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Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system
February  2019, 24(2): 467-486. doi: 10.3934/dcdsb.2018182

## Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

 1 School of Mathematical Sciences, Tongji University, Shanghai 200092, China 2 Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, USA 3 Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China

* Corresponding author: Junping Shi

Received  November 2017 Revised  January 2018 Published  June 2018

Fund Project: Partially supported by a grant from China Scholarship Council, US-NSF grant DMS-1715651, National Natural Science Foundation of China (No.11571257), Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

In this paper, we study the Hopf bifurcation and spatiotemporal pattern formation of a delayed diffusive logistic model under Neumann boundary condition with spatial heterogeneity. It is shown that for large diffusion coefficient, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady state at a critical time delay value, and the dependence of corresponding spatiotemporal patterns on the heterogeneous resource function is demonstrated via numerical simulations. Moreover, it is proved that the heterogeneous resource supply contributes to the increase of the temporal average of total biomass of the population even though the total biomass oscillates periodically in time.

Citation: Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
##### References:
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Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101. Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024. Google Scholar [8] S. S. Chen, Y. Lou and J. J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359, arXiv: 1706.02087. doi: 10.1016/j.jde.2018.01.008. Google Scholar [9] S. S. Chen and J. P. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031. Google Scholar [10] D. L. DeAngelis, W. M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254. doi: 10.1007/s00285-015-0879-y. Google Scholar [11] 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. Google Scholar [12] T. Faria and W. Z. Huang, Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay, In Differential Equations and Dynamical Systems (Lisbon, 2000), volume 31 of Fields Inst. Commun., pages 125-141. Amer. Math. Soc., Providence, RI, 2002. Google Scholar [13] S. D. Fretwell and J. S. Calver, On territorial behavior and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969), 37-44. Google Scholar [14] G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993), 89-103. doi: 10.1007/BF01063736. Google Scholar [15] S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78. doi: 10.1007/s002850100109. Google Scholar [16] S. J. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448. doi: 10.1016/j.jde.2015.03.006. Google Scholar [17] S. J. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Sci., 26 (2016), 545-580. doi: 10.1007/s00332-016-9285-x. Google Scholar [18] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1. Google Scholar [19] X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032. Google Scholar [20] X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: The general case, J. Differential Equations, 254 (2013), 4088-4108. doi: 10.1016/j.jde.2013.02.009. Google Scholar [21] X. Q. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar [22] X. Q. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calc. Var. Partial Differential Equations, 55 (2016), Art. 25, 20pp. doi: 10.1007/s00526-016-0964-0. Google Scholar [23] G. E. Hutchinson, Circular causal systems in ecology, Annals of the New York Academy of Sciences, 50 (1948), 221-246. Google Scholar [24] V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136. doi: 10.1006/jdeq.2001.4157. Google Scholar [25] K. Y. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481. Google Scholar [26] K.-L. Liao and Y. Lou, The effect of time delay in a two-patch model with random dispersal, Bull. Math. Biol., 76 (2014), 335-376. doi: 10.1007/s11538-013-9921-7. Google Scholar [27] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010. Google Scholar [28] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342. doi: 10.1007/s00285-013-0730-2. Google Scholar [29] 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. Google Scholar [30] M. Mimura, D. Terman and T. Tsujikawa, Nonlocal advection effect on bistable reactiondiffusion equations, In Patterns and Waves, volume 18 of Stud. Math. Appl., pages 507-542. North-Holland, Amsterdam, 1986. doi: 10.1016/S0168-2024(08)70144-9. Google Scholar [31] J. D. Murray, Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, third edition, 2003. Spatial models and biomedical applications. Google Scholar [32] S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theoret. Population Biol., 21 (1982), 92-113. doi: 10.1016/0040-5809(82)90008-9. Google Scholar [33] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. Google Scholar [34] Q. Y. Shi, J. P. Shi and Y. L. Song, Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition, J. Differential Equations, 263 (2017), 6537-6575. doi: 10.1016/j.jde.2017.07.024. Google Scholar [35] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. Google Scholar [36] Y. Su, J. J. Wei and J. P. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. doi: 10.1016/j.jde.2009.04.017. Google Scholar [37] 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. Google Scholar [38] O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), 439-469. Google Scholar [39] J. H. Wu, Theory and Applications of Partial Functional-Differential Equations, volume 119 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar [40] X. P. Yan and W. T. Li, Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model, Nonlinearity, 23 (2010), 1413-1431. doi: 10.1088/0951-7715/23/6/008. Google Scholar [41] 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. Google Scholar [42] B. Zhang, X. Liu, D. L. DeAngelis, W. M. Ni and G. G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54-62. doi: 10.1016/j.mbs.2015.03.005. Google Scholar

