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    Lipid metabolism of sea urchin Paracentrotus lividus in two contrasting natural habitats

    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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    Gaining insight into the assimilated diet of small bear populations by stable isotope analysis

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    Stability analysis of the coexistence equilibrium of a balanced metapopulation model

    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

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    Mapping habitat suitability for Asiatic black bear and red panda in Makalu Barun National Park of Nepal from Maxent and GARP models

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    Hygienic quality of soil in the Gemer region (Slovakia) and the impact of risk elements contamination on cultivated agricultural products

    SoilContents of risk metals in soilsLands of localities from which soil and plant samples were taken belong to agricultural lands.Soil reaction is one of the factors that most affects the behaviour of heavy metals in soil. Low pH values pose a risk of reduced nutrient intake and increase the availability of heavy metals for plants29,30.The presence of risk elements in the soil was evaluated based on their contents in bioavailable form (mobile forms), determined in soil extracts NH4NO3, and the total contents of risk elements were determined in soil extract by aqua regia (Table 1).Table 1 The contents of risk elements (Cu, Ni, Pb, Cd, Hg, Mn) in soil (mg/kg).Full size tableAccessible heavy metals for plants are those which are present in the soil solution as soluble components or those which are easily dissolved by root exudates31. The highest Cu contents determined in soil extract by NH4NO3, were in the cadastre of Gemerská Poloma (max. 0.390 mg/kg) (Table 1). However, even the highest determined concentration of Cu in its bioavailable form did not exceeded the determined critical value for this element18. Nickel is a beneficial element for plants. Elevated Ni concentrations in soils have a potential negative effect on plants32. Content of bioavailable forms of nickel is lower than the determined critical value in all analysed samples. Cadmium and lead present a risk to agricultural activity in this area. Cadmium in soil is highly bioavailable and has higher mobility in plants compared to other heavy metals. It is easily transported by roots to shoots. In contrast, lead is one of the least mobile heavy metals. It is naturally concentrated in the upper layers of the soil33. The contents of the available forms of cadmium and lead exceed the critical values for these elements. In case of lead, the determined contents are from 0.257 Henckovce to 0.676 Gemerská Poloma. Takáč et al.34 determined in 20 soil samples from the Central Spiš region 7.2–257.6 mg Cu/kg soil and 1.0–84.8 mg Pb/kg in their potentially mobilizable form and 0.4–1.4 mg Cu/kg soil and 4.3–7.1 mg Pb/kg in their mobile form. In comparison with our results, Vilček et al.35 determined a lower content of Cd (0.04), Pb (0.17), Ni (0.15) and higher Cu content (0.48) mg/kg in forms accessible to plants in 16 soil samples from locality Nižná Slaná in the years 2006–2008. However, high concentrations of metals in soil do not necessarily mean the availability of metals for plants36. As a result, extractable Mn is often a better indicator of Mn availability. Mn2+ is generally considered to be bioavailable22. The highest concentration of Mn was measured in soil samples from the cadastre of Nižná Slaná. On the contrary, the lowest concentrations were detected in samples from Gemerská Poloma cadastre, which is the furthest cadastre from the source. No critical limit is set up for manganese according to Slovak legislation, it is not possible to classify these soils as contaminated/uncontaminated. For comparison, the EDTA-extractable content of Mn ranged from 22.7 to 127 mg/kg