# American Institute of Mathematical Sciences

November  2020, 13(11): 3213-3229. doi: 10.3934/dcdss.2020162

## Bifurcations in a pollination-mutualism system with nectarless flowers

 a. School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China b. State Key Laboratory of Glassland Agro-Ecosystems, School of Life Sciences, Lanzhou University, Lanzhou, 730000, China

* Corresponding author: Yuanshi Wang

Received  October 2018 Revised  April 2019 Published  November 2020 Early access  December 2019

Fund Project: H. Wu and Y. Wang acknowledge support from NSF of China (11571382). S. Sun acknowledges support from NSF of China (31870357)

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.

Citation: Hong Wu, Shan Sun, Yuanshi Wang. Bifurcations in a pollination-mutualism system with nectarless flowers. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3213-3229. doi: 10.3934/dcdss.2020162
##### References:
 [1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866. [2] G. Bell, The evolution of empty flowers, J. Theor. Biol., 118 (1986), 253-258.  doi: 10.1016/S0022-5193(86)80057-1. [3] J. M. Biernaskie, S. C. Walker and R. J. Gegear, Bumblebees learn to forage like Bayesians, The American Naturalist, 174 (2009), 413-423.  doi: 10.1086/603629. [4] R. S. Cantrell, C. Cosner and S. G. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete and Continuous Dynamical Systems, 4 (2004), 527-546.  doi: 10.3934/dcdsb.2004.4.527. [5] J. E. Carlson and K. E. Harms, The evolution of gender-biased nectar production in hermaphroditic plants, The Botanical Review, 72 (2006), 179-205.  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 (2004), 2764-2771.  doi: 10.1890/03-0484. [7] C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307. [8] N. Davis, The Selfish Gene, Macat Library, London, 2017. doi: 10.4324/9781912281251. [9] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. [10] C. J. Essenberg, Explaining variation in the effect of floral density on pollinator visitation, The American Naturalist, 180 (2012), 153-166.  doi: 10.1086/666610. [11] M. A. Fishman and L. Hadany, Plant-pollinator population dynamics, Theoretical Population Biology, 78 (2010), 270-277.  doi: 10.1016/j.tpb.2010.08.002. [12] F. S. Gilbert, N. Haines and K. Dickson, Empty flowers, Functional Ecology, 5 (1991), 29-39.  doi: 10.2307/2389553. [13] J. Golubov, L. E. Eguiarte, M. C. Mandujano, J. L$\acute{o}$pez-Portillo and 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 (1999), 955-963.  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 (2000), 36-43.  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 (2003), 395-401.  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 (2004), 113-122.  doi: 10.1016/j.jmaa.2003.09.073. [17] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [18] A. I. Internicola, P. A. Page, G. Bernasconi and 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 (2007), 864-872.  doi: 10.1111/j.1365-2435.2007.01303.x. [19] A. I. Internicola, G. Bernasconi and 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 (2008), 1358-1365.  doi: 10.1111/j.1420-9101.2008.01565.x. [20] A. I. Internicola and 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 (2012), 1538-1543.  doi: 10.1098/rspb.2011.1849. [21] S. R.-J. Jang, Dynamics of herbivore-plant-pollinator models, J. Math. Biol., 44 (2002), 129-149.  doi: 10.1007/s002850100117. [22] J. Jersáková, S. D. Johnson and P. Kindlmann, Mechanisms and evolution of deceptive pollination in orchids, Biological Reviews, 81 (2006), 219-235. [23] S. D. Johnson and L. A. Nilsson, Pollen carryover, geitonogamy, and the evolution of deceptive pollination systems in orchids, Ecology, 80 (1999), 2607-2619.  