show all references

##### References:
 [1] N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989), 57-66. doi: 10.1016/S0022-5193(89)80189-4. Google Scholar [2] 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. Google Scholar [3] R. S. Cantrell and C. Cosner, The effects of spatial heterogeneity in population dynamics, J. Math. Biol., 29 (1991), 315-338. doi: 10.1007/BF00167155. Google Scholar [4] R. S. Cantrell and C. Cosner, On the effects of spatial heterogeneity on the persistence of interacting species, J. Math. Biol., 37 (1998), 103-145. doi: 10.1007/s002850050122. Google Scholar [5] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296. Google Scholar [6] R. S. Cantrell, C. Cosner and V. Hutson, Ecological models, permanence and spatial heterogeneity, Rocky Mountain J. Math., 26 (1996), 1-35. doi: 10.1216/rmjm/1181072101. Google Scholar [7] R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations, J. Differential Equations, 245 (2008), 3687-3703. doi: 10.1016/j.jde.2008.07.024. Google Scholar [8] S. S. Chen, Y. Lou and J. J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359, arXiv: 1706.02087. doi: 10.1016/j.jde.2018.01.008. Google Scholar [9] S. S. Chen and J. P. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470. doi: 10.1016/j.jde.2012.08.031. Google Scholar [10] D. L. DeAngelis, W. M. Ni and B. Zhang, Dispersal and spatial heterogeneity: Single species, J. Math. Biol., 72 (2016), 239-254. doi: 10.1007/s00285-015-0879-y. Google Scholar [11] 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. Google Scholar [12] T. Faria and W. Z. Huang, Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay, In Differential Equations and Dynamical Systems (Lisbon, 2000), volume 31 of Fields Inst. Commun., pages 125-141. Amer. Math. Soc., Providence, RI, 2002. Google Scholar [13] S. D. Fretwell and J. S. Calver, On territorial behavior and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969), 37-44. Google Scholar [14] G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993), 89-103. doi: 10.1007/BF01063736. Google Scholar [15] S. A. Gourley and J. W.-H. So, Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. Math. Biol., 44 (2002), 49-78. doi: 10.1007/s002850100109. Google Scholar [16] S. J. Guo, Stability and bifurcation in a reaction-diffusion model with nonlocal delay effect, J. Differential Equations, 259 (2015), 1409-1448. doi: 10.1016/j.jde.2015.03.006. Google Scholar [17] S. J. Guo and L. Ma, Stability and bifurcation in a delayed reaction-diffusion equation with Dirichlet boundary condition, J. Nonlinear Sci., 26 (2016), 545-580. doi: 10.1007/s00332-016-9285-x. Google Scholar [18] M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1. Google Scholar [19] X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅰ: Heterogeneity vs. homogeneity, J. Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032. Google Scholar [20] X. Q. He and W. M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system Ⅱ: The general case, J. Differential Equations, 254 (2013), 4088-4108. doi: 10.1016/j.jde.2013.02.009. Google Scholar [21] X. Q. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system: diffusion and spatial heterogeneity Ⅰ, Comm. Pure Appl. Math., 69 (2016), 981-1014. doi: 10.1002/cpa.21596. Google Scholar [22] X. Q. He and W. M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, Ⅱ, Calc. Var. Partial Differential Equations, 55 (2016), Art. 25, 20pp. doi: 10.1007/s00526-016-0964-0. Google Scholar [23] G. E. Hutchinson, Circular causal systems in ecology, Annals of the New York Academy of Sciences, 50 (1948), 221-246. Google Scholar [24] V. Hutson, Y. Lou and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136. doi: 10.1006/jdeq.2001.4157. Google Scholar [25] K. Y. Lam and W. M. Ni, Uniqueness and complete dynamics in heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712. doi: 10.1137/120869481. Google Scholar [26] K.