dry soil (China)29; the mobile concentrations between 0.32 and 202.0 mg/kg and the available concentrations from 5.4 to 126.3 mg/kg (Egypt)37.Based on results of statistical analysis, significant higher content of Cu, Pb and Cd can be stated in samples from Gemerská Poloma cadastre. These soils are classified as gley fluvisols, soils from the other two localities are cambisols (from medium heavy to light) and acid cambisols (Henckovce), cambisols from medium heavy to light and typically acid cambisols (Nižná Slaná). The soil profile of fluvisols is repeatedly disrupted by floods, which often enriches them with a new layer of sludge sediments2.Another method for determination of metal content in soil is mineralisation using aqua regia, which dissolves most of the soil constituents except those strongly bound in silicate minerals. This content is sometimes referred to as pseudototal (determined in aqua regia). In this way, all elements that are likely to become bioavailable in the long term are determined38.Pseudototal contents of risk metals (Table 1) determined in soil extract using aqua regia were higher than their limit value in case of Cu (Gemerská Poloma cadastre), Cd (all cadastres) and Hg (cadastre of Henckovce and Gemerská Poloma).Due to the fact that the hygienic condition of agricultural soils is assessed according to the exceeding of the limit values of at least one risk substance, the monitored plots can be classified as contaminated (Cu  > 60.0, Cd  > 0.7, Hg  > 0.5 mg/kg soil).Manganese is not classified as risk element in Slovak legislation.Tóth et al.39 classified European soils into four categories: (1) no detectable content of HM, (2) the concentration of the investigated element is above the threshold value (Hg 0.5, Cd 1, Cu 100, Pb 60 and Ni 50 mg/kg), but below the lower guideline value (Hg 2, Cd 10, Cu 150, Pb 200 and Ni 100 mg/kg), (3) concentration of one or more element exceeds the lower guideline value but is below the higher guideline value (Hg 5, Cd 20, Cu 200, Pb 750 and Ni 150 mg/kg), (4) samples having concentrations above the higher guideline value.In comparison with the threshold and guideline values, soils in cadastres of Gemerská Poloma (Cu), Henckovce, Nižná Slaná, Gemerská Poloma (Cd, Hg) represent the ecological risk. Threshold and guideline values for Mn were not defined.The Spiš region and the northern part of the Gemer region belong to the most polluted areas in Slovakia in terms of soil contamination due to mining and metallurgical activities that have been carried out here in the past. Soils near the sludge in Nižná Slaná contain 3.17–53.3 (14.2–301, 0.71–20.6, 3.33–177, 12.9–223 and 675–11,510, respectively) mg Cd (Cu, Hg, Ni, Pb and Mn, respectively)/kg of soil14. In loaded area of Dongchuan, (China), contained Cd (Cu, Hg, Ni and Pb, resp.) 0.20–3.57 (45.38–2026, 0.02–0.23, 24.06–95.9 and 6.83–146.6, resp.) mg/kg40. In contrast, in the agricultural area of Punjab of the India, the soil contamination was caused by an excessive use of agrochemicals and polluted irrigation sources. Increased Cu (Pb and Cd) contents were determined in the soil samples: 9.0–48.5 (5.5–9.67 and 0.516–1.58, resp.) mg/kg41.However, in most cases, a large portion of the total element content is not available for immediate uptake by plants. Available forms represent a small proportion of this total content which is potentially available to plants. Availability is affected by many factors, including pH, redox state, macronutrient levels, available water content and temperature29,33,36,38.Indicators of soil contaminationContamination factors and degree of contaminationThe contamination character may be described in a uniform, adequate and standardised way by means of the contamination factor and the degree of contamination. Hakanson24 reported four Contamination degrees of individual metal (({mathrm{C}}_{mathrm{f}}^{mathrm{i}})) – low (({mathrm{C}}_{mathrm{f}}^{mathrm{i}}) < 1), moderate (1 ≤ ({mathrm{C}}_{mathrm{f}}^{mathrm{i}})   More

