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Body and brain size databaseThe fossil dataset consists of the hitherto largest collection of body (n = 204) and brain size estimates (n = 166) from Homo in the past ~1.0 Ma (Fig. 1). The data on hominin body size estimates are derived from our own previous study6 plus additional estimates60 and updated chronometric ages from more recent literature. Individual body size estimates are provided by specimen in Supplementary Data 1 with data sources. The bulk of data on hominin brain sizes (endocranial volume, in cm3) is derived from recent meta-analyses7,12,13,61,62 and updated chronometric information. Specific sources of these data are indicated in Supplementary Data 2, with some assessments bearing larger errors due to the incomplete state of the crania on which they are based (e.g., Arago 21, Vértesszőlős, Zuttiyeh). Each body and brain size estimate is associated with information on estimated chronometric age (dating method and data source), geographical location (longitude and latitude), and taxonomic attribution. For the exact locations per specimen, see interactive map in Supplementary Note 1. We divided the dataset into three taxonomic units: Pleistocene H. sapiens, Neanderthals, and Mid-Pleistocene Homo. Whereas hypodigms of H. sapiens and Neanderthal remains are generally agreed upon, we use “Mid-Pleistocene Homo” as a strictly analytical unit to denote African and European Middle Pleistocene hominins that predate Neanderthals and are not assigned to Homo naledi, between ~800 and 130 ka. We refrained from further division of this group due to the often fragmentary nature of fossils, unclear alpha taxonomy, and small sample size. Analyses performed within these taxonomic units minimize phylogenetic effects of, e.g., significantly different brain sizes (e.g., ref. 2). Specimens from H. naledi and Homo floresiensis had to be excluded from this analysis as for each taxon they derive from a single location and age bracket, precluding assessment of paleoclimatic variation. Limitations to the fossil datasets (see e.g., refs. 2,3,4,6) include imprecision of brain and body size estimates due to methodical and taphonomic problems, uncertainties of absolute ages that translate into uncertainties of the associated climate, and unequal sampling of hominin fossils across time and space. These limitations were incorporated into the construction of the synthetic dataset to assess the extent of their effects on the overall results for the actual fossil dataset. For all further analyses, brain and body size values were log-transformed as they increase multiplicatively.Climate reconstructionsEach body and brain size estimate required corresponding estimates of relevant climatic variables. Our climate records are numerical model estimates based on global climate reconstructions for the past 1 Ma using the global climate model emulator GCMET27. The main idea behind GCMET is that global climate model (GCM) simulations of the past 120,000 years contain sufficient information about long-term climatic changes on time scales of ≥1000 years. Given that we know the external boundary conditions, we can reconstruct previous glacial–interglacial climatic changes. The Quaternary climate is largely determined by dynamics of the Northern Hemisphere ice sheets, which, in turn, are affected by orbital variations of the Earth around the Sun and variations of atmospheric CO2. Using these factors as external boundary conditions, GCMET can emulate the climate of the Quaternary in a similar way as a state-of-the-art GCM27.The atmospheric CO2 record of the past 1 Ma that we use in this study is a composite of the EPICA CO2 record from an Antarctic ice core63 and of output from a carbon cycle model (CYCLOPS)64. The EPICA record covers the past ~800 ka, whereas we use the CYCLOPS model output to cover the time up until 1.0 Ma. Orbital variations are based on calculations by Berger and Loutre65. Ice-sheet extents for the past 800 ka are based on numerical ice-sheet model output66. For the period before 800 ka, we assumed present-day ice-sheet configurations. This is an appropriate assumption given that all but one specimen of the fossil record before 800 ka in our datasets are within Africa or southeast Asia and thus far away from ice-sheet margins, with the local GCMET climate reconstructions not affected by this simplification.For each fossil site location from the body and brain size database, we extracted a time series of the relevant climate variables, see Table 1 (also Supplementary Figs. 10 and 11). The time series were used to attach the value of each climate variable to the fossil record, both for the actual fossil data as well as for the synthetic fossil datasets.Linear modelsThe null and two alternative linear models used throughout this manuscript are defined as follows. The null model simply estimates the mean for each taxonomic group, and we refer to this model as LM-T (linear model with taxa):$$Y={beta }_{0}+{beta }_{1}times {rm{taxon}}$$
(1)
Here Y corresponds to either body or brain size (or the log-transformed thereof), whereas β0 is the intercept, which is equivalent to the mean size of the reference taxon, and β1 is a factor that reflects the deviation from this mean size for a taxonomic group (thus giving independent intercepts for Mid-Pleistocene Homo, Neanderthals, or Pleistocene H. sapiens).The first alternative model contains the effect of the climate variable X (across all taxa):$$Y={beta }_{0}+{beta }_{1}times {rm{taxon}}+{beta }_{2}times X$$
(2)
Here β0 and β1 are the intercept terms, giving taxon-specific values, and β2 is the slope, which is the same across all taxa. We refer to this model as LM-TC (linear model with taxonomic differences plus a climate effect).The second alternative model takes taxonomic differences for the slope of the climate effect into account. This is done via an interaction term, β3, which acts as a modifier for the slopes (i.e., different intercepts, given by β0 and β1, and slopes, given by β2 and β3, for each taxonomic group):$$Y={beta }_{0}+{beta }_{1}times {rm{taxon}}+{beta }_{2}times X+{beta }_{3}times {rm{taxon}}times X$$
(3)
We refer to this model as LM-T*C (linear model with taxonomic differences plus a taxon-specific climate effect). The slopes, β2 and β3 in Eqs. (2) and (3), respectively, are presented in the main text in Tables 2 and 3 (for the log-transformed sizes) and in Supplementary Tables 1 and 2 (for the natural units of the sizes).Synthetic datasets and power analysisApart from determining the smallest sample size suitable to detect the effect of a given test at the desired level of significance, power analysis can also be used as a formal way to test whether a relationship between dependent and independent variables can be detected with the available data and proposed methods (i.e., linear models in our case) assuming that such a relationship exists. Before testing for any true association between local climate and the fossil record, we use such a power analysis to assess our power to detect relationships of different effect sizes given the uncertainties, for example, in body/brain sizes, dating, and climate reconstructions. We generated 1000 synthetic datasets for each of the ten climate variable associations (MAP, MAT, NPP, mean temperature of coldest quarter, mean precipitation of driest quarter, and the logarithm of their running standard deviation over a 10,000-year window) with body and brain size. For each association, we assumed a strong, a medium, and a weak relationship between size and climate.By strong, we refer to 1/4 of the maximum possible slope given by the range of the climate and size. Subsequently, medium is half the slope of the strong relationship, (1/8 maximum possible slope), and weak is half the slope of the medium relationship (1/16 maximum possible slope). For example, the strong association between MAT and body size (Bergmann’s rule) is −0.34 kg/°C, based on the above defined rule. This is close to the estimated association between temperature and body size of about −0.4 kg/°C found for modern humans in a recent study (ref. 31, their Fig. 5A). Unfortunately, there are no empirical data about other climatic relationships and body (or brain) size. For simplicity, we therefore applied the same rule of strong, medium, and weak associations for all other climate variables and for brain size.Before generating a synthetic dataset, we estimated the intercepts β0 and β1 and the slope β2 for the LM-TC model, Eq. (2). However, for the real fossil analysis we used the model with the interaction term, LM-T*C, Eq. (3). First, we looked up the climate record for each location and time from the climate time series and attached it to the respective empirical fossil records. We calculate the maximum slope from the X and Y ranges as β1 = range(Y)/range(X). Assigning an actual relationship factor, e.g., strong (=1/4), the intercept β0 can be calculated using the X- and Y-midpoints, β0 = Ymidpoint − 1/4β1Xmidpoint.For the synthetic fossil datasets, we assume an age uncertainty range of 10% (±5%) for radiocarbon-dated fossils, i.e., younger than 50 ka cal BP (e.g. ref. 67), and 20% (±10%) for fossils older than 50 ka coming from other dating methods with higher uncertainty such as luminescence, U-series, or ESR (e.g., ref. 68). Furthermore, we assume a standard error of 2 K for mean annual and mean temperature of coldest quarter. For all other climate variables, we assume a 20% error range (±10%). The 2 K and the 20% are in line with climate model biases as estimated in a recent study69. Within a taxonomic unit of the genus Homo, we assume a coefficient of variation (CV) of 7% for body size (average of intrapopulation means of 19 global Holocene hunter-gatherer populations, n = 510, data from JTS) and 3.5% for brain size (from ref. 28 populations, dataset: http://volweb.utk.edu/~auerbach/HOWL.htm; ref. 70, see also ref. 32). Previous research has demonstrated that the range of body size variation in Holocene human populations is larger than any taxonomic unit of earlier hominin and encompasses the range of variation found within earlier hominins6 and that sexual dimorphism in size among Mid-Pleistocene hominins is comparable to that of modern humans71. While there are significant differences in brain shape through recent hominin evolution, the range of size variation within Pleistocene hominin taxa remains comparable to that observed among modern humans60. These observations suggest that modeling the intrapopulation variation among hominin taxa upon modern human coefficients of variation provides a reasonable estimate of variation within hominin taxa that are often presented only by much smaller sample sizes. To create a synthetic dataset that has a mean and a variance as close to the fossil dataset, we introduced taxonomic size differences (β1 in Eq. (2)) that is based on the taxonomic differences in the mean size. This difference was estimated directly from the fossil dataset.The procedure to generate a single synthetic dataset is as follows. First, we selected a relationship strength, e.g., strong, and calculated the slope and intercepts. For each synthetic data point, we:

