# American Institute of Mathematical Sciences

## Modelling the evolutionary dynamics of host resistance-related traits in a susceptible-infected community with density-dependent mortality

 School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China

* Corresponding author: jianzu@xjtu.edu.cn

Received  April 2019 Revised  October 2019 Published  February 2020

Fund Project: The study was supported by grants from the National Natural Science Foundation of China (11971375, 11571272, 11201368 and 11631012), grant from the National Science and Technology Major Project of China (2018ZX10721202), grant from the Natural Science Foundation of Shaanxi Province (2019JM-273), and grant from the China Postdoctoral Science Foundation (2014M560755)

This study explores the evolutionary dynamics of host resistance based on a susceptible-infected population model with density-dependent mortality. We assume that the resistant ability of susceptible host will adaptively evolve, a different type of host differs in its susceptibility to infection, but the resistance to a pathogen involves a cost such that a less susceptible host results in a lower birth rate. By using the methods of adaptive dynamics and critical function analysis, we find that the evolutionary outcome relies mainly on the trade-off relationship between host resistance and its fertility. Firstly, we show that if the trade-off curve is globally concave, then a continuously stable strategy is predicted. In contrast, if the trade-off curve is weakly convex in the vicinity of singular strategy, then evolutionary branching of host resistance is possible. Secondly, after evolutionary branching in the host resistance has occurred, we examine the coevolutionary dynamics of dimorphic susceptible hosts and find that for a type of concave-convex-concave trade-off curve, the finally evolutionary outcome may contain a relatively higher susceptible host and a relatively higher resistant host, which can continuously stably coexist on a long-term evolutionary timescale. If the convex region of trade-off curve is relatively wider, then the finally evolutionary outcome may contain a fully resistant host and a moderately resistant host. Thirdly, through numerical simulation, we find that for a type of sigmoidal trade-off curve, after branching due to the high cost in terms of the birth rate, always the branch with stronger resistance goes extinct, the eventually evolutionary outcome includes a monomorphic host with relatively weaker resistance.

Citation: Jian Zu, Miaolei Li, Yuexi Gu, Shuting Fu. Modelling the evolutionary dynamics of host resistance-related traits in a susceptible-infected community with density-dependent mortality. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020051
##### References:
 [1] T. Ammunét, T. Klemola and K. Parvinen, Consequences of asymmetric competition between resident and invasive defoliators: A novel empirically based modelling approach, Theor. Popul. Biol., 92 (2014), 107-117.  doi: 10.1016/j.tpb.2013.12.006.  Google Scholar [2] R. M. Anderson and R. M. May, Coevolution of hosts and parasites, Parasitology, 85 (1982), 411-426.  doi: 10.1017/S0031182000055360.  Google Scholar [3] J. Antonovics and P. H. Thrall, The cost of resistance and the maintenance of genetic polymorphism in host-pathogen systems, Proc. Roy. Soc. B, 257 (1994), 105-110.  doi: 10.1098/rspb.1994.0101.  Google Scholar [4] A. Best, R. Bowers and A. White, Evolution, the loss of diversity and the role of trade-offs, Math. Biosci., 264 (2015), 86-93.  doi: 10.1016/j.mbs.2015.03.011.  Google Scholar [5] A. Best, H. Tidbury, A. White and M. Boots, The evolutionary dynamics of within-generation immune priming in invertebrate hosts, J. Royal Society Interface, 10 (2013). doi: 10.1098/rsif.2012.0887.  Google Scholar [6] A. Best, A. White and M. Boots, The implications of coevolutionary dynamics to host-parasite interactions, Amer. Naturalist, 173 (2009), 779-791.  doi: 10.1086/598494.  Google Scholar [7] A. Best, A. White and M. Boots, The evolution of host defence when parasites impact reproduction, Evolutionary Ecology Research, 18 (2017), 393-409.   Google Scholar [8] B. Boldin, S. A. H. Geritz and É. Kisdi, Superinfections and adaptive dynamics of pathogen virulence revisited: A critical function analysis, Evolutionary Ecology Research, 11 (2009), 153-175.   Google Scholar [9] M. H. Bonds, Host life-history strategy explains parasite-induced sterility, Amer. Naturalist, 168 (2006), 281-293.  doi: 10.1086/506922.  Google Scholar [10] M. Boots, A. Best, M. R. Miller and A. White, The role of ecological feedbacks in the evolution of host defence: What does theory tell us, Philos. Trans. Roy. Soc. B, 364 (2009), 27-36.  doi: 10.1098/rstb.2008.0160.  Google Scholar [11] M. Boots and M. Begon, Trade-offs with resistance to a granulosis virus in the Indian meal moth, examined by a laboratory evolution experiment, Functional Ecology, 7 (1993), 528-534.  doi: 10.2307/2390128.  Google Scholar [12] M. Boots and R. G. Bowers, Three mechanisms of host resistance to microparasites–avoidance, recovery and tolerance–show different evolutionary dynamics, J. Theoretical Biology, 201 (1999), 13-23.  doi: 10.1006/jtbi.1999.1009.  Google Scholar [13] M. Boots and R. G. Bowers, The evolution of resistance through costly acquired immunity, Proc. Roy. Soc. B, 271 (2004), 715-723.  doi: 10.1098/rspb.2003.2655.  Google Scholar [14] M. Boots and Y. Haraguchi, The evolution of costly resistance in host-parasite systems, Amer. Naturalist, 153 (1999), 359-370.  doi: 10.1086/303181.  Google Scholar [15] M. Boots, A. White, A. Best and R. Bowers, How specificity and epidemiology drive the coevolution of static trait diversity in hosts and parasites, Evolution, 68 (2014), 1594-1606.  doi: 10.1111/evo.12393.  Google Scholar [16] R. G. Bowers, A. Hoyle, A. White and M. Boots, The geometric theory of adaptive evolution: Trade-off and invasion plots, J. Theoret. Biol., 233 (2005), 363-377.  doi: 10.1016/j.jtbi.2004.10.017.  