Ecology

Systematic palaeontologyOrder Hemiptera Linnaeus, 1758Suborder Sternorrhyncha Amyot et Audinet-Serville, 1843Superfamily Protopsyllidioidea Carpenter, 1931Family Postopsyllidiidae Hakim, Azar et Huang, 2019Genus Megalophthallidion Drohojowska et Szwedo, gen. nov.LSID urn:lsid:zoobank.org:act:A6F71390-9B8E-4A19-8F30-C2A024B6EFB1Type speciesMegalophthallidion burmapateron Drohojowska et Szwedo, sp. nov.; by present designation and monotypy.EtymologyGeneric name is derived from Classic Greek megas (μέγας)—large, ophthalmos (ὀφθαλμός)—an eye and Greek form of generic name Psyllidium. Gender: masculine.Type localityNorthern Myanmar: state of Kachin, Noije bum 2001 Summit Site amber mine in the Hukawng Valley, SW of Maingkhwan.Type stratumLowermost Cenomanian, Upper Cretaceous (‘mid-Cretaceous’).DiagnosisHead capsule with 12 stiff setae on tubercles (18 setae in Postopsyllidium); fore wing without pterostigma (tiny pterostigma, widening of ScP + RA present in Postopsyllidium); vein CuP not thickened distally (distinctly thickened distally in Postopsyllidium); profemur with a row of ventral (ventrolateral) setae (two rows in Postopsyllidium).Megalophthallidion burmapateron Drohojowska et Szwedo, sp. nov.LSID urn:lsid:zoobank.org:act:F3F971F4-AE04-4F41-98B0-9A0A04470625.(Figs. 1A–F, 2A–I).Figure 1Megalophthallidion burmapteron gen. et sp. nov., holotype (MAIG 6687), imago. (A) Photo of body, ventral side; (B) photo of right antennae and (C) drawing of antenna; (D) drawing of body, dorsal side; (E) drawing of thorax structure with sclerites marked: red—pronotum; orange—mesopraescutum; yellow—mesoscutum; light green—mesoscutellum, dark green—mesopostnotum; light blue—metascutum; dark blue—metascutellum; violet—metapostnotum; (F) photo of thorax dorsal side. Scale bars: 0.5 mm (A), 0.2 mm (B–D), 0.1 mm (F).Full size imageFigure 2Megalophthallidion burmapteron gen. et sp. nov., holotype (MAIG 6687), imago. (A) Photo of right fore wing; (B) photo of right wings; (C) photo of antenna and proleg; (D) photo of proleg and mesoleg, and (E) photo of femur of proleg, and (F) photo of right metatarsus and left mesotarsus in the background, and (G) photo of right mesotarsus of mesoleg, and (H) Photo of tarsi; (I) photo of male genital block. Scale bars: 0.5 mm (A–D), 0.2 mm (B,E,F,H), 0.1 mm (G,I).Full size imageMaterialHolotype, number MAIG 6687 (BUB 96), deposited in Museum of Amber Inclusions (MAIG), University of Gdańsk, Poland. Imago, a complete and well-preserved male. Piece of amber 8 × 6 × 3 mm, cut from larger lump, polished flat on both sides.Type localityNorthern Myanmar: state of Kachin, Noije bum 2001 Summit Site amber mine in the Hukawng Valley, SW of Maingkhwan.Type stratumLowermost Cenomanian, Upper Cretaceous (‘mid-Cretaceous’).DiagnosisAs for the genus with the following additions: three ocelli distinct, antennomere IX the longest, about as long as pedicel, antennomeres III–VII and XI of similar length, antennomere XII the shortest, subconically tapered in apical portion. Paramere lobate, ventral margin with acute, small process, apical and dorsal margins rounded. Aedeagus geniculately bent at base, directed dorsally, tapered apicad.DescriptionMale (Figs. 1A–F, 2A–I). Head with compound eyes distinctly wider than pronotum (Fig. 1D–F). Compound eyes subglobular, protruding laterally. Vertex short in midline, about 2.5 times as wide as posterior margin and as long in middle; trapezoidal, anterior margin slightly arched, lateral margins diverging posteriad, posterior margin shallowly arched, disc of vertex with distinct setae on large tubercles: four setae at posterior margin, two at anterior angles of compound eyes, two medial, over the median ocellus. Three ocelli present, median ocellus distinct, visible from above, lateral ocelli near anterior angles of compound eyes. Frons about as wide as long in midline, two rows of setae on tubercles, upper row at level of median ocellus, lower one, below half of compound eye height. Clypeus, elongate, triangular, in lower portion roof-like; two setae on tubercles near upper margin. Genae very narrow. Rostrum reaching slightly beyond mesocoxae, apical segment slightly shorter than subapical one, darker. Antennae bases placed at lower margin of compound eyes; antennal fovea elevated; scapus shorter than pedicel, cylindrical; pedicel cylindrical; antennomeres IIIrd–VIIth and XIth of similar length, VIIIth slightly longer than VIIth, as long as Xth antennomere, IXth the longest, XIIth the shortest, tapered apically; rhinaria absent.Thorax (Fig. 1D–F): pronotum quadrangular, about as long as mesothorax; pronotum with anterior and posterior margins parallel, merely arcuate, disc with transverse groove in the median portion, lateral margins slightly arcuate, two distinct setae on tubercles in anterolateral angle, two setae on tubercles anterior margin at distance1/3 to median line, three distinct setae on tubercles in posterolateral angles. Mesopraescutum subtriangular, with apex widely rounded, about 0.4 times as wide as pronotum, about 0.4 times as long as wide, delicately separated from