1.

Looked up the age and added a randomly sampled error (±5% or ±10%).

2.

Looked up the fossil site and selected the climate record from the previously calculated time series for that location and sampled age.

3.

Added a randomly sampled error (i.e., S.D. of 2 K or 20%) to the climate record. This is now the X value.

4.

Multiplied X with the slope β2 and added the intercept β1 with the respective taxonomic correction. This translates the climate record X into a size estimate Y.

5.

Added a random term to Y based on the CV, i.e., 3.5% for brain and 7% for body size.

6.

Repeated steps (1)–(5) for each fossil record and saved all locations, ages, Xs, and Ys to a file. This is a single synthetic dataset in the same format as the original fossil dataset.

We repeated this N times to generate N synthetic datasets and repeated the same procedure for the other relationship strengths, i.e., medium, and weak and for all other climate variables. Panels of exemplary synthetic datasets for body and brain sizes in comparison with the original data are shown in Supplementary Figs. 10 and 11.We use the same thinning approach as described in the main text (n = 1000) for the synthetic datasets. These are then used for a power analysis to test whether the linear relationship between any climate variable and body or brain size can be detected. We fitted both the LM-TC model, Eq. (2) (in which the slope defining the relationship between climate and size is the same for the three taxonomic groups, which can differ in their intercept), and the LM-T*C model, Eq. (3) (different slopes and intercepts for the three groups). A climate effect was deemed present if the null model had a higher AIC value compared to either of the alternative models, LM-TC or LM-T*C, (ΔAIC  > 2), i.e., LM-T, Eq. (1), in which the three groups differ in size but there is no effect of climate. Figure 2 in the main text shows the power to detect a true relationship between size and climate. Individual records are color-coded according to the AIC difference between the LM-T and the alternative models, LM-TC or LM-T*C, ranging from −2 (red) to +15 (blue) with 2 as midpoint (white).All statistical tests were undertaken in Python version 3.8.5 using the following Python packages: statsmodels 0.12 (for linear models), pandas 1.1.3 (for dataframes, reading/writing CSV/Excel files), netCDF4 1.5.3 (reading NetCDF files), matplotlib 3.3.2 (for plotting), and numpy 1.19.2 (numerics).Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