doi: 10.1890/0012-9658(1999)080[2607:PCGATE. [24] T. L. de Jong and P. G. L. Klinkhamer, Evolutionary Ecology of Plant Reproductive Strategies, Cambridge University Press, Cambridge, 2005. [25] T. T. Makino and S. Sakai, Experience changes pollinator responses to floral display size: From size-based to reward-based foraging, Functional Ecology, 21 (2007), 854-863.  doi: 10.1111/j.1365-2435.2007.01293.x. [26] J. A. J. Metz, R. M. Nisbe and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?, Trends in Ecology & Evolution, 7 (1992), 198-202.  doi: 10.1016/0169-5347(92)90073-K. [27] M. R. Neiland and C. C. Wilcock, Fruit set, nectar reward, and rarity in the Orchidaceae, American Journal of Botany, 85 (1998), 1657-1671. [28] S. Neniez-Vieyra, M. Ordano, J. Fornoni, K. Boege and C. A. Domínguez, Selection on signal-reward correlation: Limits and opportunities to the evolution of deceit in Turnera ulmifolia L., J. Evol. Biol., 23 (2010), 2760-2767. [29] L. A. Nilsson, Orchid pollination biology, Trends in Ecology and Evolution, 7 (1992), 255-259. [30] L. OÑA and M. Lachmann, Ant aggression and evolutionary stability in plant-ant and plant-pollinator mutualistic interactions, J. Evol. Biol., 24 (2011), 617-629. [31] E. R. Pianka, Evolutionary ecology, Harper and Row, New York, (1974), 133–146. [32] 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 (2016), 70-76.  doi: 10.1016/j.ppees.2016.02.004. [33] L. van der Pijl and C. H. Dodson, Orchid Flowers, Their Pollination and Evolution, University of Miami Press, Coral Gables, Fla., 1966. [34] H. C. Qu, T. Seifan and M. Seifan, Effects of plant and pollinator traits on the maintenance of a food deceptive species within a plant community, Oikos, 126 (2017), 1815-1826.  doi: 10.1111/oik.04268. [35] J. L. Ren, D. D. Zhu and H. Y. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1843-1865. [36] 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. [37] T. A. Revilla, Numerical responses in resource-based mutualisms: A time scale approach, J. Theoretical Biology, 378 (2015), 39-46.  doi: 10.1016/j.jtbi.2015.04.012. [38] D. J. Rodríguez and L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bulletin of Mathematical Biology, 63 (2001), 547-571. [39] A. Smithson and L. D. B. Gigord, The evolution of empty flowers revisited, The American Naturalist, 161 (2003), 537-552.  doi: 10.1086/368347. [40] J. M. Soberon and C. Martinez del Rio, The dynamics of a plant-pollinator interaction, J. Theor. Biol., 91 (1981), 363-378.  doi: 10.1016/0022-5193(81)90238-1. [41] P. D. Taylor, Evolutionaryily stables strategies with two types of players, J. Appl. Prob., 16 (1979), 76-83.  doi: 10.2307/3213376. [42] J. D. Thakar, K. Kunte, A. K. Chauhan, A. V. Watve and M. G. Watve, Nectarless flowers: Ecological correlates and evolutionary stability, Oecologia, 136 (2003), 565-570.  doi: 10.1007/s00442-003-1304-6. [43] Y. S. Wang, Global dynamics of a competition-parasitism-mutualism model characterizing plant-pollinator-robber interactions, Physica A, 510 (2018), 26-41.  doi: 10.1016/j.physa.2018.06.068. [44] Y. S. Wang, H. Wu and S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81 (2012), 243-250. [45] Y. S. Wang, Dynamics of a plant-nectar-pollinator model and its approximate equations, Mathematical Biosciences, 307 (2019), 42-52.  doi: 10.1016/j.mbs.2018.12.001. [46] Y. S. Wang, H. Wu and D. L. DeAngelis, Global dynamics of a mutualism-competition model with one resource and multiple consumers, J. Math. Biol., 78 (2019), 683-710.  doi: 10.1007/s00285-018-1288-9.