mesoscutum. Mesoscutum as wide as pronotum at widest point, distinctly narrowed medially, anterior angles rounded, anterolateral margin sigmoid, lateral angle acute, posterior angles wide, posterior margin V-shape incised, posterolateral areas of mesoscutum disc declivent posteriorly; disc with two setae on tubercles, at 1/3 of mesoscutum width. Mesoscutellum about as long as wide, diamond-shape, anterior and lateral angles acute, posterior angle rounded. Mesopostnotum in form of transverse band, slightly widened in median portion. Metascutum narrower than mesoscutum, anterior angles widely rounded, lateral angles acute, anterolateral margin concave, posterior margin arcuate, with deep median arcuate incision. The suture between metascutum and metascutellum weakly visible, metascutellum subtriangular, longer than wide at base.Parapteron with three distinct setae.Fore wing (Fig. 2A,B) membranous, narrow, elongate, about 3.5 times as long as wide, widest at 2/3 of length. Anterior margin merely arcuate, slightly bent at very base, anteroapical angle widely arcuate, apex rounded, posteroapical angle widely arcuate, tornus arcuate, claval margin straight, with incision between terminals of Pcu (claval apex) and A1. Stem ScP + R + MP + CuA slightly arcuate, very short stalk ScP + R + MP + CuA leaving basal cell, stem ScP + R oblique, straight, forked in basal half of fore wing length, branch ScP + RA, oblique, reaching anterior margin slightly distally of half of fore wing length, slightly distally of ending of CuA2 branch; branch RP slightly arcuate, a little more curved in basal section, reaching margin at anteroapical angle; stalk MP + CuA slightly shorter than basal cell; stem MP almost straight, forked in apical half of fore wing, at about 2/3 of fore wing length, with three terminals reaching margin between apex and posteroapical angle; stem CuA shorter than branches CuA1 and CuA2, about half as long as branch CuA1; claval vein CuP weak at base, not thickened distally; claval vein Pcu straight, claval vein A1 straight. Basal cell present, subtriangular, about twice as long as wide, basal veinlet cua-cup oblique, no other veinlets present; cell r (radial) very long, longer than half of fore wing length; cell m (medial) the shortest, shorter than cell cu (areola postica). Margins of fore wing with fringe of long setae, starting on costal margin near base of fore wing, ending at level of middle of cell cu; longitudinal veins with distinct, scarcely but evenly dispersed, movable setae; terminal section of CuP with two setae; costal margin with row of short, densely distributed setae, apical margin, tornus and claval margin with rows of scaly setae.Hind wing (Fig. 2B) membranous, shorter than fore wing, 3.23 times as long as wide. Costal margin bent at base, then almost straight up to the level of ScP + RA end and wing coupling lobe, then straight to anteroapical angle, anteroapical angle widely arcuate, apex arcuate, posteroapical angle arcuate, tornus straight, claval margin merely arcuate, posteroclaval angle angulate; stem ScP + R + MP bent at base, then straight, stem ScP + R short, branch ScP + RA short, about as long as stem ScP + R, branch RP arcuate basally than straight, reaching apex; stem MP arcuate, forked slightly distad CuA1 terminus level, branch MP1+2 slightly arcuate, reaching margin at posteroapical angle, branch MP3+4 straight, reaching tornus; stem CuA slightly bent at base, then straight, forked slightly distad ScP + R forking, branch CuA1 arcuate, branch CuA2 short, straight, slightly oblique, reaching tornus; claval vein CuP weak, visible only at base, claval vein Pcu slightly arcuate; wing coupling apparatus (fold) with a few short setae.Legs slender, relatively long, profemora armed (Fig. 2C–H). Procoxa as long as profemur, narrow, flattened. Protrochanter scaphoid, elongate, with long apical and subapical setae. Profemur flattened laterally, about as long as protibia, ventrally armed with four large setae on elevated plinths; dorsal margin with row of short, decumbent setae. Protibia narrow, rounded in cross section, covered with short setae, a few longer setae in distal portion. Protarsus—single, long tarsomere, plantar surface with row of semi-erect setae; tarsal claws long, straight, directed ventrally, no arolium nor empodium.Mesocoxa elongate, narrow, slightly flattened. Mesotrochanter scaphoid. Mesofemur slender, flattened laterally, dorsal margin with short setae. Mesotibia subequal to mesofemur, slender, covered with setae, two apical setae slightly thicker and longer. Mesotarsus with three tarsomeres, basimesotarsomere the longest, shorter than cumulative length of mid- and apical mesotarsomere, plantar margins with setae, two apical setae slightly longer and thicker; midmesotarsomere the shortest, 1/3 of basimesotarsomere length, a few setae on plantar surface; apical tarsomere shorter than basimesotarsomere, twice as long as midmesotarsomere, plantar surface with a few, scarcely dispersed setae, tarsal claws long, narrow, directed ventrally, no arolium nor empodium.Metacoxa conical, narrow. Metatrochanter