day (March 14, 2021) for all air exposure and CO2 treatments, estimated at 11 dpf.Figure 3Temperature and pH experienced by the herring embryos and larvae throughout the experiment. Hourly measurements of (A) air/water temperature experienced by herring embryos in each of the tidal treatments (subtidal: continuous immersion in 9.5 °C water; low intertidal: 2 × 2 h air exposure; high intertidal: 5 + 9 h air exposure) and (B) pH levels in the tanks for each of the CO2 treatments (greens = control, 400 µatm CO2, yellows = medium, 1600 µatm CO2, reds = high, 3000 µatm CO2); dots are pH levels measured in the individual jars during larval incubation.Full size imageThe pH levels in the tanks were measured hourly and were stable over the course of the embryonic incubation period, with no overlap between treatments, although there was some overlap between individual jars. Control treatment was consistently around a pH of 8, the medium treatment had a pH of 7.4 and the high CO2 treatment had a mean pH of 7.1 (Table 1). After hatch, when the larvae were transferred to the jars, circulation and gas exchange between jars and tank were not as high and CO2 accumulated in the jars over time, leading to pH levels deviating from tank pH levels (Fig. 3B). Although oxygen levels remained high (7–9 mg/L), the pH dropped from a mean 8–7.6 in the control on two occasions, and was brought back up with a partial water exchange from the incubation tank water. The pH in the medium and high CO2 treatments were not as affected (Fig. 3B), however, final water chemistry measurements after completion of the experiment (2 days post water exchange) revealed much higher CO2 levels in all treatments (Table 1: day 15).Table 1 Mean water parameters for each treatment (mean of 3 tanks ± S.D.) at the beginning (day 1, 2021-03-06) and end (day 6, 2021-03-12) of embryonic incubation and mean parameters in the jars (N = 9) at the end of larval incubation (day 15, 2021-03-19); Temperature (T), salinity, pCO2, total CO2 (TCO2) measured at distinct sampling intervals with the BoL; total alkalinity (TA) and pH (on the total scale) calculated with CO2SYS.Full size tableEffect of air exposure and CO2 treatment during embryonic developmentNeither embryonic survival nor growth were significantly affected by treatment in our experiment. Percent daily embryonic mortality was low and not significantly affected by CO2 treatment or air exposure (CO2: p = 0.088, F2 = 2.45; Tide: p = 0.11, F2 = 2.19; CO2*Tide: p = 0.18, F2 = 1.59) . Egg diameter at 6 dpf was also not significantly affected by treatment (CO2: p = 0.38, X2 (2, N = 30) = 1.92; Tide: p = 0.83, X2 (2, N = 30) = 0.33; CO2*Tide: p = 0.08, X2 (2, N = 30) = 8.25). Metabolic rate, as indicated by embryonic heart rate, was significantly affected by air exposure at 6 dpf (p  *; 0.1  >).Full size image More

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This section is devoted to show results and to highlight eventual effects of the interplay between the nonlinearity characterizing the system dynamics and the presence of noisy fluctuations for the irradiance variable.Analysis of experimental dataThe need of taking into account noisy fluctuations of such an environmental variable is well demonstrated in Fig. 1. In the first panel (a) the experimental time behaviour of the irradiance is shown. This noisy curve is based on the experimental data (purple points) of the Boussole buoy located in the Gulf of Lion, collected over a period of nine years, precisely from 2004 to 2013. The time series of the experimental data presents quite a few gaps in time due to the malfunction of the buoy. This aspect has been remedied by merging the experimental data with those of the OASIM model validated for the Boussole site61 (yellow points). The latter is a multispectral atmospheric radiative transfer model that is in turn forced by experimental-model data based on ECMWF ERAINTERIM reanalyses which provide, for example, cloud cover data. The radiative model is partly stochastic since it considers the effects stemming from the presence of clouds, averaged along a single day (this explains why the yellow points are slightly less scattered). We see that the OASIM model accurately reproduces the profile which emerges from the experimental data. Further, we stress that the experimental data are only used in this initial analysis. In the biogeochemical simulations the irradiance signal is fully reconstructed starting from a realistic seasonal cycle combined with a range of different random fluctuations, and the information from OASIM is not used. In the second panel (b) the daily (black points) as well as the three-month (red points) running mean of the experimental series are plotted. Figure 1c shows the irradiance noisy fluctuations (INF) which have been obtained by subtracting the three-month running mean curve (3MRM, red curve in Fig. 1b) from the daily running mean one (DRM, black curve in Fig. 1b) and normalizing with respect to the mean of the 3MRM ((overline{3MRM})), namely (INF = (DRM – 3MRM) / overline{3MRM}). We see that a seasonal overall trend with higher