September  2020, 25(9): 3523-3551. doi: 10.3934/dcdsb.2020071

Dynamics of a stage-structured population model with a state-dependent delay

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China

* Corresponding author: Shangjiang Guo

Received  August 2019 Revised  November 2019 Published  September 2020 Early access  April 2020

Fund Project: The second author is supported by the NNSF of P.R. China (Grant No. 11671123)

This paper is devoted to the dynamics of a predator-prey model with stage structure for prey and state-dependent maturation delay. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. Secondly, the existence, uniqueness, and local asymptotical stability of (boundary and coexisting) equilibria are investigated by means of degree theory and Routh-Hurwitz criteria. Thirdly, the explicit bounds for the eventual behaviors of the mature population are obtained. Finally, by means of comparison principle and two auxiliary systems, it observed that the local asymptotical stability of either of the positive interior equilibrium and the positive boundary equilibrium implies that it is also globally asymptotical stable if the derivative of the delay function around this equilibrium is small enough.

Citation: Shangzhi Li, Shangjiang Guo. Dynamics of a stage-structured population model with a state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3523-3551. doi: 10.3934/dcdsb.2020071
References:
[1]

M. AdimyF. CrausteM. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.

[2]

W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U.

[3]

W. G. AielloH. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048.

[4]

J. F. M. Al-Omari, The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput., 271 (2015), 142-153.  doi: 10.1016/j.amc.2015.08.119.

[5]

J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Anal. Real World Appl., 6 (2005), 13-33.  doi: 10.1016/j.nonrwa.2004.04.002.

[6] L. S. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing, 1988. 
[7]

K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426.  doi: 10.1090/S0002-9939-96-03437-5.

[8]

K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model., 215 (2008), 69-76.  doi: 10.1016/j.ecolmodel.2008.02.019.

[9]

M. E. Fisher and B. S. Goh, Stability results for delayed-recruitment models in population dynamics, J. Math. Biol., 19 (1984), 147-156.  doi: 10.1007/BF00275937.

[10]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  doi: 10.1016/S0092-8240(86)90003-0.

[11]

R. Gambell, Birds and mammals – Antarctic whales, in Antarctica, Pergamon Press, New York, 1985, 223–241. doi: 10.1016/B978-0-08-028881-9.50022-4.

[12]

S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.  doi: 10.1007/s00285-004-0278-2.

[13]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.

[14]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.

[15]

W. S. C. Gurney and R. M. Nisbet, Fluctuating periodicity, generation separation, and the expression of larval competition, Theoret. Population Biol., 28 (1985), 150-180.  doi: 10.1016/0040-5809(85)90026-7.

[16]

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4612-9892-2.

[17]

F. Hartung and J. Turi, On the asymptotic behavior of the solutions of a state-dependent delay equation, Differential Integral Equations, 8 (1995), 1867-1872. 

[18]

A. Hou and S. Guo, Stability and Hopf bifurcation in van der Pol oscillators with state-dependent delayed feedback, Nonlinear Dynam., 79 (2015), 2407-2419.  doi: 10.1007/s11071-014-1821-3.

[19]

C. Huang, Y. Qiao and L. Huang, et al., Dynamical behaviors of a food-chain model with stage structure and time delays, Adv. Difference Equ., 2018 (2018), 26pp. doi: 10.1186/s13662-018-1589-8.

[20]

H. F. HuoW. T. Li and R. P. Agarwal, Optimal harvesting and stability for two species stage-structured system with cannibalism, Int. J. App. Math., 6 (2001), 59-79. 

[21]

Y. S. Koselsov, Properties of solutions of a class of equations with delay that describe the dynamics of change of the size of a species with account taken of age structure, Mat. Sb. (N.S.), 117 (1982), 86-94. 

[22]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.

[23]

H. D. Landahl and B. D. Hansen, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17.  doi: 10.1007/BF02463488.

[24]

S. Z. Li and S. J. Guo, Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1393-1423.  doi: 10.3934/dcdsb.2017067.

[25]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type Ⅱ functional response, J. Nonlinear Sci., 28 (2018), 1151-1187.  doi: 10.1007/s00332-018-9444-3.

[26]

M. Lloyd and H. S. Dybas, The periodical cicada problem, Evolution, 20 (1966), 133-149.  doi: 10.1111/j.1558-5646.1966.tb03350.x.

[27]

J. D. Murray, Mathematical Biology. I. An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.

[28]

A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Australian J. Zoology, 2 (1954), 1-8.  doi: 10.1071/ZO9540001.

[29] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.
[30]

Y. Qu and J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynam., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x.

[31]

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49 (1987), 253-255.  doi: 10.1016/S0092-8240(87)80045-9.