scaphoid, elongate. Metafemur slender, laterally flattened, longer than mesofemur, dorsal margin with row of short setae. Metatibia, long, slender, 1.6 times as long as metafemur, with suberect setae of different size, two larger and longer and two shorter setae subapical setae. Metatarsus slightly less than half of metatibia length, with three tarsomeres, basimetatarsomere the longest, more than twice as long as apical metatarsomere, 1.5 times as long as combined length of mid- and apical metatarsomere, plantar surface with scarce decumbent setae; mid metatarsomere the shortest, 1/4 of basimetatarsomere length, plantar surface with a few setae, two apical ones slightly thicker; apical metatarsomere about 0.4 of basimetatarsomere length, with scarcely dispersed setae on along plantar surface; tarsal claws, long, slender, other pretarsal structures absent.Abdomen (Fig. 1F) narrowly attached to thorax, tergite segment shorter, 2nd tergite distinctly longer, 3rd to 8th tergites of similar length; pygofer narrowing apicad, ventral margin strongly elongated posteriorly; anal tube short, directed posterodorsad, anal style shorter than anal tube. Paramere lobate, ventral margin with acute, small process, apical and dorsal margins rounded. Aedeagus (Fig. 2I) geniculately bent at base, directed dorsad, tapered apicad.Female. Unknown.Megalophthallidion sp. (5th instar nymph)(Figs. 3A–D, 4A–F)Figure 3Megalophthallidion sp. (MAIG 6688), nymph. (A) Photo of body, dorsal side and (B) drawing of body dorsal side; (C) photo of body dorsal side and (D) drawing of body ventral side. Scale bars: 0.5 mm (A–D).Full size imageFigure 4Megalophthallidion sp. (MAIG 6688), nymph. Photo of clypeus and (B) drawing of clypeus; (C) photo of proleg, and (D) photo of mesoleg, and (E) photo of metaleg; (F) photo of posterior part of abdomen ventral side. Scale bars: 0.1 mm (A–F).Full size imageMaterialNymph, 5th instar, MAIG 6688 (BUB 1799), deposited in Museum of Amber Inclusions (MAIG), University of Gdańsk, Poland. Piece of amber 13 × 6 × 2 mm, cut from larger lump, polished flat on one side, more convex on the other.Diagnostic charactersThe nymph of Megalophthallidion gen. nov. is similar in general body shape to the only known fossil protopsyllidioidean nymph described from Lower Cretaceous Lebanese amber—Talaya batraba Drohojowska et Szwedo, 2013. The nymph of Talaya batraba is 2nd or 3rd instar, therefore some features are difficult to compare with this last instar nymph of Megalophthallidion gen. nov. The morphological states observed in those two specimens are: head covered with strongly expanded disc and expanded disc of pronotum, however shapes and ratios of these structures differ; compound eyes on ventral side of head, shifted laterad (ommatidia on cones in T. batraba, while ventroposterior expansions are present in Megalophthallidion gen. nov.); compound eyes visible from above as short, stout cones in fissure between posterior margin of disc (hypertrophied vertex) and anterior margin of pronotum (compound eyes (?) are visible on dorsal side of Permian Aleuronympha bibulla Riek, 1974); in Megalophthallidion gen. nov. rostrum reached mesocoxa, while in Talaya batraba distinctly exceeds length of the body; abdomen with 9 segments; tergites of abdominal segments 5th–9th expanded posterolaterad in form of fan-like expansion; 9th abdominal segment short, placed ventral; anal tube short, cylindrical, epiproct (?) globular.DescriptionNymph, 5th instar (Figs. 3A–D, 4A–F). Body oval shaped, dorso-ventrally flattened, 1.5 times longer than wide with segmentation visible; on the ventral side slightly concave. Length of body c. 1.56 mm long, outline, in dorsal view, maximum width of body 0.94 mm; length of head and pronotum (cephaloprothorax) c. 0.46 mm in midline, width c. 0.83 mm; cumulative length of mesonotum + metanotum c. 0.25 mm; abdomen c. 0.8 mm long. Dorsal side (Fig. 3A,B) with distinct median line (ecdysial line), not reaching anterior or posterior margin of the body, the line distinctly roof-like in abdominal portion. Anterior margin of head (cephaloprothorax) disc arcuate, lateral angles rounded; anterior margin of pronotum arcuate, lateral margins arcuately diverging posteriad, posterior margin distinctly arcuate, anterior angles widely rounded, posterior angles acutely rounded, disc elevated, convex, lateral portions declivitous; the fissure between posterior margin of head disc and anterior margin of pronotum narrow, widened medially, with stalked compound eyes popping out.Head partly separated from prothorax, wide in ventral view. Bases of antennae protruding anterolaterally, wide, anterior margin arcuate, with a small lump extending anteriorly connecting margin with vertex expansion. Suture separating anteclypeus and postclypeus visible in ventral aspect (Fig. 4A,B). Postclypeus about three times as long as wide, oval, slightly swollen, without any setae; weak traces of salivary pump muscle attachments visible. Anteclypeus about as long as postclypeus, widened in upper section below clypeal suture, convex, carinately elevated in lower