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Spatial dynamics of a nonlocal model with periodic delay and competition

  • L. ZHANG (a1), K. H. LIU (a2) (a3), Y. J. LOU (a2) and Z. C. WANG (a1)


Each species is subject to various biotic and abiotic factors during growth. This paper formulates a deterministic model with the consideration of various factors regulating population growth such as age-dependent birth and death rates, spatial movements, seasonal variations, intra-specific competition and time-varying maturation simultaneously. The model takes the form of two coupled reaction–diffusion equations with time-dependent delays, which bring novel challenges to the theoretical analysis. Then, the model is analysed when competition among immatures is neglected, in which situation one equation for the adult population density is decoupled. The basic reproduction number $\mathcal{R}_0$ is defined and shown to determine the global attractivity of either the zero equilibrium (when $\mathcal{R}_0\leq 1$ ) or a positive periodic solution ( $\mathcal{R}_0\gt1$ ) by using the dynamical system approach on an appropriate phase space. When the immature intra-specific competition is included and the immature diffusion rate is neglected, the model is neither cooperative nor reducible to a single equation. In this case, the threshold dynamics about the population extinction and uniform persistence are established by using the newly defined basic reproduction number $\widetilde{\mathcal{R}}_0$ as a threshold index. Furthermore, numerical simulations are implemented on the population growth of two different species for two different cases to validate the analytic results.



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The work of YL and KL is supported in part by the Research Grants Council of Hong Kong (PolyU 153277/16P) and the Research Grants of Jiangsu University (4111190009). ZW and LZ are supported by NNSF of China (11371179 and 11701242) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2017-27 and lzujbky-2019-79).



