Bifurcations in a pollination-mutualism system with nectarless flowers. (English) Zbl 1469.34063
Summary: This paper considers a pollination-mutualism system in which flowering plants have strategies of secreting and cheating: secretors produce a substantial volume of nectar in flowers but cheaters produce none. Accordingly, floral visitors have strategies of neglecting and selecting: neglectors enter any flower encountered but selectors only enter full flowers since they can discriminate between secretors and cheaters. By combination of replicator equations and two-species dynamical systems, the games are described by a mathematical model in this paper. Dynamics of the model demonstrate mechanisms by which nectarless flowers can invade the secretor-pollinator system and by which a cyclic game between nectarless flowers and pollinators could occur. Criteria for the persistence of nectarless flowers are derived in terms of the given parameters (factors), including the nectar-producing cost and cheaters’ efficiency. Numerical simulations show that when parameters vary, cheaters would vary among extinction, persistence in periodic oscillations, and persistence without secretors (i.e., cheaters spread widely). We also consider the evolution of plants in a constant state of pollinator population, and the evolution of pollinators in a constant state of plant population. Dynamics of the models demonstrate conditions under which nectarless flowers (resp. selectors) could persist.
MSC:
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 92D25 | Population dynamics (general) |
| 37N25 | Dynamical systems in biology |
References:
| [1] | J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44, 331-340 (1975) · doi:10.2307/3866 |
| [2] | G. Bell, The evolution of empty flowers, J. Theor. Biol., 118, 253-258 (1986) · doi:10.1016/S0022-5193(86)80057-1 |
| [3] | J. M. Biernaskie; S. C. Walker; R. J. Gegear, Bumblebees learn to forage like Bayesians, The American Naturalist, 174, 413-423 (2009) · doi:10.1086/603629 |
| [4] | R. S. Cantrell; C. Cosner; S. G. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete and Continuous Dynamical Systems, 4, 527-546 (2004) · Zbl 1100.92061 · doi:10.3934/dcdsb.2004.4.527 |
| [5] | J. E. Carlson; K. E. Harms, The evolution of gender-biased nectar production in hermaphroditic plants, The Botanical Review, 72, 179-205 (2006) · doi:10.1663/0006-8101(2006)72[179:TEOGNP |
| [6] | R. V. Cartar, Resource tracking by bumble bees: Responses to plan-level differences in quality, Ecology, 85, 2764-2771 (2004) · doi:10.1890/03-0484 |
| [7] | C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58, 207-246 (1996) · Zbl 0878.92029 · doi:10.1007/BF02458307 |
| [8] | N. Davis, The Selfish Gene, Macat Library, London, 2017. |
| [9] | D. L. DeAngelis; R. A. Goldstein; R. V. O’Neill, A model for trophic interaction, Ecology, 56, 881-892 (1975) |
| [10] | C. J. Essenberg, Explaining variation in the effect of floral density on pollinator visitation, The American Naturalist, 180, 153-166 (2012) · doi:10.1086/666610 |
| [11] | M. A. Fishman; L. Hadany, Plant-pollinator population dynamics, Theoretical Population Biology, 78, 270-277 (2010) · Zbl 1403.92244 · doi:10.1016/j.tpb.2010.08.002 |
| [12] | F. S. Gilbert; N. Haines; K. Dickson, Empty flowers, Functional Ecology, 5, 29-39 (1991) · doi:10.2307/2389553 |
| [13] | J. Golubov; L. E. Eguiarte; M. C. Mandujano; J. L \(\acute{o}\) pez-Portillo; C. Monta \(\tilde{n}\) a, Why be a honeyless honey mesquite? Reproduction and mating system of nectarful and nectarless individuals, American Journal of Botany, 86, 955-963 (1999) · doi:10.2307/2656612 |
| [14] | A. Gumbert, Color choices by bumble bees (Bombus terrestris): Innate preferences and generalization after learning, Behavioral Ecology and Socioliology, 48, 36-43 (2000) · doi:10.1007/s002650000213 |
| [15] | T.-W. Hwang, Global analysis of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 281, 395-401 (2003) · Zbl 1033.34052 · doi:10.1016/S0022-247X(02)00395-5 |
| [16] | T.-W. Hwang, Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response, J. Math. Anal. Appl., 290, 113-122 (2004) · Zbl 1086.34028 · doi:10.1016/j.jmaa.2003.09.073 |
| [17] | A. I. Internicola; P. A. Page; G. Bernasconi; L. D. B. Gigord, Competition for pollinator visitation between deceptive and rewarding artificial inflorescences: An experimental test of the effects of floral color similarity and spatial mingling, Functional Ecology, 21, 864-872 (2007) · doi:10.1111/j.1365-2435.2007.01303.x |
| [18] | A. I. Internicola; G. Bernasconi; L. D. B. Gigord, Should a food-deceptive species flower before or after rewarding species? An experimental test of pollinator visitation behaviour under contrasting phenologies, Journal of Evolutionary Biology, 21, 1358-1365 (2008) · doi:10.1111/j.1420-9101.2008.01565.x |
