# American Institute of Mathematical Sciences

2010, 7(4): 729-737. doi: 10.3934/mbe.2010.7.729

## Stability of a delay equation arising from a juvenile-adult model

 1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010, United States

Received  May 2010 Revised  July 2010 Published  October 2010

We consider a delay equation that has been formulated from a juvenile-adult population model. We give respective conditions on the vital rates to ensure local stability of the positive equilibrium and global stability of the trivial equilibrium. We also show that under certain conditions the equation undergoes a Hopf bifurcation. We then study global asymptotic stability and present bifurcation diagrams for two special cases of the model.
Citation: Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729
##### References:
 [1] A. S. Ackleh, J. Carter, L. Cole, T. Nguyen, J. Monte and C. Pettit, Measuring and modeling the seasonal changes of an urban Green Treefrog (Hyla cinerea) population, Ecol. Modelling, 221 (2010), 281-289. doi: doi:10.1016/j.ecolmodel.2009.10.012.  Google Scholar [2] A. S. Ackleh and K. Deng, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: doi:10.1137/080723673.  Google Scholar [3] R. Bellman and K. L. Cooke, "Differential-Difference Equations," Academic Press, New York, 1963.  Google Scholar [4] J. E. Forde, "Delay Differential Equation Models in Mathematical Biology," Ph.D. Thesis, The University of Michigan, 2005. Google Scholar [5] L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Ann. New York Acad. Sci., 316 (1979), 214-235. doi: doi:10.1111/j.1749-6632.1979.tb29471.x.  Google Scholar [6] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Oscillations and global attractivity in respiratory dynamics, Dynam. Stability Systems, 4 (1989), 131-139.  Google Scholar [7] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Oscillations and global attractivity in models of haematopoiesis, J. Dynam. Differential Equations, 2 (1990), 117-132. doi: doi:10.1007/BF01057415.  Google Scholar [8] M. S. Gunzenburger and J. Travis, Evaluating predation pressure on green treefrog larvae across a habitat gradient, Oecologia, 140 (2004), 422-429. Google Scholar [9] M. S. Gunzenburger and J. Travis, Effects of multiple predator species on green treefrog (Hyla cinerea) tadpoles, Canadian J. Zoology, 83 (2005), 996-1002. doi: doi:10.1139/z05-093.  Google Scholar [10] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: doi:10.1038/287017a0.  Google Scholar [11] I. Gyori and G. Ladas, "Oscillation Theory of Delay Differential Equations with Applications," Oxford University Press, New York, 1991.  Google Scholar [12] G. Karakostas, C. G. Philos and Y. G. Sficas, Stable steady state of some population models, J. Dynam. Differential Equations, 4 (1992), 161-190. doi: doi:10.1007/BF01048159.  Google Scholar [13] T. Kostova, J. Li and M. Friedman, Two models for competition between age classes, Math. Biosci., 157 (1999), 65-89. doi: doi:10.1016/S0025-5564(98)10077-9.  Google Scholar [14] Y. Kuang, Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology, Japan J. Indust. Appl. Math., 9 (1992), 205-238. doi: doi:10.1007/BF03167566.  Google Scholar [15] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Boston, 1993.  Google Scholar [16] M. R. S. Kulenovic, G. Ladas and Y. G. Sficas, Global attractivity in population dynamics, Comput. Math. Appl., 18 (1989), 925-928. doi: doi:10.1016/0898-1221(89)90010-2.  Google Scholar [17] I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 35 (2002), 295-301. doi: doi:10.1016/S0895-7177(01)00166-2.  Google Scholar [18] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: doi:10.1126/science.267326.  Google Scholar [19] L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter and N. Pal, Population estimates of Hyla cinerea (Schneider) (H. cinerea) in an urban environment, Southeastern Naturalist, 6 (2007), 203-216. doi: doi:10.1656/1528-7092(2007)6[203:PEOHCS]2.0.CO;2.  Google Scholar [20] A. L. Skubachevskii and H. O. Walther, On Floquet multipliers for slowly oscillating periodic solutions of nonlinear functional differential equations, Tr. Mosk. Mat. Obs., 64 (2003), 3-53.  Google Scholar [21] A. L. Skubachevskii and H. O. Walther, On the Floquet multipliers of periodic solutions to non-linear functional differential equations, J. Dynam. Differential Equations, 18 (2006), 257-355. doi: doi:10.1007/s10884-006-9006-5.  Google Scholar [22] H. O. Walther, "The 2-Dimensional Attractor of $x'(t)=-\mu x(t)+f(x(t-1))$," Mem. Amer. Math. Soc., 113, 1995.  Google Scholar [23] J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007), 2483-2498. doi: doi:10.1088/0951-7715/20/11/002.  Google Scholar [24] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos., 6 (1976), 23-40.  Google Scholar [25] A. Zaghrout, A. Ammar and M. A. El-Sheikh, Oscillations and global attractivity in delay differential equations of population dynamics, Appl. Math. Comput., 77 (1996), 195-204. doi: doi:10.1016/S0096-3003(95)00213-8.  Google Scholar

