extfieldsofstudy: Biology Medicine

text:
The genus Apodrosus Marshall, 1922 in Cuba (Coleoptera, Curculionidae, Entiminae, Polydrusini)

Abstract The genus Apodrosus Marshall is newly recorded for, and revised for Cuba. Nine new species are recognized as follows: Apodrosus alberti (type locality, Granma, Parque Nacional Pico Turquino), A. alternatus (type locality, Guantánamo, El Yunque), A. franklyni (type locality, Cienfuegos, Parque Nacional Pico San Juan), A. griseus (type locality, Santiago de Cuba, Siboney-Jutici Ecological Reserve), A. mensurensis (type locality, Holguin, Parque Nacional La Mensura-Piloto), A. pseudoalternatus (type locality, Matanzas, Varahicacos), A. beckeli (type locality, Guantánamo, 8 km W. Imias), A. sandersoni (type locality, Guantánamo, Loma Lafarola), and A. zayasi (type locality, Cienfuegos, Parque Nacional Pico San Juan). A key for their identification, descriptions, summaries of natural history information and data on distributions are presented. A molecular phylogeny based on 11 species of Apodrosus from Cuba, Hispaniola and Puerto Rico is reconstructed. A sister group relationship between Polydrusus and Apodrosus is recovered with a limited sampling of the former genus. The monophyly of Apodrosus is recovered with strong support. Cuban Apodrosus are not monophyletic. Five of the six sampled Cuban species form a clade, sister to an undescribed Apodrosus species from the Dominican Republic; and, Apodrosus alternatus is sister to A. quisqueyanus Girón & Franz, 2010, a species from the Dominican Republic. Biogeographic implications for Cuban species are discussed in light of the phylogeny.

Introduction
The genus Apodrosus Marshall, 1922 was recently comprehensively revised by Girón and Franz (2010). They documented the presence of 13 species throughout the Caribbean, eleven of which they described as new. No species of the genus were known from Cuba. Following three recent (2012,2013,2014) expeditions to Cuba we have now recorded nine additional endemic species of the genus from throughout the country. These species are described herein. Girón and Franz (2010) also presented a morphological phylogeny and biogeographic analysis of the genus with two main clades represented; the Hispaniolan/ Puerto Rican A. artus-A. wolcotti clade from elevations over 1000 m and the more speciose and widespread western Caribbean A. eximius-A. empherefasciatus clade from arid, lower-elevation, coastal habitats. Here, we reinterpret their results based on a limited molecular phylogeny incorporating data from 11 species of the genus from Cuba, Hispaniola and Puerto Rico. Host plant associations continue to remain uncertain.
The phylogenetic study by Girón and Franz (2010) questioned the placement of Apodrosus in Polydrusini. At present, we do not address this issue as any changes in tribal placement would require definition of the pertinent tribes of Entiminae and this is well beyond the scope of this study.