Google Scholar [17] R. G. Bowers, The basic depression ratio of the host: The evolution of host resistance to microparasites, Proc. Roy. Soc. B, 268 (2001), 243-250.  doi: 10.1098/rspb.2000.1360.  Google Scholar [18] R. G. Bowers, A baseline model for the apparent competition between many host strains: The evolution of host resistance to microparasites, J. Theoret. Biol., 200 (1999), 65-75.  doi: 10.1006/jtbi.1999.0976.  Google Scholar [19] R. G. Bowers, M. Boots and M. Begon, Life-history trade-offs and the evolution of pathogen resistance: Competition between host strains, Proc. Roy. Soc. B, 257 (1994), 247-253.  doi: 10.1098/rspb.1994.0122.  Google Scholar [20] R. S. Cantrell, C. Cosner and K. Y. Lam, Resident-invader dynamics in infinite dimensional systems, J. Differential Equations, 263 (2017), 4565-4616.  doi: 10.1016/j.jde.2017.05.029.  Google Scholar [21] F. B. Christiansen, On conditions for evolutionary stability for a continuously varying character, Amer. Naturalist, 138 (1991), 37-50.  doi: 10.1086/285203.  Google Scholar [22] R. Cressman, CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces, J. Theoret. Biol., 262 (2010), 80-89.  doi: 10.1016/j.jtbi.2009.09.019.  Google Scholar [23] F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: The Adaptative Dynamics Approach and its Applications, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400828340.  Google Scholar [24] F. Dercole, Remarks on branching-extinction evolutionary cycles, J. Math. Biol., 47 (2003), 569-580.  doi: 10.1007/s00285-003-0236-4.  Google Scholar [25] U. Dieckmann and M. Doebeli, On the origin of species by sympatric speciation, Nature, 400 (1999), 354-357.  doi: 10.1038/22521.  Google Scholar [26] O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.  Google Scholar [27] U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.  doi: 10.1007/BF02409751.  Google Scholar [28] U. Dieckmann, J. A. J. Metz and M. W. Sabelis, Adaptive Dynamics of Infectious Diseases: In Pursuit of Virulence Management, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511525728.  Google Scholar [29] M. Doebeli and U. Dieckmann, Evolutionary branching and sympatric speciation caused by different types of ecological interactions, Amer. Naturalist, 156 (2000), S77–S101. doi: 10.1086/303417.  Google Scholar [30] I. Eshel, Evolutionary and continuous stability, J. Theoret. Biol., 103 (1983), 99-111.  doi: 10.1016/0022-5193(83)90201-1.  Google Scholar [31] C. Ferris and A. Best, The evolution of host defence to parasitism in fluctuating environments, J. Theoret. Biol., 440 (2018), 58-65.  doi: 10.1016/j.jtbi.2017.12.006.  Google Scholar [32] S. Gandon, P. Agnew and Y. Michalakis, Coevolution between parasite virulence and host life-history traits, Amer. Naturalist, 160 (2002), 374-388.  doi: 10.1086/341525.  Google Scholar [33] F. Gascuel, M. Choisy and J. M. Duplantier, et al., Host resistance, population structure and the long-term persistence of bubonic plague: Contributions of a modelling approach in the Malagasy focus, PLoS Comput. Biol., 9 (2013). doi: 10.1371/journal.pcbi.1003039.  Google Scholar [34] S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57.  doi: 10.1023/A:1006554906681.  Google Scholar [35] S. A. H. Geritz, E. van der Meijden and J. A. J. Metz, Evolutionary dynamics of seed size and seedling competitive ability, Theor. Popul. Biol., 55 (1999), 324-343.  doi: 10.1006/tpbi.1998.1409.  Google Scholar [36] S. A. H. Geritz, É. Kisdi and P. Yan, Evolutionary branching and long-term coexistence of cycling predators: Critical function analysis, Theor. Popul. Biol., 71 (2007), 424-435.  doi: 10.1016/j.tpb.2007.03.006.  Google Scholar [37] S. A. H. Geritz, M. Gyllenberg, F. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol., 44 (2002), 548-560.  doi: 10.1007/s002850100136.  Google Scholar [38] S. A. H. Geritz, Resident-invader dynamics and the coexistence of similar strategies, J. Math. Biol., 50 (2005), 67-82.  doi: 10.1007/s00285-004-0280-8.  Google Scholar [39] A. Hoyle, R. G. Bowers, A. White and M. Boots, The influence of trade-off shape on evolutionary behaviour in classical ecological scenarios, J. Theoret. Biol., 250 (2008), 498-511.  doi: 10.1016/j.jtbi.2007.10.009.  Google Scholar [40] J. Johansson, J. Ripa and N. Kuckländer, The risk of competitive exclusion during evolutionary branching: Effects of resource variability, correlation and autocorrelation, Theor. Popul. Biol., 77 (2010), 95-104.  doi: 10.1016/j.tpb.2009.10.007.  Google Scholar [41] É. Kisdi and S. A. H. Geritz, Adaptive dynamics of saturated polymorphisms, J. Math. Biol., 72 (2016), 1039-1079.  doi: 10.1007/s00285-015-0948-2.  Google Scholar [42] É. Kisdi, Evolutionary branching under asymmetric competition, J. Theoret. Biol., 197 (1999), 149-162.  doi: 10.1006/jtbi.1998.0864.  Google Scholar [43] É. Kisdi, Trade-off geometries and the adaptive dynamics of two co-evolving species, Evolutionary Ecology Research, 8 (2006), 959-973.   Google Scholar [44] É. Kisdi, F. J. A. Jacobs and S. A. H. Geritz, Red Queen evolution by cycles of evolutionary branching and extinction, Selection, 2 (2002), 161-176.  doi: 10.1556/Select.2.2001.1-2.12.  Google Scholar [45] A. R. Kraaijeveld and H. C. J. Godfray, Trade-off between parasitoid resistance and larval competitive ability in Drosophila melanogaster, Nature, 389 (1997), 278-280.  doi: 10.1038/38483.  Google Scholar [46] A. R. Kraaijeveld, S. J. Layen and P. H. Futerman, et al., Lack of phenotypic and evolutionary cross-resistance against parasitoids and pathogens in Drosophila melanogaster, PloS One, 7 (2012). doi: 10.1371/journal.pone.0053002.  Google Scholar [47] P. Landi, F. Dercole and S. Rinaldi, Branching scenarios in eco-evolutionary prey-predator models, SIAM J. Appl. Math., 73 (2013), 1634-1658.  doi: 10.1137/12088673X.  Google Scholar [48] R. Law, P. Marrow and U. Dieckmann, On evolution under asymmetric competition, Evolutionary Ecology, 11 (1997), 485-501.  doi: 10.1023/A:1018441108982.  Google Scholar [49] O. Leimar, Multidimensional convergence stability, Evolutionary Ecology Research, 11 (2009), 191-208.   