September  2020, 13(9): 2509-2535. doi: 10.3934/dcdss.2020195

Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model

1. 

Department of Mathematics and Statistics, University of Missouri-Kansas City, Kansas City, Missouri 64110-2499, USA

2. 

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland A1C 5S7, Canada

3. 

Department of Mathematics, Clarkson University, Potsdam, NY 13699-5815, USA

4. 

Centre de recherches mathématiques, Université de Montréal, Montréal, H3C 3J7, Québec

* Corresponding author: Majid Bani-Yaghoub

Received  January 2019 Revised  July 2019 Published  September 2020 Early access  December 2019

The present work investigates the effects of maturation and dispersal delays on dynamics of single species populations. Both delays have been incorporated in a single species nonlocal hyperbolic-parabolic population model, which admits traveling and stationary wave solutions. We reduce the model into various forms and obtain the corresponding analytical solutions. Analysis of the reduced models indicates that the dispersal delay can result in loss of monotonicity, where the solutions oscillate as they converge to a positive equilibrium. The stability analysis of the general model reveals that the maturation time delay admits a Hopf bifurcation threshold, which is expressed as a function of the dispersal delay. The numerical simulations of the general model suggest that the global stability of the stationary wave solutions is lost when the dispersal delay is increased from zero. In conclusion, population models with maturation and dispersal delays can give new insights into the complex dynamics of single species.

Citation: Majid Bani-Yaghoub, Chunhua Ou, Guangming Yao. Delay-induced instabilities of stationary solutions in a single species nonlocal hyperbolic-parabolic population model. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2509-2535. doi: 10.3934/dcdss.2020195
References:
[1]

S. AikioR. P. Duncan and P. E. Hulme, Lag-phases in alien plant invasions: Separating the facts from the artefacts, Oikos, 119 (2010), 370-378.  doi: 10.1111/j.1600-0706.2009.17963.x.

[2]

M. Bani-Yaghoub, Numerical simulations of traveling and stationary wave solutions arising from reaction-diffusion population models with delay and nonlocality, Int. J. Appl. Comput. Math., 4 (2018), 19pp. doi: 10.1007/s40819-017-0441-2.

[3]

M. Bani-YaghoubG. Yao and H. Voulov, Existence and stability of stationary waves of a population model with strong Allee effect, J. Comput. Appl. Math., 307 (2016), 385-393.  doi: 10.1016/j.cam.2015.11.021.

[4]

M. Bani-Yaghoub, Approximate wave solutions of delay diffusive models using a differential transform method, Appl. Math. E-Notes, 16 (2016), 99-104. 

[5]

M. Bani-Yaghoub, Approximating the traveling wavefront for a nonlocal delayed reaction-diffusion equation, J. Appl. Math. Comput., 53 (2017), 77-94.  doi: 10.1007/s12190-015-0958-7.

[6]

M. Bani-YaghoubG. YaoM. Fujiwara and D. E. Amundsen, Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model, Ecological Complexity, 21 (2015), 14-26.  doi: 10.1016/j.ecocom.2014.10.007.

[7]

M. Bani-Yaghoub and D. E. Amundsen, Oscillatory traveling waves for a population diffusion model with two age classes and nonlocality induced by maturation delay, Comput. Appl. Math., 34 (2015), 309-324.  doi: 10.1007/s40314-014-0118-y.

[8]

M. Bani-YaghoubG. Yao and A. Reed, Modeling and numerical simulations of single species dispersal in symmetrical domains, Int. J. Appl. Math., 27 (2014), 525-547. 

[9]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.

[10]

F. De MelloC. A. OliveiraR. P. RibeiroE. K. Resende and et al., Growth curve by Gompertz nonlinear regression model in female and males in tambaqui (Colossoma macropomum), An Acad Bras Cienc., 87 (2015), 2309-2315.  doi: 10.1590/0001-3765201520140315.

[11]

W. Feng and J. Hinson, Stability and pattern in two-patch predator-prey population dynamics, Discrete Contin. Dyn. Syst., 2005 (2005), 268-279.  doi: 10.3934/proc.2005.2005.268.