Materials and methods
The approach used in this study follows that of Girón and Franz (2010) for ease of comparison of species from Cuba with the rest of the genus. A differential diagnosis is presented allowing for separation from other Cuban species only. Cuban species are considered distinct based on comparison of male genitalia with figures in Girón and Franz (2010)  Phylogenies of Apodrosus species and their allies were reconstructed using molecular data. Two successive analyses were conducted, first with a broad outgroup sample, and the second focusing on just Apodrosus species (details of samples including geography, vouchering and Genbank numbers are provided in Suppl. material 1). Eleven species of Apodrosus were sampled for molecular data, with six from Cuba and five from Puerto Rico or Hispaniola (represented by the Dominican Republic). Other species of Apodrosus were not sampled due to a lack of DNA-quality specimens or failed molecular experiments. In the first analysis, sequences of eight species of Polydrusus and one species of Pachyrhinus, all Palearctic, were downloaded from Genbank. These were included to test the relationship between Apodrosus and Polydrusus, both currently placed in Polydrusini (Alonso-Zarazaga & Lyal, 1999). Representatives of New World Polydrusus were not sampled due to absence of published data or fresh specimens. Two genes (Cytochrome c Oxidase subunit I [COI], and Cytochrome c Oxidase subunit II [COII]), are available for Polydrusus and Pachyrhinus. Eleven other species of various other Neotropical entimines were also included to position Polydrusini as sampled in the current study in a broader phylogenetic context, an exercise hitherto not performed. This selection represents a broad coverage of Caribbean entimine diversity at the tribal level, but is also limited by sample availability. Girón and Franz (2010) placed Anypotactus bicaudatus as the sister of Apodrosus sampled in that study. Due to lack of DNA-quality material, this genus is not sampled in the current study. Anthonomus grandis was used to root the 33-taxon phylogeny. This phylogeny recovered a sister relationship between Polydrusus (Pachyrhinus) and Apodrosus (Suppl. material 2). In a second analysis, only species of Apodrosus and a Polydrusus (as root) were retained and this phylogeny is presented in the main text of the article. DNA extractions were performed using the Qiagen DNeasy® Blood and Tissue Kit (Qiagen). The right hind leg was excised from a specimen and extracted. Six mitochondrial, nuclear, protein-coding or ribosomal gene fragments -COI, COII, 12s, Arginine Kinase (AK), Elongation Factor 1-alpha (Ef 1-α), and 28s (D2-D4 regions) -were amplified with polymerase chain reaction using EmeraldAmp MAX PCR Master Mix in an Eppendorf vapo protect thermal cycler. Primer and PCR programming information is the same as in Zhang et al. (2017). DNA sequencing was performed at the Arizona State University DNA Laboratory on an Applied Biosystems 3730 capillary sequencer. DNA sequences were edited with the software Geneious R7 and uploaded to Genbank. The software program MAFFT (Katoh and Standley 2013) was used to align DNA sequences, applying the auto alignment strategy for protein coding genes (COI, COII, AK and Ef-1α) and L-INS-i for ribosomal genes (12s and 28s). The aligned, individual gene data sets were concatenated using SequenceMatrix (Vaidya et al. 2011). A matrix of 33 terminals and 4539 aligned nucleotide sites was generated for the first analysis and another with 13 terminals and 4494 sites for the second analysis. The maximum likelihood method was used to reconstruct phylogenies using the program RAxML (Stamatakis 2014) via the CIPRES (www.phylo.org) supercomputer cluster (Miller et al. 2010). The analysis was partitioned with each gene fragment considered a different partition.
Diagnosis. This species is easily distinguished from other Cuban species by larger eyes, elytra with all intervals of equal elevation, elytra with stria 10 continuous throughout length, body with most scales brown or copper in color, and distinctive male genitalia.
Description. Male. Body length 3.6-4.2 mm; in dorsal view about 2.3 times longer than greatest width which is at about second third of elytra; dorsal outline in lateral view quite flat. Vestiture composed of pale to dark brown scales, with very small recurved, fine brown setae and a few scattered broader white scales on elytra.. Eyes 1.4 times longer than wide, projected, separated from anterior margin of prothorax by 0.3 times greatest diameter of eye; line of anterior margin of eyes very slightly impressed; shortest distance between eyes (dorsal view) 0.35 times greatest width of pronotum; median furrow linear, narrow and deep, extending from anterior to posterior margin of eyes. Rostrum slightly longer than wide; epistoma apically with three setae situated on each side; nasal plate well defined, v-shaped, hind margin tumid, not declivious, continuing onto rostrum as an elevated median carina. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 45°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by 2.5 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eye; desmomere I about same length as II. Pronotum cylindrical, slightly wider than long, greatest width from midlength to near base; dorsal surface shallowly punctate, scales sparse, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, glabrous. Mesocoxal cavities about 5 times width of intercoxal process. Metasternum with lateral portions tumid, not posteriorly produced. Elytra in dorsal view 1.75 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins slightly divergent until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline quite flat; posterior declivity gradually descending; striae 9 and 10 complete throughout length although punctures of 10 faintly defined beyond metacoxa; intervals largely covered with scales, with dark and light areas forming an irregular pattern; all intervals equally flat, humerus angled; interval 9 very slightly tumid just anterior to metacoxa; all intervals with recurved, fine brown setae, a few scattered broader white scales on elytral declivity. Venter with scales sparse, small on ventrites, middle of ventrites 1 and 2 with moderately dense, long, fine, erect hairs, ventrites 3 and 4 subequal in length, their combined length slightly longer than ventrite 5; posterior margin of ventrite 5 widely rounded, apex at middle narrowly impressed. Tergum VII of male emarginate. Tegmen with tegminal apodeme 0.5 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 5 times longer than its greatest width; apex rounded. Endophallus extended to about midlength of aedeagal apodemes, with only a pair of inverted cshaped sclerites at about one-third length of aedeagus. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about 0.8 times length of aedeagus.
Female. Body length 3.9-4.0 mm. Differing from male as follows: elytra in lateral view with posterior declivity angulate, sutural interval very slightly inflated at about midheight declivity.
Etymology. This species is named after Albert Deler Hernández, coleopterist, of Santiago de Cuba, Cuba. Diagnosis. This species is distinguished from other Cuban species by larger eyes, elytra with intervals 3 (subbasally and discally), 5 (subbasally) and 7 (at humerus) elevated above adjacent intervals (moreso in female than in male), and distinctive male genitalia with the aedeagus length about 4.5 times maximum width.
Description. Male. Body length 3.0-3.4 mm; in dorsal view 2.1-2.3 times longer than greatest width which is between midlength and second third of elytra; dorsal outline in lateral view slightly convex. Vestiture composed of pink, pinkish-white to brown scales, with very small recurved, fine brown setae. Eyes 1.6 times longer than wide, projected, separated from anterior margin of prothorax by 0.25 times greatest diameter of eye; line of anterior margin of eyes slightly impressed; shortest distance between eyes (dorsal view) 0.3-0.4 times greatest width of pronotum; median furrow linear, narrow and deep, extending from anterior margin of eye but not reaching anterior margin of pronotum. Rostrum slightly longer than wide; epistoma apically with a single seta situated on each side; nasal plate well defined, tumid, steeply declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 45°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by 1.5 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eyes; desmomere I about same length as II. Pronotum cylindrical, about as long as wide, greatest width near base; dorsal surface shallowly punctate, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, rugose and glabrous. Mesocoxal cavities about 3 times width of intercoxal process. Metasternum with lateral portions posteriorly produced. Elytra in dorsal view 1.5 times their greatest width; anterior margin slightly sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins slightly divergent until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline slightly convex; posterior declivity gradually descending; striae 9 and 10 separate along entire length; intervals completely covered with scales, with dark and light areas forming an irregular pattern; interval 3 slightly produced at base and again at middle on disc, interval 4 tumid at base, interval 5 produced at base, then less so, then pronounced again throughout most of length, humerus distinctly angled; interval 9 slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales denser, larger on ventrites 1 and 2, small and fine, some seta-like, on ventrites 3-5; ventrite 2 about as long as ventrite 1 (medially), ventrites 3 and 4 subequal in length, their combined length slightly shorter than ventrite 5; posterior margin of ventrite 5 widely rounded, finely narrowly emarginate at middle, apex at middle narrowly impressed. Tegmen with tegminal apodeme 0.6 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 4.5 times longer than its greatest width; apex rounded. Endophallus with a pair of large asymmetrical hook-like sclerites positioned near midlength, two elongate fields of microtrichiae extended between aedeagal apodemes, and a faint apical sclerite complex. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about 0.5 times length of aedeagus.
Female. Body length 3.3-3.8 mm. Differing from male as follows: elytra with interval 3 produced at base and again more strongly so at middle on disc, interval 4 tumid at base, interval 5 strongly produced at base, then less so, then moderately pronounced again throughout most of length, humerus distinctly angled, humeral interval somewhat produced a short distance beyond humerus.
Etymology. This species is named after the Latin adjective "alternatus", referring to the alternating elevation of the elytral intervals.
Natural history. Adults were collected beating vegetation along trails in tropical wet forest.
Comments. During the initial phase of this study, this species and A. pseudoalternatus were considered as conspecific. Males can be separated by the form of the aedeagus but females at present are not separable using external features. Diagnosis. This species is difficult to distinguish from other Cuban species especially A. franklyni and A. griseus. This group of three species can be separated from other Cuban species by larger eyes, elytra with all intervals of equal elevation, and elytra with stria 10 interrupted above metacoxa. Males of the three species can be sepaarted on the basis of distinctive male genitalia. Females of this species can be separated from A. franklyni by the form of the elytral declivity in lateral view but are not separable from A. griseus using external features.

Apodrosus beckeli
Description. Male. Body length 2.2-2.3 mm; in dorsal view about 2.2 times longer than greatest width which is at about second third of elytra; dorsal outline in lateral view quite flat. Vestiture composed of grey, greyish-white to brown scales, with very small recurved, fine brown setae. Eyes 1.3 times longer than wide, projected, separated from anterior margin of prothorax by 0.6 times greatest diameter of eye; line of anterior margin of eyes very slightly impressed; shortest distance between eyes (dorsal view) 0.5 times greatest width of pronotum; median furrow linear, narrow and shallow, extending from anterior margin of eyes but not reaching anterior margin of pronotum. Rostrum slightly longer than wide; epistoma apically with three setae situated on each side; nasal plate well defined, v-shaped, slightly tumid, not declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 60°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by 1.5 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eye; desmomere I about same length as II. Pronotum cylindrical, slightly wider than long, greatest width from midlength to near base; dorsal surface shallowly punctate but largely obscured by scales, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, glabrous. Mesocoxal cavities about 3 times width of intercoxal process. Metasternum with lateral portions slightly tumid, not posteriorly produced. Elytra in dorsal view 1.7-1.8 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins subparallel until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline quite flat; posterior declivity gradually descending; stria 9 complete, stria 10 interrupted above metacoxa, resuming at suture between ventrites 1 and 2; intervals completely covered with scales, with dark and light areas forming an irregular pattern; all intervals equally flat, humerus angled; interval 9 very slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales dense, large on ventrites; ventrites 3 and 4 subequal in length, their combined length slightly shorter than ventrite 5; posterior margin of ventrite 5 widely rounded, apex at middle narrowly impressed. Tegmen with tegminal apodeme 0.5 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 4.5 times longer than its greatest width; apex rounded. Endophallus extended to about midlength of aedeagal apodemes, with only an apical hooked sclerite complex. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about same length as aedeagus.
Female. Body length 2.5-3.0 mm. Etymology. This species is named after William Edwin Beckel, PhD., Entomology, father of Margaret Beckel, President and CEO of the Canadian Museum of Nature, for his generous support of insect taxonomy.
Natural history. Adults were collected beating vegetation in dry coastal scrub.
Comments. Externally, this species is very similar to A. griseus and although males can be separated by details in the structure of the endophallus (see key), females cannot be separated using external features. Cuban species by larger eyes, elytra with all intervals of equal elevation, and elytra with stria 10 interrupted above metacoxa. Males of the three species can be separated on the basis of distinctive male genitalia. Females of this species can be separated from A. beckeli and A. griseus by the elytral profile at apical declivity distinctly angulate.