Google Scholar [50] B. Lemaitre and J. Hoffmann, The host defense of Drosophila melanogaster, Annual Rev. Immunology, 25 (2007), 697-743.  doi: 10.1146/annurev.immunol.25.022106.141615.  Google Scholar [51] S. Lion and J. A. J. Metz, Beyond $R_{0}$ Maximisation: On pathogen evolution and environmental dimensions, Trends Ecol. Evol., 33 (2018), 458-473.  doi: 10.1016/j.tree.2018.02.004.  Google Scholar [52] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982.  doi: 10.1017/CBO9780511806292.  Google Scholar [53] C. de Mazancourt and U. Dieckmann, Trade-off geometries and frequency-dependent selection, Amer. Naturalist, 164 (2004), 765-778.  doi: 10.1086/424762.  Google Scholar [54] M. A. Mealor and M. Boots, An indirect approach to imply trade-off shapes: Population level patterns in resistance suggest a decreasingly costly resistance mechanism in a model insect system, J. Evolutionary Biol., 19 (2006), 326-330.  doi: 10.1111/j.1420-9101.2005.01031.x.  Google Scholar [55] R. Medzhitov, Recognition of microorganisms and activation of the immune response, Nature, 449 (2007), 819-826.  doi: 10.1038/nature06246.  Google Scholar [56] J. A. J. Metz, R. M. Nisbet and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?, Trends Ecol. Evol., 7 (1992), 198-202.  doi: 10.1016/0169-5347(92)90073-K.  Google Scholar [57] G. Meszéna, M. Gyllenberg, F. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of Darwinian evolution, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.078105.  Google Scholar [58] M. R. Miller, A. White and M. Boots, The evolution of host resistance: Tolerance and control as distinct strategies, J. Theoret. Biol., 236 (2005), 198-207.  doi: 10.1016/j.jtbi.2005.03.005.  Google Scholar [59] M. R. Miller, A. White and M. Boots, The evolution of parasites in response to tolerance in their hosts: The good, the bad and apparent commensalism, Evolution, 60 (2006), 945-956.  doi: 10.1111/j.0014-3820.2006.tb01173.x.  Google Scholar [60] M. R. Miller, A. White and M. Boots, Host life span and the evolution of resistance characteristics, Evolution, 61 (2007), 2-14.  doi: 10.1111/j.1558-5646.2007.00001.x.  Google Scholar [61] M. A. Nowak and K. Sigmund, Evolutionary dynamics of biological games, Science, 303 (2004), 793-799.  doi: 10.1126/science.1093411.  Google Scholar [62] K. Parvinen, Evolutionary suicide, Acta Biotheoretica, 53 (2005), 241-264.  doi: 10.1007/s10441-005-2531-5.  Google Scholar [63] A. Peschel and H. G. Sahl, The co-evolution of host cationic antimicrobial peptides and microbial resistance, Nature Rev. Microbiology, 4 (2006), 529-536.  doi: 10.1038/nrmicro1441.  Google Scholar [64] O. Restif and J. C. Koella, Shared control of epidemiological traits in a coevolutionary model of host-parasite interactions, Amer. Naturalist, 161 (2003), 827-836.  doi: 10.1086/375171.  Google Scholar [65] O. Restif and J. C. Koella, Concurrent evolution of resistance and tolerance to pathogens, Amer. Naturalist, 164 (2004), E90–E102. doi: 10.1086/423713.  Google Scholar [66] D. A. Roff, Life History Evolution, Sinauer Associates, Sunderland, MA, 2002. Google Scholar [67] B. A. Roy and J. W. Kirchner, Evolutionary dynamics of pathogen resistance and tolerance, Evolution, 54 (2000), 51-63.  doi: 10.1111/j.0014-3820.2000.tb00007.x.  Google Scholar [68] J. Sardanyés and R. V. Solé, Chaotic stability in spatially-resolved host-parasite replicators: The Red Queen on a lattice, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 589-606.  doi: 10.1142/S0218127407017458.  Google Scholar [69] E. Shim and A. P. Galvani, Evolutionary repercussions of avian culling on host resistance and influenza virulence, PloS One, 4 (2009). doi: 10.1371/journal.pone.0005503.  Google Scholar [70] M. L. Simoes, E. P. Caragata and G. Dimopoulos, Diverse host and restriction factors regulate mosquito-pathogen interactions, Trends in Parasitology, 34 (2018), 603-616.  doi: 10.1016/j.pt.2018.04.011.  Google Scholar [71] S. C. Stearns, The Evolution of Life Histories, Oxford University Press, Oxford, 1992.   Google Scholar [72] T. O. Svennungsen and É. Kisdi, Evolutionary branching of virulence in a single-infection model, J. Theoret. Biol., 257 (2009), 408-418.  doi: 10.1016/j.jtbi.2008.11.014.  Google Scholar [73] A. N. Theodosopoulos, A. K. Hund and S. A. Taylor, Parasites and host species barriers in animal hybrid zones, Trends Ecol. Evol., 34 (2019), 19-30.  doi: 10.1016/j.tree.2018.09.011.  Google Scholar [74] W. Wang, Y. Li and H. W. Hethcote, Bifurcations in a host-parasite model with nonlinear incidence, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 3291-3307.  doi: 10.1142/S0218127406016793.  Google Scholar [75] J. Zu, K. F. Wang and M. Mimura, Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis, Math. Biosci., 231 (2011), 210-224.  doi: 10.1016/j.mbs.2011.03.007.  Google Scholar [76] J. Zu, J. L. Wang and J. Q. Du, Adaptive evolution of defense ability leads to diversification of prey species, Acta Biotheoretica, 62 (2014), 207-234.  doi: 10.1007/s10441-014-9218-8.  Google Scholar [77] J. Zu, B. Yuan and J. Q. Du, Top predators induce the evolutionary diversification of intermediate predator species, J. Theoret. Biol., 387 (2015), 1-12.  doi: 10.1016/j.jtbi.2015.09.024.  Google Scholar

show all references