[12]

J. Fort and V. Méndez, Time-delayed theory of the Neolithic transition in Europe, Phys. Rev. Lett., 82 (1999), 867-871.  doi: 10.1103/PhysRevLett.82.867.

[13]

M. R. Gaither, G. Aeby, M. Vignon, Y.-I. Meguro and M. Rigby, et al., An invasive fish and the time-lagged spread of its parasite across the Hawaiian archipelago, PLoS One, 8 (2013). doi: 10.1371/journal.pone.0056940.

[14]

B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. Royal Soc. London, 115 (1825), 513-583.  doi: 10.1098/rspl.1815.0271.

[15]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

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J. C. Houbolt, A recurrence matrix solution for the dynamic response of elastic aircraft, J. Aeronaut. Sci., 17 (1950), 540-550.  doi: 10.2514/8.1722.

[18]

A. Ishimaru, Diffusion of a pulse in densely distributed scatterers, J. Opt. Soc. Am., 68 (1978), 1045-1050.  doi: 10.1364/JOSA.68.001045.

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I. Jadlovska, Application of Lambert $W$ function in oscillation theory, Acta Electrotechnica et Informatica, 14 (2014), 9-17.  doi: 10.15546/aeei-2014-0002.

[20]

P. KlepacM. G. Neubert and P. van den Driessche, Dispersal delays, predator-prey stability, and the paradox of enrichment, Theor. Popul. Biol., 71 (2007), 436-444.  doi: 10.1016/j.tpb.2007.02.002.

[21]

K. Khatwani, On Ruth-Hurwitz criterion, IEEE Trans. Automat. Control, 26 (1981), 583-584.  doi: 10.1109/TAC.1981.1102670.

[22] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. 
[23]

D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.  doi: 10.1007/s00332-003-0524-6.

[24]

D. LiangJ. Wu and F. Zhang, Modelling population growth with delayed nonlocal reaction in 2-dimensions, Math. Biosci. Eng., 2 (2005), 111-132.  doi: 10.3934/mbe.2005.2.111.

[25]

A. Maignan and T. C. Scott, Fleshing out the generalized Lambert $W$ function, ACM Commun. Comput. Algebra, 50 (2016), 45-60.  doi: 10.1145/2992274.2992275.

[26]

M. C. Memory, Bifurcation and asymptotic behaviour of solutions of a delay-differential equation with diffusion, SIAM J. Math. Anal., 20 (1989), 533-546.  doi: 10.1137/0520037.

[27]

M. G. NeubertP. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theor. Popul. Biol., 61 (2002), 339-347.  doi: 10.1006/tpbi.2002.1578.

[28]

A. Otto, J. Wang and G. Radons, Delay-induced wave instabilities in single-species reaction-diffusion systems., Phys. Rev. E, 96 (2017). doi: 10.1103/PhysRevE.96.052202.

[29]

C. Ou and J. Wu, Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model, Nonlinear Anal., 63 (2005), 364-387.  doi: 10.1016/j.na.2005.05.025.

[30]

M. PelegM. Corradini and M. Normand, The logistic (Verhulst) model for sigmoid microbial growth curves revisited, Food Research International, 40 (2007), 808-818.  doi: 10.1016/j.foodres.2007.01.012.

[31]

N. Perrin and V. Mazalov, Local competition, inbreeding, and the evolution of sex-biased dispersal, Am Nat., 155 (2000), 116-127.  doi: 10.1086/303296.

[32]

R. Pinhasi, J. Fort and A. J. Ammerman, Tracing the origin and spread of agriculture in Europe, PLoS Biol., 3 (2005). doi: 10.1371/journal.pbio.0030410.

[33]

A. D. PolyaninV. G. Sorokin and A. V. Vyazmin, Exact solutions and qualitative features of nonlinear hyperbolic reaction-diffusion equations with delay, Theor. Found. Chem. Eng., 49 (2015), 622-635.  doi: 10.1134/S0040579515050243.

[34]

M. A. Schweizer, About explanations for X-ray bursts with double peaks and precursors, Canadian J. Physics, 63 (1985), 956-961.  doi: 10.1139/p85-156.