Apodrosus franklyni
Description. Male. Body length 2.8-3.2 mm; in dorsal view about 2.2 times longer than greatest width which is at about second third of elytra; dorsal outline in lateral view quite flat. Vestiture composed of grey, greyish-white to brown scales, with very small recurved, fine brown setae. Eyes 1.3 times longer than wide, projected, separated from anterior margin of prothorax by 0.5 times greatest diameter of eye; line of anterior margin of eyes very slightly impressed; shortest distance between eyes (dorsal view) 0.6 times greatest width of pronotum; median furrow linear, narrow and shallow, extending from anterior margin of eyes but not reaching anterior margin of pronotum. Rostrum slightly longer than wide; epistoma apically with three setae situated on each side; nasal plate well defined, v-shaped, slightly tumid, not declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 60°, directed posteriorly at end, not reaching anterior margin of eye, separated from it by 2.0 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to slightly beyond midlength of eye; desmomere I slightly longer than II. Pronotum cylindrical, slightly longer than wide, greatest width from midlength to near base; dorsal surface shallowly punctate, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, glabrous. Mesocoxal cavities about 5 times width of intercoxal process. Metasternum with lateral portions slightly tumid, not posteriorly produced. Elytra in dorsal view 1.4-1.5 times their greatest width; anterior margin sinuate; humeral region of elytra 1.7 times width of posterior margin of pronotum; lateral margins slightly divergent until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline quite flat; posterior declivity gradually descending; stria 9 complete, stria 10 interrupted above metacoxa, resuming at suture between ventrites 1 and 2; intervals completely covered with scales, with dark and light areas forming an irregular pattern although many specimens with a more or less distinct transverse dark macula at about posterior one-third of elytral length; all intervals equally flat, humerus angled; interval 9 very slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales dense, large on ventrites, ventrites 3 and 4 subequal in length, their combined length slightly about same length as ventrite 5; posterior margin of ventrite 5 widely rounded, apex at middle narrowly impressed. Tegmen with tegminal apodeme 0.55 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 4.7 times longer than its greatest width; apex rounded. Endophallus extended almost to apex of aedeagal apodemes, with a narrow u-shaped sclerite at midlength aedeagus and apical hooked sclerite complex. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes slightly shorter than length aedeagus.
Female. Body length 3.4-3.6 mm. Differing from male as follows: elytra in lateral view slightly tumid dorsally and with posterior declivity angulate, sutural interval very slightly inflated at about midheight of declivity.
Etymology. This species is named after Franklyn Cala Riquelme, arachnologist, of Santiago de Cuba, Cuba.
Natural history. Adults were collected beating montane vegetation along the upper part of the road to Pico San Juan.
Comments. This species and A. griseus and A. beckeli are superficially very similar. Males can be separated by the form of genitalia but females are very similar with females of A. franklyni, differing in the form of the elytral declivity in lateral view. Diagnosis. This species is difficult to distinguish from other Cuban species especially A. franklyni and A. beckeli. This group of three species can be separated from other Cuban species by larger eyes, elytra with all intervals of equal elevation, and elytra with stria 10 interrupted above metacoxa. Males of the three species can be sepaarted on the basis of distinctive male genitalia. Females of this species can be separated from A. franklyni by the form of the elytral declivity in lateral view but are not separable from A. beckeli using external features.

Apodrosus griseus
Description. Male. Body length 2.6-2.9 mm; in dorsal view 2.2-2.4 times longer than greatest width which is between first and second third of elytra; dorsal outline in lateral view quite flat. Vestiture composed of grey, greyish-white to brown scales, with very small recurved, fine brown setae. Eyes 1.3 times longer than wide, projected, separated from anterior margin of prothorax by 0.7 times greatest diameter of eye; line of anterior margin of eyes very slightly impressed; shortest distance between eyes (dorsal view) 0.3-0.4 times greatest width of pronotum; median furrow linear, narrow and shallow, extending from anterior margin of eyes but not reaching anterior margin of pronotum. Rostrum slightly longer than wide; epistoma apically with three setae situated on each side; nasal plate well defined, v-shaped, slightly tumid, not declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 60°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by 1.5 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eye; desmomere I very slightly longer than II. Pronotum cylindrical, slightly wider than long, greatest width near base; dorsal surface shallowly punctate but largely obscured by scales, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, glabrous. Mesocoxal cavities about 3 times width of intercoxal process. Metasternum with lateral portions slightly tumid, not posteriorly produced. Elytra in dorsal view 1.7-1.8 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins subparallel until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline quite flat; posterior declivity gradually descending; stria 9 complete, stria 10 interrupted above metacoxa, resuming at suture between ventrites 1 and 2; intervals completely covered with scales, with dark and light areas forming an irregular pattern; all intervals equally flat, humerus angled; interval 9 very slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales dense, large laterally on ventrites, smaller and less dense medially, ventrites 3 and 4 subequal in length, their combined length slightly shorter than ventrite 5; posterior margin of ventrite 5 widely rounded, apex at middle narrowly impressed. Tegmen with tegminal apodeme 0.8 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 6.5 times longer than its greatest width; apex rounded. Endophallus extended to apical two-thirds of aedeagal apodemes, with a narrow, scythe-like sclerite positioned near basal two-thirds of length, an elongate field of microtrichia positioned at base of aedeagus, and an apical hooked sclerite complex. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about one-half length aedeagus.
Etymology. This species is named after the Latin adjective "griseus" meaning grey, after the predominantly grey scales of the body vestiture.
Natural history. Adults were collected beating vegetation in dry thorn scrub.
Comments. Externally, this species is very similar to A. beckeli and although males can be separated by details in the structure of the endophallus (see key), females cannot be separated using external features. Diagnosis. This species is distinguished from other Cuban species by larger eyes, elytra with all intervals of equal elevation, elytra with stria 10 continuous throughout length, body with most scales brown or copper in color, and distinctive male genitalia.

Apodrosus mensurensis
Description. Male. Body length 2.7-3.0 mm; in dorsal view about 2.2 times longer than greatest width which is at about second third of elytra; dorsal outline in lateral view quite flat. Vestiture composed of predominantly light to dark brown scales, with very small recurved, fine dark brown setae. Eyes 1.3 times longer than wide, projected, separated from anterior margin of prothorax by 0.6 times greatest diameter of eye; line of anterior margin of eyes very slightly impressed; shortest distance between eyes (dorsal view) 0.4 times greatest width of pronotum; median furrow linear, narrow and shallow, extending from just behind anterior margin of eyes, not reaching anterior margin of pronotum. Rostrum slightly longer than wide; epistoma apically with four setae situated on each side; nasal plate well defined, v-shaped, slightly tumid, slightly declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by about 45°, directed posteriorly towards apex, not reaching anterior margin of eye, separated from it by about width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eye; desmomere I slightly longer than II. Pronotum cylindrical, very slightly wider than long, greatest width at about midlength; dorsal surface shallowly punctate but largely obscured by scales, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, about same width as anterior margin; scutellum subcircular, glabrous. Mesocoxal cavities about 6 times width of intercoxal process. Metasternum with lateral portions slightly tumid, not posteriorly produced. Elytra in dorsal view 1.5-1.6 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins subparallel until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline quite flat; posterior declivity gradually descending; striae 9 and 10 complete; intervals completely covered with scales, with dark and light areas forming an irregular pattern but with declivity mainly of paler scales; all intervals equally flat, humerus angled; interval 9 very slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales dense, large on ventrites, ventrites 3 and 4 subequal in length, their combined length slightly shorter than ventrite 5; posterior margin of ventrite 5 widely rounded, finely narrowly emarginate at middle, apex at middle deeply narrowly impressed. Tegmen with tegminal apodeme 0.5 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 4.5 times longer than its greatest width; tapered in apical 2/5 to narrowly rounded apex. Endophallus extended to about midlength of aedeagal apodemes, with only pair of small curved sclerites just before midlength, and a pair of narrow, faint fields of microtrichiae near base aedeagus. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about same length as aedeagus.
Female. Unknown. Etymology. This species is named after the type locality Parque Nacional La Mensura-Piloto. The specific name is an adjective.
Natural history. Adults were collected beating vegetation in a mixed pine scrub. Diagnosis. This species is distinguished from other Cuban species by larger eyes, elytra with intervals 3 (subbasally and discally), 5 (subbasally) and 7 (at humerus) elevated above adjacent intervals (more so in female than in male), and distinctive male genitalia with the aedeagus very long and slender, length about 10 times maximum width.