section, with sides distinctly declivitous, clypellus long, carinately elevated. Lora (mandibulary plates) distinct, separated from anteclypeus by shallow suture, with upper angles at half of postclypeus length, lower angles at half of anteclypeus length, about as wide as half of postclypeus width. Maxillary plates narrow. Genal portion of head enlarged, medial portion arcuately convex; lateral sections narrowing laterally, terminally encircling bases of compound eyes. Antennae short (Fig. 3C,D), placed in front of genal portion. Antennal flagellum indistinctly subdivided into four segments. Rostrum (Fig. 4A,B) three-segmented, 0.2 mm long, with apex reaching apex of mesocoxae; apical segment about 2.5 times as long as subapical one.No lateral sclerites on meso- and metathorax, only one plus one large medial sclerite on both meso- and metathorax. Mesothoracic and metathoracic wing pads distinct, wide, subtriangular, with posterior apices directed posteriorly; lateral portions of mesothoracic wing pads arcuate. Fore wing pad 0.6 mm long, with small, straight humeral lobe, forming a right angle, not protruding anteriorly. Mesothoracic tergites slightly larger than metathoracic segments (respectively c. 0.14 mm and c. 0.12 mm long in midline, 0.26 mm and 0.27 mm in lateral lines); mesothoracic tergum with distinct median elevation (low double crest with ecdysial line in between), slightly wider than long in midline, anterior margin arcuate, lateral margins straight, subparallel, posterior margin concave. Metathoracic wing pad apex slightly exceeding mesothoracic wing pad. Metathoracic tergum wider than long, slightly shorter than mesothoracic tergum, with distinct elevation in the middle.Legs relatively long (Figs. 3C,D, 4C–E). Coxae of legs placed near the median axis of the body. Prolegs: procoxal pit with margins elevated, procoxa conical (c. 0.1 mm long), protrochanter scaphoid, about as long as procoxa, profemur c. 0.13 mm long, slightly flattened laterally, merely thickened, protibia longer than profemur, c. 0.23 mm long; tarsus shorter than protibia, basiprotarsomere about as long as apical protarsomere, the latter with distinct tarsal claws, and wide arolium. Mesoleg similar to proleg, mesocoxa conical (c. 0.1 mm long), mesotrochanter scaphoid, mesofemur (c. 0.13 mm) slightly flattened laterally, mesotibia slightly longer than mesofemur (c. 0.18 mm), mesotarsus slightly shorter than mesotibia, three-segmented, basimesotarsomere the longest (c. 0.07 mm), about as long as combined length of mid- and apical mesotarsomeres (c. 0.04 mm respectively), arolium wide, tarsal claws distinct. Metaleg: metacoxa conical (c. 0.1 mm), metatrochanter scaphoid, about as long as metacoxa (c. 0.12 mm). Metafemur (c. 0.17 mm) slightly more thickened than pro- and mesofemur, metatibia slightly longer (0.19 mm) than pro- and mesotibiae. Metatarsus three-segmented: basimetatarsomere about as long (0.08 mm) as combined length of mid- and apical metatarsomeres (0.04 mm respectively), arolium lobate, wide, tarsal claws distinct, widely spread.Abdomen (Fig. 3A–D) 9-segmented, narrow at base, widening fan-shape posteriorly, 1st segment visible from above, segmentation visible, abdominal terga 5th–9th expanded posterolaterally. Tergites carinately elevated in the middle, separated by ecdysial line. 1st sternite visible in ventral view, sternites 2nd–4th fused medially, sternites 5th–9th separated; 9th abdominal segment short (Fig. 4F), placed ventrally, under tergal expansion; anal tube short, cylindrical, epiproct (?) globular. More

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The bee health modelThe conceptual model of the interactions of various stressors with honey bee health is described by the following system of ordinary differential equations (ODEs)$${{tau }_{{HB}}dot{x}}_{{HB}}= {-{delta }_{{HB}}x}_{{HB}}+{g}_{{TC}}left({x}_{{TC}}right)+{g}_{{VA}}left({x}_{{VA}}right)+{g}_{{VI}}left({x}_{{VI}}right) \ +{bar{f}}_{S,C}left({u}_{S},{u}_{C},{x}_{{TC}},{x}_{{VA}}right)+{bar{f}}_{P}left({u}_{P},{x}_{{TC}}right)+{underline{f}}_{{HB}}left({u}_{T}right)$$
(1)
$${{tau }_{{TC}}dot{x}}_{{TC}}={-{delta }_{{TC}}x}_{{TC}}+{g}_{{HB}}left({x}_{{HB}}right)$$
(2)
$${{tau }_{{VA}}dot{x}}_{{VA}}={-{delta }_{{VA}}x}_{{VA}}+{h}_{{VA}}left({{x}_{{HB}},x}_{{TC}},varepsilon {x}_{{VI}}right)+{underline{f}}_{{VA}}left({u}_{T}right)$$
(3)
$${{tau }_{{VI}}dot{x}}_{{VI}}={-{delta }_{{VI}}x}_{{VI}}+{h}_{{VI}}left({{x}_{{HB}},x}_{{TC}},{varepsilon x}_{{VI}}right)$$
(4)