[35]

T. C. ScottR. B. Mann and R. E. Martinez II, General relativity and quantum mechanics: Towards a generalization of the Lambert $W$ function, Appl. Algebra Engrg. Comm. Comput., 17 (2006), 41-47.  doi: 10.1007/s00200-006-0196-1.

[36]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of travelling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.

[37]

J. W.-H. So and J. Yu, Global attractivity for a population model with time delay, Proc. Amer. Math. Soc., 123 (1995), 2687-2694.  doi: 10.1090/S0002-9939-1995-1317052-5.

[38]

J. W.-H. SoJ. Wu and Y. Yang, Numerical Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 111 (2000), 33-51.  doi: 10.1016/S0096-3003(99)00047-8.

[39]

J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age-structure. I: Traveling wavefronts on unbounded domains, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[40]

J. W.-H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348.  doi: 10.1006/jdeq.1998.3489.

[41]

A. Solar and S. Trofimchuk, Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction, Nonlinearity, 28 (2015), 2027-2052.  doi: 10.1088/0951-7715/28/7/2027.

[42]

A. Solar and S. Trofimchuk, Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations, J. Dynam. Differential Equations, 28 (2016), 1265-1292.  doi: 10.1007/s10884-015-9482-6.

[43]

A. Soroushian and J. Farjoodi, A united starting procedure for the Houbolt method, Comm. Numer. Methods Engrg., 24 (2008), 1-13.  doi: 10.1002/cnm.949.

[44]

Y. TakeuchiJ. CuiM. Rinko and Y. Saito, Permanence of dispersal population model with time delays, J. Comput. Appl. Math., 192 (2006), 417-430.  doi: 10.1016/j.cam.2005.06.002.

[45] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.  doi: 10.2307/j.ctv301f9v.
[46]

H. R. Thieme and X. Q. Zhao, A nonlocal delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.

[47]

K. M. C. Tjrve and E. Tjrve, The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family, PLoS ONE, 12 (2017). doi: 10.1371/journal.pone.0178691.

[48]

S. Trofimchuk and V. Volpert, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 (2019), 2593-2632.  doi: 10.1088/1361-6544/ab0e23.

[49]

H. WanL. Zhang and H. Li, A single species model with symmetric bidirectional impulsive diffusion and dispersal delay, J. Appl. Math., 3 (2012), 1079-1088.  doi: 10.1155/2014/701545.

[50]

H. Wan, L. Zhang and Z. Teng, Analysis of a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay, J. Appl. Math., 2014 (2014), 11pp. doi: 10.1155/2014/701545.

[51]

P. WengD. Liang and J. Wu, Asymptotic patterns of a structured population diffusing in a two-dimensional strip, Nonlinear Anal., 69 (2008), 3931-3951.  doi: 10.1016/j.na.2007.10.027.

[52]

J. Wu, Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[53]

G. Yao, An improved localized method of approximate particular solutions for solving elliptic PDEs, Comput. Math. Appl., 71 (2016), 171-184.  doi: 10.1016/j.camwa.2015.11.008.

[54]

D. L. YoungM. H. Gu and C. M. Fan, The time-marching method of fundamental solutions for wave equations, Eng. Anal. Bound. Elem., 33 (2009), 1411-1425.  doi: 10.1016/j.enganabound.2009.05.008.

[55]

X. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146 (2002), 309-321.  doi: 10.1016/S0377-0427(02)00363-1.

show all references

References:
[1]

S. AikioR. P. Duncan and P. E. Hulme, Lag-phases in alien plant invasions: Separating the facts from the artefacts, Oikos, 119 (2010), 370-378.  doi: 10.1111/j.1600-0706.2009.17963.x.

[2]

M. Bani-Yaghoub, Numerical simulations of traveling and stationary wave solutions arising from reaction-diffusion population models with delay and nonlocality, Int. J. Appl. Comput. Math., 4 (2018), 19pp. doi: 10.1007/s40819-017-0441-2.

[3]

M. Bani-YaghoubG. Yao and H. Voulov, Existence and stability of stationary waves of a population model with strong Allee effect, J. Comput. Appl. Math., 307 (2016), 385-393.  doi: 10.1016/j.cam.2015.11.021.