Apodrosus pseudoalternatus
Description. Male. Body length 3.0 mm; in dorsal view 2.3 times longer than greatest width which is between midlength and second third of elytra; dorsal outline in lateral view moderately convex. Vestiture composed of pink, pinkish-white to brown scales, with very small recurved, fine brown setae. Eyes 1.6 times longer than wide, projected, separated from anterior margin of prothorax by 0.25 times greatest diameter of eye; line of anterior margin of eyes slightly impressed; shortest distance between eyes (dorsal view) 0.4 times greatest width of pronotum; median furrow linear, narrow and deep, extending from anterior margin of eye but not reaching anterior margin of pronotum. Rostrum slightly longer than wide; epistoma apically with single setasituated on each side; nasal plate well defined, tumid, steeply declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 45°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by 1.5 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eyes; desmomere I about same length as II. Pronotum cylindrical, about as long as wide, greatest width near base; dorsal surface shallowly punctate, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, rugose and glabrous. Mesocoxal cavities about 3 times width of intercoxal process. Metasternum with lateral portions posteriorly produced. Elytra in dorsal view 1.5 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins slightly divergent until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline slightly convex; posterior declivity gradually descending; striae 9 and 10 separate along entire length; intervals completely covered with scales, with dark and light areas forming an irregular pattern; interval 3 slightly produced at base and again at middle on disc, interval 4 tumid at base, interval 5 produced at base, then less so, then pronounced again throughout most of length, humerus distinctly angled; interval 9 slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales denser, larger on ventrites 1 and 2, small and fine, some seta-like, on ven-trites 3-5; ventrite 2 about as long as ventrite 1 (medially), ventrites 3 and 4 subequal in length, their combined length slightly shorter than ventrite 5; posterior margin of ventrite 5 widely rounded, finely narrowly emarginate at middle, apex at middle narrowly impressed. Tegmen with tegminal apodeme 0.3 times length of aedeagus; tegminal plate simple. Aedeagus in dorsal view about 10 times longer than its greatest width; apex rounded. Endophallus with a pair of asymmetrical hook-like sclerites positioned near midlength otherwise no distinct internal sclerotization. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about 0.4 times length of aedeagus.
Female. Body length 3.5 mm. Differing from male as follows: elytra with interval 3 produced at base and again more strongly so at middle on disc, interval 4 tumid at base, interval 5 strongly produced at base, then less so, then moderately pronounced again throughout most of length, humerus distinctly angled, humeral interval somewhat produced a short distance beyond humerus.
Etymology. This species is named because of the initial confusion of this species with A. alternatus. The specific name is an adjective.

Natural history. No information.
Comments. During the initial phase of this study this species and A. alternatus were placed as conspecific. Males can be separated by the form of the aedeagus but females at present are not separable using external features. Diagnosis. This species is distinguished from other Cuban species by larger eyes, elytra with all intervals of equal elevation, elytra with stria 10 continuous throughout length, body with most scales scales grey, greyish white or pearlescent in color. Only females are known.

Apodrosus sandersoni
Description. Female. Body length 3.2-3.3 mm; in dorsal view 2.3 times longer than greatest width which is at about second third of elytra; dorsal outline in lateral view slightly convex. Vestiture composed of pale grey to pearlescent scales, with very small recurved, fine greyish setae. Eyes 1.3 times longer than wide, quite flat, separated from anterior margin of prothorax by 0.5 times greatest diameter of eye; line of anterior margin of eyes not at all impressed, frons continuous with base of rostrum; shortest distance between eyes (dorsal view) 0.45 times greatest width of pronotum; median furrow linear, narrow and deep, confined to area between eyes. Rostrum slightly longer than wide; epistoma apically with three setae situated on each side; nasal plate well defined, glabrous, not declivious, continuing onto rostrum as a number of indistinct, small striae, not carinate posteriorly. Antennal insertion apicad of midpoint of ros- trum; scrobe curved downwards by 60°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by about width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly before posterior margin of eye; desmomere I very slightly longer than II. Pronotum cylindrical, slightly wider than long, greatest width from midlength to base; dorsal surface shallowly punctate, scales moderately dense, each puncture with a suberect, fine grey seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, with a few scattered scales. Mesocoxal cavities about 5 times width of intercoxal process. Metasternum with lateral portions tumid, not posteriorly produced. Elytra in dorsal view 1.6 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins slightly divergent until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline slightly convex; posterior declivity gradually descending; striae 9 and 10 complete throughout their length although punctures of 10 faintly defined beyond metacoxa; intervals largely covered with uniformly grey to pearlescent scales; all intervals equally flat, humerus angled; interval 9 very slightly tumid just anterior to metacoxa; all intervals with minuet fine grey setae. Venter with scales scattered, small on ventrites; ventrites 3 and 4 subequal in length, their combined length slightly shorter than ventrite 5; ventrite 5 setose, posterior margin widely rounded, impressed and very narrowly emarginated medially. Female not dissected.
Natural history. No information.
Comments. This species bears a superficial resemblance to A. argentatus from the Dominican Republic and Puerto Rico and can be separated by geographic distribution and the differently sculptured dorsal surface of the rostrum. Males are not available of A. sandersoni for comparison of genitalia. We have found a latitude and longitude for Loma La Farola in Guantánamo Province as 20.1592 -74.4686 and we believe these to be the approximate coordinates of specimen capture.  , road, 21.98812, -80.14632, 1086, road, 21.98812, -80.14632, m, 19.V.2013, road, 21.98812, -80.14632, , R. Anderson, 2013 collections. Paratype. Data as holotype (1 female; CMNC). Pico San Juan, near peak, 21.9886833, -80.1465833, 1105m, 19.V.2013 (1 female; ASUHIC).

