Biodiversity Journal, 2020,11 (3): 655–661 https://doi.org/10.31396/Biodiv.Jour.2020.11.3.655.661 Estimating biodiversity through extrapolation: a better function Franco Strumia Physics Department, Pisa University, Largo Pontecorvo 3, 56127 Pisa, Italy; e-mail: [email protected] ABSTRACT A more general hyperbolic function is proposed for interpolating the number of accumulated species when observations are extended to long time intervals. The suggested function take also in account the insects phenology and the rate in the species accumulation number. KEY WORDS Extrapolation, Biodiversity; Hymenoptera; Mutillidae; Chrysididae; Gasteruptiidae; Apoidea. Received 05.03.2020; accepted 26.05.2020; published online 24.07.2020 INTRODUCTION must be carefully selected in order to have the right properties, including a fast asymptotic convergence When the species richness is sampled and in- and a minimum number of parameters. vestigated by means of any kind of traps, visual The choice of the function may follows from collection or literature study, the accumulated the solution of a differential equation describing number of observed species increase monotoni- the population dynamic. In any case, the function cally as a function of investigation efforts. The S(x) must be monotonic, increasing from zero (no number of accumulated species, (Number Accu- sampling) to the asymptotic value Sm. It is worth mulated Species NAS: S(x)) is expected to ap- to note that NAS is not symmetrical in time. There proach an asymptotic value corresponding to the is a difference between past and future. Consider- extrapolated number (Sm) of species present in the ing long time intervals the extinction of one studied biotope during the investigation time in- species do not decrease the NAS, while the obser- terval. vation of a new species, truly new or occasional To estimate Sm, we need to interpolate the S(x) visitor, increases the NAS and the asymptotic data as a function of sampling numbers: from the value. This asymmetry is not present when the initial value of 0 for x=0 (no sampling) to the richness is estimated from the Lognormal distri- asymptotic value Sm. To perform interpolation we bution (Magurran 1995). In fact, the Lognormal need a function with a convenient shape to repre- distribution estimates only the number of species sent the time evolution of biodiversity. The func- present during the time interval used to perform tion is adjusted to experimental data by the a set samplings. of parameters: S(x) = f(x, A, B, C, ...). The param- In seminal papers Colwell & Coddington (1994) eters are real numbers to be calculated from re- and Colwell et al. (2004) discussed the biodiversity gression. evaluation through extrapolation. Colwell & Cod- Unfortunately, the number of available sample dington suggested to interpolate the S(x) data to an data S(x) is usually limited by constrains on avail- Hyperbolic function (the well known Michaelis- able time for fieldwork: therefore, the function S(x) Menten function): 656 FRANCO STRUMIA 1) S(x) = x/(K+x) Sm where Sm is the estimated species number and K a constant (the number of samplings needed to observe Sm /2). Function 1) increases monotonically from zero to Sm. On the contrary, the derivate of function 1) decreases monotonically from a maximum (for x=0) to zero for large x values (Fig. 1). Thus, the above eq.1) cannot take in account possible changes in the rate of new records accu- mulation following from fluctuations in phenology. Figure 1. Plot of the Michaelis-Menten function for Sm=20 Of course this is not adequate when the sampling is (red line) or of its derivative (blue line). Note the slow ap- proach to the asymptotic value of 20. performed all year around. In fact, a convenient and practical protocol to perform samplings is to cap- ture specimens continuously for at least (possibly) one full year, especially if traps are positioned on sites not easy to be visited, such as isolate islands or dangerous countries, or when special permits are required. It is well known that the individuals emer- gence rate is variable and reach its maximum in spring (at least in Palaearctic climate), depending on the species-group and influenced by seasonal fluctuations in temperature, humidity, etc. For practical reasons, at least in Palaearctic region, is convenient to continuously operate traps from early spring to late fall. In such a case, it is well Figure 2. Plot of the generalized hyperbolic function (eq.2) known that the species emergence begins slowly (red line) and of its derivative (blue line) for Sm=20 and n= 2. Note the faster approach to the asymptotic value. The de- and reaches its maximum in June-July, then sig- rivative is 0 for x=0, and it reaches a maximum and then de- nificantly decreases from July-August, especially creases monotonically