[4]

M. Bani-Yaghoub, Approximate wave solutions of delay diffusive models using a differential transform method, Appl. Math. E-Notes, 16 (2016), 99-104. 

[5]

M. Bani-Yaghoub, Approximating the traveling wavefront for a nonlocal delayed reaction-diffusion equation, J. Appl. Math. Comput., 53 (2017), 77-94.  doi: 10.1007/s12190-015-0958-7.

[6]

M. Bani-YaghoubG. YaoM. Fujiwara and D. E. Amundsen, Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model, Ecological Complexity, 21 (2015), 14-26.  doi: 10.1016/j.ecocom.2014.10.007.

[7]

M. Bani-Yaghoub and D. E. Amundsen, Oscillatory traveling waves for a population diffusion model with two age classes and nonlocality induced by maturation delay, Comput. Appl. Math., 34 (2015), 309-324.  doi: 10.1007/s40314-014-0118-y.

[8]

M. Bani-YaghoubG. Yao and A. Reed, Modeling and numerical simulations of single species dispersal in symmetrical domains, Int. J. Appl. Math., 27 (2014), 525-547. 

[9]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.

[10]

F. De MelloC. A. OliveiraR. P. RibeiroE. K. Resende and et al., Growth curve by Gompertz nonlinear regression model in female and males in tambaqui (Colossoma macropomum), An Acad Bras Cienc., 87 (2015), 2309-2315.  doi: 10.1590/0001-3765201520140315.

[11]

W. Feng and J. Hinson, Stability and pattern in two-patch predator-prey population dynamics, Discrete Contin. Dyn. Syst., 2005 (2005), 268-279.  doi: 10.3934/proc.2005.2005.268.

[12]

J. Fort and V. Méndez, Time-delayed theory of the Neolithic transition in Europe, Phys. Rev. Lett., 82 (1999), 867-871.  doi: 10.1103/PhysRevLett.82.867.

[13]

M. R. Gaither, G. Aeby, M. Vignon, Y.-I. Meguro and M. Rigby, et al., An invasive fish and the time-lagged spread of its parasite across the Hawaiian archipelago, PLoS One, 8 (2013). doi: 10.1371/journal.pone.0056940.

[14]

B. Gompertz, On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies, Phil. Trans. Royal Soc. London, 115 (1825), 513-583.  doi: 10.1098/rspl.1815.0271.

[15]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and its Applications, 74, Kluwer Academic Publishers Group, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9.

[16]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.

[17]

J. C. Houbolt, A recurrence matrix solution for the dynamic response of elastic aircraft, J. Aeronaut. Sci., 17 (1950), 540-550.  doi: 10.2514/8.1722.

[18]

A. Ishimaru, Diffusion of a pulse in densely distributed scatterers, J. Opt. Soc. Am., 68 (1978), 1045-1050.  doi: 10.1364/JOSA.68.001045.

[19]

I. Jadlovska, Application of Lambert $W$ function in oscillation theory, Acta Electrotechnica et Informatica, 14 (2014), 9-17.  doi: 10.15546/aeei-2014-0002.

[20]

P. KlepacM. G. Neubert and P. van den Driessche, Dispersal delays, predator-prey stability, and the paradox of enrichment, Theor. Popul. Biol., 71 (2007), 436-444.  doi: 10.1016/j.tpb.2007.02.002.

[21]

K. Khatwani, On Ruth-Hurwitz criterion, IEEE Trans. Automat. Control, 26 (1981), 583-584.  doi: 10.1109/TAC.1981.1102670.

[22] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993. 
[23]

D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.  doi: 10.1007/s00332-003-0524-6.

[24]

D. LiangJ. Wu and F. Zhang, Modelling population growth with delayed nonlocal reaction in 2-dimensions, Math. Biosci. Eng., 2 (2005), 111-132.  doi: 10.3934/mbe.2005.2.111.

[25]

A. Maignan and T. C. Scott, Fleshing out the generalized Lambert $W$ function, ACM Commun. Comput. Algebra, 50 (2016), 45-60.  doi: 10.1145/2992274.2992275.