oscillations during the winter time can be seen, implying that the characteristics of the noise may change over the year. Moreover, a slight imbalance between positive and negative values of the noisy fluctuations (that is, different values of the maximum fluctuation intensity) is present. The physical reason for the occurrence of such an aspect can be ascribed to the fact that the maximum value of solar irradiance corresponds to that measured during a sunny day. Conversely, the minimum level tends to zero corresponding to a dense darkness. While the former is close to the mean value of the solar irradiance (most of all in summer), the latter is much further away and then a natural asymmetry arises in the random fluctuations. However, it should be noted that, apart from the intense spikes, the asymmetry is not so pronounced, as proved by the mean value (red line in Fig. 1c) which is practically zero, namely (0.4%) of the (overline{3MRM}). Therefore, basing on this last observation, to model the noise affecting the irradiance dynamics, as a first approximation we consider a symmetric Gaussian autocorrelated noise as described in the next subsection.On the basis of such experimental results, we postulate the hypothesis that random fluctuations of light cannot be neglected, most of all in the study of ecological systems where light profoundly determines the system dynamics, governing fundamental processes at the basis of of the food web.Figure 1(a) Experimental data (purple points) of the stochastic solar irradiance collected by the Boussole buoy in a time-window of 9 years (2004-2013); the yellow points are the data generated by the OASIM model used to fill the gaps present in the experimental time-series due to malfunctioning of the buoy. (b) Daily (black points) and three-month (red points) running mean of the light curve in panel (a). (c) Irradiance noisy fluctuations (INF), obtained by subtracting the three-month running mean curve (3MRM) from the daily running mean one (DRM) and normalizing with respect to the mean value of 3MRM ((overline{3MRM})), namely (INF = (DRM – 3MRM) / overline{3MRM}); the red line represents the mean value of such fluctuations. Data already presented and validated in61.Full size imageSolar irradianceThe solar irradiance forcing is derived considering a deterministic seasonal oscillation combined with an Ornstein-Uhlenbeck process. The coefficient of variation (CV) of simulated light forcing, Fig. 2, (CV=sigma / mu) ((mu) and (sigma) being mean value and standard deviation calculated over both time and numerical realizations), is shown for 231 (D-tau) pairs. D and (tau) represent the intensity of a Gaussian noise source and the auto-correlation time of the fluctuations, respectively (see Eqs. (2) and (3)).Each pixel represents the mean value on time of CV calculated with respect to 1000 different stochastic realizations. Figure 2Coefficient of variation ((CV=sigma / mu)) of irradiance resulting from numerical integration of model equations for 231 (D-tau) different scenarios.Full size imageIt is easy to see the agreement between the results obtained from the numerical integration and the theoretical ones derivable from Eq. (5) by putting (text {var}{F_L(0)}=0) and (t gg 1), getting (sigma ^2_L=D / 2tau). In Fig. 2, indeed, the maximum values of (sigma) lie in the upper left part of the plot corresponding to small (high) values of (tau) (D). As it is clear the values of D have been chosen in order to obtain a relative standard deviation ranging from (5%mu) to (60%mu). We underline that, in this case, it is possible to interchangeably consider (sigma) and CV since the dependence of CV on D and (tau) does not differ from that of (sigma) (meaning that the dependence of (sigma) is not altered by dividing by (mu)) (results not shown).Effects on population dynamicsIn this section the noise-induced effects on the population dynamics are examined. The nine planktonic populations present a different qualitative behaviour of the CV, compared to that of the irradiance. In this case, the CV is characterized by a strong non-monotonic dependence on the parameter (tau). This aspect can be appreciated in Fig. 3 where different curves of CV versus the time correlation parameter are shown for different fixed values of D.Figure 3Coefficient of variation ((CV=sigma / mu)) of the nine planktonic populations resulting from numerical integration of model equations plotted versus the considered values of (tau); the different curves are related to different values of the noise intensity D.Full size imageThe existence of a maximum value for CV can be appreciated for each species. Although the qualitative behaviour is the same for all strains, particular attention has to be payed on diatoms and nanoflagellates. All the other species, indeed, present a percent variation of standard deviation between (2%) and (15%). In the case of nanoflagellates, instead, the D-dependent range is (20-90%), while diatoms reach values over the (100%) for the highest values of D. Therefore, these two species, in particular, and the whole system, in general, are extremely