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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020162
References:
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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.  Google Scholar

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G. Bell, The evolution of empty flowers, J. Theor. Biol., 118 (1986), 253-258.  doi: 10.1016/S0022-5193(86)80057-1.  Google Scholar

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J. M. BiernaskieS. C. Walker and R. J. Gegear, Bumblebees learn to forage like Bayesians, The American Naturalist, 174 (2009), 413-423.  doi: 10.1086/603629.  Google Scholar

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R. S. CantrellC. 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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307.  Google Scholar

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N. Davis, The Selfish Gene, Macat Library, London, 2017. doi: 10.4324/9781912281251.  Google Scholar

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D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[12]

F. S. GilbertN. Haines and K. Dickson, Empty flowers, Functional Ecology, 5 (1991), 29-39.  doi: 10.2307/2389553.  Google Scholar

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J. GolubovL. E. EguiarteM. C. MandujanoJ. 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.  Google Scholar

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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.  Google Scholar

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A. I. InternicolaP. A. PageG. 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.  Google Scholar

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A. I. InternicolaG. 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.  Google Scholar

[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.  Google Scholar

[21]

S. R.-J. Jang, Dynamics of herbivore-plant-pollinator models, J. Math. Biol., 44 (2002), 129-149.  doi: 10.1007/s002850100117.  Google Scholar

[22]

J. JersákováS. D. Johnson and P. Kindlmann, Mechanisms and evolution of deceptive pollination in orchids, Biological Reviews, 81 (2006), 219-235.   Google Scholar

[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.  Google Scholar

[24] T. L. de Jong and P. G. L. Klinkhamer, Evolutionary Ecology of Plant Reproductive Strategies, Cambridge University Press, Cambridge, 2005.   Google Scholar
[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.  Google Scholar

[26]

J. A. J. MetzR. 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.  Google Scholar

[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.   Google Scholar

[28]

S. Neniez-VieyraM. OrdanoJ. FornoniK. 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.   Google Scholar

[29]

L. A. Nilsson, Orchid pollination biology, Trends in Ecology and Evolution, 7 (1992), 255-259.   Google Scholar

[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.   Google Scholar

[31]

E. R. Pianka, Evolutionary ecology, Harper and Row, New York, (1974), 133–146. Google Scholar

[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.  Google Scholar

[33] L. van der Pijl and C. H. Dodson, Orchid Flowers, Their Pollination and Evolution, University of Miami Press, Coral Gables, Fla., 1966.   Google Scholar
[34]

H. C. QuT. 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.  Google Scholar

[35]

J. L. RenD. 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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[38]

D. J. Rodríguez and L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bulletin of Mathematical Biology, 63 (2001), 547-571.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

P. D. Taylor, Evolutionaryily stables strategies with two types of players, J. Appl. Prob., 16 (1979), 76-83.  doi: 10.2307/3213376.  Google Scholar

[42]

J. D. ThakarK. KunteA. K. ChauhanA. 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.  Google Scholar

[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.  Google Scholar

[44]

Y. S. WangH. Wu and S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81 (2012), 243-250.   Google Scholar

[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.  Google Scholar

[46]

Y. S. WangH. 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.  Google Scholar

show all references

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.  Google Scholar

[2]

G. Bell, The evolution of empty flowers, J. Theor. Biol., 118 (1986), 253-258.  doi: 10.1016/S0022-5193(86)80057-1.  Google Scholar

[3]

J. M. BiernaskieS. C. Walker and R. J. Gegear, Bumblebees learn to forage like Bayesians, The American Naturalist, 174 (2009), 413-423.  doi: 10.1086/603629.  Google Scholar

[4]

R. S. CantrellC. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. Cosner, Variability, vagueness and comparison methods for ecological models, Bull. Math. Biol., 58 (1996), 207-246.  doi: 10.1007/BF02458307.  Google Scholar

[8]

