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: |
Figure 1. 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
Figure 2. 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
N. F. Britton
, Aggregation and the competitive exclusion principle, J. Theoret. Biol., 136 (1989)
, 57-66.
doi: 10.1016/S0022-5193(89)80189-4.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
S. D. Fretwell
and J. S. Calver
, On territorial behavior and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969)
, 37-44.
![]() |
|
G. Friesecke
, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993)
, 89-103.
doi: 10.1007/BF01063736.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
G. E. Hutchinson
, Circular causal systems in ecology, Annals of the New York Academy of Sciences, 50 (1948)
, 221-246.
![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
J. D. Murray, Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, third edition, 2003. Spatial models and biomedical applications.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1967.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
O. Vasilyeva
and F. Lutscher
, Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010)
, 439-469.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
The non-homogeneous steady states of Eq (2) when
The non-homogeneous steady states of Eq. (2) when
The non-homogeneous steady states of Eq. (2) when
The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that
The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that
The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that