-L. Liao and Y. Lou, The effect of time delay in a two-patch model with random dispersal, Bull. Math. Biol., 76 (2014), 335-376. doi: 10.1007/s11538-013-9921-7. Google Scholar [27] Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, J. Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010. Google Scholar [28] Y. Lou and F. Lutscher, Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), 1319-1342. doi: 10.1007/s00285-013-0730-2. Google Scholar [29] 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. Google Scholar [30] M. Mimura, D. Terman and T. Tsujikawa, Nonlocal advection effect on bistable reactiondiffusion equations, In Patterns and Waves, volume 18 of Stud. Math. Appl., pages 507-542. North-Holland, Amsterdam, 1986. doi: 10.1016/S0168-2024(08)70144-9. Google Scholar [31] J. D. Murray, Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, third edition, 2003. Spatial models and biomedical applications. Google Scholar [32] S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theoret. Population Biol., 21 (1982), 92-113. doi: 10.1016/0040-5809(82)90008-9. Google Scholar [33] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967. Google Scholar [34] Q. Y. Shi, J. P. Shi and Y. L. Song, Hopf bifurcation in a reaction-diffusion equation with distributed delay and Dirichlet boundary condition, J. Differential Equations, 263 (2017), 6537-6575. doi: 10.1016/j.jde.2017.07.024. Google Scholar [35] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3. Google Scholar [36] Y. Su, J. J. Wei and J. P. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184. doi: 10.1016/j.jde.2009.04.017. Google Scholar [37] 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. Google Scholar [38] O. Vasilyeva and F. Lutscher, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), 439-469. Google Scholar [39] J. H. Wu, Theory and Applications of Partial Functional-Differential Equations, volume 119 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar [40] X. P. Yan and W. T. Li, Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model, Nonlinearity, 23 (2010), 1413-1431. doi: 10.1088/0951-7715/23/6/008. Google Scholar [41] 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. Google Scholar [42] B. Zhang, X. Liu, D. L. DeAngelis, W. M. Ni and G. G. Wang, Effects of dispersal on total biomass in a patchy, heterogeneous system: Analysis and experiment, Math. Biosci., 264 (2015), 54-62. doi: 10.1016/j.mbs.2015.03.005. Google Scholar
The non-homogeneous steady states of Eq (2) when $m(x)$ is a cosine function: (a) $m(x) = \cos(x)+2$; (b) $m(x) = \cos(1.5x)+2$; (c) $m(x) = \cos(2x)+2$. Here $d = 2$ (which is equivalent to $\lambda = 0.5$), $\tau = 0.71<\tau_{0\lambda }\approx0.785$ and initial value $u_{0} = 2$ for all three cases, and the solution converges to the non-homogeneous steady state
The non-homogeneous steady states of Eq. (2) when $m(x)$ is a sine function: (a) $m(x) = \sin(x)+2$; (b) $m(x) = \sin(1.5x)+1.788$; (c) $m(x) = \sin(2x)+2$. The parameters are the same as in Figure 1, and here $\tau = 0.73<\tau_{0\lambda }\approx0.785$. The solution converges to the non-homogeneous steady state for each case
The non-homogeneous steady states of Eq. (2) when $m(x)$ is a monotone linear function: (a) $m(x) = 1+x/\pi$; (b) $m(x) = 3-x/\pi$. Here $d = 2$ and $\tau = 0.73<\tau_{0\lambda }$. The solution converges to the positive monotone steady state
The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that $m(x)$ is cosine function: (a) $m(x) = \cos(x)+2$; (b) $m(x) = \cos(1.5x)+2$; (c) $m(x) = \cos(2x)+2$. Here $d = 2$, and $\tau = 0.82>\tau_{0\lambda }\approx 0.785$
The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that $m(x)$ is sine function: (a) $m(x) = \sin(x)+2$; (b) $m(x) = \sin(1.5x)+1.788$; (c) $m(x) = \sin(2x)+2$. Here $d = 2$, and $\tau = 0.82>\tau_{0\lambda }\approx 0.785$
The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that $m(x)$ is monotone linear function: (a) $m(x) = 1+x/\pi$; (b) $m(x) = 3-x/\pi$. Here $d = 2$, and $\tau = 0.82>\tau_{0\lambda }\approx 0.785$
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