for the state variables ({x}_{{HB}}) representing honey bee health, ({x}_{{TC}}) the stress due to toxic compounds (e.g., neonicotinoid insecticides), ({x}_{{VA}}) the stress due to parasites (e.g., V. destructor) and ({x}_{{VI}}) the stress due to pathogens (e.g., DWV). The system includes the effects of external inputs as sugar ({u}_{S}), pollen ({u}_{P}), absolute deviation from desired temperature ({u}_{T}) and sub-optimal temperature ({u}_{C}). All the inputs and possible parameters are non-negative; the coefficients (tau) denote the time constants; the coefficients (delta) denote the self-regulation parameters; (varepsilon) in the last two equations allows to account for pathogens that can ((varepsilon , > , 0)) or cannot ((varepsilon=0)) impair the immune system (through link m in Fig. 1). We assume that the functions (g) are smooth, bounded, positive, convex and decreasing to 0; the functions (bar{f}) are smooth, bounded, non-negative, concave and increasing with respect to (w.r.t.) (u) arguments (vanishing only when the first (u) argument vanishes) while convex and decreasing to 0 w.r.t. (x) arguments; the functions ({underline{f}}) are smooth, bounded, non-positive and decreasing (vanishing only when (u=0)); the functions (h) are smooth, bounded, positive, convex and decreasing to 0 w.r.t. the first argument while concave and increasing w.r.t. all the other arguments. For a detailed description of the various functions, together with a summary of the biological effects they account for and a reference to the conceptual model in Fig. 1, see Supplementary Table 3.Structural analysis of the bee health modelWe describe here the structural considerations and computations that yield the structural influence matrix for the honey bee health system.The structural influence matrix (M) is defined as follows. (M) is a symbolic matrix with entries ({M}_{{ij}}) chosen among: +,−,0,?, according to the criteria described below. Consider an equilibrium point (bar{x}) and a constant perturbation (u) applied on the (j)-th system variable (small enough not to compromise the stability of the equilibrium). The equilibrium value will be modified as (bar{x}+delta bar{x}). Consider the sign of the perturbation of the (i)-th variable, (delta bar{{x}_{i}}). Then ({M}_{{ij}}) = + if (delta bar{{x}_{i}}) always has the same sign as (u); ({M}_{{ij}}=) − if (delta bar{{x}_{i}}) always has the opposite sign as (u); ({M}_{{ij}}) = 0 if always (delta bar{{x}_{i}}=0); regardless of the system parameters. Conversely, if the sign does depend on the system parameters, we set ({M}_{{ij}}) = ?.In this section we prove that the influence matrix of the honey bee health system is structurally determined, i.e., there are no “?”‘ entries in (M).We start with the following proposition.
Proposition 1
Assume that a matrix
(J)
is Hurwitz stable (i.e., all its eigenvalues have negative real part) and has the sign pattern
$${sign}left(Jright)=left[begin{array}{cccc}- & – & – & -\ – & – & 0 & 0\ – &+& – &+\ – &+& 0 & -end{array}right]$$
Then, the sign pattern of
({adj}left(-Jright))
, the adjoint of
(-J)
, is
$${sign}left({adj}left(-Jright)right)=left[begin{array}{cccc}+& – & – & -\ – &+&+&+\ – &+&+&+\ – &+&+&+end{array}right]$$
Proof To prove the statement, we just change the sign of the first variable, hence we change sign to the first row and column of matrix (J). The resulting matrix (M) is such that$${sign}left(Mright)=left[begin{array}{cccc}- &+&+&+\+& – & 0 & 0\+&+& – &+\+&+& 0 & -end{array}right]$$We observe that (M) is a Metzler matrix, namely, all its off-diagonal entries are non-negative. Moreover, the matrix is Hurwitz stable. Then, we can proceed as in the proof of Proposition 4 in a previous report16. Given a Metzler matrix that is Hurwitz stable, its inverse has non-positive entries; hence, the inverse of (-M) has non-negative entries: ({left(-Mright)}^{-1}ge 0) elementwise. Moreover, we observe that(,M) is an irreducible matrix, i.e., there is no variable permutation that brings the matrix in a block (either upper or lower) triangular form. This implies that the inverse of (-M) has strictly positive entries: ({left(-Mright)}^{-1} , > , 0) elementwise. Also, stability implies that the determinant of (-M) is positive: ({det }left(-Mright) , > , 0). Then, ({adj}left(-Mright)={left(-Mright)}^{-1}{det }left(-Mright) > 0), hence the adjoint of (-M) is also positive elementwise. To consider again the original sign of the variables, we change sign to the first row and column of ({adj}left(-Mright)), and we get the signature above for ({adj}left(-Jright)).The next step is the characterization of the structural influence matrix, which corresponds to the sign pattern of the adjoint of the negative Jacobian matrix in Proposition 1.To this aim, we first consider the linearized system and write it in a matrix-vector form$$dot{x}left(tright)={Jx}left(tright)+{e}_{j}u$$where (dot{x}left(tright)) is the time derivative of the four-dimensional vector (xleft(tright)) and ({e}_{k}), (k={{{{mathrm{1,2,3,4}}}}}), is an input vector, constant in time, with a single non-zero component, the (k)-th, equal to 1, while the scalar (u , > , 0) is the magnitude of the input. We wish to assess the (i)-th component of (xleft(tright)), ({x}_{i}left(tright)={e}_{i}^{T}xleft(tright)). If (J) is Hurwitz, as assumed, the steady-state value of variable ({x}_{i}left(tright)) due to the input perturbation ({e}_{k}) applied to the equation of variable ({x}_{k}left(tright)) is achieved for$$0=Jbar{x}+{e}_{k}u,$$namely$${x}_{i}=-{e}_{i}^{T}{J}^{-1}{e}_{k}u,$$which implies that the sign of the steady-state value ({bar{x}}_{i}) of variable ({x}_{i}) due to a persistent positive input acting on the (k)-th