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##### References:
 [1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol., 44 (1975), 331-340.  doi: 10.2307/3866. [2] G. Bell, The evolution of empty flowers, J. Theor. Biol., 118 (1986), 253-258.  doi: 10.1016/S0022-5193(86)80057-1. [3] J. M. Biernaskie, S. C. Walker and R. J. Gegear, Bumblebees learn to forage like Bayesians, The American Naturalist, 174 (2009), 413-423.  doi: 10.1086/603629. [4] R. S. Cantrell, C. Cosner and S. G. Ruan, Intraspecific interference and consumer-resource dynamics, Discrete and Continuous Dynamical Systems, 4 (2004), 527-546.  doi: 10.3934/dcdsb.2004.4.527. [5] J. E. Carlson and K. E. Harms, The evolution of gender-biased nectar production in hermaphroditic plants, The Botanical Review, 72 (2006), 179-205.  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 (2004), 2764-2771.  doi: 10.1890/03-0484. [7] C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307. [8] N. Davis, The Selfish Gene, Macat Library, London, 2017. doi: 10.4324/9781912281251. [9] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. [10] C. J. Essenberg, Explaining variation in the effect of floral density on pollinator visitation, The American Naturalist, 180 (2012), 153-166.  doi: 10.1086/666610. [11] M. A. Fishman and L. Hadany, Plant-pollinator population dynamics, Theoretical Population Biology, 78 (2010), 270-277.  doi: 10.1016/j.tpb.2010.08.002. [12] F. S. Gilbert, N. Haines and K. Dickson, Empty flowers, Functional Ecology, 5 (1991), 29-39.  doi: 10.2307/2389553. [13] J. Golubov, L. E. Eguiarte, M. C. Mandujano, J. L$\acute{o}$pez-Portillo and 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 (1999), 955-963.  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 (2000), 36-43.  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 (2003), 395-401.  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 (2004), 113-122.  doi: 10.1016/j.jmaa.2003.09.073. [17] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179. [18] A. I. Internicola, P. A. Page, G. Bernasconi and 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 (2007), 864-872.  doi: 10.1111/j.1365-2435.2007.01303.x. [19] A. I. Internicola, G. Bernasconi and 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 (2008), 1358-1365.  doi: 10.1111/j.1420-9101.2008.01565.x. [20] A. I. Internicola and 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 (2012), 1538-1543.  doi: 10.1098/rspb.2011.1849. [21] S. R.-J. Jang, Dynamics of herbivore-plant-pollinator models, J. Math. Biol., 44 (2002), 129-149.  doi: 10.1007/s002850100117. [22] J. Jersáková, S. D. Johnson and P. Kindlmann, Mechanisms and evolution of deceptive pollination in orchids, Biological Reviews, 81 (2006), 219-235. [23] S. D. Johnson and L. A. Nilsson, Pollen carryover, geitonogamy, and the evolution of deceptive pollination systems in orchids, Ecology, 80 (1999), 2607-2619.  doi: 10.1890/0012-9658(1999)080[2607:PCGATE. [24] T. L. de Jong and P. G. L. Klinkhamer, Evolutionary Ecology of Plant Reproductive Strategies, Cambridge University Press, Cambridge, 2005. [25] T. T. Makino and S. Sakai, Experience changes pollinator responses to floral display size: From size-based to reward-based foraging, Functional Ecology, 21 (2007), 854-863.  doi: 10.1111/j.1365-2435.2007.01293.x. [26] J. A. J. Metz, R. M. Nisbe and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?, Trends in Ecology & Evolution, 7 (1992), 198-202.  doi: 10.1016/0169-5347(92)90073-K. [27] M. R. Neiland and C. C. Wilcock, Fruit set, nectar reward, and rarity in the Orchidaceae, American Journal of Botany, 85 (1998), 1657-1671. [28] S. Neniez-Vieyra, M. Ordano, J. Fornoni, K. Boege and C. A. Domínguez, Selection on signal-reward correlation: Limits and opportunities to the evolution of deceit in Turnera ulmifolia L., J. Evol. Biol., 23 (2010), 2760-2767. [29] L. A. Nilsson, Orchid pollination biology, Trends in Ecology and Evolution, 7 (1992), 255-259. [30] L. OÑA and M. Lachmann, Ant aggression and evolutionary stability in plant-ant and plant-pollinator mutualistic interactions, J. Evol. Biol., 24 (2011), 617-629. [31] E. R. Pianka, Evolutionary ecology, Harper and Row, New York, (1974), 133–146. [32] 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 (2016), 70-76.  