sensitive to the auto-correlation time which characterizes the noise.We note that the different curves related to the different selected values of D approach the horizontal axis, tending asymptotically to vanish as (tau) increases. Such a behaviour can be explained by the fact that high values of (tau) give rise to a more correlated dynamics, so that (tau rightarrow infty) implies fully correlated time-behaviours corresponding to the deterministic case. In this instance, then, all the different realizations give the same results, making the standard deviation vanish. The same happens, independently of the value of (tau), for low values of noise intensity for which the corresponding curves approach the same almost vanishing value (see orange, gray and yellow lines). Differently from the previous case, when (tau rightarrow 0) the noise tends to a delta-correlated noise, that is a white noise; for (tau ne 0), instead, the noise spectrum is not flat, being characterized by a Cauchy-Lorentz distribution. The strong nonmonotonicity of CV with respect to (tau), emerging when there are relatively high values of CV, implies a greater variability of the system biomass. Lower values of CV indicate that the system dynamics is less influenced by the presence of noise where very little or no differences with respect to the deterministic case are present. Conversely, high values of CV clearly demonstrate the remarkable signature of the presence of an impacting noise source. It is interesting to note that the noise influence on the ecosystem strongly depends on both (tau) and D, that is, just an intense noise is not enough to generate a greater response of the ecosystem. In particular, experimental data are characterized by a CV approximately equal to 0.361, which corresponds to values of D and (tau) lying on the diagonal strip in Fig. 2 ranging from ((tau ,D)=(0.5,10^4)) to ((tau ,D)=(365,10^7)). Finally we note the presence of a noise suppression effect. High values of D, indeed, can generate slight effects when the correlation time (tau) does not take on suitable values.The results shown here are an extension of the previous work by Benincà et al.56. There, the authors analyse a simpler, less realistic model of two interacting populations, whose dynamics is affected by a randomly fluctuating temperature. In that case, moreover, the deterministic oscillations of the temperature are suppressed, and the system exhibits intrinsic Lotka-Volterra oscillations whose frequency match with the characteristic one(s) of the noise. On the contrary, here, the observed maximum response (see Fig. 3) cannot be interpreted as a synchronization effect, since our model does not present intrinsic Lotka-Volterra-like oscillations and the periodic population variability is only due to the deterministic forcing(s).The nonmonotonic behaviour of the CV can be then interpreted as the signature of the intimate interplay between the ecological system and the noise. This interplay, indeed, has a pivotal role in both determining the dynamics of the populations and defining the characteristics of the ecosystem.In Fig. 3 it can be observed that the value of (tau) for which CV is maximum strongly depends on the noise intensity D. In particular, it is possible to note that the peaks in Fig. 3 move towards higher values of (tau) as the noise intensity increases. Thus, Fig. 3 demonstrates that the maximum-response effect to the random fluctuations is sensitive to the noise intensity D.However, it is important to underline that the response of the system to the noisy signal does not depend on the yearly oscillations induced by the deterministic forcings. Indeed, by considering constant the deterministic part of all external forcings (temperature, irradiance, wind and salinity), the non monotonic behaviour of CV with respect to both (tau) and D is still present, provided that the populations are not extinct (plot not shown). In this scenario indeed, besides dinoflagellates, diatoms and nanoflagellates are practically extinct as well, exhibiting thus a constant vanishing variance. All the other strains, instead, present qualitatively the same nonmonotonicity with only slight differences (shift of the peaks and different mean values of the CV curves), probably due to the extinction of diatoms and nanoflagellates which causes relevant differences in the system dynamics. More specifically, the system’s response seems to depend on both the noise intensity and the correlation time (see Fig. 3).In this scenario (absence of seasonal driving) we have studied the dependence on both parameters D and (tau) of the probability density functions (PDFs) of the non-vanishing populations. In Fig. 4, the PDFs of bacteria (B1), picophytoplankton (P3), microzooplankton (Z5) and etherotrophic nanoflagellates (Z6) are plotted for (tau =0.5) and eight different values of the parameter D.Figure 4Dependence of the probability density functions of non-vanishing populations on the parameter D for (tau =0.5). The curves are normalized within the interval taken into account. For this reason the relative peaks of the curves in