Stage duration distributions and intraspecific competition: a review of continuous stage-structured models

Yijun Lou; Bei Sun; Yijun Lou; Bei Sun

doi:10.3934/mbe.2022355

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 8: 7543-7569. doi: 10.3934/mbe.2022355

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Stage duration distributions and intraspecific competition: a review of continuous stage-structured models

Yijun Lou ^,,
Bei Sun

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong SAR, China

^†In memory of Stephen A. Gourley. This work was partially supported by the National Natural Science Foundation of China (Grant No. 12071393) and a general research fund from The Hong Kong Research Grants Council (Grant No. 15304821).
Academic Editor: Rongsong Liu

Received: 25 February 2022 Revised: 30 April 2022 Accepted: 05 May 2022 Published: 20 May 2022

Stage structured models, by grouping individuals with similar demographic characteristics together, have proven useful in describing population dynamics. This manuscript starts from reviewing two widely used modeling frameworks that are in the form of integral equations and age-structured partial differential equations. Both modeling frameworks can be reduced to the same differential equation structures with/without time delays by applying Dirac and gamma distributions for the stage durations. Each framework has its advantages and inherent limitations. The net reproduction number and initial growth rate can be easily defined from the integral equation. However, it becomes challenging to integrate the density-dependent regulations on the stage distribution and survival probabilities in an integral equation, which may be suitably incorporated into partial differential equations. Further recent modeling studies, in particular those by Stephen A. Gourley and collaborators, are reviewed under the conditions of the stage duration distribution and survival probability being regulated by population density.
- stage structured model,
- stage duration distribution,
- intraspecific competition,
- renew equation,
- age-structured model
Citation: Yijun Lou, Bei Sun. Stage duration distributions and intraspecific competition: a review of continuous stage-structured models[J]. Mathematical Biosciences and Engineering, 2022, 19(8): 7543-7569. doi: 10.3934/mbe.2022355

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Abstract

Stage structured models, by grouping individuals with similar demographic characteristics together, have proven useful in describing population dynamics. This manuscript starts from reviewing two widely used modeling frameworks that are in the form of integral equations and age-structured partial differential equations. Both modeling frameworks can be reduced to the same differential equation structures with/without time delays by applying Dirac and gamma distributions for the stage durations. Each framework has its advantages and inherent limitations. The net reproduction number and initial growth rate can be easily defined from the integral equation. However, it becomes challenging to integrate the density-dependent regulations on the stage distribution and survival probabilities in an integral equation, which may be suitably incorporated into partial differential equations. Further recent modeling studies, in particular those by Stephen A. Gourley and collaborators, are reviewed under the conditions of the stage duration distribution and survival probability being regulated by population density.