doi: 10.1016/j.ppees.2016.02.004. [33] L. van der Pijl and C. H. Dodson, Orchid Flowers, Their Pollination and Evolution, University of Miami Press, Coral Gables, Fla., 1966. [34] H. C. Qu, T. Seifan and M. Seifan, Effects of plant and pollinator traits on the maintenance of a food deceptive species within a plant community, Oikos, 126 (2017), 1815-1826.  doi: 10.1111/oik.04268. [35] J. L. Ren, D. D. Zhu and H. Y. Wang, Spreading-vanishing dichotomy in information diffusion in online social networks with intervention, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 1843-1865. [36] 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. [37] T. A. Revilla, Numerical responses in resource-based mutualisms: A time scale approach, J. Theoretical Biology, 378 (2015), 39-46.  doi: 10.1016/j.jtbi.2015.04.012. [38] D. J. Rodríguez and L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bulletin of Mathematical Biology, 63 (2001), 547-571. [39] A. Smithson and L. D. B. Gigord, The evolution of empty flowers revisited, The American Naturalist, 161 (2003), 537-552.  doi: 10.1086/368347. [40] J. M. Soberon and C. Martinez del Rio, The dynamics of a plant-pollinator interaction, J. Theor. Biol., 91 (1981), 363-378.  doi: 10.1016/0022-5193(81)90238-1. [41] P. D. Taylor, Evolutionaryily stables strategies with two types of players, J. Appl. Prob., 16 (1979), 76-83.  doi: 10.2307/3213376. [42] J. D. Thakar, K. Kunte, A. K. Chauhan, A. V. Watve and M. G. Watve, Nectarless flowers: Ecological correlates and evolutionary stability, Oecologia, 136 (2003), 565-570.  doi: 10.1007/s00442-003-1304-6. [43] Y. S. Wang, Global dynamics of a competition-parasitism-mutualism model characterizing plant-pollinator-robber interactions, Physica A, 510 (2018), 26-41.  doi: 10.1016/j.physa.2018.06.068. [44] Y. S. Wang, H. Wu and S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81 (2012), 243-250. [45] Y. S. Wang, Dynamics of a plant-nectar-pollinator model and its approximate equations, Mathematical Biosciences, 307 (2019), 42-52.  doi: 10.1016/j.mbs.2018.12.001. [46] Y. S. Wang, H. Wu and D. L. DeAngelis, Global dynamics of a mutualism-competition model with one resource and multiple consumers, J. Math. Biol., 78 (2019), 683-710.  doi: 10.1007/s00285-018-1288-9.
Phase-plane panels for population dynamics of pollinator-plant systems (1)-(4), in which $l_N$ and $l_M$ denote the isoclines of pollinators ($N$) and plants ($M$), respectively. Solid and open circles denote stable and unstable equilibria. Vector fields are displayed by gray arrows, which show the direction and speed of population trajectories. Let $r_1 = 0.5, d_1 = 0.01, \alpha = 0.15, \beta = 0.8, $$d_2 = 0.55, a_{12} = 0.8, a_{21} = \tilde{a}_{21} = 0.9, b_{12} = 0.2, \tilde{d}_2 = 0.5, \tilde{r}_1 = 0.45, \tilde{\alpha} = 0.1. (a) When secretors' efficiency in translating neglector-secretor interactions into fitness is large, the two species can coexist at a steady state E_{11}(u_{11},v_{11}) . (b) When cheaters' efficiency in translating neglector-cheater interactions into fitness is large, neglectors and cheaters can coexist at a steady state E_{12} . (c) When the secretors' efficiency in translating selector-secretor interactions into fitness is large, the two species coexist at a steady state E_{21}(u_{21},v_{12}) with u_{21}<u_{11} . (d) In the selector-cheater system, selectors approach the carrying capacity and cheaters go to extinction, which is shown by the stable equilibrium E_{22} Solutions of equations (5). Let r_1 = 0.5, d_1 = 0.01, \alpha = 0.15, \beta = 0.8,$$ d_2 = 0.55, a_{12} = 0.002, a_{21} = \tilde{a}_{21} = 0.9,$ $b_{12} = 0.05, \tilde{d}_2 = 0.5,$ $\tilde{r}_1 = 0.45, \tilde{\alpha} = 0.1.$ Then $u_{11}>u_{21}, u_{12}<u_{22},$ $v_{11}<v_{21}, v_{12}>v_{22}$, which implies that condition (16) holds and selectors and cheaters persist in periodic oscillations. Numerical simulations demonstrate that when the frequency of secretors is small, neglectors decrease monotonically. Otherwise, neglectors increase monotonically. On the other hand, when the frequency of neglectors is small, secretors increase monotonically. Otherwise, selectors decrease monotonically
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