[26]

M. C. Memory, Bifurcation and asymptotic behaviour of solutions of a delay-differential equation with diffusion, SIAM J. Math. Anal., 20 (1989), 533-546.  doi: 10.1137/0520037.

[27]

M. G. NeubertP. Klepac and P. van den Driessche, Stabilizing dispersal delays in predator-prey metapopulation models, Theor. Popul. Biol., 61 (2002), 339-347.  doi: 10.1006/tpbi.2002.1578.

[28]

A. Otto, J. Wang and G. Radons, Delay-induced wave instabilities in single-species reaction-diffusion systems., Phys. Rev. E, 96 (2017). doi: 10.1103/PhysRevE.96.052202.

[29]

C. Ou and J. Wu, Existence and uniqueness of a wavefront in a delayed hyperbolic-parabolic model, Nonlinear Anal., 63 (2005), 364-387.  doi: 10.1016/j.na.2005.05.025.

[30]

M. PelegM. Corradini and M. Normand, The logistic (Verhulst) model for sigmoid microbial growth curves revisited, Food Research International, 40 (2007), 808-818.  doi: 10.1016/j.foodres.2007.01.012.

[31]

N. Perrin and V. Mazalov, Local competition, inbreeding, and the evolution of sex-biased dispersal, Am Nat., 155 (2000), 116-127.  doi: 10.1086/303296.

[32]

R. Pinhasi, J. Fort and A. J. Ammerman, Tracing the origin and spread of agriculture in Europe, PLoS Biol., 3 (2005). doi: 10.1371/journal.pbio.0030410.

[33]

A. D. PolyaninV. G. Sorokin and A. V. Vyazmin, Exact solutions and qualitative features of nonlinear hyperbolic reaction-diffusion equations with delay, Theor. Found. Chem. Eng., 49 (2015), 622-635.  doi: 10.1134/S0040579515050243.

[34]

M. A. Schweizer, About explanations for X-ray bursts with double peaks and precursors, Canadian J. Physics, 63 (1985), 956-961.  doi: 10.1139/p85-156.

[35]

T. C. ScottR. B. Mann and R. E. Martinez II, General relativity and quantum mechanics: Towards a generalization of the Lambert $W$ function, Appl. Algebra Engrg. Comm. Comput., 17 (2006), 41-47.  doi: 10.1007/s00200-006-0196-1.

[36]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of travelling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.

[37]

J. W.-H. So and J. Yu, Global attractivity for a population model with time delay, Proc. Amer. Math. Soc., 123 (1995), 2687-2694.  doi: 10.1090/S0002-9939-1995-1317052-5.

[38]

J. W.-H. SoJ. Wu and Y. Yang, Numerical Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation, Appl. Math. Comput., 111 (2000), 33-51.  doi: 10.1016/S0096-3003(99)00047-8.

[39]

J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age-structure. I: Traveling wavefronts on unbounded domains, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.

[40]

J. W.-H. So and Y. Yang, Dirichlet problem for the diffusive Nicholson's blowflies equation, J. Differential Equations, 150 (1998), 317-348.  doi: 10.1006/jdeq.1998.3489.

[41]

A. Solar and S. Trofimchuk, Asymptotic convergence to pushed wavefronts in a monostable equation with delayed reaction, Nonlinearity, 28 (2015), 2027-2052.  doi: 10.1088/0951-7715/28/7/2027.

[42]

A. Solar and S. Trofimchuk, Speed selection and stability of wavefronts for delayed monostable reaction-diffusion equations, J. Dynam. Differential Equations, 28 (2016), 1265-1292.  doi: 10.1007/s10884-015-9482-6.

[43]

A. Soroushian and J. Farjoodi, A united starting procedure for the Houbolt method, Comm. Numer. Methods Engrg., 24 (2008), 1-13.  doi: 10.1002/cnm.949.

[44]

Y. TakeuchiJ. CuiM. Rinko and Y. Saito, Permanence of dispersal population model with time delays, J. Comput. Appl. Math., 192 (2006), 417-430.  doi: 10.1016/j.cam.2005.06.002.

