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August  2020, 25(8): 3049-3086. doi: 10.3934/dcdsb.2020051

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  August 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (8) : 3049-3086. doi: 10.3934/dcdsb.2020051
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show all references

References:
[1]

T. AmmunétT. 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.

[2]

R. M. Anderson and R. M. May, Coevolution of hosts and parasites, Parasitology, 85 (1982), 411-426.  doi: 10.1017/S0031182000055360.

[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.

[4]

A. BestR. 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.

[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.

[6]

A. BestA. White and M. Boots, The implications of coevolutionary dynamics to host-parasite interactions, Amer. Naturalist, 173 (2009), 779-791.  doi: 10.1086/598494.

[7]

A. BestA. White and M. Boots, The evolution of host defence when parasites impact reproduction, Evolutionary Ecology Research, 18 (2017), 393-409. 

[8]

B. BoldinS. 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. 

[9]

M. H. Bonds, Host life-history strategy explains parasite-induced sterility, Amer. Naturalist, 168 (2006), 281-293.  doi: 10.1086/506922.

[10]

M. BootsA. BestM. 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.

[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.

[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.

[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.

[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.

[15]

M. BootsA. WhiteA. 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.

[16]

R. G. BowersA. HoyleA. 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.

[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.

[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.

[19]

R. G. BowersM. 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.

[20]

R. S. CantrellC. 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.

[21]

F. B. Christiansen, On conditions for evolutionary stability for a continuously varying character, Amer. Naturalist, 138 (1991), 37-50.  doi: 10.1086/285203.

[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.

[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.
[24]

F. Dercole, Remarks on branching-extinction evolutionary cycles, J. Math. Biol., 47 (2003), 569-580.  doi: 10.1007/s00285-003-0236-4.

[25]

U. Dieckmann and M. Doebeli, On the origin of species by sympatric speciation, Nature, 400 (1999), 354-357.  doi: 10.1038/22521.

[26]

O. DiekmannP. E. JabinS. 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.

[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.

[28] U. DieckmannJ. 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.
[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.

[30]

I. Eshel, Evolutionary and continuous stability, J. Theoret. Biol., 103 (1983), 99-111.  doi: 10.1016/0022-5193(83)90201-1.

[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.

[32]

S. GandonP. Agnew and Y. Michalakis, Coevolution between parasite virulence and host life-history traits, Amer. Naturalist, 160 (2002), 374-388.  doi: 10.1086/341525.

[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.

[34]

S. A. H. GeritzÉ. KisdiG. 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.

[35]

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Figure 1.  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 $
Figure 2.  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
Figure 3.  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 $
Figure 4.  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.
Figure 5.  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
Figure 6.  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 $
Figure 12.  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 $
Figure 7.  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
Figure 8.  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
Figure 9.  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 $
Figure 10.  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
Figure 11.  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
Table 1.  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<b^{''}(\beta^{*})<b_{crit}^{''}(\beta^{*}) $ Concave-convex-concave or sigmoidal
CSS $ (\beta_{1}^{*}, \beta_{2}^{*}) $ $ b^{''}(\beta_{i}^{*})<0, (i=1, 2) $ Concave-convex-concave
Extinction of one branch No attracting singular strategies Sigmoidal
Note: CSS means continuously stable strategy; EBP means evolutionary branching point.
Evolutionary outcomes Evolutionary conditions Shape of trade-off function
CSS $ \beta^{*} $ $ b^{''}(\beta^{*})<0 $ Globally concave
EBP $ \beta^{*} $ $ 0<b^{''}(\beta^{*})<b_{crit}^{''}(\beta^{*}) $ Concave-convex-concave or sigmoidal
CSS $ (\beta_{1}^{*}, \beta_{2}^{*}) $ $ b^{''}(\beta_{i}^{*})<0, (i=1, 2) $ Concave-convex-concave
Extinction of one branch No attracting singular strategies Sigmoidal
Note: CSS means continuously stable strategy; EBP means evolutionary branching point.
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