November  2020, 40(11): 6547-6573. doi: 10.3934/dcds.2020290

Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments

1. 

School of Mathematical Sciences, East China Normal University, Shanghai 200241, China

2. 

School of Mathematical Sciences and Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

3. 

Chinese University of Hong Kong – Shenzhen, Shenzhen, China

4. 

School of Mathematics, University of Minnesota, MN 55455, USA

* Corresponding author: Xiaoqing He

Received  May 2020 Revised  May 2020 Published  July 2020

Previously in [14], we considered a diffusive logistic equation with two parameters, $ r(x) $ – intrinsic growth rate and $ K(x) $ – carrying capacity. We investigated and compared two special cases of the way in which $ r(x) $ and $ K(x) $ are related for both the logistic equations and the corresponding Lotka-Volterra competition-diffusion systems. In this paper, we continue to study the Lotka-Volterra competition-diffusion system with general intrinsic growth rates and carrying capacities for two competing species in heterogeneous environments. We establish the main result that determines the global dynamics of the system under a general criterion. Furthermore, when the ratios of the intrinsic growth rate to the carrying capacity for each species are proportional — such ratios can also be interpreted as the competition coefficients — this criterion reduces to what we obtained in [18]. We also study the detailed dynamics in terms of dispersal rates for such general case. On the other hand, when the two ratios are not proportional, our results in [14] show that the criterion in [18] cannot be fully recovered as counterexamples exist. This indicates the importance and subtleties of the roles of heterogeneous competition coefficients in the dynamics of the Lotka-Volterra competition-diffusion systems. Our results apply to competition-diffusion-advection systems as well. (See Corollary 5.1 in the last section.)

Citation: Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete and Continuous Dynamical Systems, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290
References:
[1]

I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017) doi: 10.1090/memo/1161.

[2]

I. AverillD. Munther and Y. Lou, On several conjectures from evolution of dispersal, J. Biol. Dyn., 6 (2012), 117-130.  doi: 10.1080/17513758.2010.529169.

[3]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35pp. doi: 10.1007/s00526-018-1419-6.

[4]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[5]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318.  doi: 10.1017/S030821050001876X.

[6]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.

[7]

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.

[8]

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.

[9]

R. S. CantrellC. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36.  doi: 10.3934/mbe.2010.7.17.

[10]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.

[11]

D. DeAngelisW.-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.

[12]

D. DeAngelisW.-M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.

[13]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

[14]

Q. Guo, X. He and W.-M. Ni, On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments, J. Math. Biol., (2020). doi: 10.1007/s00285-020-01507-9.

[15]

X. HeK.-Y. LamY. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model, J. Math. Biol., 78 (2019), 1605-1636.  doi: 10.1007/s00285-018-1321-z.

[16]

X. 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.

[17]

X. 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.

[18]

X. 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.

[19]

X. 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), 20pp. doi: 10.1007/s00526-016-0964-0.

[20]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations, 56 (2017), 26pp. doi: 10.1007/s00526-017-1234-5.

[21]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; copublished in the US with John Wiley & Sons, Inc., New York), 1991.

[22]

P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions, in Nonlinear Analysis and Optimization, Lecture Notes in Math., 1107, Springer, Berlin, 1984,106–114. doi: 10.1007/BFb0101496.

[23]

M. W. Hirsch and H. L. Smith, Asymptotically stable equilibria for monotone semiflows, Discrete Contin. Dyn. Syst., 14 (2006), 385-398.  doi: 10.3934/dcds.2006.14.385.

[24]

V. HutsonY. Lou and and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.

[25]

M. G. Kre${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Matem. Nauk (N. S.), 3 (1948), 3-95. 

[26]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067.  doi: 10.3934/dcds.2010.28.1051.

[27]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.

[28]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.

[29]

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.

[30]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5.

[31]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595.

[32]

Y. LouD. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), 953-969.  doi: 10.3934/dcds.2016.36.953.

[33]

Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl. (9), 121 (2019) 47–82. doi: 10.1016/j.matpur.2018.06.010.

[34]

J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution,and speciation, Evolutionary Ecology Research, 14 (2012), 627-665. 

[35]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[36]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.  doi: 10.1512/iumj.1972.21.21079.

[37]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981/82), 459-470.  doi: 10.1007/BF01453979.

[38]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[39]

D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003.

[40]

B. Zhang, D. DeAngelis, W.-M. Ni, Y. Wang, L. Zhai, A. Kula, S. Xu and J. D. Van Dyken, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, Amer. Naturalist, (2020). doi: 10.1086/709293.

[41]

B. ZhangA. KulaK. MackL. ZhaiA. RyceW.-M. NiD. DeAngelis and J. D. Van Dyken, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Lett., 20 (2017), 1118-1128.  doi: 10.1111/ele.12807.

[42]

B. ZhangX. LiuD. L. DeAngelisW.-M. Ni and G. Wang, Effects of dispersal on total biomass in a patchy heterogeneous environment: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.