Results of our research showed that lipid accumulation in sea urchin gonads follows a periodic fluctuation, in agreement with previous observations22. The analysis of Fig. 1 suggests the key role of photoperiod in triggering and then modulating fat utilization and storage mechanisms in P. lividus gonads, while the effect of temperature in gametogenesis and spawning in echinoderms still remains uncertain5,22,28. In fact, a change in photoperiod anticipated the corresponding change in gonad total lipids content in both habitats, while the role of temperature was not very clear, since lipid changes seemed not to be associated with changes in temperature. Most likely, the combined effect of both parameters regulates reproductive cycle of sea urchins. Similar periodical trends in total lipid content were also observed in recent studies on P. lividus gonads19,20 collected in different geographical areas. For example, Rocha et al.20 reported that gonadal lipid content are likely influenced by the environmental conditions characterizing the harvest site in the Praia Norte (Portugal). In this work and in the abovementioned studies, total lipid content in gonads changed as a function of gametogenic cycle, i.e. increased until the recovery/growing stage (I–II) and then progressively decreased until the premature/mature stage (III–IV)27. In another detailed characterization of Arbacia dufresnii, Dìaz de Vivar et al.29 observed a marked dependence of the total lipid content with gonad maturation, with a significant decrease in lipid content in spawned compared to intact gonads, especially in female sea urchins.Assuming P. oceanica and H. scoparia were the main dietary sources of lipids in our study, gonad lipid content was relatively independent from dietary lipid intake, in agreement with data from other authors19,20,22. Indeed, total lipids in P. oceanica and H. scoparia were very low (approximately 1% D.W.) and seasonal variations of lipid levels in these main dietary substrates were definitely negligible. These results are further supported by the literature30,31. For a better understanding of the comparison between different scientific reports19,20,22, it should be recalled here that the displacement of periodical gametogenic cycles is strongly influenced by several environmental factors and is, therefore, dependent on the growing habitat16,32.As far as the commercial value of sea urchin gonads is considered, several reports20,33 suggest that the best harvesting period is when gonads are in the growing stage, when nutrient contents (i.e. proteins, lipids and carbohydrates) are at their highest levels, and when sensorial characteristics are optimal. In fact, gonad maturation decreases the overall quality of roe, and make them more bitter and less pleasant33,34. However, it is striking that very often the official regulation on the harvest of P. lividus in Sardinia allowed collection of sea urchins in the period from November to April, when products are nutrient-poor and in the late stages of gametogenesis (i.e. pre-mature, mature and spawning stages)35.Our observations suggest that total lipids from dietary sources concentrate in the gut. The amount of lipids in these latter samples is actually always much higher than in the sea grass and macroalgae analyzed. The concentration of lipids in the gut has been already observed in other echinoderms as well36,37. These evidences suggest that digestion phenomena occurring in the gut may include the concentration of nutrients. Moreover, our data show that lipid fatty acid composition in gut is considerably consistent, regardless dietary lipid. While further studies are needed, most recent findings strongly suggest that gut flora have a role in assisting digestion and absorption of nutrients in sea urchins38. De novo synthesis of fatty acids by microbiotes, an interesting hypothesis that would especially concern the modulation of short chain fatty acids levels, should be further and specifically investigated. Based on most recent findings, a possible role of bacteria in nutrient production and processing has been postulated39. However, it should be also reckoned that other lipids may come from other dietary sources beyond the main sea grass and macroalgae (“Supplementary Material”). This latter hypothesis, however, would not explain the substantial increase observed in gut lipids, since other possible sources do not have very high lipid contents and were taken in small percentage. For example, it was previously observed in adult Strongylocentrotus intermedius that algal pellets exceeded 80–90% (wet weight) of gut contents, complemented by detritus, small animals (e.g. small crustaceans and mollusks) and non-foods (e.g. sand, shell fragments)40. Moreover, in P. lividus sampled from natural conditions in Corsica (France), 95% of the total gut content was represented by plant material41. Similarly, animal taxa in our study represented a very low percentage of the gut content, and species populating the rocky bottom, other than H. scoparia, have low lipid content and likely had little relevance on sea urchins diet. Also Murillo-Navarro and Jimenez-Guirado25, in a yearlong investigation, found that H. scoparia was the most abundant brown alga in gut contents of P. lividus.Brown algae and leaves of P. oceanica are in fact generally considered among the primary components of adult P. lividus diets1,24,25. It has been also observed by other authors that sea urchins consume all parts of P. oceanica and preferentially green leaves colonised by epiphytes1,26,42,43,44. Epiphytes were not removed from our samples before analysis.The role of gut and stomach as nutrient storage organs is generally acknowledged41,45. This is demonstrated by the almost double lipid contents found in gut than in food sources in the present investigation and by other studies36,37. As a later digestion step, lipids are selectively stored in gonads, where almost three or even four times the lipids contents found in the gut were detected. This supports the hypothesis of lipid relocation from gut to gonads, thus confirming the role of gonads as an important storage tissue for P. lividus, as was previously established by other authors22,46 and correspondingly a role in lipid metabolism can be ascribed to the digestive tract. It also further proves that the amount of fat daily introduced with diet has only a limited influence on the seasonal evolution of total lipids in gonads. Of course, nutrients and especially lipids stored in gonads serve during gametogenesis, as an energy source for developing embryos and are mobilized during pre-feeding development of larvae5. In echinoderms, indeed, nutrients provided in the eggs are needed by developing embryos and larvae.In two recent investigations on P. lividus collected along the Atlantic coast of Portugal, Rocha et al.19,20 evidenced slightly different seasonal trends. They observed both a maximum lipid content and an increase in PUFA content in gonads during the fall season. In contrast, we observed a peak in total lipids during summer, and an increase in PUFA during winter. Likely, the different climatic and environmental conditions of the Atlantic coast with respect to the Mediterranean basin (especially seawater temperatures) induce different gametogenesis cycles16, which in turn modulate the lipid balance in gonads. Gametogenic stages are in fact differently distributed along the year in ours and the cited works by Rocha et al.19,20. In general, lipid content in gonads seem to increase during the recovery (stage I) and growing (stage II) gametogenic stages27, when gonads are packed with nutritive phagocytes and only few germ cells are present.Other studies suggested that specific fatty acids found in the gonads of sea urchins may be synthesized by other tissues such as the intestine and then mobilized to the gonads47.Regardless the different food availability in the two analyzed sites, our results show a remarkable robustness of the fatty acids profile of gut contents. This is particularly interesting since they show a regulation of physiologically essential C 20:5 n-3 and C 20:4 n-6 at gut level, which seem to quite finely level out according to season, regardless the dietary contents of these fatty acids.The increase in gonad PUFA observed in both habitats during winter did not seem to correlate with substantial changes in the main taxa isolated in the gut content of the sea urchin sampled in the P. oceanica meadow, nor to relevant changes in the specimens populating the rocky bottom habitat (“Supplementary Material”). This is consistent with our previous studies21,22, which linked the phenomenon to both the cold acclimatization effect and gametogenesis. Raise in PUFA in lower temperatures allows maintaining cell membrane fluidity and, consequently, supports its functionality.The questions arise whether the lipid species contained in the food sources can be directly and selectively absorbed by sea urchin gonads and how much food habits affect gonads composition. In order to answer these questions, discussion should be directed to each relevant fatty acid.The fatty acids of glycerolipids of higher-plants chloroplasts are highly unsaturated, and the most represented fatty acid is C 18:3 (n-3)48. Instead, brown algae, such as Phaeophyceae, contain a large amount of C 20:4 (n-6) and C 20:5 (n-3)49. During our studies, the most significant difference between the fatty acid profiles of P. oceanica and H. scoparia was related to C 18:3 (n-3). According to our data, P. oceanica contained, on average, more than ten times the amount of this FA in H. scoparia.The fatty acid profile of P. oceanica described in the present study is in agreement with previous reports50,51 and confirms that lipids of P. oceanica are mainly represented by the C 18:3 (n-3), C 18:2 (n-6), and C 16:051. On