##### References:
 [1] T. Ammunét, T. Klemola and K. Parvinen, Consequences of asymmetric competition between resident and invasive defoliators: A novel empirically based modelling approach, Theor. Popul. Biol., 92 (2014), 107-117.  doi: 10.1016/j.tpb.2013.12.006.  Google Scholar [2] R. M. Anderson and R. M. May, Coevolution of hosts and parasites, Parasitology, 85 (1982), 411-426.  doi: 10.1017/S0031182000055360.  Google Scholar [3] J. Antonovics and P. H. Thrall, The cost of resistance and the maintenance of genetic polymorphism in host-pathogen systems, Proc. Roy. Soc. B, 257 (1994), 105-110.  doi: 10.1098/rspb.1994.0101.  Google Scholar [4] A. Best, R. Bowers and A. White, Evolution, the loss of diversity and the role of trade-offs, Math. Biosci., 264 (2015), 86-93.  doi: 10.1016/j.mbs.2015.03.011.  Google Scholar [5] A. Best, H. Tidbury, A. White and M. Boots, The evolutionary dynamics of within-generation immune priming in invertebrate hosts, J. Royal Society Interface, 10 (2013). doi: 10.1098/rsif.2012.0887.  Google Scholar [6] A. Best, A. White and M. Boots, The implications of coevolutionary dynamics to host-parasite interactions, Amer. Naturalist, 173 (2009), 779-791.  doi: 10.1086/598494.  Google Scholar [7] A. Best, A. White and M. Boots, The evolution of host defence when parasites impact reproduction, Evolutionary Ecology Research, 18 (2017), 393-409.   Google Scholar [8] B. Boldin, S. A. H. Geritz and É. Kisdi, Superinfections and adaptive dynamics of pathogen virulence revisited: A critical function analysis, Evolutionary Ecology Research, 11 (2009), 153-175.   Google Scholar [9] M. H. Bonds, Host life-history strategy explains parasite-induced sterility, Amer. Naturalist, 168 (2006), 281-293.  doi: 10.1086/506922.  Google Scholar [10] M. Boots, A. Best, M. R. Miller and A. White, The role of ecological feedbacks in the evolution of host defence: What does theory tell us, Philos. Trans. Roy. Soc. B, 364 (2009), 27-36.  doi: 10.1098/rstb.2008.0160.  Google Scholar [11] M. Boots and M. Begon, Trade-offs with resistance to a granulosis virus in the Indian meal moth, examined by a laboratory evolution experiment, Functional Ecology, 7 (1993), 528-534.  doi: 10.2307/2390128.  Google Scholar [12] M. Boots and R. G. Bowers, Three mechanisms of host resistance to microparasites–avoidance, recovery and tolerance–show different evolutionary dynamics, J. Theoretical Biology, 201 (1999), 13-23.  doi: 10.1006/jtbi.1999.1009.  Google Scholar [13] M. Boots and R. G. Bowers, The evolution of resistance through costly acquired immunity, Proc. Roy. Soc. B, 271 (2004), 715-723.  doi: 10.1098/rspb.2003.2655.  Google Scholar [14] M. Boots and Y. Haraguchi, The evolution of costly resistance in host-parasite systems, Amer. Naturalist, 153 (1999), 359-370.  doi: 10.1086/303181.  Google Scholar [15] M. Boots, A. White, A. Best and R. Bowers, How specificity and epidemiology drive the coevolution of static trait diversity in hosts and parasites, Evolution, 68 (2014), 1594-1606.  doi: 10.1111/evo.12393.  Google Scholar [16] R. G. Bowers, A. Hoyle, A. White and M. Boots, The geometric theory of adaptive evolution: Trade-off and invasion plots, J. Theoret. Biol., 233 (2005), 363-377.  doi: 10.1016/j.jtbi.2004.10.017.  Google Scholar [17] R. G. Bowers, The basic depression ratio of the host: The evolution of host resistance to microparasites, Proc. Roy. Soc. B, 268 (2001), 243-250.  doi: 10.1098/rspb.2000.1360.  Google Scholar [18] R. G. Bowers, A baseline model for the apparent competition between many host strains: The evolution of host resistance to microparasites, J. Theoret. Biol., 200 (1999), 65-75.  doi: 10.1006/jtbi.1999.0976.  Google Scholar [19] R. G. Bowers, M. Boots and M. Begon, Life-history trade-offs and the evolution of pathogen resistance: Competition between host strains, Proc. Roy. Soc. B, 257 (1994), 247-253.  doi: 10.1098/rspb.1994.0122.  Google Scholar [20] R. S. Cantrell, C. Cosner and K. Y. Lam, Resident-invader dynamics in infinite dimensional systems, J. Differential Equations, 263 (2017), 4565-4616.  doi: 10.1016/j.jde.2017.05.029.  Google Scholar [21] F. B. Christiansen, On conditions for evolutionary stability for a continuously varying character, Amer. Naturalist, 138 (1991), 37-50.  doi: 10.1086/285203.  Google Scholar [22] R. Cressman, CSS, NIS and dynamic stability for two-species behavioral models with continuous trait spaces, J. Theoret. Biol., 262 (2010), 80-89.  doi: 10.1016/j.jtbi.2009.09.019.  Google Scholar [23] F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: The Adaptative Dynamics Approach and its Applications, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2008.  doi: 10.1515/9781400828340.  Google Scholar [24] F. Dercole, Remarks on branching-extinction evolutionary cycles, J. Math. Biol., 47 (2003), 569-580.  doi: 10.1007/s00285-003-0236-4.  Google Scholar [25] U. Dieckmann and M. Doebeli, On the origin of species by sympatric speciation, Nature, 400 (1999), 354-357.  doi: 10.1038/22521.  Google Scholar [26] O. Diekmann, P. E. Jabin, S. Mischler and B. Perthame, The dynamics of adaptation: An illuminating example and a Hamilton-Jacobi approach, Theor. Popul. Biol., 67 (2005), 257-271.  doi: 10.1016/j.tpb.2004.12.003.  Google Scholar [27] U. Dieckmann and R. Law, The dynamical theory of coevolution: A derivation from stochastic ecological processes, J. Math. Biol., 34 (1996), 579-612.  doi: 10.1007/BF02409751.  Google Scholar [28] U. Dieckmann, J. A. J. Metz and M. W. Sabelis, Adaptive Dynamics of Infectious Diseases: In Pursuit of Virulence Management, Cambridge University Press, Cambridge, 2005.  doi: 10.1017/CBO9780511525728.  Google Scholar [29] M. Doebeli and U. Dieckmann, Evolutionary branching and sympatric speciation caused by different types of ecological interactions, Amer. Naturalist, 156 (2000), S77–S101. doi: 10.1086/303417.  Google Scholar [30] I. Eshel, Evolutionary and continuous stability, J. Theoret. Biol., 103 (1983), 99-111.  doi: 10.1016/0022-5193(83)90201-1.  Google Scholar [31] C. Ferris and A. Best, The evolution of host defence to parasitism in fluctuating environments, J. Theoret. Biol., 440 (2018), 58-65.  doi: 10.1016/j.jtbi.2017.12.006.  Google Scholar [32] S. Gandon, P. Agnew and Y. Michalakis, Coevolution between parasite virulence and host life-history traits, Amer. Naturalist, 160 (2002), 374-388.  doi: 10.1086/341525.  Google Scholar [33] F. Gascuel, M. Choisy and J. M. Duplantier, et al., Host resistance, population structure and the long-term persistence of bubonic plague: Contributions of a modelling approach in the Malagasy focus, PLoS Comput. Biol., 9 (2013). doi: 10.1371/journal.pcbi.1003039.  Google Scholar [34] S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree, Evolutionary Ecology, 12 (1998), 35-57.  doi: 10.1023/A:1006554906681.  Google Scholar [35] S. A. H. Geritz, E. van der Meijden and J. A. J. Metz, Evolutionary dynamics of seed size and seedling competitive ability, Theor. Popul. Biol., 55 (1999), 324-343.  