Apodrosus zayasi
Diagnosis. This species is distinguished from other Cuban species by the eyes small, rounded, the distance from posterior margin of eye to posterior margin of head about the same as greatest diameter of an eye, and by distinctive male genitalia. It is the only Cuban species with such small, rounded eyes.
Description. Male. Body length 3.6 mm; in dorsal view about 2.3 times longer than greatest width which is at about second third of elytra; dorsal outline in lateral view quite tumid. Vestiture composed of pale to dark brown scales, with very small recurved, fine brown setae. Eyes 1.1 times longer than wide, projected, separated from anterior margin of prothorax by about greatest diameter of eye; line of anterior margin of eyes very slightly impressed; shortest distance between eyes (dorsal view) 0.5 times greatest width of pronotum; median furrow linear, narrow and shallow, extending from anterior margin of eyes but not reaching anterior margin of pronotum, partially obscured by scales. Rostrum slightly longer than wide; epistoma apically with two setae situated on each side; nasal plate well defined, v-shaped, slightly tumid, not declivious. Antennal insertion apicad of midpoint of rostrum; scrobe curved downwards by 45°, directed posteriorly at end, barely reaching anterior margin of eye, separated from it by 2.0 times width of scrobe. Mandibles with 2 lateral setae. Antennae reddish brown; antennal scape extending to just slightly beyond posterior margin of eye; desmomere I about same length as II. Pronotum cylindrical, slightly wider than long, greatest width at midlength; dorsal surface shallowly punctate but largely obscured by scales, each puncture with a curved, fine brown seta; posterior margin slightly bisinuate, slightly wider than anterior margin; scutellum subcircular, glabrous. Mesocoxal cavities about 5 times width of intercoxal process. Metasternum with lateral portions slightly tumid, not posteriorly produced. Elytra in dorsal view 1.8 times their greatest width; anterior margin sinuate; humeral region of elytra 1.5 times width of posterior margin of pronotum; lateral margins slightly divergent until second third, thereafter convergent; apex acutely rounded; in lateral view with dorsal outline tumid; posterior declivity gradually descending; stria 9 complete, stria 10 interrupted above metacoxa (appearing to merge with stria 9), resuming at suture between ventrites 1 and 2; intervals completely covered with scales, with dark and light areas forming an irregular pattern; all intervals equally flat, humerus angled; interval 9 very slightly tumid above metacoxa; all intervals with recurved, fine brown setae. Venter with scales very sparse, linear and hair-like on all ventrites; ventrites 3 and 4 subequal in length, their combined length shorter than ventrite 5; posterior margin of ventrite 5 widely rounded, finely narrowly emarginate at middle, apex at middle narrowly impressed. Tegmen with tegminal apodeme 0.8 times length of aedeagus; tegminal plate simple. Aedeagus short and robust, in dorsal view about 3.0 times longer than its greatest width; apex rounded, deflexed ventrally. Endophallus extended to just beyond base of aedeagus, with no visible internal sclerotization. Aedeagus in lateral view slightly evenly convex. Aedeagal apodemes about same length as aedeagus.
Etymology. This species is named after Fernando de Zayas (1912-1983, entomologist, Cuban Academy of Sciences. Natural history. Adults were collected beating vegetation along the upper part of the road to Pico San Juan. Apodrosus   1 Eyes small, rounded, the distance from posterior margin of eye to posterior margin of head about the same as greatest diameter of an eye (Fig. 24); male with aedeagus very short and stout, length about 3 times maximum width (Fig. 26) Male with aedeagus very long and slender, length about 10 times maximum width (Fig. 21); endophallus lacking any fields of microtrichiae (Fig. 21) Male with aedeagus with endophallus extended from about midlength of aedeagal apodemes to apical two-thirds of aedeagal apodemes, with either a narrow, scythe-like sclerite positioned near basal two-thirds of length, an elongate field of microtrichia positioned at base of aedeagus, and an apical hooked sclerite complex (Fig. 15), or with only an apical hooked sclerite complex (Fig. 9); female with elytral profile at apical declivity evenly rounded .........8 8

Key to Cuban species of
Male with aedeagus with endophallus extended from about midlength of aedeagal apodemes to apical two-thirds of aedeagal apodemes, with either a narrow, scythe-like sclerite positioned near basal two-thirds of length, an elongate field of microtrichia positioned at base of aedeagus, and an apical hooked sclerite complex (Fig. 15)

Phylogeny and biogeographic implications
A distribution map of Cuban Apodrosus is shown in Fig. 27. The 33-taxon molecular phylogeny (Suppl. material 2), reconstructed using a maximum likelihood method, recovers the monophyly of Apodrosus (bootstrap value 100%) represented by 11 species in the current study. Apodrosus and Polydrusus are recovered as sister groups, although with poor nodal support. In other words, the monophyly of Polydrusini as sampled in the current study is recovered. It is worth pointing out that numerous genera of this tribe are not sampled here and the phylogenetic coherence or monophyly of Polydrusini remain to be fully tested. Most critically, Anypotactus, sister group of Apodrosus recovered by Girón and Franz (2010) was not sampled here. It would be premature to draw a conclusion on the placement of Apodrosus. Pachyrhinus lethierryi is nested within Polydrusus, necessitating the synonymy of the two genera, but this question is left for future investigations. The 13-taxon Apodrosus focal phylogeny is congruent with the 33-taxon phylogeny in the "deeper" relationships within Apodrosus, but the two differ in relationships among five Cuban species (A. zayasi-A. mensurensis clade in Fig. 28). We will base our phylogenetic and biogeographic discussions on the 13-taxon phylogeny, as our biogeographic discussions primarily concern inter-island patterns and the phylogenetic resolution within Cuban species is inconsequential in that regard. We recognize that bootstrap nodal support values are low for several clades, reflecting potential inconsistency in the data. We opted for presenting a fully dichotomous phylogeny, as is the case in many publications. Biogeographic scenarios are discussed in light of this dichotomous phylogeny (which is the best phylogeny available), but it is understood that they also are subject to future testing and revisions when additional data is acquired. Apodrosus argentatus Wolcott, 1924, a widespread species distributed in both Puerto Rico and the Dominican Republic, is recovered as sister to the remaining species of Apodrosus. The six Cuban species sampled here are not monophyletic. Five of the Cuban species form a clade (A. zayasi-A. mensurensis), which is sister to an unidentified species from Dominican Republic (which might represent a new species as it could not be keyed out using Girón and Franz 2010). A single species, Apodrosus alternatus, forms a sister relationship with A. quisqueyanus Girón & Franz, 2010, a spe- cies from the Dominican Republic. Together, they are part of a larger clade that also includes two Puerto Rican species, A. wolcotti and A. epipolevatus Girón & Franz, 2010. The current molecular phylogeny (Fig. 28) and the morphological phylogeny by Girón and Franz (2010) differ in taxon sampling, but congruence and conflict between these phylogenies can be observed and are presented as follows. Apodrosus wolcotti and A. epipolevatus, both from Puerto Rico, are sister species in both analyses. In Girón and Franz (2010), A. quisqueyanus is more closely related to A. argentatus than either is to the two Puerto Rican species. In contrast, the current phylogeny posits a closer relationship between A. quisqueyanus and the Puerto Rican species, which are in the same clade, whereas A. argentatus is outside that clade.
According to the current phylogenetic hypothesis (Fig. 28), species of Apodrosus from the same island are distributed in multiple clades and some sister groups are formed by species from different islands. This pattern of the geographic distributions of species of Apodrosus with regards to the phylogeny is reminiscent of other Caribbean entimines (Zhang et al. 2017), and is indicative of multiple successive dispersal events between islands. A "stepping-stone" scenario of an origin in Puerto Rico and subsequent dispersal to Hispaniola and Cuba is plausible. An alternative explanation of the biogeographic pattern is island to island vicariance. According to the geological evolution model proposed by Iturralde-Vinent and MacPhee (1999), the three Greater Antillean islands, Cuba, Hispaniola and Puerto Rico, were connected to form a landspan in the Late Eocene and Early Oligocene (35-33 million-years ago [Ma]). Eastern Cuba and northern Hispaniola were physically connected during the Early Oligocene, which split apart later in that epoch. On the other hand, central Hispaniola and Puerto Rico remained connected probably until late in the Miocene (14 Ma). A sister relationship between Cuba and the Dominican Republic appears twice in our phylogeny, which is in accordance with the vicariance of the two islands. However, in the clade "A.epipolevatus-A. alternatus", Puerto Rico is sister to Cuba and the Dominican Republic, implying an early split of Puerto Rico from the two larger islands, which is at odds with the vicariance model proposed by Iturralde-Vinent and MacPhee (1999) (i.e., a late split between Puerto Rico and Hispaniola).