| [19] | A. I. Internicola; L. D. Harder, Bumble-bee learning selects for both early and long flowering in food-deceptive plants, Proceedings of the Royal Society B: Biological Sciences, 279, 1538-1543 (2012) · doi:10.1098/rspb.2011.1849 |
| [20] | S. R.-J. Jang, Dynamics of herbivore-plant-pollinator models, J. Math. Biol., 44, 129-149 (2002) · Zbl 0991.92031 · doi:10.1007/s002850100117 |
| [21] | J. Jersáková; S. D. Johnson; P. Kindlmann, Mechanisms and evolution of deceptive pollination in orchids, Biological Reviews, 81, 219-235 (2006) |
| [22] | S. D. Johnson; L. A. Nilsson, Pollen carryover, geitonogamy, and the evolution of deceptive pollination systems in orchids, Ecology, 80, 2607-2619 (1999) · doi:10.1890/0012-9658(1999)080[2607:PCGATE |
| [23] | T. T. Makino; S. Sakai, Experience changes pollinator responses to floral display size: From size-based to reward-based foraging, Functional Ecology, 21, 854-863 (2007) · doi:10.1111/j.1365-2435.2007.01293.x |
| [24] | J. A. J. Metz; R. M. Nisbe; S. A. H. Geritz, How should we define ’fitness’ for general ecological scenarios?, Trends in Ecology & Evolution, 7, 198-202 (1992) · doi:10.1016/0169-5347(92)90073-K |
| [25] | M. R. Neiland; C. C. Wilcock, Fruit set, nectar reward, and rarity in the Orchidaceae, American Journal of Botany, 85, 1657-1671 (1998) |
| [26] | S. Neniez-Vieyra; M. Ordano; J. Fornoni; K. Boege; C. A. Domínguez, Selection on signal-reward correlation: Limits and opportunities to the evolution of deceit in Turnera ulmifolia L., J. Evol. Biol., 23, 2760-2767 (2010) |
| [27] | L. A. Nilsson, Orchid pollination biology, Trends in Ecology and Evolution, 7, 255-259 (1992) |
| [28] | L. Oña; M. Lachmann, Ant aggression and evolutionary stability in plant-ant and plant-pollinator mutualistic interactions, J. Evol. Biol., 24, 617-629 (2011) |
| [29] | E. R. Pianka, Evolutionary ecology, Harper and Row, New York, (1974), 133-146. |
| [30] | G. H. Pyke, Plant-pollinator co-evolution: It’s time to reconnect with optimal foraging theory and evolutionarily stable strategies, Perspectives in Plant Ecology Evolution and Systematics, 19, 70-76 (2016) · doi:10.1016/j.ppees.2016.02.004 |
| [31] | H. C. Qu; T. Seifan; M. Seifan, Effects of plant and pollinator traits on the maintenance of a food deceptive species within a plant community, Oikos, 126, 1815-1826 (2017) · doi:10.1111/oik.04268 |
| [32] | J. L. Ren; D. D. Zhu; H. Y. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24, 1843-1865 (2019) · Zbl 1411.35139 |
| [33] | S. Renner, Rewardless Flowers in the Angiosperms and the Role of Insect Cognition in Their Evolution. In Plant-Pollinator Interactions: From Specialization to Generalization, Nickolas Merritt Waser editor, 2006. |
| [34] | T. A. Revilla, Numerical responses in resource-based mutualisms: A time scale approach, J. Theoretical Biology, 378, 39-46 (2015) · Zbl 1343.92435 · doi:10.1016/j.jtbi.2015.04.012 |
| [35] | D. J. Rodríguez; L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bulletin of Mathematical Biology, 63, 547-571 (2001) · Zbl 1323.92210 |
| [36] | A. Smithson; L. D. B. Gigord, The evolution of empty flowers revisited, The American Naturalist, 161, 537-552 (2003) · doi:10.1086/368347 |
| [37] | J. M. Soberon; C. Martinez del Rio, The dynamics of a plant-pollinator interaction, J. Theor. Biol., 91, 363-378 (1981) · doi:10.1016/0022-5193(81)90238-1 |
| [38] | P. D. Taylor, Evolutionaryily stables strategies with two types of players, J. Appl. Prob., 16, 76-83 (1979) · Zbl 0398.90120 · doi:10.2307/3213376 |
| [39] | J. D. Thakar; K. Kunte; A. K. Chauhan; A. V. Watve; M. G. Watve, Nectarless flowers: Ecological correlates and evolutionary stability, Oecologia, 136, 565-570 (2003) · doi:10.1007/s00442-003-1304-6 |
| [40] | Y. S. Wang, Global dynamics of a competition-parasitism-mutualism model characterizing plant-pollinator-robber interactions, Physica A, 510, 26-41 (2018) · Zbl 1514.92102 · doi:10.1016/j.physa.2018.06.068 |
| [41] | Y. S. Wang; H. Wu; S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81, 243-250 (2012) · Zbl 1322.92064 |
| [42] | Y. S. Wang, Dynamics of a plant-nectar-pollinator model and its approximate equations, Mathematical Biosciences, 307, 42-52 (2019) · Zbl 1409.92273 · doi:10.1016/j.mbs.2018.12.001 |
| [43] | Y. S. Wang; H. Wu; D. L. DeAngelis, Global dynamics of a mutualism-competition model with one resource and multiple consumers, J. Math. Biol., 78, 683-710 (2019) · Zbl 1410.34142 · doi:10.1007/s00285-018-1288-9 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