[43]

P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.

show all references

References:
[1]

I. Averill, K.-Y. Lam and Y. Lou, The role of advection in a two-species competition model: A bifurcation approach, Mem. Amer. Math. Soc., 245 (2017) doi: 10.1090/memo/1161.

[2]

I. AverillD. Munther and Y. Lou, On several conjectures from evolution of dispersal, J. Biol. Dyn., 6 (2012), 117-130.  doi: 10.1080/17513758.2010.529169.

[3]

X. Bai and F. Li, Classification of global dynamics of competition models with nonlocal dispersals I: Symmetric kernels, Calc. Var. Partial Differential Equations, 57 (2018), 35pp. doi: 10.1007/s00526-018-1419-6.

[4]

K. J. Brown and S. S. Lin, On the existence of positive eigenfunctions for an eigenvalue problem with indefinite weight function, J. Math. Anal. Appl., 75 (1980), 112-120.  doi: 10.1016/0022-247X(80)90309-1.

[5]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A, 112 (1989), 293-318.  doi: 10.1017/S030821050001876X.

[6]

R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: Population models in disrupted environments. II, SIAM J. Math. Anal., 22 (1991), 1043-1064.  doi: 10.1137/0522068.

[7]

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.

[8]

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.

[9]

R. S. CantrellC. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Math. Biosci. Eng., 7 (2010), 17-36.  doi: 10.3934/mbe.2010.7.17.

[10]

X. ChenK.-Y. Lam and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete Contin. Dyn. Syst. A, 32 (2012), 3841-3859.  doi: 10.3934/dcds.2012.32.3841.

[11]

D. DeAngelisW.-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.

[12]

D. DeAngelisW.-M. Ni and B. Zhang, Effects of diffusion on total biomass in heterogeneous continuous and discrete-patch systems, Theoretical Ecology, 9 (2016), 443-453.  doi: 10.1007/s12080-016-0302-3.

[13]

J. DockeryV. HutsonK. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83.  doi: 10.1007/s002850050120.

[14]

Q. Guo, X. He and W.-M. Ni, On the effects of carrying capacity and intrinsic growth rate on single and multiple species in spatially heterogeneous environments, J. Math. Biol., (2020). doi: 10.1007/s00285-020-01507-9.

[15]

X. HeK.-Y. LamY. Lou and W.-M. Ni, Dynamics of a consumer-resource reaction-diffusion model, J. Math. Biol., 78 (2019), 1605-1636.  doi: 10.1007/s00285-018-1321-z.

[16]

X. 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.

[17]

X. 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.

[18]

X. 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.

[19]

X. 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), 20pp. doi: 10.1007/s00526-016-0964-0.

[20]

X. He and W.-M. Ni, Global dynamics of the Lotka-Volterra competition-diffusion system with equal amount of total resources, III, Calc. Var. Partial Differential Equations, 56 (2017), 26pp. doi: 10.1007/s00526-017-1234-5.

[21]

P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics Series, 247. Longman Scientific & Technical, Harlow; copublished in the US with John Wiley & Sons, Inc., New York), 1991.

[22]

P. Hess and S. Senn, Another approach to elliptic eigenvalue problems with respect to indefinite weight functions, in Nonlinear Analysis and Optimization, Lecture Notes in Math., 1107, Springer, Berlin, 1984,106–114. doi: 10.1007/BFb0101496.

[23]

M. W. Hirsch and H. L. Smith, Asymptotically stable equilibria for monotone semiflows, Discrete Contin. Dyn. Syst., 14 (2006), 385-398.  doi: 10.3934/dcds.2006.14.385.

[24]

V. HutsonY. Lou and and K. Mischaikow, Spatial heterogeneity of resources versus Lotka-Volterra dynamics, J. Differential Equations, 185 (2002), 97-136.  doi: 10.1006/jdeq.2001.4157.

[25]

M. G. Kre${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$n and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Uspekhi Matem. Nauk (N. S.), 3 (1948), 3-95. 

[26]

K.-Y. Lam and W.-M. Ni, Limiting profiles of semilinear elliptic equations with large advection in population dynamics, Discrete Contin. Dyn. Syst., 28 (2010), 1051-1067.  doi: 10.3934/dcds.2010.28.1051.

[27]

K.-Y. Lam and W.-M. Ni, Uniqueness and complete dynamics in the heterogeneous competition-diffusion systems, SIAM J. Appl. Math., 72 (2012), 1695-1712.  doi: 10.1137/120869481.

[28]

K.-Y. LamY. Lou and F. Lutscher, Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), 188-212.  doi: 10.1080/17513758.2014.969336.

[29]

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.

[30]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences, Lecture Notes in Math., 1922, Springer, Berlin, 2008,171–205. doi: 10.1007/978-3-540-74331-6_5.

[31]

Y. Lou and E. Yanagida, Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics, Japan J. Indust. Appl. Math., 23 (2006), 275-292.  doi: 10.1007/BF03167595.