extfieldsofstudy:
Computer Science
Medicine

text:
Modeling cooperating micro-organisms in antibiotic environment

Recent experiments with the bacteria Paenibacillus vortex reveal a remarkable strategy enabling it to cope with antibiotics by cooperating with a different bacterium—Escherichia coli. While P. vortex is a highly effective swarmer, it is sensitive to the antibiotic ampicillin. On the other hand, E. coli can degrade ampicillin but is non-motile when grown on high agar percentages. The two bacterial species form a shared colony in which E. coli is transported by P. vortex and E. coli detoxifies the ampicillin. The paper presents a simplified model, consisting of coupled reaction-diffusion equations, describing the development of ring patterns in the shared colony. Our results demonstrate some of the possible cooperative movement strategies bacteria utilize in order to survive harsh conditions. In addition, we explore the behavior of mixed colonies under new conditions such as antibiotic gradients, synchronization between colonies and possible dynamics of a 3-species system including P. vortex, E. coli and a carbon producing algae that provides nutrients under illuminated, nutrient poor conditions. The derived model was able to simulate an asymmetric relationship between two or three micro-organisms where cooperation is required for survival. Computationally, in order to avoid numerical artifacts due to symmetries within the discretizing grid, the model was solved using a second order Vectorizable Random Lattices method, which is developed as a finite volume scheme on a random grid.

Introduction
Cooperation between bacterial species is an exciting demonstration of how micro-organisms can facilitate dispersal by forming multispecies swarms with mutual benefits [1]. For example, experiments show that different bacterial species can use collective migration to cross barriers and reach new habitats [2][3][4][5][6]. From a modeling point of view, multispecies swarms and colonies present new theoretical and computational challenges, in particular due to a rich phase diagram with a multitude of possible emerging patterns.
Here, we focus on explaining recent experiments by Finkelshtein et al. [7] in which the bacterial species Paenibacillus vortex (P. vortex) and Escherichia coli (E. coli) were shown to cooperate in order to survive and grow in an antibiotic-rich environment. It is known that P. vortex is a highly effective swarmer that can move rapidly over surfaces [8] and move towards nutrients [9] or away from antibiotics [2,10]. P. vortex is sensitive to high concentrations of PLOS  ampicillin, a beta-lactam antibiotic that damages the cell wall, leading to death [7]. In contrast, E. coli cannot produce the lubricant needed for efficient movement across surfaces on high-percentage agar plates that permit P. vortex swarming. However, the strain studied carries the gene encoding a beta-lactamase enzyme that can degrade ampicillin [7]. Surprisingly, Finkelshtein et al. [7] show that a mixed colony of P. vortex and E. coli can expand the colony size and grow even at high antibiotic concentrations in which neither species can grow on its own-P. vortex alone dies because of antibiotics while E. coli alone cannot move toward other regions on the plate when nutrients available locally are completely consumed. The mixed colony develops a ring-shaped pattern with intermittent switching between low and high bacterial densities (see Fig 1). It has been hypothesized that P. vortex carry E. coli towards the edge of the colony so that the latter can inactivate the antibiotics. Once the antibiotic has been locally deactivated, the colony continues its expansion. Thus, the ringed pattern is formed by sequential transitions between two growth states: the first is a high-density, cooperating (or expanding) state in which the colony expands rapidly [7], while the second is a low-density, competing (or building) state, in which the different types of bacteria compete for the limited nutrient supply. This two-species cooperation changes the usual, continuous expansion of a P. vortex colony observed under rich conditions, into a periodic, oscillating growth dynamics. Following the experimental observations, one of the main assumptions underlying our model is that P. vortex can appear in two forms or phases [10]. The first, termed a "builder" phase, occurs when the bacterial density is relatively low. Since movement depends on the ability of bacteria to extract liquids from the substrate, motility in this phase is low and depends on the local bacterial density. On the other hand, when the bacterial population is large enough to support the colony's expansion then the cells transition to a morphologically different phase. This subpopulation, termed "explorer", is characterized by cell elongation and rapid movement into new territories [10]. We assume that transitions between the phases is a reversible global event that encompasses the entire colony once the average bacterial concentration reaches a critical high or low threshold. See the methods section for details on how these observations are implemented.
In this paper, we present a mathematical model describing the bacterial density for the two bacterial species in an antibiotic environment. Our results are compared with experimental observations. In particular, we confirm the biological hypotheses on which the model is based and shed light on the way in which different bacterial species cooperate in order to survive and develop a shared colony in a hostile environment.
The literature shows a wide variety of approaches to model development of bacterial colonies [1][2][3][4][5]9,[11][12][13][14][15][16][17][18][19][20][21][22][23][24]. The proposed models can be divided into two main categories: discrete models [3,4,11,15,19,22] and continuous models [3][4][5]9,[11][12][13][14][15][16][17][18][19][20][21]23,24]. In the discrete approach, the characteristics of individuals or groups in a colony are described, for instance, the location of individuals, nutrient consumption rates and mortality rates. The advantage of these models is the differentiation between the characteristics of different individuals and the ability to follow an individual or specific group for a length of time. However, simulations of realistic bacteria numbers in a colony are not possible. On the other hand, continuous models describe characteristics shared by a large number of cells, for example, local density averages, mean nutrient consumption rates etc. The advantage of these models is their relative simplicity. However, they are not able to focus on a certain bacterial group and the long-term processes influencing it.
A special emphasis was given to the Paenibacillus bacteria, which were found to develop unique patterns, branching out from the center of the colony in different directions [17,18]. Experiments have shown that different pattern characteristics, for example, the number and density of branches and the growth rate, depend on the environment: the growth medium, humidity, temperature, the amount of nutrients and the initial bacterial concentration [20,21].
We will focus and generalize the continuous model suggested by Schwarcz et al. [23], involving coupled partial differential equations. This two-dimensional model was shown to successfully reproduce all possible patterns observed for different colonies and with realistic growth rates. See also [12,13,19,24] for similar approaches. One of the main computational challenges in simulating the growth of bacterial colonies and other complicated reaction-diffusion processes showing pattern formation, is that solutions (or solution interfaces) may become unstable. As a result, the numerical methods and, in particular, the grid used for discretization is highly important, as symmetries in the underlying computational lattice can induce similar symmetries in the solution. To overcome this difficulty, we applied a Vectorizable Random Lattice (VRL), in which vertices are a set of randomly chosen points with uniform distribution. See Fig 2 for a simple numerical example. The method, originally proposed in [25], was adapted for simulating the growth of bacterial colonies in [23]. Here, the numerical scheme is re-derived using a finite-volume approach [26], which facilitates generalization to different time-stepping schemes and conservative high-order methods.