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    Heterodissemination: precision insecticide delivery to mosquito larval habitats by cohabiting vertebrates

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    An evolutionary perspective on kin care directed up the generations

    ParticipantsData were drawn from the NCDS, which is a nationally representative study that has followed a cohort of participants all born in a single week in the United Kingdom since 1958. Since birth, they have been followed up a total of 11 times at ages 7, 11, 16, 23, 33, 42, 44, 46, 50 and 55. As data on time spent caring for grandchildren is only available from the most recent interview, all analyses here are cross-sectional, with all women included in the sample being aged either 55 or 56 (depending on whether the interview was conducted in 2013 or 2014) and representing the third generation of women in Fig. 1. The sample was limited to women who had at least one parent alive and at least one grandchild (n = 934). Data from the NCDS are available from the UK Data Service, and the participant characteristics shown in Supplementary Table S1.VariablesHours spent helping parents per weekInformation regarding parental caregiving was included as a count variable. In the most recent interviews, participants were asked whether they ever do various activities for their parents (e.g. shopping for them, helping with basic personal needs, giving them lifts, etc.), and if they do, how many hours on average per week do they spend doing so. Any women who reported not helping their parents do any of the activities were coded as helping their parents for zero hours per week.Hours spent caring for grandchildren per monthThe number of hours spent caring for grandchildren per month was also included as a count variable. Women were asked whether they ever look after their grandchildren without the grandchild’s parents being present, and if they do, at what frequency and for how many hours. Women who stated that they did not care for their grandchildren or did so less often than monthly were coded as caring for their grandchildren for zero hours per month. This measure also includes overnight stays.Fecundity status at age 55Fecundity status was derived from information on age, year and reason for last menstrual period, which was collected at ages 44, 50, and 55. Based on this, a binary categorical variable was derived where women were coded as either ‘Still menstruating’ or ‘No longer menstruating’. The latter category comprised of women who were post-menopausal or who had stopped menstruating for another reason, such as a surgical menopause. Women who had stopped menstruating due to menopause or other reasons were grouped together as the direct fitness implications of no longer menstruating are the same, regardless of the reason for it.Control variablesCovariates included were selected based on their expected effect on the woman’s ability to help other family members. As a proxy of socioeconomic status, the age at which the woman left education was included. Employment status was utilised to give an indication of the woman’s time constraints (i.e. if she was employed, it can be expected she had less time to care for kin)24, with women being coded as either employed, unemployed, or other, with the latter category including those who are doing something other than formal employment but do not classify themselves as unemployed (e.g. retired, volunteering, studying, etc.). Self-perceived health was used as a measure of how physically able the woman is to help family members25, and number of grandchildren was also included to adjust for how many grandparenting responsibilities a woman had. We also included information on the mortality status of the woman’s parents (i.e. whether she had both parents alive or not), which was derived from interviews at ages 7, 11, 16, 23, 42, 46, 50 and 55. The focal woman’s mother’s and father’s age at birth (collected in the perinatal interview) were also included to control for the amount of help her parents may need, as older parents would expected to be more in need of assistance. Finally, in models predicting hours spent caring for parents, time spent caring for grandchildren was adjusted for, and vice versa for models where hours spent caring for grandchildren was the outcome.AnalysesTime spent helping parents and caring for grandchildren were both modelled using zero-inflated negative binomial regression (ZINB). This modelling procedure was selected both due to the over-dispersed nature of the data with excess zeros, and because zero-inflated models allow for zeros to be generated through two distinct processes. Here, the model distinguishes between excess zeroes, which occur when the event could not have happened, and true zeros, which occur when there could have been an event. Therefore, the model estimates a binary outcome (does not care versus does care) and a count outcome (the number of hours spent caring). This method is theoretically appropriate, as there are many different reasons people would offer no care to kin: while some people may choose to invest less, for some people the choice is out of their control, with external factors influencing caring behaviours, such as living far away from kin26. In addition to this, ZINB was found to better fit the data than negative binomial regression (Supplementary Table S2).Time spent helping parents was first modelled. A ‘base’ model was first made containing the age the woman left education, employment status, marital status, self-perceived health, number of grandchildren, parent mortality status, age of parents, and time spent caring for grandchildren. Fecundity status was subsequently added, and model fitting then carried out on these two models, utilising their Akaike Information Criterion (AIC) value to understand whether a model including fecundity better fit the data than one without. The model with the lowest AIC value is taken to best fit the data. As AIC values penalise models for complexity, it means the model with the most terms will not automatically be selected as the best. The ΔAIC was also calculated, which is the difference between the candidate models AIC and the AIC value of the best fitting candidate model. If the ΔAIC value is ≤ 2, then it indicates that there is still good evidence to support the candidate model, meaning that a candidate model with a ΔAIC of ≤ 2 is almost as good as the best fitting model. A ΔAIC value of between 4 and 7 is taken to indicate the candidate model has considerably less support, and a ΔAIC of greater than 10 indicates there is no support for the candidate model27. The Akaike weights (wi) were also calculated to evaluate model fit, which give the probability that the candidate model is the best among the set of presented candidate models27. The same procedure was then used to model time spent caring for grandchild per month: a model including just the covariates was first made, but this time adjusting for time spent helping parents rather than time caring for grandchildren, with fecundity status then being added, and model fitting was once again carried out using the methods outlined above. All analyses were carried in R using the zeroinfl function with a negative binomial distribution specified28, and model fitting carried out with the package AICcmodavg29. All visualisations were created using ggplot230. More