the contrary, the fatty acid composition of H. scoparia seems to be quite variable considering previously published reports, although literature generally agrees on the most abundant fatty acids (i.e. C 16:0, C 18:2 n-6, C 20:5 n-3 and C 20:4 n-6)31,52.Both in rocky bottom and in P. oceanica meadows, gonadal C 18:3 (n-3) decreased when sea urchins metabolism is mainly influenced by production of gametes (from November), i.e. when gonads reached premature/mature stages, as previously observed15,22. Our data showed a decrease of C 18:3 (n-3) in gut roughly corresponding to an increase of the same FA in gonads (Fig. 4), suggesting that dietary C 18:3 (n-3) was not selectively and directly retained in gonads from the diet, but likely took active part to metabolic processes of bioconversion or is catabolized during β-oxidation of lipids.Also C 18:2 (n-6) showed a similar behaviour in our study and in other previous investigations15,20.Remarkably, C 20:5 (n-3) and C 20:4 (n-6) were the most abundant LC-PUFA in both gut and gonads, in contrast with the composition of the main dietary sources of lipids in the two habitats. In fact, while high percentages of these fatty acids were found in the brown algae H. scoparia, they were present only in very low percentages in the P. oceanica samples. In sea urchins, the fatty acid profile of diet is often scarcely reflected in gut contents and gonads53. From July to March we detected higher percentages of C 20:5 (n-3) in gonad samples collected from P. oceanica meadow than in the corresponding samples from rocky bottom. Moreover, our data clearly show that the C 20:5 (n-3) contained in either gonads and gut does not reflect seasonal variations of this FA in the main sea grass and macroalgae populating the two sites. This result supports earlier observations5,21.Beyond P. oceanica, green algae, especially C. cylindracea, represented additional dietary sources of C 20:5 (n-3) in the P. oceanica meadow. P. lividus usually feeds on brown algae and only less frequently on green algae1,15. In fact, green algae represented less than 5% of the gut content in P. oceanica meadow all year long but from October to December, when they increased from 10 to 25%. In this period, C 20:4 (n-6) and C 20:5 (n-3) in gonads reached their lowest values, but the C 20:5 (n-3) content in gut noticeably increased. After January, when sea urchin reduced feeding in green algae and again less than 5% of green algae was found in the gut content, C 20:5 (n-3) and C 20:4 (n-6) content in sea urchins gut started increasing. To explain this observation, we recall that it was found in S. droebachiensis that dietary FA were not incorporated in sea urchin tissues after short feeding experiments54, but longer experiments allowed to observe diet-related modifications in tissues36. Therefore, it is reasonable to think that nutrients are transferred from gut to gonads. Among other dietary sources of lipids, brown algae in P. oceanica meadow likely did not significantly contribute to increase LC PUFA in gut contents and gonads prior to gametogenesis, being brown algae intake almost always low in the present study.The observed increase of C 20:4 (n-6) in gonads in December was less correlated to the dietary availability of this FA, but was likely associated to cold adaptation and to the growth and maturation of gametes21. In fact, even when the main dietary source of lipids, P. oceanica, was almost completely devoid of this FA, the percentage of C 20:4 (n-6) in gonads was 10–15% and not significant increase of this FA was observed in gut contents from October to December.As for most aquatic consumers, C 20:5 (n-3) and C 20:4 (n-6) can be selectively retained in gonads from dietary sources or accumulated through the conversion of other essential 18-carbon FA.Since we found similar amount of C 20:5 (n-3) C 20:5 (n-3) and C 20:4 (n-6) in P. lividus gonads and gut contents and these values were much higher than in dietary sources, retention or biosynthesis should have occurred already at intestinal level, as previously suggested for other echinoderms36,37,47. As previously hypothesized for Strongylocentrotus intermedius, likely these FA were transferred to gonads after being processed and stored in the digestive tract47.Recently, Kabeya et al.55 found that P. lividus possesses desaturases that are able to convert C 18:3 (n-3) and C 18:2 (n-6) into C 20:5 (n-3) and C 20:4 (n-6), respectively. Han et al.47 characterized the expression of fatty acid desaturases (SiFad1) in different tissues of S. intermedius and concluded that the highest expression is in the intestine, while gonads have lower expression level. Therefore, while retention from diet and biosynthesis from C18 precursors of essential lipid species such as C 20:5 (n-3) and C 20:4 (n-6) might occur already in the gut36,37,41,45, also gonads might possess some, likely lower, biosynthetic functions. Kabeya et al.55 did not specifically quantify the expression of desaturases in different tissues of P. lividus, therefore further research in this sense would be beneficial.It should be mentioned that sex-induced difference of fatty acid profiles of sea urchin gonads were not studied in the present work, but males and females specimens were pooled together. Some previous reports have evidenced differences in lipid classes and fatty acids profiles between sexes15,29, while other studies did not spot statistically significant gender-related discrepancies5. Fatty acids profiles of gonads are likely to be related by sea urchin gender, but it is reasonable to believe that such differences would not disprove the aforementioned considerations on lipid storage and metabolism at gut and at gonad level. In particular, the differences in C 18:3 n-3, C 18:2 n-6, C 20:4 n-6 and C 20:5 n-3 found in previous studies between male and female gonads were quite low (maximum 2–4% of total FAME). Gender differences are ascribable to the increasing presence of lipid-rich gametes (oocytes or sperm) during the gonad maturation period. Also differences in lipid classes are expected in this period, being triglycerides mainly present in female gametes29,56. According to previous reports, during the reproductive period females of both P. lividus and Arbacia lixula showed lower proportions of 20:4n-6, while 20:5n-3 was higher in males of P. lividus and in females of A. lixula56. In P. lividus, such differences were found to be very limited for 20:4n-6 and 20:5n-3 (0.1% and 1.3%, respectively, between mean values of total FAME percentage)56. Also in Arbacia dufresnii the differences between male and female intact gonads for 20:4n-6, while 20:5n-3 were found to be not very important, but both fatty acids seemed to be slightly more concentrated in male tissues29.In any case, the present study confirms that during maturation stages of gonads, when their nutritive content decreases20,22, C 20:5 (n-3) and C 20:4 (n-6) levels increase, and so does their nutritional quality. C 20:5 (n-3) consumption is in fact associated to reduced risk of several chronic diseases57. At the same time, previous reports showed that the best commercial value of sea urchin gonads is before the onset of gametogenesis20,33. These results are quite relevant not only because they allow to deepen the knowledge of the metabolic response of sea urchin P. lividus to season and diet, but also for both improving echinoculture practices and guiding relevant policies directed to regulate the harvest of wild populations. Changes in the concentration of biochemical components in the gonads of sea urchins impact their sensory quality20,33,34. In particular, gonads in their mature stages were described as more bitter34 and of lower quality overall33 than when they are in the growing stage. On the other hand, gonads in the growing stage reach the highest contents of nutrients (protein, fat, carbohydrates)20. Harvest of wild sea urchin during the reproductive time should be avoided, and this is particularly important for an endangered species such as P. lividus. Echinoculture could provide sea urchin roe for which the harvest time should be carefully scheduled as a function of analytical quality parameters and based on expected use.In conclusion, P. oceanica and H. scoparia, primarily constituted P. lividus diet in two contrasting sites within the same geographical area. Green algae, especially C. cylindracea, supplemented sea urchin diet in the P. oceanica meadow prior to gametogenesis, demonstrating the ability of P. lividus to select their diet according to requirements. Total lipid content in gonads changed periodically as a function of gametogenic cycle, being relatively independent from dietary lipid intake and showing a maximum during the growing stage and a minimum in mature gonads. Fatty acid profiles of P. oceanica and H. scoparia were significantly different from each other throughout the year. C 18:3 (n-3) was the main differential dietary marker in P. lividus gonads and gut contents. The main PUFA of P. lividus gonads, C 20:5 (n-3) and C 20:4 (n-6) were associated to increased consumption of green algae in P. oceanica meadow. LC-PUFA were selectively allocated in gonads as a function of reproductive cycle. Conversion of C 18:3 (n-3) to C 20:5 (n-3) and of C 18:2 (n-6) to C 20:4 (n-6) at gut level cannot be excluded, although further research in this sense is desirable. It is worth to note that harvest is generally allowed in Sardinia during gonad maturation, when main nutrients (lipids, carbohydrates, proteins) are at lowest level and also the sensory quality of roe is low, but gonads are rich in healthy LC-PUFA. Our results suggest that rearing of P. lividus would be possible with diets very poor in LC-PUFA given a supplement of this nutrients is provided prior to gametogenesis, when gonads are in the growing/premature stages. More