doi: 10.1006/tpbi.1998.1409.  Google Scholar [36] S. A. H. Geritz, É. Kisdi and P. Yan, Evolutionary branching and long-term coexistence of cycling predators: Critical function analysis, Theor. Popul. Biol., 71 (2007), 424-435.  doi: 10.1016/j.tpb.2007.03.006.  Google Scholar [37] S. A. H. Geritz, M. Gyllenberg, F. J. A. Jacobs and K. Parvinen, Invasion dynamics and attractor inheritance, J. Math. Biol., 44 (2002), 548-560.  doi: 10.1007/s002850100136.  Google Scholar [38] S. A. H. Geritz, Resident-invader dynamics and the coexistence of similar strategies, J. Math. Biol., 50 (2005), 67-82.  doi: 10.1007/s00285-004-0280-8.  Google Scholar [39] A. Hoyle, R. G. Bowers, A. White and M. Boots, The influence of trade-off shape on evolutionary behaviour in classical ecological scenarios, J. Theoret. Biol., 250 (2008), 498-511.  doi: 10.1016/j.jtbi.2007.10.009.  Google Scholar [40] J. Johansson, J. Ripa and N. Kuckländer, The risk of competitive exclusion during evolutionary branching: Effects of resource variability, correlation and autocorrelation, Theor. Popul. Biol., 77 (2010), 95-104.  doi: 10.1016/j.tpb.2009.10.007.  Google Scholar [41] É. Kisdi and S. A. H. Geritz, Adaptive dynamics of saturated polymorphisms, J. Math. Biol., 72 (2016), 1039-1079.  doi: 10.1007/s00285-015-0948-2.  Google Scholar [42] É. Kisdi, Evolutionary branching under asymmetric competition, J. Theoret. Biol., 197 (1999), 149-162.  doi: 10.1006/jtbi.1998.0864.  Google Scholar [43] É. Kisdi, Trade-off geometries and the adaptive dynamics of two co-evolving species, Evolutionary Ecology Research, 8 (2006), 959-973.   Google Scholar [44] É. Kisdi, F. J. A. Jacobs and S. A. H. Geritz, Red Queen evolution by cycles of evolutionary branching and extinction, Selection, 2 (2002), 161-176.  doi: 10.1556/Select.2.2001.1-2.12.  Google Scholar [45] A. R. Kraaijeveld and H. C. J. Godfray, Trade-off between parasitoid resistance and larval competitive ability in Drosophila melanogaster, Nature, 389 (1997), 278-280.  doi: 10.1038/38483.  Google Scholar [46] A. R. Kraaijeveld, S. J. Layen and P. H. Futerman, et al., Lack of phenotypic and evolutionary cross-resistance against parasitoids and pathogens in Drosophila melanogaster, PloS One, 7 (2012). doi: 10.1371/journal.pone.0053002.  Google Scholar [47] P. Landi, F. Dercole and S. Rinaldi, Branching scenarios in eco-evolutionary prey-predator models, SIAM J. Appl. Math., 73 (2013), 1634-1658.  doi: 10.1137/12088673X.  Google Scholar [48] R. Law, P. Marrow and U. Dieckmann, On evolution under asymmetric competition, Evolutionary Ecology, 11 (1997), 485-501.  doi: 10.1023/A:1018441108982.  Google Scholar [49] O. Leimar, Multidimensional convergence stability, Evolutionary Ecology Research, 11 (2009), 191-208.   Google Scholar [50] B. Lemaitre and J. Hoffmann, The host defense of Drosophila melanogaster, Annual Rev. Immunology, 25 (2007), 697-743.  doi: 10.1146/annurev.immunol.25.022106.141615.  Google Scholar [51] S. Lion and J. A. J. Metz, Beyond $R_{0}$ Maximisation: On pathogen evolution and environmental dimensions, Trends Ecol. Evol., 33 (2018), 458-473.  doi: 10.1016/j.tree.2018.02.004.  Google Scholar [52] J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982.  doi: 10.1017/CBO9780511806292.  Google Scholar [53] C. de Mazancourt and U. Dieckmann, Trade-off geometries and frequency-dependent selection, Amer. Naturalist, 164 (2004), 765-778.  doi: 10.1086/424762.  Google Scholar [54] M. A. Mealor and M. Boots, An indirect approach to imply trade-off shapes: Population level patterns in resistance suggest a decreasingly costly resistance mechanism in a model insect system, J. Evolutionary Biol., 19 (2006), 326-330.  doi: 10.1111/j.1420-9101.2005.01031.x.  Google Scholar [55] R. Medzhitov, Recognition of microorganisms and activation of the immune response, Nature, 449 (2007), 819-826.  doi: 10.1038/nature06246.  Google Scholar [56] J. A. J. Metz, R. M. Nisbet and S. A. H. Geritz, How should we define 'fitness' for general ecological scenarios?, Trends Ecol. Evol., 7 (1992), 198-202.  doi: 10.1016/0169-5347(92)90073-K.  Google Scholar [57] G. Meszéna, M. Gyllenberg, F. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of Darwinian evolution, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.078105.  Google Scholar [58] M. R. Miller, A. White and M. Boots, The evolution of host resistance: Tolerance and control as distinct strategies, J. Theoret. Biol., 236 (2005), 198-207.  doi: 10.1016/j.jtbi.2005.03.005.  Google Scholar [59] M. R. Miller, A. White and M. Boots, The evolution of parasites in response to tolerance in their hosts: The good, the bad and apparent commensalism, Evolution, 60 (2006), 945-956.  doi: 10.1111/j.0014-3820.2006.tb01173.x.  Google Scholar [60] M. R. Miller, A. White and M. Boots, Host life span and the evolution of resistance characteristics, Evolution, 61 (2007), 2-14.  doi: 10.1111/j.1558-5646.2007.00001.x.  Google Scholar [61] M. A. Nowak and K. Sigmund, Evolutionary dynamics of biological games, Science, 303 (2004), 793-799.  doi: 10.1126/science.1093411.  Google Scholar [62] K. Parvinen, Evolutionary suicide, Acta Biotheoretica, 53 (2005), 241-264.  doi: 10.1007/s10441-005-2531-5.  Google Scholar [63] A. Peschel and H. G. Sahl, The co-evolution of host cationic antimicrobial peptides and microbial resistance, Nature Rev. Microbiology, 4 (2006), 529-536.  doi: 10.1038/nrmicro1441.  Google Scholar [64] O. Restif and J. C. Koella, Shared control of epidemiological traits in a coevolutionary model of host-parasite interactions, Amer. Naturalist, 161 (2003), 827-836.  doi: 10.1086/375171.  Google Scholar [65] O. Restif and J. C. Koella, Concurrent evolution of resistance and tolerance to pathogens, Amer. Naturalist, 164 (2004), E90–E102. doi: 10.1086/423713.  Google Scholar [66] D. A. Roff, Life History Evolution, Sinauer Associates, Sunderland, MA, 2002. Google Scholar [67] B. A. Roy and J. W. Kirchner, Evolutionary dynamics of pathogen resistance and tolerance, Evolution, 54 (2000), 51-63.  doi: 10.1111/j.0014-3820.2000.tb00007.x.  