Materials and methods
For a single species, the model is described in terms of a two-dimensional system of reactiondiffusion equations as follows (see also [12,13,19,23,24]).
• The first Eq describes the density of bacteria b(x,y,t). The first term is a nonlinear diffusion with a coefficient that depends on b. The exponent k describes the mobility of bacteria within the medium [12,13]. Following [23], we take k = 0 for explorers and k = 1 for builders (see details below). The second and third terms describe the bacterial reproduction and death rates, which may depend on the local nutrient concentration [27]. • The second Eq describes the nutrient density n(x,y,t), where, the first term is diffusion and the second describes nutrient consumption.
For the rest of the paper, bacteria densities is measured in terms of the local area fraction that cells occupy.
The main purpose of the paper is to generalize the model (1) to the two-species case with antibiotics.
• The first Eq. describes the density of P. vortex, denoted b 1 (x,y,t). This equation is similar in form to the equation presented in model (1), except that the death term μ 1 (n,a)b 1 depends also on the antibiotics concentration, a(x,y,t).
• The second Eq. describes the density of E. coli, denoted b 2 (x,y,t). The first term describes the (low) self-movement of E. coli as a linear diffusion with a small parameter, D 2 ( D 1 . The second term describes advection with the flow of P. vortex, i.e., in the direction of rb 1 . This component is the main movement component for b 2 , i.e., D 2 ( C 2 . The last two terms are reproduction and death. Note that E. Coli is not sensitive to the antibiotics used in the experiments and, accordingly, its death rate does not depend on a. • The third Eq. describes the nutrient concentration n(x,y,t), similar to (1b).
• The last Eq describes the antibiotic concentration a(x,y,t), where, the first term describes diffusion and the second term describes the inactivation of the antibiotics by E. coli.
We assume no-flux conditions on the domain boundary. In experiments, at the initial inoculation spot the two species are homogenously mixed (normally 50:50). The next stage is the formation of swarming groups of P. vortex but the system rapidly (24 h) corrects to approximately 50:50 even if the initial inoculation is not balanced. The antibiotic and nutrients are initially distributed homogeneously. Accordingly, nutrient and antibiotics initial concentrations are constants, denoted n 0 and a 0 . The initial bacterial concentrations are a smoothed circle (see below). Following the biological considerations described above, transitions between the exploring and building states depend on the average density of P. vortex bacteria in the entire colony, Initially, all bacteria are builders. In this phase, motility depends on the ability of cells to extract liquids from the substrate which, in turn, depends on the local bacterial density. Hence, we take k = 1. It has been shown that with a nonlinear diffusion, the colony's boundary is sharp, which is consistent with the biological picture of a relatively stationary growth stage. Once b 1 becomes larger than a prescribed threshold b max , the entire colony transitions to a new state of explorers in which k = 0. In this phase, the expansion rate of the colony is significantly larger. As a result, the average bacterial density reduces. The colony will switch back to a building state if b 1 becomes smaller than a second threshold b min . Selected parameters values are detailed in Table 1.
The value of b max also sets the physical scale for the bacterial concentrations. Assuming that b max corresponds to an area fraction of approximately 50%, then, unit concentration b i = 1 corresponds to an area fraction of about 1=2b max .

The numerical method
As discussed above, one of the main difficulties in solving equations of the form of (1) and (2) is that symmetries in the discretizing lattice may become manifested in the numerical solution [16,23,24]. Different approaches have been suggested to overcome this problem. For example, [16] suggest adding some low-amplitude random noise to solutions on regular lattices. Instead, we will apply a random grid method, which does not exhibit any artifact symmetries. We will follow the Vectorizable Random Lattice (VRL) approach developed in [23,25]. One of the main goals of this paper is to further develop the numerical analysis of nonlinear reaction-diffusion equations on the VRL.
The random lattice was first presented by Christ et al. [28]. Its main feature is the random selection of points in the domain. According to [28], the number of vertices on a random lattice in a given area or volume is a random variable with a Poisson distribution (which is called a Poisson random lattice). The randomness decreases the measurable influence of the lattice on the numerical solution and increases isotropy. Despite the significant advantage of the Poisson random lattice, its use is numerically complex. In particular, the use of efficient numerical methods such as implicit schemes or implementing parallel computation is difficult [29][30][31][32]. In order to take advantage of the random lattice on one hand, but enable the development of efficient numerical methods on the other, Moukarzel and Hermann (1992) suggested the VRL [25]. The VRL is also composed of random nodes, however, as opposed to the Poisson Random Lattice, the nodes are drawn in a particular way, giving it additional structure. See Appendix A for further details.

Results
The model (2) was solved with various initial conditions for the functions b 1 and b 2 , reproducing different experimental setups. Accordingly, the simulated domain corresponds to 90 mm and every simulation time unit corresponds to about 0.5 hours. For example, units for diffusion constants are about 1.5Á10 −6 m/sec 2 (recall densities are measured in terms of area fraction).
First, we consider each species on its own, i.e., either b 1 = 0 and b 2 ¼ 1 r<r 0 or the other way around. The function 1 r<r 0 is a smoothed (twice differentiable) indicator function of the unit disk {r < r 0 }. Fig 3 shows that, as seen in experiments, neither P. vortex nor E. coli can grow on its owna colony of E. coli cannot expand because of its low motility, while P. vortex dies due to antibiotics. In order to depict the colony and its history, the figure shows the accumulated bacterial density s i + b i , where ds i /dt = μ i b i , i.e., the sum of live and dead bacteria at location (x,y) up to time t.
In contrast to these results, Fig 4 shows that a mixed colony (initial conditions b 1 ¼ b 2 ¼ 1 r<r 0 ) can grow successfully, showing a ring-like pattern, qualitatively similar to experiments. The rings are of intermittently high and low density and each one represents a different state of the P. vortex bacteria-exploring or building. Fig 5 depicts the radius of the colonies in all three cases described above. Similar to the experimental observations (Fig 5A), the simulated colony ( Fig 5B) shows a non-continuous expansion rate, corresponding to the alternating building and exploring states. Fig  6 shows a cross section of the colonies. While the cross section of the mutual colony is periodic across the radius, P. vortex alone shows a low density decreasing towards the edge (due to antibiotic). On the other hand, the density of E. coli increases towards the edge, where nutrients are higher. It is interesting to note that the periodicity is in the radius rather than the area. This is in accordance with theories describing the advancing edge as a traveling wave propagating at constant speed along the colony's radius [1][2][3][4][5]9,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][33][34][35]. Our simulations suggest that the builder/explorer phases expand with distinct wave propagation speeds and that the transitions are entrained to changes in the concentration of antibiotics at the front.
In addition, we note that the results are not specific to the parameters detailed in Table 1. See S1 Fig for a     In order to further examine the coupling between the builder/explorer transitions with the multi-species interaction, we study the dynamics of our model under initial conditions which were not realized in the experiments of [7] . Fig 7 depicts simulation results with initial two separated colonies, each one containing both P. vortex and E. coli. While separated, each colony determines the builder/explorer phase on its own. Once the two colonies overlap, the phase is the same and the colonies synchronize. Fig 8 depicts simulation results with an initial sharp gradient in the antibiotic concentration, a(x,y,t) = 1 {x<0} , i.e., only the left half of the domain contains antibiotics. All other parameters are exactly the same. Interestingly, we see that although the mixed colony can expand in all directions, the colony only grows towards the antibiotics-free half.
Finally, we begin to explore more complicated systems and dynamics. We consider a joint P. vortex and E. coli colony that also includes a microalgae, such as Chlorella vulgaris. This algae is non-motile but can produce carbon that acts as additional nutrients to the bacteria (Finkelshtein and Polikovsky, unpublished). It has been suggested that P. vortex can transport and use the algae to fix carbon. See Appendix B for the precise formulation of the model and simulation details. organisms. Our preliminary results show that not only is such a system stable, it can facilitate a more rapid colony expansion than in the absence of C. vulgaris.