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We now derive the metapopulation model used in this paper. We start by deriving a general metapopulation model that is based on the seminal work of Levin50. Assuming that the inter-patch migrations are detailed-balanced, we make use of the formulation in Eq. (8) to derive a balanced metapopulation model. We then show that the balanced model admits a unique coexistence equilibrium that is asymptotically stable if the dispersal network is heterogeneous, whereas the same equilibrium is neutrally stable in the case of a homogeneous network.General metapopulation modelMathematical models based on traditional metapopulation theory usually assume that the metapopulation is made up of many neighboring spatially homogeneous habitat patches connected via dispersal. Consider an interconnected network of m discrete patches each being inhabited by the same n species. In addition, assume that species can migrate from one patch to some or all of the other patches. The rate of migration of each species between two patches is directly proportional to the proportion of the particular species in the originating patch, with a (nonnegative) constant of proportionality being the same across species. This constant of proportionality will be referred to as the rate constant associated with the migration. It is assumed that if there is migration between two given patches, then it is bidirectional, i.e., the rate constant of migration from j to k is strictly positive if and only the same holds for the migration from k to j. Just like in the case of a reversible single-species chemical reaction network, inter-patch migrations may be described by a weighted symmetric directed graph (G_2=(V_2,E_2)) where (V_2={1,ldots ,m}) is the set of patches (vertices) and an edge ((j,k)in E_2) means that every species can migrate from patch j to patch k. Finally, it is also assumed that the graph (G_2) corresponding to the inter-patch migration is connected, i.e., there is a path between every two distinct vertices of the graph.The flow of species between the patches can be summarized in a weighted (mtimes m) adjacency matrix ({mathbf {A}}) with entry (A_{jk}) being equal to the rate constant of migration of species from the (j{ {text {th}}}) to the (k{ {text {th}}}) patch. The diagonal elements of ({mathbf {A}}) are hence equal to 0. Due to the bidirectional nature of migration, it holds that (A_{jk} >0 Leftrightarrow A_{kj} >0) and (A_{jk}=0 Leftrightarrow A_{kj}=0), for any (jne k). Let (Delta =text {diag}(delta _1,ldots ,delta _m)) denote the m-dimensional diagonal matrix whose (j{ {text {th}}}) entry is given by$$begin{aligned} delta _{j}=sum _{k=1}^{m} A_{jk}. end{aligned}$$Define ({mathbf {L}}:=Delta -{mathbf {A}}^top ). Note that$$begin{aligned} (mathbb {1}^m)^{top }{mathbf {L}}=(mathbb {1}^m)^{top }Delta -big ({mathbf {A}}mathbb {1}^mbig )^{top }=({mathbf {0}}^m)^top . end{aligned}$$Let ({mathbf {x}}in S^{mn}), with (x_{i,j}) the proportion of species i in patch j across the entire metapopulation, then the net migration rate (psi _{i,j}) of species i from other patches to patch j is given by$$begin{aligned} psi _{i,j}=sum _{k=1}^{m}A_{kj}x_{i,k}-sum _{k=1}^{m}A_{jk}x_{i,j}=sum _{k=1}^{m}A_{kj}x_{i,k}-delta _{j}x_{i,j}=-sum _{k=1}^{m}L_{jk}x_{i,k}. end{aligned}$$Let us denote (Psi _i:=left( psi _{i,1},psi _{i,2},ldots ,psi _{i,m}right) ^{top }) and ({mathbf {r}}_i:=left( x_{i,1},x_{i,2},ldots ,x_{i,m}right) ^{top }), then$$begin{aligned} Psi _{i}=-{mathbf {L}}{mathbf {r}}_{i}. end{aligned}$$
(9)
Within each patch, the proportions of species are affected by other patches only via migration. Let (phi _{i,j}) denote the rate of change of the proportion of species i in patch j in the absence of migration. Since the dominance relationships among the species (described by a tournament matrix ({mathbf {T}})) are assumed to be the same for all patches and since the habitat patches are spatially homogeneous, the expression for (phi _{i,j}) is given by the right-hand side of System (1):$$begin{aligned} phi _{i,j}=x_{i,j}left( {mathbf {T}}{mathbf {p}}_{j}right) _{i}, end{aligned}$$
(10)
where ({mathbf {p}}_j:=left( x_{1,j},x_{2,j},ldots , x_{n,j}right) ^{top }), (i=1,ldots ,n) and (j=1,ldots ,m). Assuming migration among the patches, the proportion of a species within a patch is influenced by two factors: the first is the interaction with other species within the patch and the second is the migration of that particular species to or from other patches. Thus, the metapopulation model describing the dynamics of the n species in the m-patch network is described by the system of mn differential equations;$$begin{aligned} {dot{x}}_{i,j}=phi _{i,j}+psi _{i,j}=x_{i,j}left( {mathbf {T}}{mathbf {p}}_{j}right) _{i}-left( {mathbf {L}}{mathbf {r}}_{i}right) _{j},qquad i=1,ldots ,n,quad j=1,ldots ,m . end{aligned}$$
(11)
This system evolves on the unit simplex (S^{mn}).
Proposition 2