Google Scholar [68] J. Sardanyés and R. V. Solé, Chaotic stability in spatially-resolved host-parasite replicators: The Red Queen on a lattice, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 589-606.  doi: 10.1142/S0218127407017458.  Google Scholar [69] E. Shim and A. P. Galvani, Evolutionary repercussions of avian culling on host resistance and influenza virulence, PloS One, 4 (2009). doi: 10.1371/journal.pone.0005503.  Google Scholar [70] M. L. Simoes, E. P. Caragata and G. Dimopoulos, Diverse host and restriction factors regulate mosquito-pathogen interactions, Trends in Parasitology, 34 (2018), 603-616.  doi: 10.1016/j.pt.2018.04.011.  Google Scholar [71] S. C. Stearns, The Evolution of Life Histories, Oxford University Press, Oxford, 1992.   Google Scholar [72] T. O. Svennungsen and É. Kisdi, Evolutionary branching of virulence in a single-infection model, J. Theoret. Biol., 257 (2009), 408-418.  doi: 10.1016/j.jtbi.2008.11.014.  Google Scholar [73] A. N. Theodosopoulos, A. K. Hund and S. A. Taylor, Parasites and host species barriers in animal hybrid zones, Trends Ecol. Evol., 34 (2019), 19-30.  doi: 10.1016/j.tree.2018.09.011.  Google Scholar [74] W. Wang, Y. Li and H. W. Hethcote, Bifurcations in a host-parasite model with nonlinear incidence, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16 (2006), 3291-3307.  doi: 10.1142/S0218127406016793.  Google Scholar [75] J. Zu, K. F. Wang and M. Mimura, Evolutionary branching and evolutionarily stable coexistence of predator species: Critical function analysis, Math. Biosci., 231 (2011), 210-224.  doi: 10.1016/j.mbs.2011.03.007.  Google Scholar [76] J. Zu, J. L. Wang and J. Q. Du, Adaptive evolution of defense ability leads to diversification of prey species, Acta Biotheoretica, 62 (2014), 207-234.  doi: 10.1007/s10441-014-9218-8.  Google Scholar [77] J. Zu, B. Yuan and J. Q. Du, Top predators induce the evolutionary diversification of intermediate predator species, J. Theoret. Biol., 387 (2015), 1-12.  doi: 10.1016/j.jtbi.2015.09.024.  Google Scholar
Continuously stable strategy of host resistance when the initial trait value $\beta (0) = 0.0009$. (a) A trade-off curve $b(\beta)$ described in (17) with $b_{0} = 0.01, b_{1} = 2.0$ and $b_{2} = 0.1$ and a tangential critical function curve $b_{crit}(\beta)$. The filled circle $\beta^{*} = 0.000646$ is an evolutionarily singular strategy. The grey area indicates combinations of $\beta$ and $b(\beta)$ for which the endemic equilibrium, $(S^{*}(\beta), I^{*}(\beta))$, is globally asymptotically stable. (b) A pairwise invasibility plot (PIP) obtained with the trade-off function in (a). The grey areas indicate the mutant host can invade, while in green areas the mutant host becomes extinct. $\beta^{*} = 0.000646$ is a continuously stable strategy. (c) An evolutionary time series plot of transmission rate $\beta$ obtained through simulation of model (11) with initial condition $\beta (0) = 0.0009$. (d) The basic reproduction number $R_{0}$ against evolutionary time. (e) Equilibrium population densities of susceptible hosts $S^{*}(\beta)$ and infected hosts $I^{*}(\beta)$ when the transmission rate $\beta$ evolves. Other parameter values: $\alpha = 0.01, m_{0} = 0.005, m_{1} = 0.0001, \mu = 0.01, \sigma = 0.00001$
Continuously stable strategy of host resistance when the initial trait value $\beta (0) = 0.0003$. (a) An evolutionary time series plot of transmission rate $\beta$ obtained through simulation of model (11) with initial condition $\beta (0) = 0.0003$. (b) The basic reproduction number $R_{0}$ against evolutionary time. (c) Equilibrium population densities of susceptible hosts $S^{*}(\beta)$ and infected hosts $I^{*}(\beta)$ when the transmission rate $\beta$ evolves. (d) The basic reproduction number $R_{0}$ against transmission rate $\beta$. Other parameter values are the same as in Fig. 1
Evolutionary branching strategy of host resistance. (a) A trade-off curve $b(\beta)$ described in (18) with $b_0 = 0.055, b_{1} = 4.0, b_{2} = 0.05, b_{3} = 0.05, \beta_0 = 0.0005, \sigma_c = 0.0002$ and a tangential critical function curve $b_{crit}(\beta)$. The filled circle $A = 0.000419$ is an evolutionarily singular strategy. The grey area indicates combinations of $\beta$ and $b(\beta)$ for which the endemic equilibrium, $(S^{*}(\beta), I^{*}(\beta))$, is globally asymptotically stable. (b) A pairwise invasibility plot (PIP) obtained with the trade-off function in (a). The grey regions indicate the mutant host can invade, while in green regions the mutant host becomes extinct. $A, B, C$ are three evolutionarily singular strategies. (c) A mutual invasibility plot obtained with the trade-off function in (a). In the grey regions, both $f(\beta_{m}, \beta)>0$ and $f(\beta, \beta_{m})>0$. Other parameter values: $\alpha = 0.075, m_{0} = 0.006, m_{1} = 0.00014, \mu = 0.01, \sigma = 0.00001$
Bifurcation diagram of evolutionary dynamics when the parameter of tarde-off function is changed. (a) Evolutionarily singular strategy $\beta^{*}$ versus the maximum birth rate $b_{1}$ when the trade-off function is the same as in Fig. 1a. (b) Evolutionarily singular strategy $\beta^{*}$ versus the maximum birth rate $b_{1}$ when the trade-off function is the same as in Fig. 3a. (c) Evolutionarily singular strategy $\beta^{*}$ versus the trade-off strength $b_{3}$ when the trade-off function is the same as in Fig. 3a. (d) Evolutionarily singular strategy $\beta^{*}$ versus the shape parameter $\sigma_c$ when the trade-off function is the same as in Fig. 3a. In (a), (b), (c) and (d), arrows indicate the direction of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $CSS$ denotes the continuously stable strategy, $EBP$ denotes the evolutionary branching point. The grey region indicates combination of $\beta$ and $b(\beta)$ for which the endemic equilibrium $(S^{*}(\beta), I^{*}(\beta))$ is globally asymptotically stable. In non-grey region, $I^{*}(\beta)$ equals to zero, the evolution of host resistance can not occur. In (a), other parameter values are the same as in Fig. 1. In (b), (c) and (d), other parameter values are the same as in Fig. 3.