N. Davis, The Selfish Gene, Macat Library, London, 2017. doi: 10.4324/9781912281251.  Google Scholar

[9]

D. L. DeAngelisR. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

F. S. GilbertN. Haines and K. Dickson, Empty flowers, Functional Ecology, 5 (1991), 29-39.  doi: 10.2307/2389553.  Google Scholar

[13]

J. GolubovL. E. EguiarteM. C. MandujanoJ. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.  doi: 10.1017/CBO9781139173179.  Google Scholar
[18]

A. I. InternicolaP. A. PageG. 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.  Google Scholar

[19]

A. I. InternicolaG. 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.  Google Scholar

[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.  Google Scholar

[21]

S. R.-J. Jang, Dynamics of herbivore-plant-pollinator models, J. Math. Biol., 44 (2002), 129-149.  doi: 10.1007/s002850100117.  Google Scholar

[22]

J. JersákováS. D. Johnson and P. Kindlmann, Mechanisms and evolution of deceptive pollination in orchids, Biological Reviews, 81 (2006), 219-235.   Google Scholar

[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.  Google Scholar

[24] T. L. de Jong and P. G. L. Klinkhamer, Evolutionary Ecology of Plant Reproductive Strategies, Cambridge University Press, Cambridge, 2005.   Google Scholar
[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.  Google Scholar

[26]

J. A. J. MetzR. 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.  Google Scholar

[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.   Google Scholar

[28]

S. Neniez-VieyraM. OrdanoJ. FornoniK. 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.   Google Scholar

[29]

L. A. Nilsson, Orchid pollination biology, Trends in Ecology and Evolution, 7 (1992), 255-259.   Google Scholar

[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.   Google Scholar

[31]

E. R. Pianka, Evolutionary ecology, Harper and Row, New York, (1974), 133–146. Google Scholar

[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.  Google Scholar

[33] L. van der Pijl and C. H. Dodson, Orchid Flowers, Their Pollination and Evolution, University of Miami Press, Coral Gables, Fla., 1966.   Google Scholar
[34]

H. C. QuT. 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.  Google Scholar

[35]

J. L. RenD. 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.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[38]

D. J. Rodríguez and L. Torres-Sorando, Models of infectious diseases in spatially heterogeneous environments, Bulletin of Mathematical Biology, 63 (2001), 547-571.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

P. D. Taylor, Evolutionaryily stables strategies with two types of players, J. Appl. Prob., 16 (1979), 76-83.  doi: 10.2307/3213376.  Google Scholar

[42]

J. D. ThakarK. KunteA. K. ChauhanA. 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.  Google Scholar

[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.  Google Scholar

[44]

Y. S. WangH. Wu and S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81 (2012), 243-250.   Google Scholar

[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.  Google Scholar

[46]

Y. S. WangH. 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.  Google Scholar

Figure 1.  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, $$ d_2 = 0.55, a_{12} = 0.8, a_{21} = \tilde{a}_{21} = 0.9, $ $ b_{12} = 0.2, \tilde{d}_2 = 0.5, $ $ \tilde{r}_1 = 0.45, \tilde{\alpha} = 0.1. $ (a) When secretors' efficiency in translating neglector-secretor interactions into fitness is large, the two species can coexist at a steady state $ E_{11}(u_{11},v_{11}) $. (b) When cheaters' efficiency in translating neglector-cheater interactions into fitness is large, neglectors and cheaters can coexist at a steady state $ E_{12} $. (c) When the secretors' efficiency in translating selector-secretor interactions into fitness is large, the two species coexist at a steady state $ E_{21}(u_{21},v_{12}) $ with $ u_{21}<u_{11} $. (d) In the selector-cheater system, selectors approach the carrying capacity and cheaters go to extinction, which is shown by the stable equilibrium $ E_{22} $
Figure 2.  Solutions of equations (5). Let $ r_1 = 0.5, d_1 = 0.01, \alpha = 0.15, \beta = 0.8, $$ 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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