the bottom panels have different values compared to those of the top panels. However, the figure aims at showing the existence of the value of the noise intensity for which the system is more sensitive as well as the generation of a stationary out-of-equilibrium state induced by the noise.Full size imageWe see that the mean value and the variance of these populations are strongly affected by the presence of random fluctuations in the irradiance. Specifically, as the noise intensity increases the mean values of picophytoplankton and bacteria concentrations exhibit a shift. In particular, the results indicate that picophytoplankton is disavantaged by the presence of a noisy component in the irradiance, which indeed tends to inhibit its ability to absorbe the solar light, slowing down its growth. As a consequence, since phytoplankton and bacteria compete for the same resources, as the former declines the latter are favoured, with a compensation mechanism which allows their predators (zooplankton populations) to be almost not affected by the noisy behaviour of the irradiance. Further, we note that for intermediate values of the noise intensity ((D = 10^4 – 10^5)) a maximum of the variance occurs (the PDFs are clearly spread on a wider range of values). Such an effect indicates that the noisy behaviour of irradiance strongly influences the whole ecosystem dynamics. Moreover, the nonmonotonic behaviour of the variance (its PDFs become larger and then tighter again as the noise intensity increases) indicates that the noise pushes the ecosystem away from equilibrium, driving it towards a non-equilibrium steady state. Finally, we note that the nonmonotonic behaviour of CV as a function of the noise intensity remains also in the presence of seasonal driving.Figure 5Coefficient of variation ((CV=sigma / mu)) of nine planktonic populations resulting from numerical integration of model equations plotted versus the considered values of D; different curves correspond to different values of the correlation time (tau).Full size imageFigure 5 shows indeed the nonmonotonic response of the ecosystem to the change of D when the deterministic seasonal cycling of the four environmental parameters (temperature, irradiance, wind and salinity) is present. It is easy to observe that also in this instance the major noise-induced effect appears in nanoflagellates and diatoms with a percent standard deviation of 50(%) and 100(%), respectively. The coalescence of different curves (related to different values of (tau)), as D decreases, is due to the fact that for (D rightarrow 0) the impact of the noise is negligible and the evolution of the system practically resembles the deterministic one. On the contrary, for higher values of D remarkable differences arise and clear peaks of CV appear in the considered range of variation.These plots show that, for a fixed value of (tau), there exists a value of the noise intensity for which the planktonic concentrations are maximally spread around their mean values (corresponding to the maximum value of CV and then of the variance). Moreover, such a nonmonotonic behaviour suggests the presence of a resonance, which can be interpreted as the effect of the interplay between the nonlinearity of the system and the environmental random fluctuations.Also in this case, the interplay between the two parameters D and (tau) in determining and characterizing the dynamics of the ecosystem transparently emerges. The value of D corresponding to the maximum value of CV, indeed, basically depends on the specific value of (tau).Finally, we point out that the different dynamic scenarios identified by the D-(tau) couples can be experienced by the system during the year, since the two parameters may seasonally vary depending on the different weather conditions. In other words, a seasonally varying noise (see Fig. 1c) may cause the nine populations explore different regions of the D-(tau) space during the year. Therefore, the results reported in this paper can highlight the detectable yearly variability of a marine ecosystem which does not stem from the deterministic seasonal variation of environmental parameters.Effects on the organic carbonIn this subsection the effects of the irradiance noise on the biogechemistry are analysed. In Fig. 6 the dependence on (tau) of both the CV [panel (a)] and the mean value concentration [panel (b)] of detritus, labile dissolved organic carbon (L-DOC), semi-labile dissolved organic carbon (SL-DOC) and gross primary production (GPP) are shown. All these biogeochemical properties are correlated with carbon cycling. Gross primary production is related to the amount of carbon entering in the ecosystem, and is related to the maximum energy available in the ecosystem progressively dissipated in the trophic web. Gross primary production is directly affected by light fluctuation and its CV shape is very similar to that of the irradiance, Fig. 2. We selected also detritus and DOC because they are important indicators for the carbon cycling dynamics and are related to the cycling of chemicals like heavy metals62. The different curves, related