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 & 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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[21]

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

[22]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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 $\alpha=2,\gamma=0.1,\mu=0.4,\beta=0.365$
[1]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

[2]

Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445

[3]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169

[4]

Hans-Otto Walther. On Poisson's state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365

[5]

Alexander Rezounenko. Stability of a viral infection model with state-dependent delay, CTL and antibody immune responses. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1547-1563. doi: 10.3934/dcdsb.2017074

[6]

Alexander Rezounenko. Viral infection model with diffusion and state-dependent delay: Stability of classical solutions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1091-1105. doi: 10.3934/dcdsb.2018143

[7]

Xiuli Sun, Rong Yuan, Yunfei Lv. Global Hopf bifurcations of neutral functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 667-700. doi: 10.3934/dcdsb.2018038

[8]

Igor Chueshov, Alexander V. Rezounenko. Finite-dimensional global attractors for parabolic nonlinear equations with state-dependent delay. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1685-1704. doi: 10.3934/cpaa.2015.14.1685

[9]

Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039

[10]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

[11]

Benjamin B. Kennedy. Multiple periodic solutions of state-dependent threshold delay equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1801-1833. doi: 10.3934/dcds.2012.32.1801

[12]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993

[13]

Odo Diekmann, Karolína Korvasová. Linearization of solution operators for state-dependent delay equations: A simple example. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 137-149. doi: 10.3934/dcds.2016.36.137

[14]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167

[15]

A. R. Humphries, O. A. DeMasi, F. M. G. Magpantay, F. Upham. Dynamics of a delay differential equation with multiple state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2701-2727. doi: 10.3934/dcds.2012.32.2701

[16]

Hermann Brunner, Stefano Maset. Time transformations for state-dependent delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (1) : 23-45. doi: 10.3934/cpaa.2010.9.23

[17]

Matthias Büger, Marcus R.W. Martin. Stabilizing control for an unbounded state-dependent delay equation. Conference Publications, 2001, 2001 (Special) : 56-65. doi: 10.3934/proc.2001.2001.56

[18]

Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167

[19]

Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931

[20]

Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913

2018 Impact Factor: 1.008

Metrics

  • PDF downloads (12)
  • HTML views (3)
  • Cited by (1)

Other articles
by authors

[Back to Top]