equation has the same sign as ({(-{J}^{-1})}_{{ik}}), the (left(i,kright)) entry of matrix ({left(-Jright)}^{-1}). Since we assume Hurwitz stability, we have that ({det }left(-Jright)) is positive, hence the sign pattern of the inverse ({left(-Jright)}^{-1}) corresponds to the sign pattern of the adjoint, ({adj}left(-Jright)). In fact, ({adj}left(-Jright)={left(-Jright)}^{-1}{det }left(-Jright)).We next consider the nonlinear system under investigation, which we write in the form$$dot{x}left(tright)=fleft(xleft(tright)right)$$and without restriction we assume that the zero vector is an equilibrium point: (0=fleft(0right)). This condition can be always achieved, without loss of generality, by a translation of coordinates. We also consider a stable equilibrium: we assume that the linearized system at the equilibrium is asymptotically stable, namely its Jacobian (J), which has the sign pattern considered in Proposition 1 above, is Hurwitz. We also assume that a constant input perturbation of magnitude (u) is applied to the system, affecting the (k)-th equation, i.e.,$$dot{x}left(tright)=fleft(xleft(tright)right)+{e}_{k}u,$$and that the perturbation is small enough to keep the state in the domain of attraction of the considered equilibrium. Due to this perturbation, a new steady state (bar{x}left(uright)) is reached that satisfies the condition$$0=fleft(bar{x}left(uright)right)+{e}_{k}u$$To determine the sign of the new equilibrium components (bar{x}left(uright)), we consider this new equilibrium vector as a function of (u) in a small interval (left[0,{x}_{{MAX}}right]). Adopting the implicit function theorem yields$$frac{d}{{dx}}bar{x}left(uright)=-J{left(uright)}^{-1}{e}_{k}u,$$where we have denoted by (Jleft(uright)) the Jacobian matrix computed at the perturbed equilibrium (bar{x}left(uright)). Hence, for (u) small enough, the sign of the derivatives of the entries of the new, perturbed equilibrium are, structurally, the same as those in the (k)-th column of matrix (-{J}^{-1}). Since, by construction, (xleft(0right)=0), this is also the sign of the elements of vector (bar{x}left(uright)), for (u) in the interval (left[0,{x}_{{MAX}}right]).We have therefore proved that the original nonlinear system describing honey bee health admits the following structural influence matrix:$$left[begin{array}{cccc}+& – & – & -\ – &+&+&+\ – &+&+&+\ – &+&+&+end{array}right]$$System equilibriaThe results concerning the system equilibria were obtained through a standard analytical treatment of the nonlinear equations describing the equilibrium conditions of the system of differential Eqs. (1), (2), (3), (4). A detailed description of methods is reported in Supplementary Methods.Laboratory experiments using honey beesTo confirm the bistability of the system representing honey bee health as affected by multiple stressors, we used data from several survival experiments, carried out in a laboratory environment according to the same standardized method, over a 6-year period (Source data file).All experiments involved Apis mellifera worker bees, sampled at the larval stage or before eclosion, from the hives of the experimental apiary of the University of Udine (46°04′54.2″N, 13°12′34.2″E). Previous studies indicated that the local bee population consists of hybrids between A. mellifera ligustica and A.m. carnica62,63. Ethical approval was not required for this study.We considered experiments on the effect of the following stressors: infection with 1000 DWV genome copies administered through the diet before pupation, feeding with a 50 ppm nicotine in a sugar solution at the adult stage, exposition to a sub-optimal temperature of 32 °C at the adult stage. All experiments were replicated 3 to 13 times, using, in total, the number of bees reported in Table 1.For the artificial infection with DWV, we collected with soft forceps individual L4 larvae from the brood cells of several combs. Groups of 20–30 of such larvae were placed in Petri dishes with an artificial diet made of 50% royal jelly, 37% distilled water, 6% glucose, 6% fructose, and 1% yeast. 25 DWV copies per mg of diet were added or not to the diet according to the experimental group (note that a bee larva at this stage consumes about 40 mg of larval food per day, thus the viral infection per bee was 1000 viral copies). After 24 h larvae were transferred onto a piece of filter paper to remove the residues of the diet and then into a clean Petri dish, where they were maintained until eclosion. At the day of emergence, bees were transferred to plastic cages in a thermostatic cabinet, where they were kept until death. The DWV extract was prepared according to previously described protocols64 and quantified according to standard methods.For the treatment with nicotine, 10 µL of pure nicotine were added to 200 g of the sugar solution used for the feeding of the caged bees, to reach the concentration of 50 ppm.Finally, to expose bees to a 32 °C temperature, the plastic cages with the adult bees were kept in a thermostatic cabinet whose temperature was set accordingly.To monitor the survival of the adult