[32]

Y. Satio and Y. Takeuchi, A time-delay model for prey-predator growth with stage structure, Can. Appl. Math. Q., 11 (2003), 293-302. 

[33]

X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 423-430.  doi: 10.1007/s102550200042.

[34]

X. Song and L. Chen, A predator-prey system with stage structure and harvesting for predator, Ann. Differential Equations, 18 (2002), 264-277. 

[35]

K. Tognetti, The two stage stochastic model, Math. Biosci., 25 (1975), 195-204.  doi: 10.1016/0025-5564(75)90002-4.

[36]

P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.  doi: 10.2307/1932137.

[37]

P. J. Wangersky and W. J. Cunningham, On time lags in equations of growth, Proc. Natl. Acad. Sci. U.S.A., 42 (1956), 699-702.  doi: 10.1073/pnas.42.9.699.

[38]

S. N. WoodS. P. BlytheW. S. C. Gurney and R. M. Nisbet, Instability in mortality estimation schemes related to stage-structure population models, IMA J. Math. Appl. Med. Biol., 6 (1989), 47-68.  doi: 10.1093/imammb/6.1.47.

[39]

S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059.

[40]

Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay, Appl. Math. Comput., 214 (2009), 228-235.  doi: 10.1016/j.amc.2009.03.078.

[41] Q. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990. 
[42]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for a nonlinear second order differential equation with state-dependent delay, Proc. Amer. Math. Soc., 145 (2017), 4893-4903.  doi: 10.1090/proc/13714.

[43]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state-dependent delay, Math. Methods Appl. Sci., 40 (2017), 5307-5331.  doi: 10.1002/mma.4388.

show all references

References:
[1]

M. AdimyF. CrausteM. Hbid and R. Qesmi, Stability and Hopf bifurcation for a cell population model with state-dependent delay, SIAM J. Appl. Math., 70 (2010), 1611-1633.  doi: 10.1137/080742713.

[2]

W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci., 101 (1990), 139-153.  doi: 10.1016/0025-5564(90)90019-U.

[3]

W. G. AielloH. I. Freedman and J. Wu, Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. Math., 52 (1992), 855-869.  doi: 10.1137/0152048.

[4]

J. F. M. Al-Omari, The effect of state dependent delay and harvesting on a stage-structured predator-prey model, Appl. Math. Comput., 271 (2015), 142-153.  doi: 10.1016/j.amc.2015.08.119.

[5]

J. F. M. Al-Omari and S. A. Gourley, Dynamics of a stage-structured population model incorporating a state-dependent maturation delay, Nonlinear Anal. Real World Appl., 6 (2005), 13-33.  doi: 10.1016/j.nonrwa.2004.04.002.

[6] L. S. Chen, Mathematical Models and Methods in Ecology, Science Press, Beijing, 1988. 
[7]

K. L. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426.  doi: 10.1090/S0002-9939-96-03437-5.

[8]

K. Das and S. Ray, Effect of delay on nutrient cycling in phytoplankton-zooplankton interactions in estuarine system, Ecol. Model., 215 (2008), 69-76.  doi: 10.1016/j.ecolmodel.2008.02.019.

[9]

M. E. Fisher and B. S. Goh, Stability results for delayed-recruitment models in population dynamics, J. Math. Biol., 19 (1984), 147-156.  doi: 10.1007/BF00275937.

[10]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  doi: 10.1016/S0092-8240(86)90003-0.

[11]

R. Gambell, Birds and mammals – Antarctic whales, in Antarctica, Pergamon Press, New York, 1985, 223–241. doi: 10.1016/B978-0-08-028881-9.50022-4.

[12]

S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.  doi: 10.1007/s00285-004-0278-2.

[13]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations, Applied Mathematical Sciences, 184, Springer, New York, 2013. doi: 10.1007/978-1-4614-6992-6.

[14]

W. S. C. GurneyS. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.  doi: 10.1038/287017a0.

[15]

W. S. C. Gurney and R. M. Nisbet, Fluctuating periodicity, generation separation, and the expression of larval competition, Theoret. Population Biol., 28 (1985), 150-180.  doi: 10.1016/0040-5809(85)90026-7.

[16]

J. Hale, Theory of Functional Differential Equations, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977. doi: 10.1007/978-1-4612-9892-2.

[17]

F. Hartung and J. Turi, On the asymptotic behavior of the solutions of a state-dependent delay equation, Differential Integral Equations, 8 (1995), 1867-1872. 

[18]

A. Hou and S. Guo, Stability and Hopf bifurcation in van der Pol oscillators with state-dependent delayed feedback, Nonlinear Dynam., 79 (2015), 2407-2419.  doi: 10.1007/s11071-014-1821-3.