References

[1]	T. Okuyama, Stage duration distributions in matrix population models, Ecol. Evol., 8 (2018), 7936–7945. https://doi.org/10.1017/S0956792515000418 doi: 10.1017/S0956792515000418
[2]	P. de Valpine, K. Scranton, J. Knape, K. Ram, N. J. Mills, The importance of individual developmental variation in stage-structured population models, Ecol. Lett., 17 (2014), 1026–1038. https://doi.org/10.1111/ele.12290 doi: 10.1111/ele.12290
[3]	H. J. Wearing, P. Rohani, T. C. Cameron, S. M. Sait, The dynamical consequences of developmental variability and demographic stochasticity for host-parasitoid interactions, Am. Nat., 164 (2004), 543–558. https://doi.org/10.1086/424040 doi: 10.1086/424040
[4]	O. Gilad, Competition and competition models, Encycl. Ecol., 2008 (2008), 707–712. https://doi.org/10.1016/B978-008045405-4.00666-2 doi: 10.1016/B978-008045405-4.00666-2
[5]	F. J. Richards, Flexible growth function for empirical use, J. Exp. Bot., 10 (1959), 290–301. https://doi.org/10.1093/jxb/10.2.290 doi: 10.1093/jxb/10.2.290
[6]	X. Wang, J. Wu, Y. Yang, Richards model revisited: Validation by and application to infection dynamics, J. Theor. Biol., 313 (2012), 12–19. https://doi.org/10.1016/j.jtbi.2012.07.024 doi: 10.1016/j.jtbi.2012.07.024
[7]	A. Tsoularis, J. Wallace, Analysis of logistic growth models, Math. Biosci., 179 (2002), 21–55. https://doi.org/10.1016/S0025-5564(02)00096-2 doi: 10.1016/S0025-5564(02)00096-2
[8]	J. Wang, S. W. McCue, M. J. Simpson, Extended logistic growth model for heterogeneous populations, J. Theor. Biol., 445 (2018), 51–61. https://doi.org/10.1016/j.jtbi.2018.02.027 doi: 10.1016/j.jtbi.2018.02.027
[9]	J. Fang, Y, Lou, J. Wu, Can pathogen spread keep pace with its host invasion?, SIAM J. Appl. Math., 76 (2016), 1633–1657. https://doi.org/10.1137/15M1029564 doi: 10.1137/15M1029564
[10]	R. Arditi, C. Lobry, T. Sari, Asymmetric dispersal in the multi-patch logistic equation, Theor. Popul. Biol., 120 (2018), 11–15. https://doi.org/10.1016/j.tpb.2017.12.006 doi: 10.1016/j.tpb.2017.12.006
[11]	D. L. DeAngelis, W. Ni, B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theor. Ecol., 9 (2016), 443–453. https://doi.org/10.1007/s12080-016-0302-3 doi: 10.1007/s12080-016-0302-3
[12]	K. Nagahara, E. Yanagida, Maximization of the total population in a reaction-diffusion model with logistic growth, Calc. Var. Partial. Differ. Equation, 57 (2018), 1–14. https://doi.org/10.1007/s00526-018-1353-7 doi: 10.1007/s00526-018-1353-7
[13]	B. Zhang, X. Liu, D. L. DeAngelis, W. Ni, G. G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54–62. https://doi.org/10.1016/j.mbs.2015.03.005 doi: 10.1016/j.mbs.2015.03.005
[14]	F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical Models in Epidemiology, Springer-Verlag, 2019.
[15]	D. Champredon, J. Dushoff, D. J. D. Earn, Equivalence of the Erlang-distributed SEIR epidemic model and the renewal equation, SIAM J. Appl. Math., 78 (2018), 3258–3278. https://doi.org/10.1137/18M1186411 doi: 10.1137/18M1186411
[16]	Z. Feng, Applications of Epidemiological Models to Public Health Policymaking: The Role of Heterogeneity in Model Ppredictions, World Scientific, 2014.
[17]	Z. Feng, H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅰ: Fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803–833. https://doi.org/10.1137/S0036139998347834 doi: 10.1137/S0036139998347834
[18]	Z. Feng, H. R. Horst, Endemic models with arbitrarily distributed periods of infection Ⅱ: Fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61 (2000), 983–1012. https://doi.org/10.1137/S0036139998347846 doi: 10.1137/S0036139998347846