to different values of D, approach the same (vanishing) value for large (tau). As previously discussed for the CV [Fig. 6(a)] of biomass concentrations, this circumstance is due to the fact that, in this case, the system dynamics tends to the deterministic case, characterized by a unique possible realization implying a vanishing standard deviation. For high correlation times thus the system is insensitive to the noise intensity. On the contrary, for small values of (tau), different values of D lead to significant differences of the variance. In particular, detritus, L-DOC and SL-DOC exhibit a clear non-monotonic behaviour whose maximum value depends on the combined values of D-(tau). Only the GPP presents a decreasing monotonic behaviour.The dependence of the mean value concentration on (tau), instead, is qualitatively the same for all the four parameters. Also in this case we can note a diversification with respect to D occurring at small (tau) and a (deterministic) constant value arising for low (high) values of D ((tau)).These results manifest that not only the population dynamics, but also all the biogeochemical processes are profoundly affected by the presence of stochastic environmental variables. The values and the behaviour of the examined quantities are indeed determined by the intimate interplay between the intensity and the time correlation of the noise fluctuations.Figure 6(a) Coefficient of variation ((CV=sigma / mu)) and (b) mean value concentration ((mu)) of detritus, labile dissolved organic carbon (L-DOC), semi-labile dissolved organic carbon (SL-DOC) and gross primary production (GPP) resulting from numerical integration of model equations plotted versus the considered values of (tau); the different curves are related to different values of the correlation time D.Full size image More

Growth conditions and plant materialsTwo experiments were conducted concurrently (sites A and B) in the same screen house in 2019 between the months of May and July at the Landmark University Greenhouse and Hydroponic Technology Center, a section of the Teaching and Research Farm of the University in Omu-Aran, Kwara State Nigeria. Experiment at site B was conducted simultaneously as A so as to validate the results of experiment A. Landmark University lies within Latitude 8° 7′ 26.21388″ and 5° 5′ 0.1788″. Both experiments (A & B) involved tomato (Solanum lycopersicum L. variety cherry) and cucumber (Cucumis sativus L. variety marketer) crops. For each crop, seeds were sown into a separate seed tray filled with coco peat (Coco peat, SRIMATHI EXPORT, INDIA). Cocopeat is the mesocarp tissue or husk after the grinding of coconut fruit. It has a lightweight and high water and nutrient holding capacities, it has an acceptable pH, electrical conductivity, and other chemical attributes27. Rice husk is the by-product of rice after milling. The rice husk used was collected from the rice processing mill of Landmark University. Rice husk is a highly porous and light weighted material with a very high specific area28.Two sets of seed trays (one for organic and another for inorganic fertilizers) were used each for tomato and cucumber crops in the nursery. Both were raised in the nursery for two weeks before transplanting. Black grow bags (30 × 17 cm) filled with a coco peat/rice husk (1:4 ratio by volume) mixture with a weight of about 10 kg were arranged in a screen house. Both the nursery and establishment of crop proper take place in a screen house. The screen house has a galvanized iron as the frame, a UV covering on top, side net for screening insect pests the floor fairly covered with granite. Temperature and relative humidity within the screen house during the period of the experiment was monitored using a Thermograph and a Barograph, and they were at an average of 31 °C and 75%, respectively.The grow bags were randomly placed in the screen house for the unbiased application of amendments. For both tomato and cucumber crops, the treatment comprised of six (6) levels of liquid organic fertilizer (5, 15, 25, 35, 45, 55 mL), in-organic fertilizer, and a control (ordinary borehole water). Levels of organic fertilizers were selected based on the recommendation of 20 mL of liquid organic fertilizer by29. The eight (8) treatments both for tomato and cucumber were arranged in a Completely Randomized Design replicated three times. One healthy plant was maintained per grow bag and four grow bags represent a treatment and there were 32 plants per block each for tomato and cucumber. For both crops, the experiment lasted for 90 days.Organic and in-organic nutrient solutionsThe liquid organic fertilizer used was obtained from the biomass of Mexican sunflower (Tithonia diversifolia). Fresh biomass (mainly leaves and stems) of the plant was collected from the Teaching and Research Farms of Landmark University, Nigeria. After rinsing, they were cut with a sterile knife into pieces of ≤ 1 cm size. A sample was taken for initial physicochemical analyses by grinding in a sterile mortal, diluted with sterile water and analyzed. The biomass was then soaked in sterile water inside