[45] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, Princeton University Press, Princeton, NJ, 2003.  doi: 10.2307/j.ctv301f9v.
[46]

H. R. Thieme and X. Q. Zhao, A nonlocal delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.  doi: 10.1016/S0362-546X(00)00112-7.

[47]

K. M. C. Tjrve and E. Tjrve, The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family, PLoS ONE, 12 (2017). doi: 10.1371/journal.pone.0178691.

[48]

S. Trofimchuk and V. Volpert, Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities, Nonlinearity, 32 (2019), 2593-2632.  doi: 10.1088/1361-6544/ab0e23.

[49]

H. WanL. Zhang and H. Li, A single species model with symmetric bidirectional impulsive diffusion and dispersal delay, J. Appl. Math., 3 (2012), 1079-1088.  doi: 10.1155/2014/701545.

[50]

H. Wan, L. Zhang and Z. Teng, Analysis of a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay, J. Appl. Math., 2014 (2014), 11pp. doi: 10.1155/2014/701545.

[51]

P. WengD. Liang and J. Wu, Asymptotic patterns of a structured population diffusing in a two-dimensional strip, Nonlinear Anal., 69 (2008), 3931-3951.  doi: 10.1016/j.na.2007.10.027.

[52]

J. Wu, Theory and Applications of Partial Functional-Differential Equations, Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.

[53]

G. Yao, An improved localized method of approximate particular solutions for solving elliptic PDEs, Comput. Math. Appl., 71 (2016), 171-184.  doi: 10.1016/j.camwa.2015.11.008.

[54]

D. L. YoungM. H. Gu and C. M. Fan, The time-marching method of fundamental solutions for wave equations, Eng. Anal. Bound. Elem., 33 (2009), 1411-1425.  doi: 10.1016/j.enganabound.2009.05.008.

[55]

X. Zou, Delay induced traveling wave fronts in reaction diffusion equations of KPP-Fisher type, J. Comput. Appl. Math., 146 (2002), 309-321.  doi: 10.1016/S0377-0427(02)00363-1.