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##### References:
 [1] A. S. Ackleh, J. Carter, L. Cole, T. Nguyen, J. Monte and C. Pettit, Measuring and modeling the seasonal changes of an urban Green Treefrog (Hyla cinerea) population, Ecol. Modelling, 221 (2010), 281-289. doi: doi:10.1016/j.ecolmodel.2009.10.012.  Google Scholar [2] A. S. Ackleh and K. Deng, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: doi:10.1137/080723673.  Google Scholar [3] R. Bellman and K. L. Cooke, "Differential-Difference Equations," Academic Press, New York, 1963.  Google Scholar [4] J. E. Forde, "Delay Differential Equation Models in Mathematical Biology," Ph.D. Thesis, The University of Michigan, 2005. Google Scholar [5] L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Ann. New York Acad. Sci., 316 (1979), 214-235. doi: doi:10.1111/j.1749-6632.1979.tb29471.x.  Google Scholar [6] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Oscillations and global attractivity in respiratory dynamics, Dynam. Stability Systems, 4 (1989), 131-139.  Google Scholar [7] K. Gopalsamy, M. R. S. Kulenovic and G. Ladas, Oscillations and global attractivity in models of haematopoiesis, J. Dynam. Differential Equations, 2 (1990), 117-132. doi: doi:10.1007/BF01057415.  Google Scholar [8] M. S. Gunzenburger and J. Travis, Evaluating predation pressure on green treefrog larvae across a habitat gradient, Oecologia, 140 (2004), 422-429. Google Scholar [9] M. S. Gunzenburger and J. Travis, Effects of multiple predator species on green treefrog (Hyla cinerea) tadpoles, Canadian J. Zoology, 83 (2005), 996-1002. doi: doi:10.1139/z05-093.  Google Scholar [10] W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: doi:10.1038/287017a0.  Google Scholar [11] I. Gyori and G. Ladas, "Oscillation Theory of Delay Differential Equations with Applications," Oxford University Press, New York, 1991.  Google Scholar [12] G. Karakostas, C. G. Philos and Y. G. Sficas, Stable steady state of some population models, J. Dynam. Differential Equations, 4 (1992), 161-190. doi: doi:10.1007/BF01048159.  Google Scholar [13] T. Kostova, J. Li and M. Friedman, Two models for competition between age classes, Math. Biosci., 157 (1999), 65-89. doi: doi:10.1016/S0025-5564(98)10077-9.  Google Scholar [14] Y. Kuang, Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology, Japan J. Indust. Appl. Math., 9 (1992), 205-238. doi: doi:10.1007/BF03167566.  Google Scholar [15] Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Boston, 1993.  Google Scholar [16] M. R. S. Kulenovic, G. Ladas and Y. G. Sficas, Global attractivity in population dynamics, Comput. Math. Appl., 18 (1989), 925-928. doi: doi:10.1016/0898-1221(89)90010-2.  Google Scholar [17] I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 35 (2002), 295-301. doi: doi:10.1016/S0895-7177(01)00166-2.  Google Scholar [18] M. C. Mackey and L. Glass, Oscillation and chaos in physiological control system, Science, 197 (1977), 287-289. doi: doi:10.1126/science.267326.  Google Scholar [19] L. Pham, S. Boudreaux, S. Karhbet, B. Price, A.S. Ackleh, J. Carter and N. Pal, Population estimates of Hyla cinerea (Schneider) (H. cinerea) in an urban environment, Southeastern Naturalist, 6 (2007), 203-216. doi: doi:10.1656/1528-7092(2007)6[203:PEOHCS]2.0.CO;2.  Google Scholar [20] A. L. Skubachevskii and H. O. Walther, On Floquet multipliers for slowly oscillating periodic solutions of nonlinear functional differential equations, Tr. Mosk. Mat. Obs., 64 (2003), 3-53.  Google Scholar [21] A. L. Skubachevskii and H. O. Walther, On the Floquet multipliers of periodic solutions to non-linear functional differential equations, J. Dynam. Differential Equations, 18 (2006), 257-355. doi: doi:10.1007/s10884-006-9006-5.  Google Scholar [22] H. O. Walther, "The 2-Dimensional Attractor of $x'(t)=-\mu x(t)+f(x(t-1))$," Mem. Amer. Math. Soc., 113, 1995.  Google Scholar [23] J. Wei, Bifurcation analysis in a scalar delay differential equation, Nonlinearity, 20 (2007), 2483-2498. doi: doi:10.1088/0951-7715/20/11/002.  Google Scholar [24] M. Wazewska-Czyzewska and A. Lasota, Mathematical problems of the dynamics of a system of red blood cells, Mat. Stos., 6 (1976), 23-40.  Google Scholar [25] A. Zaghrout, A. Ammar and M. A. El-Sheikh, Oscillations and global attractivity in delay differential equations of population dynamics, Appl. Math. Comput., 77 (1996), 195-204. doi: doi:10.1016/S0096-3003(95)00213-8.  Google Scholar
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