The unit simplex (S^{mn}) is positively invariant for System (11).

Proof
To show the invariance of the unit simplex (S^{mn}) under the flow of System (11), it suffices to show that each of the faces of the simplex cannot be crossed, i.e., the vector field points inward from the faces of (S^{mn}).
On the one hand, if (x_{i,j}=0) for some i, j, then$$begin{aligned}{dot{x}}_{i,j}=sum _{k=1}^{m}A_{kj}x_{i,k}ge 0,end{aligned}$$which implies that (x_{i,j}=0) cannot be crossed from positive to negative. In an ecological context, this condition simply states the obvious fact that an extinct species is in no danger of declining. On the other hand, if (x_{i,j}=1) for some i, j, then obviously (x_{l,k}=0) for any (lne i) or (kne j) and$$begin{aligned}{dot{x}}_{i,j}=-delta _{j}< 0.end{aligned}$$Hence, the vector field associated with System (11) points inward from the faces of (S^{mn}). So, (S^{mn}) is positively invariant under the flow of System (11). (square ) Note that Proposition 2 does not exclude the solution trajectories of System (11) from approaching the boundary equilibria of the system as (trightarrow infty ). We call metapopulation model (11) persistent if for every ({mathbf {x}}_0in S^{mn}_{+}), the (omega )-limit set (omega ({mathbf {x}}_0)) does not intersect the boundary of (S^{mn}). In other words, a metapopulation model is persistent if the initial existence of all the species implies that none of the species goes extinct with the passage of time.Balanced homogeneous and heterogeneous metapopulation modelsWe say that the inter-patch migration of a metapopulation model is detailed balanced if the overall migration rate of any species between any two patches is zero for a certain positive set of proportions of that species in the different patches. From the theory of detailed-balanced reaction networks described in “Detailed-balanced single species mass action reaction networks” section, it follows that a detailed-balanced inter-patch migration network corresponds to a detailed-balanced single species mass action reaction network. Let B denote the incidence matrix corresponding to the directed graph (G_2) describing the inter-patch migrations and let r denote the number of edges in (G_2). Comparing Eqs. (8) and (9), it follows that if the inter-patch migration is detailed balanced, then there exist diagonal matrices ({mathcal {K}}in {mathbb {R}}^{rtimes r}) and ({mathbf {Z}}^*in {mathbb {R}}^{mtimes m}) with positive diagonal entries such that ((mathbb {1}^m)^{top }{mathbf {Z}}^*mathbb {1}^m=1) and$$begin{aligned} {mathbf {L}}={mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }({mathbf {Z}}^*)^{-1}. end{aligned}$$Let ({mathbf {Z}}^*=text { diag}({mathbf {z}}^*)). Equation (9) can now be rewritten as$$begin{aligned} Psi _{i}=-{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) . end{aligned}$$ (12) Henceforth in this manuscript, we restrict our analysis to metapopulation models of type (11) for which the interactions within each patch correspond to a tournament with a completely mixed optimal strategy and whose inter-patch migration is detailed balanced. Such metapopulation models will be referred to as balanced metapopulation models.We have seen earlier in “Species interactions and tournament matrices” section that if the interactions within every patch correspond to a tournament with a completely mixed optimal strategy, then the corresponding mean-field model admits a unique coexistence equilibrium ({mathbf {y}}^*in S^{n}_{+}) with ({mathbf {T}}{mathbf {y}}^*={mathbf {0}}^n). Thus, for a balanced metapopulation model, System (10) can be rewritten as$$begin{aligned} phi _{i,j}=x_{i,j}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i, end{aligned}$$ (13) where ({mathbf {Y}}^*:=) diag(({mathbf {y}}^*)). Consequently, from Eqs. (11)–(13), it follows that the dynamics of a balanced metapopulation model containing n species and m patches can be described by mn differential equations$$begin{aligned} {dot{x}}_{i,j}=x_{i,j}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) right) _{j} ,qquad i=1,ldots ,n,quad j=1,ldots ,m . end{aligned}$$ (14) If all the elements of ({mathbf {z}}^*) in the above equation are equal, i.e., if (z_j^*=frac{1}{m}) for (j=1,ldots ,m), then we say that the balanced metapopulation model is homogeneous, otherwise we call it heterogeneous. Whether a balanced metapopulation model is homogeneous or not can be checked from the adjacency matrix ({mathbf {A}}) corresponding to its inter-patch migration graph (G_2). If ({mathbf {A}}) is symmetric, then the model is homogeneous, otherwise it is heterogeneous. Remark 3 In35, the authors assume that migrations from one patch to other patches are random with a probability of migration (or migration constant) equal to the reciprocal of the number of dispersal links from a patch to other patches. They thus define a dispersal graph to be homogeneous if all nodes have the same degree (number of links), otherwise the graph is heterogeneous. With this definition, homogeneity, in general, is equivalent to the existence of cycles in the dispersal graph, whereas heterogeneity is equivalent to their absence. However, with our new definition, it is clear that this is not necessary. An example of such a case is shown in Fig. 2.Figure 2Left: A heterogeneous dispersal graph according to35. Right: A homogeneous dispersal graph according to our definition.Full size image Coexistence equilibrium and its uniquenessIn this section, we present a theorem that gives an expression for a coexistence equilibrium of a balanced metapopulation model. Before we state our main theorem in this section, we need the following lemma. Lemma 4 Let ({mathbf {B}}in {mathbb {R}}^{mtimes r}) denote the incidence matrix of a finite connected directed graph (G_2) and let ({mathcal {K}}in {mathbb {R}}^{rtimes r}) denote a diagonal matrix with positive diagonal entries. For any ({mathbf {w}}in {mathbb {R}}_+^m), it holds that (-{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) ge 0). Moreover (-{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) = 0) if and only if ({mathbf {w}}=qmathbb {1}^m), where (qin {mathbb {R}}_+). Proof Assume that the (p{ {text {th}}}) edge of the graph (G_2) is directed from vertex (i_p) to vertex (j_p). Hence, (B_{i_pp}=-1), (B_{j_pp}=1) and (B_{kp}=0) for (i_pne kne j_p). Thus,$$begin{aligned} -{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) =sum _{p=1}^m(w_{j_p}-w_{i_p})kappa _pleft( frac{1}{w_{i_p}}-frac{1}{w_{j_p}}right) =sum _{p=1}^mfrac{kappa _p}{w_{i_p}w_{j_p}}left( w_{j_p}-w_{i_p}right) ^2ge 0. end{aligned}$$Moreover, (-{mathbf {w}}^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{mathbb {1}^m}{{mathbf {w}}}right) =0) if and only if (w_{j_p}=w_{i_p}) for (p=1,ldots ,m), which is equivalent with ({mathbf {B}}^{top }{mathbf {w}}={mathbf {0}}^r). Since the graph (G_2) is connected, we recall from48 that (text {rank}({mathbf {B}})=m-1) and furthermore (text {ker}({mathbf {B}}^{top })=mathbb {1}^m). Therefore ({mathbf {B}}^{top }{mathbf {w}}={mathbf {0}}^r) if and only if ({mathbf {w}}=qmathbb {1}^m), where (qin {mathbb {R}}_+). This completes the proof. (square ) We now state the main theorem of this section. Theorem 5 A balanced metapopulation model described by System (14) admits a unique coexistence equilibrium ({mathbf {x}}^*in S^{mn}_{+}). The proportion (x_{i,j}^{*}) of species i in patch j at the unique coexistence equilibrium is given by$$begin{aligned} x_{i,j}^*=y^{*}_iz^{*}_j. end{aligned}$$ (15) for (i=1,ldots ,n) and (j=1,ldots ,m). Proof We divide the proof into two parts. In the first part we prove that System (15) indeed yields an equilibrium for the model. In the second part, we prove that this coexistence equilibrium is unique. Let us define$$begin{aligned} {mathbf {p}}_{j}^*:=left( x_{1,j}^*, x_{2,j}^*, ldots , x_{n,j}^*right) ^top =z_j^*{mathbf {y}}^*; quad {mathbf {r}}_{i}^{*}:=left( x_{i,1}^{*}, x_{i,2}^{*}, ldots , x_{i,m}^{*}right) ^top =y_i^*{mathbf {z}}^*. end{aligned}$$For ({mathbf {x}}^*) to be an equilibrium of System (14), it should render the right-hand side equal to zero. Note that$$begin{aligned} mathbf {TY}^*left( frac{{mathbf {p}}_{j}^*}{{mathbf {y}}^*}right) =z_j^*mathbf {TY}^*mathbb {1}^n=z_j^*{mathbf {T}}{mathbf {y}}^{*}={mathbf {0}}^n end{aligned}$$and$$begin{aligned} {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^*}{{mathbf {z}}^{*}}right) =y_i^*{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }mathbb {1}^m={mathbf {0}}^m. end{aligned}$$In addition,$$begin{aligned} (mathbb {1}^{mn})^{top }{mathbf {x}}^*=sum _{i=1}^nsum _{j=1}^mx_{i,j}^*=sum _{i=1}^{n}y_i^*sum _{j=1}^mz_j^{*}=1. end{aligned}$$Thus, ({mathbf {x}}^*) is a coexistence equilibrium of System (14). Assume that there exists another coexistence equilibrium ({mathbf {x}}^{**}in , S^{mn}_{+}). Let (x_{i,j}^{**}) denote the corresponding proportion of species i in patch j and define$$begin{aligned} {mathbf {p}}_{j}^{**}:=left( x_{1,j}^{**}, x_{2,j}^{**}, ldots , x_{n,j}^{**}right) ^top ; qquad {mathbf {r}}_{i}^{**}:=left( x_{i,1}^{**}, x_{i,2}^{**}, ldots , x_{i,m}^{**}right) ^top . end{aligned}$$It follows that for any i, j it holds that$$begin{aligned} x_{i,j}^{**}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=0. end{aligned}$$ (16) Multiplying both sides of this equality with (frac{x_{i,j}^*}{x_{i,j}^{**}}), we get$$begin{aligned} x_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i- frac{x_{i,j}^*}{x_{i,j}^{**}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=0. end{aligned}$$Summing the left-hand side of the above expression over the different species and patches, we get$$begin{aligned} sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i- sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}^{**}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=0. end{aligned}$$ (17) Now consider the two terms in the left-hand side of the above equality separately. For the first term, note that for any j it holds that$$begin{aligned} sum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i= & {} sum _{i=1}^nx_{i,j}^*left( {mathbf {T}}{mathbf {p}}_{j}^{**}right) _{i} = sum _{i=1}^{n}x_{i,j}^{*}left( sum _{l=1}^{n}T_{il}x_{l,j}^{**}right) =-sum _{l=1}^{n}x_{l,j}^{**}left( sum _{i=1}^{n}T_{li}x_{i,j}^{*}right) \= & {} -sum _{l=1}^{n}x_{l,j}^{**}left( sum _{i=1}^nT_{li}y_i^{*}z_j^*right) =-z_j^*sum _{l=1}^{n}x_{l,j}^{**}({mathbf {T}}{mathbf {y}}^*)_l=0. end{aligned}$$Hence,$$begin{aligned} sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i=0. end{aligned}$$For the second term, we find$$begin{aligned} -sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}^{**}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=-sum _{i=1}^ny_i^*sum _{j=1}^mfrac{z_j^*}{x_{i,j}^{**}}left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=-sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}^{**}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) . end{aligned}$$Thus, Eq. (17) can be simplified as$$begin{aligned} -sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}^{**}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) =0. end{aligned}$$Since (y_i^* >0) for (i=1,ldots ,n), it holds for any (i=1,ldots ,n) that$$begin{aligned} -left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}^{**}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) =0. end{aligned}$$
(18)
From Eq. (18) and Lemma 4, it follows that ({mathbf {r}}_{i}^{**}=q_i{mathbf {z}}^*) with (q_iin {mathbb {R}}_+) for (i=1,ldots ,n). Thus, (x_{i,j}^{**}=q_iz_{j}^*) and ({mathbf {p}}_{j}^{**}=z_j^*{mathbf {q}}) for (i=1,ldots ,n) and (j=1,ldots ,m). Substituting the latter in the left-hand side of Eq. (16), we get$$begin{aligned} x_{i,j}^{**}left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}^{**}}{{mathbf {y}}^*}right) right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{**}}{{mathbf {z}}^{*}}right) right) _{j}=q_i{z_j^*}^2left( {mathbf {T}}{mathbf {Y}}^*left( frac{{mathbf {q}}}{{mathbf {y}}^*}right) right) _i-q_ileft( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }mathbb {1}^mright) _j=q_i{z_j^*}^2(mathbf {Tq})_i. end{aligned}$$Since (q_i >0) for (i=1,ldots ,n), for Eq. (16) to hold, we should have (mathbf {Tq}={mathbf {0}}^n). Also note that$$begin{aligned} (mathbb {1}^{mn})^{top }{mathbf {x}}^{**}=sum _{i=1}^nsum _{j=1}^{m}x_{i,j}^{**}=sum _{i=1}^nq_isum _{j=1}^mz_j^* =sum _{i=1}^nq_i=1. end{aligned}$$Since the metapopulation model is balanced, it follows that ({mathbf {q}}={mathbf {y}}^*). Thus, (x_{i,j}^{**}=y_i^*z_j^*=x_{i,j}^*) for (i=1,ldots ,n) and (j=1,ldots ,m). This proves the uniqueness of the coexistence equilibrium ({mathbf {x}}^*). (square )
We now give examples of two balanced metapopulation models.