Figure 1.  A schematic representation of the delayed HPRD model (26). Individuals can randomly move between infinitely many habitats on a straight line. Considering habitats centered at $ x_{i-1}, x_{i} $ and $ x_{i+1} $, the effects of birth function $ b(w) $, mortality rate $ d_{m}, $ survival rate $ \varepsilon $, dispersal delay $ r $, migration between habitats $ f_{\alpha}\ast b(w) $, and migration from other regions $ k_{i} $ are indicated in the diagram
Figure 2.  Oscillation due to dispersal delay. (a) without dispersal delay (i.e. when r = 0), the solution of the reduced model (30) converges monotonically to the constant solution $ k/d_{m} = 2 $ as $ t \rightarrow \infty. $ (b) when $ r>0 $, the convergence occurs non-monotonically and the solution $ w(x, t) $ oscillates around $ k/d_{m} = 2. $ See the video clips "monotonic.gif" and "oscillatory.gif" in the supplementary document corresponding to panels (a) and (b) respectively
Figure 3.  Graph of the Hopf bifurcation value $ \hat{\tau}(r) $ as an increasing function of the dispersal delay $ r $. See the text for the specific parameter values
Figure 4.  The destabilizing effect of dispersal delay on the stationary wave pulse of the general model (26). Panels (a) - (d) show that the stability of the stationary pulse is lost as the value of dispersal delay $ r $ is increased from zero. The values of $ r $ are indicated in each panel. See the video clip "Pulse.gif" in the supplementary document animating the solutions in panels (b) - (d)
Figure 5.  The destabilizing effect of dispersal delay on the stationary wavefront of the general model (26). Panels (a) - (d) show that the stability of the stationary wavefront is lost as the value of dispersal delay $ r $ is increased from zero. The values of $ r $ are indicated in each panel. See the video clip "Front.gif" in the supplementary document animating the solutions in panels (b) - (d) with respect to time $ t $
Table 1.  A summary of the parameters and functions used in the general model (26)
Symbol Description Dimension
$ \tau $ maturation time delay $ [T] $
$ r $ dispersal delay $ [T] $
$ k $ migration rate (source) $ [T]^{-1} $
$ D_m $ diffusion rate of the mature population $ [L]^{2}[T]^{-1} $
$ D_I(a) $ diffusion rate of the immature population at age $ a $ $ [L]^{2}[T]^{-1} $
$ d_I(a) $ mortality rate of the immature population at age $ a $ $ [T]^{-1} $
$ d_m $ mortality rate of the mature population $ [T]^{-1} $
$ \varepsilon $ survival rate of the mature population $ - $
$ b(w) $ birth function (birth rate) $ - $
$ f_{\alpha}(x) $ survival probability after traveling x units (Gaussian function) $ [L]^{-1} $
Symbol Description Dimension
$ \tau $ maturation time delay $ [T] $
$ r $ dispersal delay $ [T] $
$ k $ migration rate (source) $ [T]^{-1} $
$ D_m $ diffusion rate of the mature population $ [L]^{2}[T]^{-1} $
$ D_I(a) $ diffusion rate of the immature population at age $ a $ $ [L]^{2}[T]^{-1} $
$ d_I(a) $ mortality rate of the immature population at age $ a $ $ [T]^{-1} $
$ d_m $ mortality rate of the mature population $ [T]^{-1} $
$ \varepsilon $ survival rate of the mature population $ - $
$ b(w) $ birth function (birth rate) $ - $
$ f_{\alpha}(x) $ survival probability after traveling x units (Gaussian function) $ [L]^{-1} $
Table 2.  A summary of the description and outcomes of the reduced models (26)
Model $ D_{I} $ $ D_{M} $ $ r $ $ \varepsilon $ $ \tau $ Model Outcomes Refs
(28) 0 0 0 0 0 globally stable equilibrium at $ w* =k/d_m $ [45]
(29) 0 0 + 0 0 oscillations about $ w* = k/d_m $ [15,22]
(30) 0 + + 0 0 spacial oscillations about $ w* $ [45]
(31) 0 0 0 + 0 survival-extinction, survival of the species [45]
(32) 0 0 0 + + delay-induced, bifurcating solutions [15,22]
(33) 0 + 0 + 0 target waves, stationary & traveling waves [43,18]
(34) 0 + 0 + + periodic solution in the spatial domain [18]
(36) + + 0 + + wave approximation, spatial survival [54,5,34]
Model Descriptions: Equation (28): spatially homogeneous without delay or survival of immature population; Equation (29): spatially homogenous with dispersal delay; Equation (30): dispersal delay and diffusion are present; Equation (31): spatially homogeneous without any delay term; Equation (32): spatially homogeneous with maturation time delay; Equation (33): local reaction-diffusion model; Equation (34): delayed local reaction-diffusion; Equation (36): nonlocal delayed RD equation.
Model $ D_{I} $ $ D_{M} $ $ r $ $ \varepsilon $ $ \tau $ Model Outcomes Refs
(28) 0 0 0 0 0 globally stable equilibrium at $ w* =k/d_m $ [45]
(29) 0 0 + 0 0 oscillations about $ w* = k/d_m $ [15,22]
(30) 0 + + 0 0 spacial oscillations about $ w* $ [45]
(31) 0 0 0 + 0 survival-extinction, survival of the species [45]
(32) 0 0 0 + + delay-induced, bifurcating solutions [15,22]
(33) 0 + 0 + 0 target waves, stationary & traveling waves [43,18]
(34) 0 + 0 + + periodic solution in the spatial domain [18]
(36) + + 0 + + wave approximation, spatial survival [54,5,34]
Model Descriptions: Equation (28): spatially homogeneous without delay or survival of immature population; Equation (29): spatially homogenous with dispersal delay; Equation (30): dispersal delay and diffusion are present; Equation (31): spatially homogeneous without any delay term; Equation (32): spatially homogeneous with maturation time delay; Equation (33): local reaction-diffusion model; Equation (34): delayed local reaction-diffusion; Equation (36): nonlocal delayed RD equation.
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