[19]	S. L. Robertson, K. A. Caillouët, A host stage-structured model of enzootic West Nile virus transmission to explore the effect of avian stage-dependent exposure to vectors, J. Theor. Biol., 399 (2016), 33–42. https://doi.org/10.1016/j.jtbi.2016.03.031 doi: 10.1016/j.jtbi.2016.03.031
[20]	R. S. Cantrell, C. Cosner, S. Martínez, Persistence for a two-stage reaction-diffusion system, Mathematics, 8 (2020), 396. https://doi.org/10.3390/math8030396 doi: 10.3390/math8030396
[21]	L. Mari, R. Casagrandi, E. Bertuzzo, A. Rinaldo, M. Gatto, Metapopulation persistence and species spread in river networks, Ecol. Lett., 17 (2014), 426–434. https://doi.org/10.1111/ele.12242 doi: 10.1111/ele.12242
[22]	K. Best, A. S. Perelson, Mathematical modeling of within-host Zika virus dynamics, Immunol. Rev., 285 (2018), 81–96. https://doi.org/10.1111/imr.12687 doi: 10.1111/imr.12687
[23]	D. L. Chao, M. P. Davenport, S. Forrest, A. S. Perelson, Stochastic stage-structured modeling of the adaptive immune system, in Proceedings of the IEEE Computer Society Bioinformatics Conference, (2003), 124–131. https://doi.org/10.1109/CSB.2003.1227311
[24]	A. Feng, U. Obolski, L. Stone, D. He, Modelling COVID-19 vaccine breakthrough infections in highly vaccinated Israel-The effects of waning immunity and third vaccination dose, preprint, medRxiv: 2022.01.08.22268950. https://doi.org/10.1101/2022.01.08.22268950
[25]	J. Li, F. Brauer, Continuous-time age-structured models in population dynamics and epidemiology, in Mathematical Epidemiology, Springer-Verlag, 2008. https://doi.org/10.1007/978-3-540-78911-6
[26]	H. R. Thieme, Mathematics in Population Biology, Princeton University Press, 2003.
[27]	A. J. Lotka, Relation between birth rates and death rates, Science, 26 (1907), 21–22. https://doi.org/10.1007/978-3-642-81046-6 doi: 10.1007/978-3-642-81046-6
[28]	A. P. Farrell, J. P. Collins, A. L. Greer, H. R. Thieme, Times from infection to disease-induced death and their influence on final population sizes after epidemic outbreaks, Bull. Math. Biol., 80 (2018), 1937–1961. https://doi.org/10.1007/s11538-018-0446-y doi: 10.1007/s11538-018-0446-y
[29]	H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer Singapore, 2017.
[30]	M. Gyllenberg, Mathematical aspects of physiologically structured populations: the contributions of JAJ Metz, J. Biol. Dyn., 1 (2007), 3–44. https://doi.org/10.1080/17513750601032737 doi: 10.1080/17513750601032737
[31]	A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 14 (1926), 98–130. https://doi.org/10.1017/S0013091500034428 doi: 10.1017/S0013091500034428
[32]	J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, 1998.
[33]	M. Iannelli, F. Milner, The Basic Approach to Age-structured Population Dynamics: Models, Methods and Numerics, Springer-Verlag, 2017.
[34]	P. Magal, S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer-Verlag, 2018.
[35]	S. Ruan, Delay differential equations in single species dynamics, in Delay Differential Equations and Applications, Springer-Verlag, 2006. https://doi.org/10.1007/1-4020-3647-7_11
[36]	O. Diekmann, M. Gyllenberg, J. A. J. Metz, On models of physiologically structured populations and their reduction to ordinary differential equations, J. Math. Biol., 80 (2020), 189–204. https://doi.org/10.1007/s00285-019-01431-7 doi: 10.1007/s00285-019-01431-7
[37]	O. Diekmann, Dynamics of structured populations, JSMB Newslett., 93 (2021), 6–15. https://doi.org/10.1007/978-3-662-13159-6 doi: 10.1007/978-3-662-13159-6
[38]	S. Gourley, R. Liu, Delay equation models for populations that experience competition at immature life stages, J. Differ. Equation, 259 (2015), 1757–1777. https://doi.org/10.1016/j.jde.2015.03.012 doi: 10.1016/j.jde.2015.03.012