[19]

C. Huang, Y. Qiao and L. Huang, et al., Dynamical behaviors of a food-chain model with stage structure and time delays, Adv. Difference Equ., 2018 (2018), 26pp. doi: 10.1186/s13662-018-1589-8.

[20]

H. F. HuoW. T. Li and R. P. Agarwal, Optimal harvesting and stability for two species stage-structured system with cannibalism, Int. J. App. Math., 6 (2001), 59-79. 

[21]

Y. S. Koselsov, Properties of solutions of a class of equations with delay that describe the dynamics of change of the size of a species with account taken of age structure, Mat. Sb. (N.S.), 117 (1982), 86-94. 

[22]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics, Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.

[23]

H. D. Landahl and B. D. Hansen, A three stage population model with cannibalism, Bull. Math. Biol., 37 (1975), 11-17.  doi: 10.1007/BF02463488.

[24]

S. Z. Li and S. J. Guo, Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1393-1423.  doi: 10.3934/dcdsb.2017067.

[25]

Q. LiuD. JiangT. Hayat and A. Alsaedi, Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type Ⅱ functional response, J. Nonlinear Sci., 28 (2018), 1151-1187.  doi: 10.1007/s00332-018-9444-3.

[26]

M. Lloyd and H. S. Dybas, The periodical cicada problem, Evolution, 20 (1966), 133-149.  doi: 10.1111/j.1558-5646.1966.tb03350.x.

[27]

J. D. Murray, Mathematical Biology. I. An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.

[28]

A. J. Nicholson, Compensatory reactions of populations to stresses, and their evolutionary significance, Australian J. Zoology, 2 (1954), 1-8.  doi: 10.1071/ZO9540001.

[29] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.
[30]

Y. Qu and J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynam., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x.

[31]

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49 (1987), 253-255.  doi: 10.1016/S0092-8240(87)80045-9.

[32]

Y. Satio and Y. Takeuchi, A time-delay model for prey-predator growth with stage structure, Can. Appl. Math. Q., 11 (2003), 293-302. 

[33]

X. Song and L. Chen, Optimal harvesting and stability for a predator-prey system with stage structure, Acta Math. Appl. Sin. Engl. Ser., 18 (2002), 423-430.  doi: 10.1007/s102550200042.

[34]

X. Song and L. Chen, A predator-prey system with stage structure and harvesting for predator, Ann. Differential Equations, 18 (2002), 264-277. 

[35]

K. Tognetti, The two stage stochastic model, Math. Biosci., 25 (1975), 195-204.  doi: 10.1016/0025-5564(75)90002-4.

[36]

P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.  doi: 10.2307/1932137.

[37]

P. J. Wangersky and W. J. Cunningham, On time lags in equations of growth, Proc. Natl. Acad. Sci. U.S.A., 42 (1956), 699-702.  doi: 10.1073/pnas.42.9.699.

[38]

S. N. WoodS. P. BlytheW. S. C. Gurney and R. M. Nisbet, Instability in mortality estimation schemes related to stage-structure population models, IMA J. Math. Appl. Med. Biol., 6 (1989), 47-68.  doi: 10.1093/imammb/6.1.47.

[39]

S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059.

[40]

Y. Yang, Hopf bifurcation in a two-competitor, one-prey system with time delay, Appl. Math. Comput., 214 (2009), 228-235.  doi: 10.1016/j.amc.2009.03.078.

[41] Q. YeZ. LiM. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Foundations of Modern Mathematics Series, Science Press, Beijing, 1990. 
[42]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for a nonlinear second order differential equation with state-dependent delay, Proc. Amer. Math. Soc., 145 (2017), 4893-4903.  doi: 10.1090/proc/13714.

[43]

L. Zhang and S. J. Guo, Slowly oscillating periodic solutions for the Nicholson's blowflies equation with state-dependent delay, Math. Methods Appl. Sci., 40 (2017), 5307-5331.  doi: 10.1002/mma.4388.

Figure 1.  Simulations of system (41) with $ \tau(x) = 4-2e^{-0.1x} $ and $ (r, c) = (0.1, 1) $ illustrate that the positive equilibrium is globally asymptotically stable
Figure 2.  Simulations of system (41) with $ \tau(x) = 4-2e^{-0.1x} $ and $ (r, c) = (0.6, 0.1) $ illustrate that the positive equilibrium is globally asymptotically stable
Figure 3.  Simulations of system (41) with $ \tau(x)\equiv 4 $ and $ (r, c) = (0.1, 1) $ illustrate that the positive equilibrium is globally asymptotically stable
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