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[1]Alto, B. W., Lounibos, L. P., Higgs, S. & Juliano, S. A. (2005) Larval competition differentially affects arbovirus infection in Aedes mosquitoes. Ecology 86(12), 32793288.
[2]Alto, B. W., Lounibos, L. P., Mores, C. N., & Reiskind, M. H. (2008) Larval competition alters susceptibility of adult Aedes mosquitoes to dengue infection. Proc. R. Soc. B 275(1633), 463471.
[3]Altwegg, R. (2002) Trait-mediated indirect effects and complex life-cycles in two European frogs. Evol. Ecol. Res. 4(4), 519536.
[4]Brambell, F. W. R. (2010) The reproduction of the wild rabbit Oryctolagus cuniculus (L.). J. Zool. 114(1–2), 145.
[5]Cantrell, R. S. & Cosner, C. (2004) Spatial Ecology via Reaction-Diffusion Equations. John Wiley & Sons, New York.
[6]Craig, L. E., Norris, D. E., Sanders, M. L., Glass, G. E. & Schwartz, B. S. (1996) Acquired resistance and antibody response of raccoons (Procyon lotor) to sequential feedings of Ixodes scapularis (Acari: Ixodidae). Vet. Parasitol. 63(3–4), 291301.
[7]De Valdez, M. R. W. (2017) Mosquito species distribution across urban, suburban, and semi-rural residences in San Antonio, Texas. J. Vector Ecol. 42(1), 184.
[8]Dziminski, M. A. (2009) Intraspecific competition in the larvae of quacking frogs (Crinia georgiana). Copeia 2009(4), 724726.
[9]Ewing, D. A., Cobbold, C. A., Purse, B. V., Nunn, M. A. & White, S. M. (2016) Modelling the effect of temperature on the seasonal population dynamics of temperate mosquitoes. J. Theor. Biol. 400, 6579.
[10]Fang, J., Gourley, S. A. & Lou, Y. (2016) Stage-structured models of intra-and inter-specific competition within age classes. J. Differ. Equ. 260(2), 19181953.
[11]Friedman, A. (2008) Partial Differential Equations of Parabolic Type. Courier Dover Publications, New York.
[12]Gaines, M. S. & McClenaghan, L. R Jr., (1980) Dispersal in small mammals. Ann. Rev. EcoL Syst. 11(1), 163196.
[13]Gourley, S. A. & Kuang, Y. (2003) Wavefronts and global stability in a time-delayed population model with stage structure. Proc. R. Soc. Lond. A 459(2034), 15631579.
[14]Gourley, S. A., Liu, R. & Lou, Y. (2017) Intra-specific competition and insect larval development: A model with time-dependent delay. P. Roy. Soc. Edinb. A 147(2), 353369.
[15]Gourley, S. A., Liu, R. & Wu, J. (2008) Spatiotemporal patterns of disease spread: Interaction of physiological structure, spatial movements, disease progression and human intervention. In: Structured Population Models in Biology and Epidemiology, vol. 1936. Springer, Berlin, pp. 165208.
[16]Hale, J. K. (1988) Asymptotic Behavior of Dissipative Systems, vol. 25. American Mathematical Society.
[17]Iannelli, M. (1995) Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori e Stampatori, Pisa.
[18]Ido, T., Alon, S., Ofer, O., Leon, B. & Yoel, M. (2013) Inter- and intra-specific density-dependent effects on life history and development strategies of larval mosquitoes. Plos One 8(3), e57875.
[19]Jin, Y. & Zhao, X.-Q. (2009) Spatial dynamics of a nonlocal periodic reaction-diffusion model with stage structure. SIAM J. Math. Anal. 40(6), 24962516.
[20]Kontsiotis, V. J., Bakaloudis, D. E., Xofis, P., Konstantaras, N., Petrakis, N. & Tsiompanoudis, A. (2013) Modeling the distribution of wild rabbits (Oryctolagus cuniculus) on a Mediterranean island. Ecol. Res. 28(2), 317325.
[21]Künkele, J. & Von Holst, D. (1996) Natal dispersal in the European wild rabbit. Anim. Behav. 51(5), 10471059.
[22]Lees, A. C. & Bell, D. J. (2008) A conservation paradox for the 21st century: The European wild rabbit Oryctolagus cuniculus, an invasive alien and an endangered native species. Mammal Rev. 38(4), 304320.
[23]Legros, M., Lloyd, A. L., Huang, Y. & Gould, F. (2009) Density-dependent intraspecific competition in the larval stage of Aedes aegypti (Diptera: Culicidae): Revisiting the current paradigm. J. Med. Entomol. 46(3), 409419.
[24]Levin, S. A. (1974) Dispersion and population interactions. Am. Nat. 108(960), 207228.
[25]Li, J. & Brauer, F. (2008) Continuous-time age-structured models in population dynamics and epidemiology. In: Mathematical Epidemiology, Springer, Berlin, Heidelberg, pp. 205227.
[26]Liang, X., Zhang, L. & Zhao, X.-Q. (2019) Basic reproduct ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J. Dyn. Differ. Equ. 31, 12471278.
[27]Liang, X. & Zhao, X.-Q. (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Commun. Pure Appl. Math. 60(1), 140.