bees treated as above, they were maintained from eclosion until death in plastic cages in a dark incubator at 34.5 °C (or 32 °C, according to the experiment), 75% R.H.; two syringes were used to supply a sugar solution made of 2.4 mol/L of glucose and fructose (61% and 31%, respectively) and water, respectively; dead bees were counted daily.All the results of these experiments are reported in Source data file.All experiments were carried out during the summer months, from June to September for 6 consecutive years. Previous data indicated that, in this region, virus prevalence increases along the active season starting from very low levels in spring and reaching 100% of virus-infected honey bees by the end of the summer; virus abundance in infected honey bees follows a similar trend28. For this reason, it can be assumed that bees sampled early in the season are either uninfected or they bear only a very low viral infection level, whereas bees sampled later in the season are likely to be virus-infected, bearing moderate to high viral infections. To confirm this assumption and identify a method for filtering our data according to viral infection, we assessed viral infection in a sample of bees from the untreated control group of each experiment, by means of qRT-PCR. According to standard practice, we assumed that Ct values below 30 are indicative of an effective viral infection, whereas Ct above that threshold are more likely in virus negative bees. As expected, we found that virus prevalence increases from June to September (Supplementary Figure 1a), in such a way that up to mid July only the minority of bees can be considered as viral infected (Supplementary Figure 1b). Therefore, we classified as “early” all the samples collected up to mid July and assumed that viral infection in those samples was low; on the other hand, samples collected from mid July till September were classified as “late” and we assumed that viral infection in those samples was high.qRT-PCR analysis of viral infection was carried out as follows. At the beginning of every experiment (i.e., at day 0), two to five bees for each replication were sampled in liquid nitrogen and transferred in a −80 °C refrigerator. After defrosting of samples in RNA later, the gut of each honey bee was eliminated to avoid the clogging of the mini spin column used after. The whole body of sampled bees was homogenized using a TissueLyser (Qiagen®, Germany). Total RNA was extracted from each bee according to the procedure provided with the RNeasy Plus mini kit (Qiagen®, Germany). The amount of RNA in each sample was quantified with a NanoDrop® spectrophotomer (ThermoFisher™, USA). cDNA was synthetized starting from 500 ng of RNA following the manufacturer specifications (PROMEGA, Italy). Additional negative control samples containing no RT enzyme were included. DWV presence was verified by qRT-PCR considering as positive all samples with a Ct value lower than 30. The following primers were adopted: DWV (F: GGTAAGCGATGGTTGTTTG, R: CCGTGAATATAGTGTGAGG65). 10 ng of cDNA from each sample were analyzed using SYBR®green dye (Ambion®) according to the manufacturer specifications, on a BioRad CFX96 Touch™ Real time PCR Detector. Primer efficiency was calculated according to the formula (E={10}^{left(-1/{{{{{{rm{slope}}}}}}}-1right)*100}). The following thermal cycling profiles were adopted: one cycle at 95 °C for 10 min, 40 cycles at 95 °C for 15 s and 60 °C for 1 min, and one cycle at 68 °C for 7 min.Individual survival and colony stabilityTo investigate how the death rate of forager bees affects colony growth, a compartment model of honey bee colony population dynamics was proposed50. This model showed that death rates over a critical threshold led to colony failure. Here we modified this model to include premature death of bees at younger age, as predicted by our model of individual bee health in the presence of an immuno-suppressive virus. We show that the critical threshold found in the previously published model50 becomes a decreasing function of the death rate of the younger individuals, so that premature death (and, in turn, immune-suppression) favors colony collapse.In more details, we first summarize the results of the previously published model50 where two populations (F) (forager) and (H) (hive) of bees are considered and where conditions are provided on the mortality (m) of (F) under which the whole population collapses: namely, mathematically stated, the system admits the zero equilibrium only. Here we extend the model partitioning (H) in two categories, (Y) (younger hive bees) and (O) (older hive bees), asintroducing an early mortality factor (n) for the young population, showing how such a factor worsens the collapsing condition.The previously published model50 concerns the interaction between hive bees (H) and forager bees (F) and is described by the ODEs$$dot{H}=Lfrac{H+F}{w+H+F}-Hleft(alpha -sigma frac{F}{H+F}right)$$$$dot{F}=Hleft(alpha -sigma frac{F}{H+F}right)-{mF}.