Example 1

It is easy to verify that the network shown in Fig. 3 corresponds to a balanced metapopulation model governed by System (14) with$$begin{aligned} {mathbf {T}} = left[ begin{array}{rrr} 0 &{}quad 1 &{}quad -1\ -1 &{}quad 0 &{}quad 1\ 1 &{}quad -1 &{}quad 0 end{array}right] ; quad {mathbf {B}} = left[ begin{array}{rrr} -1 &{}quad 0 &{}quad 1\ 1 &{}quad -1 &{}quad 0\ 0 &{}quad 1 &{}quad -1 end{array}right] ; end{aligned}$$({mathbf {y}}^*=left( frac{1}{3}, frac{1}{3}, frac{1}{3} right) ^{top }), ({mathbf {z}}^*=left( frac{1}{5}, frac{2}{5}, frac{2}{5} right) ^{top }) and ({mathcal {K}}=text { diag}left( frac{1}{10},frac{3}{10},frac{1}{10}right) ). Note that this metapopulation model is heterogeneous. From Theorem 5, it follows that the species proportions at the unique coexistence equilibrium for this model are given by (x_{i,1}^*=frac{1}{15}) and (x_{i,2}^*=x_{i,3}^*= frac{2}{15}).Figure 3A metapopulation network composed of three patches. Each patch contains a local population composed of three species (1, 2 and 3), in cyclic competition, as shown by the black arrows. The red arrows denote migrations among the patches in the directions shown.Full size image

Example 2

It is easy to verify that the network shown in Fig. 4 corresponds to a balanced metapopulation model governed by System (14) with$$begin{aligned} {mathbf {T}} = left[ begin{array}{rrr} 0 &{}quad 1 &{} quad -1\ -1 &{}quad 0 &{} quad 1\ 1 &{}quad -1 &{}quad 0 end{array}right] ; quad {mathbf {B}} = left[ begin{array}{rrr} 1 &{}quad -1\ 0 &{}quad 1\ -1 &{}quad 0 end{array}right] ; end{aligned}$$({mathbf {y}}^{*}={mathbf {z}}^*=left( frac{1}{3}, frac{1}{3}, frac{1}{3} right) ^{top }) and ({mathcal {K}}=frac{1}{3}text { diag}(mathbb {1}_2)). Note that this metapopulation model is homogeneous. From Theorem 5, it follows that the species proportions at the unique coexistence equilibrium in this case are all given by (x_{i,j}^*=frac{1}{9}) for (i,j= 1,2,3).Figure 4A metapopulation network composed of three patches. Species can migrate from patch 1 to the other two patches and vice versa. However, there exists no migrations between patches 2 and 3.Full size image
StabilityWe now prove the local stability of the unique coexistence equilibrium corresponding to the balanced metapopulation model (14). For the proof, we make use of the same Lyapunov function as in “Neutral stability” section, coupled with LaSalle’s invariance principle51, (52, Section 4.2), (53, pp. 188–189).

Theorem 6

Consider the balanced metapopulation model (14) with coexistence equilibrium ({mathbf {x}}^*in , S^{mn}_{+}).

1.

If the model is heterogeneous, then ({mathbf {x}}^*) is locally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}) in the neighbourhood of ({mathbf {x}}^*). Furthermore, if the model is persistent, then ({mathbf {x}}^*) is globally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}).

2.

