June  2017, 22(4): 1393-1423. doi: 10.3934/dcdsb.2017067

Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China

* Corresponding author: S. Guo

Received  May 2016 Revised  November 2016 Published  February 2017

Fund Project: The second author is supported by NSF of China (Grants No. 11671123 & 11271115).

This paper is devoted to a cooperative model composed of two species withstage structure and state-dependent maturation delays. Firstly, positivity and boundedness of solutions are addressed to describe the population survival and the natural restriction of limited resources. It is shown that for a given pair of positive initial functions, the two mature populations are uniformly bounded away from zero and that the two mature populations are bounded above only if the the coupling strength is small enough. Moreover, if the coupling strength is large enough then the two mature populations tend to infinity as the time tends to infinity. In particular, the positivity of the two immature populations has been established under some additional conditions. Secondly, the existence and patterns of equilibria are investigated by means of degree theory and Lyapunov-Schmidt reduction. Thirdly, the local stability of the equilibria is also discussed through a formal linearization. Fourthly, the global behavior of solutions is discussed and the explicit bounds for the eventual behaviors of the two mature populations and two immature populations are obtained. Finally, global asymptotical stability is investigated by using the comparison principle of the state-dependent delay equations.

Citation: Shangzhi Li, Shangjiang Guo. Dynamics of a two-species stage-structured model incorporating state-dependent maturation delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1393-1423. doi: 10.3934/dcdsb.2017067
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 and S. A. Gourley, Dynamics of stage-structure population model incorporating a state-dependent maturation delay, Nonl. Anal., 6 (2005), 13-33.  doi: 10.1016/j.nonrwa.2004.04.002.

[5]

J. F. M. Al-Omari and S. A. Gourley, Stability and traveling fronts in Lotka-Volterra competition models with stage structure, SIAM J. Appl. Math., 63 (2003), 2063-2086.  doi: 10.1137/S0036139902416500.

[6]

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.

[7]

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.

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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.

[9]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  doi: 10.1007/BF02462319.

[10]

R. Gambell, Birds and mammals – Antarctic whales, in Antarctica, in Contributions to Nonlinear Functional Analysis (eds. W. N. Bonner and D. W. H. Walton), Pergamon Press, (1985), 223–241.

[11]

B. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.  doi: 10.1007/BF00275063.

[12]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

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

[14]

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

[15]

J. Hale, Theory of Functional Differential Equations Springer-Verlag, New York, 1977.

[16]

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

[17]

A. Hou and S. Guo, Stability and bifurcation in a state-dependent delayed predator-prey system, International Journal of Bifurcation and Chaos, 26 (2016), 1650060, 15pp.  doi: 10.1142/S0218127416500607.

[18]

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

[19]

Q. Hu and X. Zhao, Global dynamics of a state-dependent delay model with unimodal feedback, J. Math. Anal. Appl., 399 (2013), 133-146.  doi: 10.1016/j.jmaa.2012.09.058.

[20]

Y. S. Koslesov, Properties of solutions of a class of equations with lag which describe the dynamics of change in the population of a species with age structure taken into account, Math. USSR. Sbornik, 45 (1983), 91-100. 

[21]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics Academic, New York, 1993.

[22]

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

[23]

Z. Lu and W. Wang, Permanence and global attractivity for Lotka-Volterra difference systems, J. Math. Biol., 39 (1999), 269-282.  doi: 10.1007/s002850050171.

[24]

Y. Lv and R. Yuan, Global stability and wavefronts in a cooperation model with state-dependent time delay, J. Math. Anal. Appl., 415 (2014), 543-573.  doi: 10.1016/j.jmaa.2014.01.086.

[25]

Y. Muroya, Uniform persistence for Lotka-Volterra-type delay differential systems, Nonlinear Anal. Real World Appl., 4 (2003), 689-710.  doi: 10.1016/S1468-1218(02)00072-X.

[26]

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

[27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations 2nd edition, Plenum Press, New York, 1994.  doi: 10.1007/978-1-4612-0873-0.
[28]

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49 (1987), 253-255.  doi: 10.1007/BF02459701.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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.

[34]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[35]

A. Zaghrout and S. Attalah, Analysis of a model of stage-structured population dynamics growth with time state-dependent time delay, Appl. Math. Comput., 77 (1996), 185-194.  doi: 10.1016/S0096-3003(95)00212-X.