$$Above, (L) is the queen’s eggs laying rate, (w) is the rate at which (L) is reached as the total population (H+F) gets large, (alpha) is the maximum rate at which hive bees become forager bees in the absence of the latter, (sigma) measures the reduction of recruitment of hive bees in the presence of forager bees and, finally, (m) is the death rate of forager bees (while the death rate of hive bees is assumed to be negligible).We first summarize the main results in terms of a threshold value for (m) in view of colony collapse, as our further analysis will follow a similar approach. All the parameters are assumed to be positive.The search for the equilibria of the above ODEs leads to the unique nontrivial equilibrium (beyond the trivial one)$$bar{H}=frac{L}{{mJ}}-frac{w}{1+J}$$$$bar{F}=Jbar{H}$$for$$J=Jleft(mright):=frac{alpha -sigma -m+sqrt{{left(alpha -sigma -mright)}^{2}+4malpha }}{2m}.$$Note that (J) is alway positive (and, moreover, it is independent of (L) and (w)). It follows that (bar{F}) and (bar{H}) have the same sign, so that the existence of the nontrivial equilibrium is equivalent to (bar{F}+bar{H} , > , 0). It is not difficult to recover that$$bar{F}+bar{H}=frac{w}{m}left(lfrac{1+J}{J}-mright)$$where (l:=L/w) is introduced for brevity. Then if (alpha le l) we get$$bar{F}+bar{H}=frac{w}{m}left(lfrac{1+J}{J}-mright)ge frac{w}{m}left(alpha frac{1+J}{J}-mright)=frac{w}{m}left(sigma+{mJ}right) , > , 0,$$with the last equality following from$$alpha -sigma frac{J}{1+J}-{mJ}=0,$$which in turn comes from annihilating the right-hand side of the second ODE and from using (J=bar{F}/bar{H}) while searching for equilibria. We conclude that, independently of (m), the colony never collapses if the recruitment rate (alpha) of forager bees is sufficiently low.Hence, we assume (alpha , > , l). Observe that$$bar{F}+bar{H}iff l , > , Jleft(m-lright)$$guarantees existence whenever (m) is sufficiently small, viz. (mle l). Assume then (m , > , l), so that the above condition reads$$J , < , frac{l}{m-l}$$leading to the threshold condition$$m , < , bar{m}:=frac{l}{2}frac{alpha+sigma+sqrt{{left(alpha -sigma right)}^{2}+4sigma l}}{alpha -l}$$by using the definition of (J), see Eq. (2) the previously published model50.A standard stability analysis shows that, assuming (alpha,m , > , l), the nontrivial equilibrium is (globally) asymptotically stable whenever it exists (positive), i.e., whenever (m , < , bar{m}). Otherwise, the only (globally) attracting equilibrium is the trivial one, corresponding to colony collapse (see Fig. 5 for the previously published model50 or Fig. 4 for (n=0)). In the mathematical jargon, the disappearance of the positive equilibrium, for (m) exceeding (bar{m}), is referred to as a transcritical bifurcation43.Now, in view of the outcome of the analysis of our model of individual bee health, we introduce a mortality term for the younger bees. As forager bees are recruited from adult hive bees, we divide the class of hive bees (H) in younger (Y) and older (O), assuming that the former die at a rate (n), while the death rate of the latter remains negligible according to the previously published model50. Obviously, (H=Y+O). The original ODEs are consequently modified as$$dot{Y}=Lfrac{H+F}{w+H+F}-Y$$$$dot{O}=left(1-nright)Y-Hleft(alpha -sigma frac{F}{H+F}right)$$$$dot{F}=Hleft(alpha -sigma frac{F}{H+F}right)-{mF}.$$Note that the sum of the first two equations above gives$$dot{H}=Lfrac{H+F}{w+H+F}-Hleft(alpha -sigma frac{F}{H+F}right)-{nY}.$$The new negative mortality term for younger hive bees, (-{nY}), models the fact that only the younger hive bees die prematurely while the rest of the dynamics is unchanged with respect to the original model.The search for equilibria soon gives$$bar{Y}=Lfrac{bar{H}+bar{F}}{w+bar{H}+bar{F}}$$from the first ODE above, so that the remaining two equilibrium conditions lead to$$bar{H}=frac{{L}_{n}}{{mJ}}-frac{w}{1+J}$$$$bar{F}=Jbar{H}$$for the same (J) originally defined and ({L}_{n}:=Lleft(1-nright)) (note that (nin left({{{{mathrm{0,1}}}}}right)), and the case (n=0) brings us back to the original model). From this point on the analysis is the same as that previously summarized for the original model, but for replacing (L) with ({L}_{n}) and (l) with (l:=lleft(1-nright)). Consequently, by assuming (alpha,m , > , {l}_{n}) (which is less restrictive when (n , > , 0)), the threshold condition (m < bar{m}) becomes$$m , < , bar{m}left(nright):=frac{{l}_{n}}{2}frac{alpha+sigma+sqrt{{left(alpha -sigma right)}^{2}+4sigma {l}_{n}}}{alpha -{l}_{n}},$$which clearly returns the original threshold condition when (n=0). Since$$frac{dbar{m}}{{dn}}left(nright) , < , 0$$as it can be immediately verified, it follows that the critical value for (m), (bar{m}left(nright)), beyond which the colony system admits only the zero equilibrium, i.e., the transcritical bifurcation value, decreases with (n) (Fig. 4). We thus conclude that colony collapse is favored by the premature death of younger hive bees, possibly caused by a virus impairing the immune system as shown by the analysis of our model of individual bee health.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

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