If the model is homogeneous and persistent, then as (trightarrow infty ), the solution trajectories converge to a limit cycle satisfying the equation ({dot{x}}_{i,j}=x_{i,j}({mathbf {T}}{mathbf {p}}_{j})_i) with (x_{i,j}=x_{i,k}), for (i=1,ldots ,n) and (j,k=1,ldots ,m).

Proof
Let (x_{i,j}) denote the proportion of species i in patch j. Assuming that ({mathbf {x}}in S^{mn}_{+}), consider the Lyapunov function$$begin{aligned} V({mathbf {x}})=-(mathbf {x^{*}})^{top }text {Ln}left( frac{{mathbf {x}}}{{mathbf {x}}^*}right) . end{aligned}$$
(19)
By Gibbs inequality, V(x) is positive on (S^{mn}_{+}) and is equal to zero only if ({mathbf {x}}={mathbf {x}}^*). Taking the time derivative of V, we have$$begin{aligned} {dot{V}}({mathbf {x}})=-sum _{j=1}^msum _{i=1}^{n}left( frac{x_{i,j}^{*}}{x_{i,j}}right) {dot{x}}_{i,j}. end{aligned}$$From Eq. (14), it follows that$$begin{aligned} {dot{V}}({mathbf {x}})= -sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i+ sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) right) _{j}. end{aligned}$$As in the proof of Theorem 5, it can be verified that$$begin{aligned} sum _{j=1}^msum _{i=1}^nx_{i,j}^*left( mathbf {TY}^*left( frac{{mathbf {p}}_{j}}{{mathbf {y}}^*}right) right) _i=0 end{aligned}$$and$$begin{aligned} sum _{i=1}^nsum _{j=1}^mfrac{x_{i,j}^*}{x_{i,j}} left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) right) _{j}=sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) . end{aligned}$$Thus,$$begin{aligned} {dot{V}}({mathbf {x}})=sum _{i=1}^ny_i^*left( frac{{mathbf {z}}^{*}}{{mathbf {r}}_{i}}right) ^{top }{mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}}{{mathbf {z}}^{*}}right) . end{aligned}$$Since (y_i^* >0) for (i=1,ldots ,n), it follows from Lemma 4 that ({dot{V}}({mathbf {x}})le 0) and ({dot{V}}({mathbf {x}})=0) if and only if ({mathbf {r}}_i=q_i{mathbf {z}}^*) with (q_iin {mathbb {R}}_+), for (i=1,ldots ,n). Thus,$$begin{aligned} x_{i,j}=q_iz_j^*, end{aligned}$$
(20)
for (i= 1,ldots ,n) and (j=1,ldots ,m). Since ((mathbb {1}^{mn})^{top }{mathbf {x}}=1), we obtain$$begin{aligned} sum _{i=1}^nsum _{j=1}^{m}x_{i,j}=sum _{i=1}^nq_isum _{j=1}^mz_j^* =sum _{i=1}^nq_i=1. end{aligned}$$Let ({mathcal {E}}subset S^{mn}_{+}) be the set of all vectors ({mathbf {x}}) for which condition (20) is satisfied with ((mathbb {1}^n)^{top }{mathbf {q}}=1). We now determine the largest subset of ({mathcal {E}}) that is positively invariant w.r.t. System (14). Assume that ({mathbf {x}}) continuously takes values from ({mathcal {E}}) and satisfies System (14). Since ({mathbf {x}}) takes values from ({mathcal {E}}), we have ({dot{x}}_{i,j}=z_j^*{dot{q}}_i). Since ({mathbf {x}}) also satisfies System (14), we have$$begin{aligned} {dot{x}}_{i,j}=x_{i,j}left( {mathbf {T}}{mathbf {p}}_{j}right) _i-left( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }left( frac{{mathbf {r}}_{i}^{*}}{{mathbf {z}}^{*}}right) right) _{j}=q_i{z_j^*}^2(mathbf {Tq})_i-q_ileft( {mathbf {B}}{mathcal {K}}{mathbf {B}}^{top }mathbb {1}^mright) _j=q_i{z_j^*}^2(mathbf {Tq})_i. end{aligned}$$Thus, (z_j^*{dot{q}}_i=q_i{z_j^*}^2(mathbf {Tq})_i) which implies that$$begin{aligned} {dot{q}}_i=z_j^*q_i(mathbf {Tq})_i, end{aligned}$$
(21)
for (i=1,ldots ,n) and (j=1,ldots ,m). We now consider two cases.
Case 1: The model is heterogeneous, i.e., the vector ({mathbf {z}}^*) is not parallel to (mathbb {1}^m).
In this case, Eq. (21) will be satisfied only if (q_i(mathbf {Tq})_i=0) for (i=1,ldots ,n). Since (q_iin {mathbb {R}}_+) for (i=1,ldots ,n), it follows that (mathbf {Tq}={mathbf {0}}^n). Since ((mathbb {1}^n)^{top }{mathbf {q}}=1), we have ({mathbf {q}}={mathbf {y}}^*). This implies that (x_{i,j}=y_i^*z_j^*=x_{i,j}^*) for (i=1,ldots ,n) and (j= 1,ldots ,m). Thus, the largest subset of ({mathcal {E}}) that is positively invariant w.r.t. System (14) consists of just the unique equilibrium ({mathbf {x}}^*in S^{mn}_{+}). By LaSalle’s invariance principle, it follows that the equilibrium ({mathbf {x}}^*) is locally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}) in the neighbourhood of ({mathbf {x}}^*), and globally asymptotically stable w.r.t. all initial conditions in (S^{mn}_{+}) provided that System (14) is persistent.

Case 2: The model is homogeneous, i.e.
({mathbf {z}}^*=frac{1}{m}mathbb {1}^m)

In this case, Eq. (21) takes the form ({dot{q}}_i=frac{q_i}{m}(mathbf {Tq})_i). We have (x_{i,j}=q_iz_j^*=frac{q_i}{m}) and$$begin{aligned} {dot{x}}_{i,j}=frac{{dot{q}}_i}{m}=frac{q_i}{m^2}(mathbf {Tq})_i=x_{i,j}({mathbf {T}}{mathbf {p}}_{j})_i. end{aligned}$$Consequently, the largest subset of ({mathcal {E}}) that is positively invariant w.r.t. System (14) consists of all vectors ({mathbf {x}}(t)in , S^{mn}_{+}) satisfying ({dot{x}}_{i,j}=x_{i,j}({mathbf {T}}{mathbf {p}}_{j})_i) with (x_{i,j}=x_{i,k}) for (i=1,ldots ,n) and (j,k=1,ldots ,m). The proof for Case 2 again follows from LaSalle’s invariance principle. (square )
The above results can be illustrated by simulating System (14) for the metapopulation models shown in Fig. 3 and 4 in Examples 1 and 2, respectively. The results of the simulations are shown in Figs. 5 and 6, respectively.Figure 5Left: Dynamics of the metapopulation model in Fig. 3 for patches 1 and 3 showing asymptotic stability of the coexistence equilibrium. Right: The time evolution of the proportion of species 1 in the three patches.Full size imageFigure 6Left: Dynamics of the metapopulation model in Fig. 4 for patches 1 and 3 showing a limit cycle arising from the neutral stability of the coexistence equilibrium. Right: Time evolution of the proportion of species 1 in the three patches. Note that the dynamics in all patches are the same and thus the three graphs overlap.Full size image More