a clean container, and allowed to ferment spontaneously for a period of 14 days. During the fermentation, samples were taken every 4 days for microbial analyses of the major players during the fermentation. At the end of fermentation, the mixture was separated using a sieve of mesh size ≤ 2 mm. The liquid portion was then refrigerated prior to the planting regime while another sample was taken to ascertain the physicochemical and microbial qualities of the produced liquid fertilizer. The chemical analysis is presented in Table 4. For inorganic fertilizer, Water soluble fertilizers employed in hydroponics were used (Hydroponics fertilizer, Anmol chemicals, India); calcium nitrate 650 mg L−1, potassium nitrate 450 mg L−1, magnesium 400 mg L−1, chelate 20 mg L−1, mono-ammonium phosphate 400 mg L−1. The electrical conductivity (EC) of the solution was 1.9 dS m-1.Irrigation and fertigationThe tomato and cucumber plants were fertigated morning and evening daily for one hour on each occasion according to the treatments. Preparation of the nutrient solution was with borehole water and was supplied to plants by an online pressure drip irrigation system set at 2.0 L h-1 using an arrowhead on each tomato and cucumber plant. Different tanks (250 L) were installed according to the various treatments making a total of 8 tanks. The organic fertilizer was diluted according to the various treatments equivalent to 1.25, 3.75, 6.25, 8.75, 11.25, and 13.75 L per 250 L of water respectively for 5, 15, 25, 35, 45, and 55 mL treatments. The nutrient solutions were refilled when the consumption is less than 20% of the initial volume (250 L) in the tank. One day per week, crops were irrigated with ordinary water to wash out pipes and prevent deposits of salts. The same concentration of nutrient was used from transplanting to the termination of the study for both tomato and cucumber crops, however, at the flowering of the crops, the volume of fertigation was increased to 3.0 L h-1 to be able to cope with the size of the plants.Trellising, pest and diseases controlFor both tomato and cucumber crops, plant vines were supported by twisting them around a wire that is- attached to the roof of the screen house and 2 m from the ground. Lateral outgrowths were cut off every week to ensure a sturdy single stem. Pests and diseases were scouted every day. Whiteflies, aphids, and other insects were controlled with orizon (Producer, location of producer) (active ingredient, acetamiprid, and abamectin) using 0.133% v/v. Fungi were controlled using ridomil gold (Producer, Location of producer) at 2% w/v.Determination of growth and yield of tomato and cucumberThree tomato and cucumber plants were randomly selected for each treatment for the determination of growth parameters (plant height, leaf area, number of leaves per plant, and stem diameter) at mid the flowering stage of tomato and cucumber plants.The leaf area of tomato was calculated using the model (A = KL2) developed by Lyon30, where L = Length of tomato leaf, K = constant which is 0.1551, and A = leaf area of tomato. Similarly, the leaf area of cucumber was calculated using A = 0.88LW – 4.27, where L = cucumber leaf length and W = cucumber leaf width, A = leaf area of cucumber31.Tomato fruits were ready for harvest from 65 days after transplanting, harvestings were done twice every week (Mondays and Fridays) for up to 85 days after transplanting. Similarly, harvesting of cucumber fruits started 35 days after transplanting and harvestings were also done twice a week (Mondays and Fridays), harvesting was carried out till 60 days after transplanting. Tomato and cucumber fruit yields were counted and weighed at each harvest.Analysis of tomato and cucumber leaves and fruitsAt the 50% flowering stage of tomato and cucumber plants, ten leaf samples were collected from each treatment. The leaf samples were oven-dried at 75 °C for 24 h and thereafter grounded. The grounded samples were later analyzed for nitrogen (N), phosphorous (P), potassium (K), calcium (Ca), and magnesium (Mg) content using the method of described by32. At harvest, four matured tomato and cucumber fruits of uniform size were selected per treatment, and their nutrient compositions were determined using the method of33.Statistical analysisAll data collected on the growth, yield, leaf, and fruit nutrient contents of tomato and cucumber were subjected to analysis of variance (ANOVA). The SPSS V 21.0 (New York, USA) software was used to perform ANOVA and Duncan’s multiple range test (DMRT) was used to compare means at a 5% probability level.
Ethical approvalI confirm that all the research meets ethical guidelines and adheres to the legal requirements of the study country.Compliance with international, national and/or institutional guidelinesExperimental research (either cultivated or wild), comply with relevant institutional, national, and international guidelines and legislation. Experimental studies were carried out in accordance with relevant institutional, national or international guidelines or regulation. More

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