# American Institute of Mathematical Sciences

November  2020, 13(11): 3139-3155. doi: 10.3934/dcdss.2020134

## Dirichlet problem for a diffusive logistic population model with two delays

 1 Tongji Zhejiang College, Jiaxing, Zhejiang 314051, China 2 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China 3 Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5, Canada

* Corresponding author

Received  December 2018 Revised  February 2019 Published  November 2020 Early access  November 2019

In this paper, we investigate a diffusive logistic equation with non-zero Dirichlet boundary condition and two delays. We first exclude the existence of positive heterogeneous steady states, which implies the uniqueness of constant positive steady state. Then, we analyze the local stability and local Hopf bifurcation at the positive steady state. We show that multiple delays can induce multiple stability switches. Furthermore, we prove global stability of the positive steady state under certain conditions and obtain global Hopf bifurcation results. Our theoretical results are illustrated with numerical simulations.

Citation: Xuejun Pan, Hongying Shu, Yuming Chen. Dirichlet problem for a diffusive logistic population model with two delays. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3139-3155. doi: 10.3934/dcdss.2020134
##### References:
 [1] B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation, SIAM J. Appl. Dyn. Syst., 13 (2014), 537-563.  doi: 10.1137/120904226. [2] R. D. Braddock and P. van den Driessche, On a two-lag differential delay equation, J. Austral. Math. Soc. Ser. B, 24 (1982/83), 292-317.  doi: 10.1017/S0334270000002939. [3] S. Busenberg and W. Z. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.  doi: 10.1006/jdeq.1996.0003. [4] S. S. Chen and J. J. Wei, Stability and bifurcation in a diffusive logistic population model with multiple delays, Internat. J. Bifur. Chaos, 25 (2015), 1550107, 9 pp. doi: 10.1142/S0218127415501072. [5] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7. [6] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787.  doi: 10.1016/j.jde.2010.11.011. [7] K. Gopalsamy, The Delay Logistic Equation, in Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications, vol. 74, Springer, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. [8] G. E. Hutchison, Circular causal systems in ecology, Ann. N. Y. Acad. Sci., 50 (1948), 221-246. [9] R. L. Kitching, Time resources and population dynamics in insects, Austral. J. Ecol., 2 (1977), 31-42.  doi: 10.1111/j.1442-9993.1977.tb01125.x. [10] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. [11] J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, with Applications, Mathematics in Science and Engineering, Vol. 4 Academic Press, New York-London 1961 [12] G. Lin, Spreading speed of the delayed Fisher equation without quasimonotonicity, Nonlinear Anal. Real World Appl., 12 (2011), 3713-3718.  doi: 10.1016/j.nonrwa.2011.07.004. [13] M. C. Memory, Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion, SIAM J. Math. Anal., 20 (1989), 533-546.  doi: 10.1137/0520037. [14] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [15] S. Ruan, Delay differential equations in single species dynamics, Delay Differential Equations and Applications, NATO Sci. Ser. Ⅱ Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517.  doi: 10.1007/1-4020-3647-7_11. [16] Y. Su, J. J. Wei and J. P. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dynam. Differential Equations, 24 (2012), 897-925.  doi: 10.1007/s10884-012-9268-z. [17] P. F. Verhulst, Notice sur la loi que la population pursuit dans son accroissement, Corresp. Math. Phys., 10 (1838), 113-121. [18] J. H. 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. [19] J. H. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2. [20] J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [21] X. P. Yan and J. P. Shi, Stability switches in a logistic population model with mixed instantaneous and delayed density dependence, J. Dynam. Differential Equations, 29 (2017), 113-130.  doi: 10.1007/s10884-015-9432-3. [22] K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J., 12 (1982), 321-348.  doi: 10.32917/hmj/1206133754. [23] X. F. 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.

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##### References:
 [1] B. Bánhelyi, T. Csendes, T. Krisztin and A. Neumaier, Global attractivity of the zero solution for Wright's equation, SIAM J. Appl. Dyn. Syst., 13 (2014), 537-563.  doi: 10.1137/120904226. [2] R. D. Braddock and P. van den Driessche, On a two-lag differential delay equation, J. Austral. Math. Soc. Ser. B, 24 (1982/83), 292-317.  doi: 10.1017/S0334270000002939. [3] S. Busenberg and W. Z. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.  doi: 10.1006/jdeq.1996.0003. [4] S. S. Chen and J. J. Wei, Stability and bifurcation in a diffusive logistic population model with multiple delays, Internat. J. Bifur. Chaos, 25 (2015), 1550107, 9 pp. doi: 10.1142/S0218127415501072. [5] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7. [6] A. Gomez and S. Trofimchuk, Monotone traveling wavefronts of the KPP-Fisher delayed equation, J. Differential Equations, 250 (2011), 1767-1787.  doi: 10.1016/j.jde.2010.11.011. [7] K. Gopalsamy, The Delay Logistic Equation, in Stability and Oscillations in Delay Differential Equations of Population Dynamics, Mathematics and Its Applications, vol. 74, Springer, Dordrecht, 1992. doi: 10.1007/978-94-015-7920-9. [8] G. E. Hutchison, Circular causal systems in ecology, Ann. N. Y. Acad. Sci., 50 (1948), 221-246. [9] R. L. Kitching, Time resources and population dynamics in insects, Austral. J. Ecol., 2 (1977), 31-42.  doi: 10.1111/j.1442-9993.1977.tb01125.x. [10] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering, 191. Academic Press, Inc., Boston, MA, 1993. [11] J. LaSalle and S. Lefschetz, Stability by Lyapunov's Direct Method, with Applications, Mathematics in Science and Engineering, Vol. 4 Academic Press, New York-London 1961 [12] G. Lin, Spreading speed of the delayed Fisher equation without quasimonotonicity, Nonlinear Anal. Real World Appl., 12 (2011), 3713-3718.  doi: 10.1016/j.nonrwa.2011.07.004. [13] M. C. Memory, Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion, SIAM J. Math. Anal., 20 (1989), 533-546.  doi: 10.1137/0520037. [14] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [15] S. Ruan, Delay differential equations in single species dynamics, Delay Differential Equations and Applications, NATO Sci. Ser. Ⅱ Math. Phys. Chem., Springer, Dordrecht, 205 (2006), 477-517.  doi: 10.1007/1-4020-3647-7_11. [16] Y. Su, J. J. Wei and J. P. Shi, Hopf bifurcation in a diffusive logistic equation with mixed delayed and instantaneous density dependence, J. Dynam. Differential Equations, 24 (2012), 897-925.  doi: 10.1007/s10884-012-9268-z. [17] P. F. Verhulst, Notice sur la loi que la population pursuit dans son accroissement, Corresp. Math. Phys., 10 (1838), 113-121. [18] J. H. 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. [19] J. H. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2. [20] J. H. Wu and X. F. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dynam. Differential Equations, 13 (2001), 651-687.  doi: 10.1023/A:1016690424892. [21] X. P. Yan and J. P. Shi, Stability switches in a logistic population model with mixed instantaneous and delayed density dependence, J. Dynam. Differential Equations, 29 (2017), 113-130.  doi: 10.1007/s10884-015-9432-3. [22] K. Yoshida, The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology, Hiroshima Math. J., 12 (1982), 321-348.  doi: 10.32917/hmj/1206133754. [23] X. F. 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.
The phase portrait of (9)
The root distribution of (14)
Left: $u_*$ is stable at $\tau = 5$. Right: $u_*$ is unstable at $\tau = 12$
Left: $u_*$ is stable at $\tau = 18$. Right: $u_*$ is unstable at $\tau = 30$
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