Bifurcation diagram of evolutionary dynamics when the demographic parameter is changed. (a) Evolutionarily singular strategy $\beta^{*}$ versus the pathogen-induced mortality rate $\alpha$ when the trade-off function is the same as in Fig. 1a. (b) Evolutionarily singular strategy $\beta^{*}$ versus the natural death rate $m_{0}$ when the trade-off function is the same as in Fig. 1a. (c) Evolutionarily singular strategy $\beta^{*}$ versus the strength of density-dependent mortality $m_{1}$ when the trade-off function is the same as in Fig. 1a. (d) Evolutionarily singular strategy $\beta^{*}$ versus the pathogen-induced mortality rate $\alpha$ when the trade-off function is the same as in Fig. 3a. (e) Evolutionarily singular strategy $\beta^{*}$ versus the natural death rate $m_{0}$ when the trade-off function is the same as in Fig. 3a. (f) Evolutionarily singular strategy $\beta^{*}$ versus the strength of density-dependent mortality $m_{1}$ when the trade-off function is the same as in Fig. 3a. In (a), (b), (c), (d), (e) and (f), the arrows indicate the directions of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $CSS$ denotes the continuously stable strategy, $EBP$ denotes the evolutionary branching point. The grey region indicates combination of $\beta$ and $b(\beta)$ for which the endemic equilibrium is globally asymptotically stable. In (a), (b) and (c), other parameter values are the same as in Fig. 1. In (d), (e) and (f), other parameter values are the same as in Fig. 3
Continuously stable coexistence of dimorphic susceptible hosts. The trade-off function is given by (18) with parameters $b_{0} = 0.055, b_{1} = 4.0, b_{2} = 0.05, b_{3} = 0.05, \beta_{0} = 0.0005, \sigma_{c} = 0.0002$. (a) A trait evolution plot. The singular strategy $A = 0.000419$ is an evolutionary branching point. After the branching has occurred, there exist an interior dimorphic singular strategy $E_{1} = (0.000932, 0.000103)$ and its reflection point $E_{2} = (0.000103, 0.000932)$. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $\beta_{1}$ and $\beta_{2}$. The black curves and red curves indicate the isoclines of traits $\beta_{1}$ and $\beta_{2}$, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable, whereas the dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. (b) A fitness landscape plot when $(\beta_{1}^{*}, \beta_{2}^{*}) = (0.000932, 0.000103)$. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with the initial condition $\beta(0) = 0.0005$. At evolutionary time $\tau_1$ evolutionary branching occurs. (d) Equilibrium population densities of susceptible hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve. (e) Equilibrium population density of infected hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve. Other parameter values: $\alpha = 0.075, m_{0} = 0.006, m_{1} = 0.00014, \mu = 0.01, \sigma = 0.00001$
The trade-off curve $b(\beta)$. The trade-off function is given by (18). (a) The trade-off curve $I$ is concave-convex-concave when $\sigma_c = 0.0002$. The trade-off curve $II$ is also concave-convex-concave, but with a wider convex region when $\sigma_c = 0.0005$. Other parameter values: $b_{0} = 0.055, b_{1} = 4.0, b_{2} = 0.05, b_{3} = 0.05, \beta_{0} = 0.0005$. (b) The trade-off curve $I$ looks like a sigmoidal curve when $b_{1} = 5.5$ and $\sigma_c = 0.008$. The trade-off curve $II$ looks also like a sigmoidal curve, but with a larger maximum birth rate when $b_{1} = 10.0$ and $\sigma_c = 0.0085$. Other parameter values: $b_{0} = 0.05, b_{2} = 0.05, b_{3} = 0.8, \beta_{0} = 0.0018$
Continuously stable coexistence of boundary dimorphism. The trade-off function and parameter values are the same as in Fig. 6 except for $\sigma_{c} = 0.0005$. (a) A trait evolution plot. The singular strategy $A = 0.000492$ is an evolutionary branching point. After the branching has occurred, there is no interior dimorphic singular strategy. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $\beta_{1}$ and $\beta_{2}$. The black curves and red curves indicate the isoclines of traits $\beta_{1}$ and $\beta_{2}$, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable. (b) A fitness landscape plot when $(\beta_{1}^{*}, \beta_{2}^{*}) = (0.001113, 0.0)$. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with the initial condition $\beta(0) = 0.0006$. (d) Equilibrium population densities of susceptible hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve. (e) Equilibrium population density of infected hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve
Bifurcation diagram of monomorphic evolutionary dynamics when one of the parameters in the model is changed. The trade-off function is given by (18) with parameters $b_{0} = 0.05, b_{1} = 5.5, b_{2} = 0.05, b_{3} = 0.8, \beta_{0} = 0.0018$ and $\sigma_c = 0.008$. (a) Evolutionarily singular strategy $\beta^{*}$ versus the maximum birth rate $b_{1}$. (b) Evolutionarily singular strategy $\beta^{*}$ versus the shape parameter $\sigma_c$. (c) Evolutionarily singular strategy $\beta^{*}$ versus the natural death rate $m_{0}$. (d) Evolutionarily singular strategy $\beta^{*}$ versus the strength of density-dependent mortality $m_{1}$. In (a), (b), (c) and (d), arrows indicate the direction of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $CSS$ denotes the continuously stable strategy, $EBP$ denotes the evolutionary branching point. The grey region indicates combination of $\beta$ and $b(\beta)$ for which the endemic equilibrium $(S^{*}(\beta), I^{*}(\beta))$ is globally asymptotically stable
Evolutionary extinction of one emerging branch. The trade-off function is given by (18) with parameters $b_{0} = 0.05, b_{1} = 5.5, b_{2} = 0.05, b_{3} = 0.8, \beta_{0} = 0.0018$ and $\sigma_c = 0.008$. (a) A pairwise invasibility plot. (b) A trait evolution plot. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $\beta_{1}$ and $\beta_{2}$. The black curves and red curves indicate the isoclines of traits $\beta_{1}$ and $\beta_{2}$, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable, whereas the dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with initial condition $\beta(0) = 0.0015$. At evolutionary time $\tau_1$ evolutionary branching occurs and at evolutionary time $\tau_2$ the emerging branch $\beta_{2}$ become extinct. (d) Equilibrium population densities of susceptible hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve. At evolutionary time $\tau_2$, the equilibrium population density of susceptible hosts $S_{2}$ becomes zero. The dashed curve indicates the total equilibrium population density of the two types of susceptible hosts. (e) Equilibrium population density of infected hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve. Other parameter values: $\alpha = 0.075, m_{0} = 0.0055, m_{1} = 0.000098, \mu = 0.01, \sigma = 0.00005$