[28]Liu, K., Lou, Y. & Wu, J. (2017) Analysis of an age structured model for tick populations subject to seasonal effects. J. Differ. Equ. 263(4), 20782112.
[29]Lou, Y. & Zhao, X.-Q. (2017) A theoretical approach to understanding population dynamics with seasonal developmental durations. J. Nonlinear Sci. 27(2), 573603.
[30]Martin, R. H. & Smith, H. L. (1990) Abstract functional-differential equations and reaction-diffusion systems. Trans. Am. Math. Soc. 321(1), 144.
[31]Mcgrady, M. J., Ueta, M., Potapov, E. R., Utekhina, I., Masterov, V., Ladyguine, A., Zykov, V., Cibor, J., Fuller, M. & Seegar, W. S. (2003) Movements by juvenile and immature Steller’s Sea Eagles Haliaeetus pelagicus tracked by satellite. Ibis 145(2), 318328.
[32]Metz, J. A. J. & Diekmann, O. (1986) The Dynamics of Physiologically Structured Populations. Springer-Verlag, Berlin, Heidelberg.
[33]Nisbet, R. M. (1997) Delay-differential equations for structured populations. In: Structured-Population Models in Marine, Terrestrial, and Freshwater Systems. Springer, pp. 89118.
[34]Ogden, N. H., Bigras-Poulin, M., O’Callaghan, C. J., Barker, I. K., Lindsay, L. R., Maarouf, A., Smoyer-Tomic, K. E., Waltner-Toews, D. & Charron, D. (2005) A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick ixodes scapularis. Int. J. Parasitol. 35(4), 375389.
[35]Reiskind, M. H. & Lounibos, L. P. (2009) Effects of intraspecific larval competition on adult longevity in the mosquitoes Aedes aegypti and Aedes albopictus. Med. Vet. Entomol. 23(1), 6268.
[36]Silver, J. B. (2007) Mosquito Ecology: Field Sampling Methods. Springer Science & Business Media.
[37]Simoy, M. I., Simoy, M. V. & Canziani, G. A. (2015) The effect of temperature on the population dynamics of Aedes aegypti. Ecol. Model. 314(5), 100110.
[38]Smith, D. L., Perkins, T. A., Reiner, R. C., Barker, C. M., Niu, T., Chaves, L. F., Ellis, A. M., George, D. B., Menach, A. L. & Pulliam, J. R. C. (2014) Recasting the theory of mosquito-borne pathogen transmission dynamics and control. Trans. R. Soc. Trop. Med. Hyg. 108(4), 185197.
[39]So, J. W.-H., Wu, J. & Zou, X. (2001) Structured population on two patches: Modelling dispersal and delay. J. Math. Biol. 43(1), 3751.
[40]Thieme, H. R. & Zhao, X.-Q. (2003) Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction–diffusion models. J. Differ. Equ. 195(2), 430470.
[41]Wang, X. & Zou, X. (2018) Threshold dynamics of a temperature-dependent stage-structured mosquito population model with nested delays. Bull. Math. Biol. 80(7), 19621987.
[42]Webb, G. F. (1985) Theory of Nonlinear Age-Dependent Population Dynamics. Marcel Dekker, New York.
[43]Willadsen, P. & Jongejan, F. (1999) Immunology of the tick–host interaction and the control of ticks and tick-borne diseases. Parasitol. Today 15(7), 258262.
[44]Wilson, M. L., Litwin, T. S., Gavin, T. A., Capkanis, M. C., Maclean, D. C. & Spielman, A. (1990) Host-dependent differences in feeding and reproduction of Ixodes dammini (Acari: Ixodidae). J. Med. Entomol. 27(6), 945954.
[45]Wu, J. (1996) Theory and Applications of Partial Functional Differential Equations, vol. 119. Springer-Verlag, New York.
[46]Wu, X., Magpantay, F. M. G., Wu, J. & Zou, X. (2015) Stage-structured population systems with temporally periodic delay. Math. Methods Appl. Sci. 38(16), 34643481.
[47]Xu, D. & Zhao, X.-Q. (2003) A nonlocal reaction-diffusion population model with stage structure. Canad. Appl. Math. Quart. 11(3), 303319.
[48]Xu, D. & Zhao, X.-Q. (2005) Dynamics in a periodic competitive model with stage structure. J. Math. Anal. Appl. 311(2), 417438.
[49]Yang, H., Boldrini, J., Fassoni, A., Lima, K., Freitas, L., Gomez, M., Andrade, V. & Freitas, A. (2014) Abiotic effects on population dynamics of mosquitoes and their influence on dengue transmission. In: Ecological Modelling Applied to Entomology, Springer, pp. 3980.
[50]Yi, T. & Zou, X. (2011) Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain. J. Differ. Equ. 251(9), 25982611.
[51]Zhang, L., Wang, Z.-C. & Zhao, X.-Q. (2015) Threshold dynamics of a time periodic reaction–diffusion epidemic model with latent period. J. Differ. Equ. 258(9), 30113036.
[52]Zhao, X.-Q. (2017) Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. 29, 6782.
[53]Zhao, X.-Q. (2017) Dynamical Systems in Population Biology. Springer-Verlag, New York.


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Spatial dynamics of a nonlocal model with periodic delay and competition

  • L. ZHANG (a1), K. H. LIU (a2) (a3), Y. J. LOU (a2) and Z. C. WANG (a1)


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