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Invasion by an exotic species in a three-species competition-diffusion system
<p>We consider the situation where an exotic species <em>w</em> invades an ecosystem inhabited by two native species <em>u</em> and <em>v</em>. All species are competing for the same limited resource. Supposing that <em>u</em> and <em>v</em> are not able to coexist in the absence of the invader, we want to determine whether a successful invasion by <em>w</em> may allow all species to coexist (competitor-mediated coexistence). Mathematically, this problem can be modelled by the following three-species competition-diffusion system<br> <span class="math-tex">\( \left\{ \begin{alignedat}{6} u_t &= d_1 \, \Delta u &&+ (r_1 &&- u &&- b_{12} \, v &&- b_{13} \, w &&)\,u, \\ v_t &= d_2 \, \Delta v &&+ (r_2 &&- v &&- b_{21} \, u &&- b_{23} \, w &&)\,v, \\ w_t &= d_3 \, \Delta w &&+ (r_3 &&- w &&- b_{31} \, u &&- b_{32} \, v &&)\,w, \end{alignedat} \right.\)</span><br> where all parameters are positive constants.</p> <p>We are interested in the case in which the invading species is weaker than the native ones, i.e., it is not able to survive in the diffusion-free system obtained by setting <em>d</em><sub>1</sub> = <em>d</em><sub>2</sub> = <em>d</em><sub>3</sub> = 0.<br> We fix all parameters as<br> <span class="math-tex">\( \begin{aligned} & d_1 = d_2 = d_3 = 1, \\ & r_1 = r_2 = 28, \\ & \begin{aligned} b_{12} &= 22/21, & b_{13} &= 4, \\ b_{21} &= 1.87, & b_{23} &= 3/4, \\ b_{31} &= 26/21, & b_{32} &= 22/21, \\ \end{aligned} \end{aligned}\)</span><br> and leave <em>r</em><sub>3</sub>, which measures the strength of the exotic species, as a free parameter. Depending on the value of <em>r</em><sub>3</sub>, the invasion can be either successful or not and competitor-mediated coexistence may or may not occur, as can be seen in the movies here presented. The species <em>u</em>, <em>v</em> and <em>w</em> are denoted by the red, green and blue colours respectively. The yellow line marks the interface between the species <em>u</em> and <em>v</em>. We remark that competitor-mediated coexistence only occurs for intermediate values of <em>r</em><sub>3</sub>.</p>
FIGURE 8. Cultivated species. A–B. Pleiostachya pruinosa. C. Stachyphrynium repens. D in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 8. Cultivated species. A–B. Pleiostachya pruinosa. C. Stachyphrynium repens. D. Stromanthe sanguinea 'Tricolor'. E–F. Thalia dealbata. G–H. Thalia geniculata. (Photos: Jana Leong-Škorničková)
FIGURE 6. Schumannianthus benthamianus. A. Habit. B in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 6. Schumannianthus benthamianus. A. Habit. B. Detail of flowers. Phrynium sp. C. Habit. Thaumatococcus daniellii. D. Habit. E. Inflorescence and flowers. F. Fruits. (Photos: Jana Leong-Škorničková)
FIGURE 5. Phrynium villosulum. A. Habit. B in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 5. Phrynium villosulum. A. Habit. B. Detail of inflorescence and flower. C. Leaf blades showing the typical ornamentation of this species. Based on SNG-326 (A–B) and an ornamental planting at Singapore Botanic Gardens (C). (Photos: Matti Niissalo-A & B; Jana Leong-Škorničková-C)
FIGURE 7. Cultivated species. A–B. Calathea lutea. C–D. Calathea crotalifera. E. Goeppertia majestica. F. Goeppertia makoyana. G. Goeppertia undulata. H. Goeppertia warszewiczii. I. Ctenanthe setosa. J. Maranta arundinacea. K–L. Maranta leuconeura. M–N. Marantochloa mannii. O–P in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 7. Cultivated species. A–B. Calathea lutea. C–D. Calathea crotalifera. E. Goeppertia majestica. F. Goeppertia makoyana. G. Goeppertia undulata. H. Goeppertia warszewiczii. I. Ctenanthe setosa. J. Maranta arundinacea. K–L. Maranta leuconeura. M–N. Marantochloa mannii. O–P. Marantochloa purpurea. (Photos: Jana Leong-Škorničková)
FIGURE 3. Stachyphrynium latifolium. A in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 3. Stachyphrynium latifolium. A. Base of the plant with inflorescences. B. Habit, detail of lower side of the lamina in inset. C. Detail of inflorescence. D. Detail of flowers in bracts (front view). E. Detail of flowers in bracts (side view). F. Detail of fruits and seeds. G. Fruit, still positioned in bract. H. Flower pair in bract. Based on SNG-019 and SNG-084. (Photos: Jana Leong-Škorničková)
FIGURE 4. Stachyphrynium parvum. A. Habit. B in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 4. Stachyphrynium parvum. A. Habit. B. Detail of leaf blade bases showing pulvini. C. Base of plant with an inflorescence. D. Detail of inflorescence and flower. E. Detail of flower (front view). F. Detail of flower (side view). Based on SNG-189. (Photos: Jana Leong-Škorničková)
FIGURE 2. Phrynium hirtum. A–B in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 2. Phrynium hirtum. A–B. Habit, detail of lower side of the lamina in inset. C. Detail of inflorescence (side view). D. Detail of flowers (front view). E. Detail of fruit and seeds (scale in mm). F. Detail of infructescence. Based on SNG-348. (Photos: Jana LeongŠkorničková)
FIGURE 1. Donax canniformis. A. Habit. B. Inflorescence. C in Notes on Singaporean native Zingiberales II: revision of Marantaceae, with a new generic record and notes on naturalised and commonly cultivated exotic species
FIGURE 1. Donax canniformis. A. Habit. B. Inflorescence. C. Detail of flowers (side view). D. Detail of flowers (front view). E. Infructescence with detail of seeds (in inset, scale in mm). F. Detail of flowers in dorsal view, also showing the bracteoles. Based on SNG- 333. (Photos: Jana Leong-Škorničková)
Data from: Exotic species drive patterns of plant species diversity in 93 restored tallgrass prairies
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Data from: Correlation of native and exotic species richness: a global meta-analysis finds no invasion paradox across scales
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Data from: Diversity patterns of native and exotic fish species suggest homogenization processes, but partly fail to highlight extinction threats
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Data from: A basin-scale application of environmental DNA assessment for rare endemic species and closely related exotic species in rivers: a case study of giant salamanders in Japan
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Does urbanization favor exotic bee species? Implications for the conservation of native bees in cities
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Data from: Seasonality of precipitation interacts with exotic species to alter composition and phenology of a semi-arid grassland
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Data from: Rapid genetic adaptation precedes the spread of an exotic plant species
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Data from: Species pools and differential performance generate variation in leaf nutrients between native and exotic species in succession
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Data from: Shrubs as ecosystem engineers across an environmental gradient: effects on species richness and exotic plant invasion
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Data from: Stronger effect of gastropods than rodents on seedling establishment, irrespective of exotic or native plant species origin
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Data from: Revisiting the ichthyodiversity of Java and Bali through DNA barcodes: taxonomic coverage, identification accuracy, cryptic diversity and identification of exotic species
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