to zero. in the case of hot and dry summers as in mediter- ranean sites. This heuristically suggests the need of a simple hyperbolic function, with a derivate not monotonic. We suggest a generalized hyper- bolic function as: 2) S(x) = xn/(K+xn) Sm where “n” is an additional constant and a real positive number. When n>1, eq. 2) has the needed inflexion point (a derivative maximum) and, in ad- dition, approaches faster the asymptotic value Sm (Fig. 2). This is also an additional favorable prop- erty because in a year interval, the samplings num- Figure 3. Plot of eq. 2). for four n values, from bottom to ber may be reduced due to constrains in operating top: n= 1, 1.5, 2 and 3 respectively. The asymptotic value is the trap, possibly in remote locations or not easy to fixed at Sm= 20. For n=3 the asymptote is reasonably ap- be visited (Fig. 3). proached after only 4 to 6 samples. Estimating biodiversity through extrapolation: a better function 657 EXPERIMENTAL RESULTS the Michaelis-Menten function is unreasonable. On the contrary, the interpolation with eq.2 is in- Some examples are discussed to illustrate the side the acceptable range and shows that the better interpolations available from eq.2), when the Caselli reserve hosts a rich Gasteruptiidae biodi- constant n is let free to better adjust the function to versity, as already observed for others insects the experimental data. When the experimental points are too few the interpolation with eq.2) (three free parameters instead of two) may become inad- equate and the calculated Sm value becomes unsta- ble. In such a case, a better and easer interpolation can be obtained by fixing n to have any value in the range 1.5<n<2.5, thus reducing the free parameters to two. In such a case, a good extrapolation is al- ways available, as it will be shown. In the following examples, the non-linear interpolation is calculated with the fast converging Marquardt method and commercial software. In recent years, a number of Malaise traps oper- ated in Tuscany, Sardinia, Sicily, Piedmont and small Tyrrhenian islands in order to investigate Hy- menoptera biodiversity. Major evidence was given to Aculeate Hymenoptera, mainly for the relevance as impollinators (Abrol, 2012). In 2016, two Malaise traps were positioned in the Figure 4 . Number of Mutillidae species captured in the “Ca- Monterufoli-Caselli Nature Reserve in southern Pisa selli Nature Reserve” in 2016 by Malaise traps; the blue line Province (“Colline metallifere” Tuscany), in order to shows the best interpolation from eq.2 and the red line the best interpolation from eq.1; the broken lines show the 90% investigate the influence of a humid zone on ento- confidence range of eq.2; eq. 1 is clearly unable to properly mocoenosis (Strumia et al., 2017). The trap captured interpolate the experimental data. continuously from April 28th to September 8th and the material was sorted and identified in nine sam- ples. The observed number of the accumulated species of Hymenoptera Mutillidae is shown in fig- ure 4, together with the best interpolations available from eqs.2 and 1, respectively. The predicted asymp- totic values are respectively 16< Sm=18 <19 (95% confidence interval, n= 2.83) and 12<S=33<54 with the Michaelis-Menten function (n=1, 95% confi- dence interval, n=1). Not only the interpolation with the Michaelis-Menten function is clearly unable to properly interpolate the experimental data, but the predicted richness is too large and in disagreement with the results already observed in Tuscany (Stru- mia, 2007; Strumia & Pagliano, 2007). In the same site, the species of Hymenoptera Gasteruptiidae were also counted. The interpolated data gives a similar result as shown in figure 5. The calculated Sm values are respectively Sm =19 Figure 5. Number of the accumulated Gateruptiidae species +/- 6 (n=1) and Sm =14 +/-1 (n=4). In Italy, only observed in the “Caselli Nature. Reserve” in 2016 and cap- 20 species of Gasteruptiidae are known (Pagliano tured by a Malaise trap (blue dots); blue line shows the best & Scaramozzino, 2000), thus the prediction with interpolation with eq.2; red line those with eq.1. 658 FRANCO STRUMIA groups: Apoidea, Chrysididae, Pompilidae (Filippi If all data are combined and ordered following & Strumia, 2018). the beginning date of each sampling interval, a Malaise traps operated in the “San Rossore Na- larger number of S(x) data points becomes avail- ture Reserve” near Pisa town. San Rossore extends able, and a more accurate interpolation is possible. along the Mediterranean coast and has several humid This is illustrated in the following study on the and wooded zones on the back of coastal sand dunes. Hymenoptera Chrysididae biodiversity in a vine- These sites are favorable for sand nesting insects. yard of southern Piedmont (north-western Italy).