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 and S. A. Gourley, Dynamics of stage-structure population model incorporating a state-dependent maturation delay, Nonl. Anal., 6 (2005), 13-33.  doi: 10.1016/j.nonrwa.2004.04.002.

[5]

J. F. M. Al-Omari and S. A. Gourley, Stability and traveling fronts in Lotka-Volterra competition models with stage structure, SIAM J. Appl. Math., 63 (2003), 2063-2086.  doi: 10.1137/S0036139902416500.

[6]

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.

[7]

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.

[8]

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.

[9]

H. I. Freedman and K. Gopalsamy, Global stability in time-delayed single species dynamics, Bull. Math. Biol., 48 (1986), 485-492.  doi: 10.1007/BF02462319.

[10]

R. Gambell, Birds and mammals – Antarctic whales, in Antarctica, in Contributions to Nonlinear Functional Analysis (eds. W. N. Bonner and D. W. H. Walton), Pergamon Press, (1985), 223–241.

[11]

B. Goh, Global stability in two species interactions, J. Math. Biol., 3 (1976), 313-318.  doi: 10.1007/BF00275063.

[12]

S. Guo and J. Wu, Bifurcation Theory of Functional Differential Equations Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[13]

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

[14]

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

[15]

J. Hale, Theory of Functional Differential Equations Springer-Verlag, New York, 1977.

[16]

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

[17]

A. Hou and S. Guo, Stability and bifurcation in a state-dependent delayed predator-prey system, International Journal of Bifurcation and Chaos, 26 (2016), 1650060, 15pp.  doi: 10.1142/S0218127416500607.

[18]

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

[19]

Q. Hu and X. Zhao, Global dynamics of a state-dependent delay model with unimodal feedback, J. Math. Anal. Appl., 399 (2013), 133-146.  doi: 10.1016/j.jmaa.2012.09.058.

[20]

Y. S. Koslesov, Properties of solutions of a class of equations with lag which describe the dynamics of change in the population of a species with age structure taken into account, Math. USSR. Sbornik, 45 (1983), 91-100. 

[21]

Y. Kuang, Delay Differential Equation with Applications in Population Dynamics Academic, New York, 1993.

[22]

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

[23]

Z. Lu and W. Wang, Permanence and global attractivity for Lotka-Volterra difference systems, J. Math. Biol., 39 (1999), 269-282.  doi: 10.1007/s002850050171.

[24]

Y. Lv and R. Yuan, Global stability and wavefronts in a cooperation model with state-dependent time delay, J. Math. Anal. Appl., 415 (2014), 543-573.  doi: 10.1016/j.jmaa.2014.01.086.

[25]

Y. Muroya, Uniform persistence for Lotka-Volterra-type delay differential systems, Nonlinear Anal. Real World Appl., 4 (2003), 689-710.  doi: 10.1016/S1468-1218(02)00072-X.

[26]

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

[27] C. V. Pao, Nonlinear Parabolic and Elliptic Equations 2nd edition, Plenum Press, New York, 1994.  doi: 10.1007/978-1-4612-0873-0.
[28]

G. Rosen, Time delays produced by essential nonlinearity in population growth models, Bull. Math. Biol., 49 (1987), 253-255.  doi: 10.1007/BF02459701.

[29]

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

[30]

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

[31]

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

[32]

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

[33]

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.

[34]

Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[35]

A. Zaghrout and S. Attalah, Analysis of a model of stage-structured population dynamics growth with time state-dependent time delay, Appl. Math. Comput., 77 (1996), 185-194.  doi: 10.1016/S0096-3003(95)00212-X.

Figure 1.  Simulations of system (3) illustrate that the synchronous equilibrium is globally asymptotically stable, where $\alpha=2,\gamma=0.1,\mu=0.1,\beta=0.365$
Figure 2.  Simulations of system (3) illustrate that the synchronous equilibrium is globally asymptotically stable, where $\alpha=1.5,\gamma=0.2,\mu=0.1,\beta=0.365$
Figure 3.  Simulations of system (3) illustrate that every solution of (3) is asymptotically synchronous and tends to infinity as $t$ tends to infinity, where $\alpha=2,\gamma=0.1,\mu=0.4,\beta=0.365$
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