Bifurcation diagram of monomorphic evolutionary dynamics when one of the parameters in the model is changed. The trade-off function and parameter values are the same as in Fig. 9 except for $b_{1} = 10.0$ and $\sigma_c = 0.0085$. (a) Evolutionarily singular strategy $\beta^{*}$ versus the maximum birth rate $b_{1}$. (b) Evolutionarily singular strategy $\beta^{*}$ versus the shape parameter $\sigma_c$. (c) Evolutionarily singular strategy $\beta^{*}$ versus the natural death rate $m_{0}$. (d) Evolutionarily singular strategy $\beta^{*}$ versus the strength of density-dependent mortality $m_{1}$. In (a), (b), (c) and (d), arrows indicate the direction of evolutionary change. The black solid curves indicate the evolutionarily singular strategies which are both convergence stable and evolutionarily stable; the red dashed curves indicate the evolutionarily singular strategies which are not evolutionarily stable. $CSS$ denotes the continuously stable strategy, $EBP$ denotes the evolutionary branching point. The grey region indicates combination of $\beta$ and $b(\beta)$ for which the endemic equilibrium $(S^{*}(\beta), I^{*}(\beta))$ is globally asymptotically stable
Evolutionary extinction of one existing branch. The trade-off function and parameter values are the same as in Fig. 9 except for $b_{1} = 10.0$ and $\sigma_c = 0.0085$. (a) A pairwise invasibility plot. (b) A trait evolution plot. The vector fields obtained from the deterministic model (29) indicate the directions of evolutionary change of traits $\beta_{1}$ and $\beta_{2}$. The black curves and red curves indicate the isoclines of traits $\beta_{1}$ and $\beta_{2}$, respectively. The solid curves indicate the evolutionarily singular strategies which are evolutionarily stable, whereas the dashed curve indicates the evolutionarily singular strategies which are not evolutionarily stable. (c) A simulated evolutionary tree obtained through simulation of models (11) and (29) with initial condition $\beta(0) = 0.0006$. At evolutionary time $\tau_1$ evolutionary branching occurs and at evolutionary time $\tau_2$ the existing branch $\beta_{1}$ become extinct. (d) Equilibrium population densities of susceptible hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve. At evolutionary time $\tau_2$, the equilibrium population density of susceptible hosts $S_{1}$ becomes zero. The dashed curve indicates the total equilibrium population density of the two types of susceptible hosts. (e) Equilibrium population density of infected hosts when the traits $\beta_{1}$ and $\beta_{2}$ evolve
Summary of the evolutionary outcomes with a trade-off function $b(\beta)$
 Evolutionary outcomes Evolutionary conditions Shape of trade-off function CSS $\beta^{*}$ $b^{''}(\beta^{*})<0$ Globally concave EBP $\beta^{*}$ $0  Evolutionary outcomes Evolutionary conditions Shape of trade-off function CSS$ \beta^{*}  b^{''}(\beta^{*})<0 $Globally concave EBP$ \beta^{*}  0
 [1] Feng-Bin Wang, Junping Shi, Xingfu Zou. Dynamics of a host-pathogen system on a bounded spatial domain. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2535-2560. doi: 10.3934/cpaa.2015.14.2535 [2] Jeremias Epperlein, Stefan Siegmund, Petr Stehlík, Vladimír  Švígler. Coexistence equilibria of evolutionary games on graphs under deterministic imitation dynamics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 803-813. doi: 10.3934/dcdsb.2016.21.803 [3] Marcello Delitala, Tommaso Lorenzi. Evolutionary branching patterns in predator-prey structured populations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2267-2282. doi: 10.3934/dcdsb.2013.18.2267 [4] Bruno Buonomo, Marianna Cerasuolo. The effect of time delay in plant--pathogen interactions with host demography. Mathematical Biosciences & Engineering, 2015, 12 (3) : 473-490. doi: 10.3934/mbe.2015.12.473 [5] Caichun Chai, Tiaojun Xiao, Eilin Francis. Is social responsibility for firms competing on quantity evolutionary stable?. Journal of Industrial & Management Optimization, 2018, 14 (1) : 325-347. doi: 10.3934/jimo.2017049 [6] Kaifa Wang, Yang Kuang. Fluctuation and extinction dynamics in host-microparasite systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1537-1548. doi: 10.3934/cpaa.2011.10.1537 [7] Tzy-Wei Hwang, Yang Kuang. Host Extinction Dynamics in a Simple Parasite-Host Interaction Model. Mathematical Biosciences & Engineering, 2005, 2 (4) : 743-751. doi: 10.3934/mbe.2005.2.743 [8] Minette Herrera, Aaron Miller, Joel Nishimura. Altruistic aging: The evolutionary dynamics balancing longevity and evolvability. Mathematical Biosciences & Engineering, 2017, 14 (2) : 455-465. doi: 10.3934/mbe.2017028 [9] Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643 [10] Amy Veprauskas, J. M. Cushing. Evolutionary dynamics of a multi-trait semelparous model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 655-676. doi: 10.3934/dcdsb.2016.21.655 [11] Alex Potapov, Ulrike E. Schlägel, Mark A. Lewis. Evolutionarily stable diffusive dispersal. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3319-3340. doi: 10.3934/dcdsb.2014.19.3319 [12] Xiulan Lai, Xingfu Zou. Dynamics of evolutionary competition between budding and lytic viral release strategies. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1091-1113. doi: 10.3934/mbe.2014.11.1091 [13] William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485 [14] Amy Veprauskas, J. M. Cushing. Errata: Evolutionary dynamics of a multi-trait semelparous model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 1023-1026. doi: 10.3934/dcdsb.2016.21.1023 [15] Hassan Najafi Alishah, Pedro Duarte. Hamiltonian evolutionary games. Journal of Dynamics & Games, 2015, 2 (1) : 33-49. doi: 10.3934/jdg.2015.2.33 [16] Jian Zu, Wendi Wang, Bo Zu. Evolutionary dynamics of prey-predator systems with Holling type II functional response. Mathematical Biosciences & Engineering, 2007, 4 (2) : 221-237. doi: 10.3934/mbe.2007.4.221 [17] Shui-Nee Chow, Kening Lu, Yun-Qiu Shen. Normal forms for quasiperiodic evolutionary equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (1) : 65-94. doi: 10.3934/dcds.1996.2.65 [18] Alexey Cheskidov, Landon Kavlie. Pullback attractors for generalized evolutionary systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 749-779. doi: 10.3934/dcdsb.2015.20.749 [19] Andrzej Swierniak, Michal Krzeslak. Application of evolutionary games to modeling carcinogenesis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 873-911. doi: 10.3934/mbe.2013.10.873 [20] Jing-Jing Xiang, Yihao Fang. Evolutionarily stable dispersal strategies in a two-patch advective environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1875-1887. doi: 10.3934/dcdsb.2018245

2018 Impact Factor: 1.008

## Tools

Article outline

Figures and Tables