Discussion
In nature, bacterial cultures are typically highly heterogenous, involving thousands of bacterial species and other micro-organisms. Within such diverse systems, cooperation is a major survival strategy. This includes cooperative movement (e.g. swarming) in multi-species suspensions [1]. Taking the first steps in understanding such cultures, several recent experiments considered simple mixtures of 2 or 3 micro-organisms, for example, two bacteria or bacteria and fungi, revealing highly complex dynamics and patterns.
Here, we present a model describing cooperation between P. vortex and E. coli species. The model describes the joint movement of bacteria in a hostile environment containing antibiotics and sparse nutrients. Our results demonstrate that, on one hand, the bacteria must cooperate in order to survive this environment while at the same time compete over the available nutrients. The model was solved on a Vectorizable Random Lattice, using new high order schemes that conserve mass. Similar to experiments, simulations show a ring-like pattern consistent with intermittent builder and explorer bacterial phases.
Our exploration into new possible system setups offer several predictions that may be tested experimentally. For example, with a sharp antibiotics gradient, our simulations suggest that the mixed colony only grows towards the low-antibiotics region. This occurs although our model does not include explicit terms that enable P. vortex to "unload" E. coli if it is not needed  (to save energy). Simulations with two colonies suggest that the ring pattern should synchronize once the colonies are in contact. Such an experiment can validate our assumption that that builder/explorer phase is a global colony-wide property.
Finally, to the best of our knowledge, our three-species model is the first attempt to model realistic three-species colonies including phenotype heterogeneity (P. vortex builder/explorers) with a particular function or objective in mind. An additional important factor, which is not currently taken into account in our model, is the ability of P. vortex to deposit, or unload E. coli or algae. This is particularly important if the larger algae could overgrow. For example, this could be modelled as an upper limit on transport capacity.
Understanding the dynamics of multi-species colonies is a rapidly growing field as microbiologist are increasingly aware of swarming organisms composed of multiple subpopulations [36]. This work takes another step towards simulating a 'moving ecosystem' or the construction of 'multispecies consortia' within the growing field of synthetic ecology [37].

Appendix A: The numerical method
In this appendix, we review the construction of the Vectorial Random Lattice (VRL). In addition, we present a new systematic method for discretizing nonlinear reaction-diffusion equations such as (1) or (2) using the finite volume approach [26]. We apply both forward-Euler and Crank-Nicolson time steps and discuss solution of the resulting implicit equations using iterative methods.
The VRL is constructed by first choosing a single point drawn randomly in each cell of a uniform rectangular lattice with grid spacing h (called the reference lattice). Let 0 d i h denote a minimal distance between two adjacent random nodes in direction i. If the distance between two points in cells that are adjacent in the direction i are closer than a distance d i , then the two points are redrawn. Therefore, the values of d i determine the randomness of the lattice: for d i = 0 randomness is maximal, while with d i = h the lattice is rectangular. The set of random points chosen within the cells of the reference lattice are the nodes of the VRL. Consider the Voronoi diagram associated with this set of points. A pair of nodes {v i ,v j } with i 6 ¼ j are called natural neighbors in the Voronoi partition if v i and v j share a common face. The Delaunay triangulation, which is the VRL, is obtained by linking all pairs of natural neighbors. The process is depicted in Fig 10. An equivalent method for obtaining the Delaunay triangulation for the randomly drawn nodes consists of finding all the trios of nodes located on the perimeter of a circle that does not contain (inside or on the edge) additional nodes. The three nodes will be natural neighbors with each other [30,31,32].
In contrast to uniform lattices, in which all interior nodes have the same number of neighbors, the number of neighbors in VRL varies. The mean number of neighbors is roughly six (identically, the average number of faces in each polygon in a Voronoi diagram is approximately six). The closer two cells on the reference lattice are to each other, the probability that the Voronoi nodes in those cells will be natural neighbors increases. However, there is a nonzero probability that a pair of nodes from non-adjacent cells on the reference lattice are also natural neighbors, thus enabling isotropic connections of the nodes. The probabilities of natural neighbors being found in different distances was calculated in [25]. In particular, it drops sharply (below 10 −4 ) for the next-next-nearest neighbors.

Calculating the Laplacian on a VRL
The Laplacian operator in the diffusion equation is composed of the divergence of the flux of a conserved quantity. In general, for a given flux vector J, the diffusion term takes the form, r Á J ¼ r Á ½cðx; y; φÞrφ; Where φ(x,y) and ψ(x,y,φ) are given function. For example, φ = b and ψ = b k for a nonlinear diffusion, with k = 0 for the Laplacian.
Lemma 1: Let φ(x,y) denote a continuously twice differentiable function and let μ i be the set of Voronoi nodes which are natural neighbors of a node v i . Then, the Laplacian is given by where w ij = f ij /(l ij A i ) and, • f ij is the length of the mutual face connecting nodes v i and v j .
• l ij is the Euclidean distance between v i and v j .
• A i is the area of V i , the Voronoi cell to which v i belongs.
Proof: We will prove lemma 1 using the finite volume approach. See [23,32] for an alternative derivation.
In order to numerically approximate Δφ(v i ), we calculate the average value of the Laplacian in each Voronoi cell, where ds is an area element, @V i is the boundary of V i , dl a length element,n an external normal to V i and we have used the divergence theorem. For a uniform grid with spacing h, the error in approximating Δφ by its average in a cell is of the order h 2 [26]. This is not as simple in the random grid because there is some small probability that the distance to one of the natural neighbors is large. Nonetheless, the probability of such an event decreases rapidly. For example, the probability to have a neighbor with distance larger than 4h is smaller than 0.01 [25]. Moreover, for any > 0, there exists a constant D independent of h such that the probability to have a neighbor with distance larger than Dh is smaller than . We therefore neglect this event and assume that the accuracy of the finite volume approximation is of order h 2 .
If v i has m natural neighbors, its boundary @V i can be written as the union of m polygonial faces. Denoting the common face of the V i and V j cells by K ij , the integral on the RHS of Eq (4) can be written as which concludes the proof of Lemma 1. Lemma 2: The proof is similar to Lemma 1.
Using the formulas derived above, different numerical schemes such as forward-Euler, backward-Euler and Crank-Nicolson can be easily developed. For example, with forward-Euler, the numerical flux from cell V i to cell V i is given by where φ n i ¼ φðv i ; nDtÞ. In particular, the flux is skew-symmetric, which implies that the discretized concentration φ is conserved. Similarly, using Crank-Nicolson, the numerical flux is given by, which is again skew-symmetric.

Iterative solutions for implicit methods
In our model, we take ψ = φ k , and the Crank-Nicolson scheme reduces to With linear diffusion, k = 0, solving for φ nþ1 i amounts to a sparse linear set of equations. However, with a nonlinear diffusion, k ! 1, the system becomes non-linear. For simplicity, we consider the case k = 1, which is relevant to our model. Larger values of k are treated similarly. First, rewrite the Crank-Nicolson scheme as, Using matrix notation, the system takes the form, where, assuming a N × N reference lattice, F n 2 R N 2 denotes the solution vector W 2 M N 2 denotes the N 2 × N 2 weighted adjacency matrix, and Z 2 M N 2 is a diagonal matrix with elements Z ii ¼ X j2m i w ij . In the above, diag(F n ) denotes the N 2 × N 2 diagonal matrix with F n on the diagonal. Noting that all matrices are sparse, the system (5) is solved iteratively using a splitting method (we applied Gauss-Seidel). For each n, we formally write, and begin with an initial guess, V (0) = F n . Then, at each iteration, the new value of approximate solution for F n+1 is substituted into V (r) . Additional numerical details as well as solution of toy models and error analysis are detailed in [24].

Appendix B: Three species simulations
Our 3-species simulations added another component to the system-an algae. The algae donates fixed Carbon to P vortex and E coli, which effectively acts as additional nutrients. On the other hand, the algae compete with the bacteria on available nutrients.