[39]	R. Liu, G. Röst, S. Gourley, Age-dependent intra-specific competition in pre-adult life stages and its effects on adult population dynamics, Eur. J. Appl. Math., 27 (2016), 131–156. https://doi.org/10.1017/S0956792515000418 doi: 10.1017/S0956792515000418
[40]	J. Arino, L. Wang, G. S. Wolkowicz, An alternative formulation for a delayed logistic equation, J. Theor. Biol., 241 (2006), 109–119. https://doi.org/10.1016/j.jtbi.2005.11.007 doi: 10.1016/j.jtbi.2005.11.007
[41]	C. J. Lin, L. Wang, G. S. Wolkowicz, An alternative formulation for a distributed delayed logistic equation, Bull. Math. Biol., 80 (2018), 1713–1735. https://doi.org/10.1007/s11538-018-0432-4 doi: 10.1007/s11538-018-0432-4
[42]	J. Fang, S. Gourley, Y. Lou, Stage-structured models of intra-and inter-specific competition within age classes, J. Differ. Equ., 260 (2016), 1918–1953. https://doi.org/10.1016/j.jde.2015.09.048 doi: 10.1016/j.jde.2015.09.048
[43]	S. Gourley, R. Liu, Y. Lou, Intra-specific competition and insect larval development: a model with time-dependent delay, P. Roy. Soc. Edinb. A., 147 (2017), 353–369. https://doi.org/10.1017/S0308210516000159 doi: 10.1017/S0308210516000159
[44]	K. L. Cooke, Functional Differential Equations, Some Models and Perturbation Problems, Differential Equations and Dynamical Systems (eds. J. K. Hale and J. P. LaSalle), Academic Press, (1967), 167–183.
[45]	F. C. Hoppensteadt, P. Waltman, A problem in the theory of epidemics, Math. Biosci., 9 (1970), 71–91. https://doi.org/10.1016/0025-5564(70)90094-5 doi: 10.1016/0025-5564(70)90094-5
[46]	F. C. Hoppensteadt, P. Waltman, A problem in the theory of epidemics, Ⅱ, Math. Biosci., 12 (1971), 133–145. https://doi.org/10.1016/0025-5564(71)90078-2 doi: 10.1016/0025-5564(71)90078-2
[47]	H. Brunner, S. Gourley, R. Liu, Y. Xiao, Pauses of larval development and their consequences for stage-structured populations, SIAM J. Appl. Math., 77 (2017), 977–994. https://doi.org/10.1137/16M1105475 doi: 10.1137/16M1105475
[48]	H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011.
[49]	K. J. Brown, Y. Zhang, On a system of reaction-diffusion equations describing a population with two age groups, J. Math. Anal. Appl., 282 (2003), 444–452. https://doi.org/10.1016/S0022-247X(02)00374-8 doi: 10.1016/S0022-247X(02)00374-8
[50]	S. A. Gourley, J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, in Nonlinear Dynamics and Evolution Equations, 48 (2006), 137–200. https://doi.org/10.1090/fic/048
[51]	J. Al-Omari, S. A. Gourley, Monotone travelling fronts in an age-structured reaction-diffusion model of a single species, J Math. Biol., 45 (2002), 294–312. https://doi.org/10.1007/s002850200159 doi: 10.1007/s002850200159
[52]	J. Al-Omari, S. A. Gourley, A nonlocal reaction-diffusion model for a single species with stage structure and distributed maturation delay, Eur. J. Appl. Math., 16 (2005), 37–51. https://doi.org/10.1017/S0956792504005716 doi: 10.1017/S0956792504005716
[53]	Y. Kuang, S. A. Gourley, Wavefronts and global stability in a time-delayed population model with stage structure, Proc. Roy. Soc. Lon. A: Math. Phy. Eng. Sci., 459 (2003), 1563–1579. https://doi.org/10.1098/rspa.2002.1094 doi: 10.1098/rspa.2002.1094
[54]	Y. Lou, K. Liu, D. He, D. Gao, S. Ruan, Modelling diapause in mosquito population growth, J. Math. Biol., 78 (2019), 2259–2288. https://doi.org/10.1007/s00285-019-01343-6 doi: 10.1007/s00285-019-01343-6
[55]	Y. Lou, X. Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573–603. https://doi.org/10.1007/s00332-016-9344-3 doi: 10.1007/s00332-016-9344-3

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