[32]

Y. LouD. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), 953-969.  doi: 10.3934/dcds.2016.36.953.

[33]

Y. Lou, X.-Q. Zhao and P. Zhou, Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl. (9), 121 (2019) 47–82. doi: 10.1016/j.matpur.2018.06.010.

[34]

J. Mallet, The struggle for existence: How the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution,and speciation, Evolutionary Ecology Research, 14 (2012), 627-665. 

[35]

W.-M. Ni, The Mathematics of Diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972.

[36]

D. H. Sattinger, Monotone methods in nonlinear elliptic and parabolic boundary value problems, Indiana Univ. Math. J., 21 (1971/72), 979-1000.  doi: 10.1512/iumj.1972.21.21079.

[37]

S. Senn and P. Hess, On positive solutions of a linear elliptic eigenvalue problem with Neumann boundary conditions, Math. Ann., 258 (1981/82), 459-470.  doi: 10.1007/BF01453979.

[38]

H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41, American Mathematical Society, Providence, RI, 1995.

[39]

D. Tang and P. Zhou, On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), 1570-1599.  doi: 10.1016/j.jde.2019.09.003.

[40]

B. Zhang, D. DeAngelis, W.-M. Ni, Y. Wang, L. Zhai, A. Kula, S. Xu and J. D. Van Dyken, Effect of stressors on the carrying capacity of spatially-distributed metapopulations, Amer. Naturalist, (2020). doi: 10.1086/709293.

[41]

B. ZhangA. KulaK. MackL. ZhaiA. RyceW.-M. NiD. DeAngelis and J. D. Van Dyken, Carrying capacity in a heterogeneous environment with habitat connectivity, Ecology Lett., 20 (2017), 1118-1128.  doi: 10.1111/ele.12807.

[42]

B. ZhangX. LiuD. L. DeAngelisW.-M. Ni and G. Wang, Effects of dispersal on total biomass in a patchy heterogeneous environment: Analysis and experiment, Math. Biosci., 264 (2015), 54-62.  doi: 10.1016/j.mbs.2015.03.005.

[43]

P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.

Figure 1.  Global dynamics of system (4) with $ r_i/\xi_i\not\equiv{\rm{const}} $, $ i = 1, 2 $. See Theorem 1.3
Figure 2.  Shapes of $ \Sigma_U $ and $ \Sigma_- $ for Theorem 1.4 as illustration
Figure 3.  Another scenario for Theorem 1.4. See also Theorem 4.2(ⅰ) for details
Figure 4.  Shapes of $ \Sigma_U $, $ \Sigma_V $ and $ \Sigma_- $ for Theorem 1.5 as illustration
[1]

Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237

[2]

Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4739-4753. doi: 10.3934/dcdsb.2019028

[3]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124

[4]

Yanan Wang, Tao Xie, Xiaowen Jie. A mathematical analysis for the forecast research on tourism carrying capacity to promote the effective and sustainable development of tourism. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 837-847. doi: 10.3934/dcdss.2019056

[5]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059

[6]

Fengqi Yi, Hua Zhang, Alhaji Cherif, Wenying Zhang. Spatiotemporal patterns of a homogeneous diffusive system modeling hair growth: Global asymptotic behavior and multiple bifurcation analysis. Communications on Pure and Applied Analysis, 2014, 13 (1) : 347-369. doi: 10.3934/cpaa.2014.13.347

[7]

J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054

[8]

W. E. Fitzgibbon, M.E. Parrott, Glenn Webb. Diffusive epidemic models with spatial and age dependent heterogeneity. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 35-57. doi: 10.3934/dcds.1995.1.35

[9]

Yu-Xia Wang, Wan-Tong Li. Combined effects of the spatial heterogeneity and the functional response. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 19-39. doi: 10.3934/dcds.2019002

[10]

Yuan-Hang Su, Wan-Tong Li, Fei-Ying Yang. Effects of nonlocal dispersal and spatial heterogeneity on total biomass. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4929-4936. doi: 10.3934/dcdsb.2019038

[11]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic and Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

[12]

Fang Liu, Zhen Jin, Cai-Yun Wang. Global analysis of SIRI knowledge dissemination model with recalling rate. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3099-3114. doi: 10.3934/dcdss.2020116

[13]

Mohammad A. Safi, Abba B. Gumel. Global asymptotic dynamics of a model for quarantine and isolation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 209-231. doi: 10.3934/dcdsb.2010.14.209

[14]

Kentarou Fujie. Global asymptotic stability in a chemotaxis-growth model for tumor invasion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 203-209. doi: 10.3934/dcdss.2020011

[15]

Roger M. Nisbet, Kurt E. Anderson, Edward McCauley, Mark A. Lewis. Response of equilibrium states to spatial environmental heterogeneity in advective systems. Mathematical Biosciences & Engineering, 2007, 4 (1) : 1-13. doi: 10.3934/mbe.2007.4.1

[16]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182

[17]

Seung-Yeal Ha, Se Eun Noh, Jinyeong Park. Practical synchronization of generalized Kuramoto systems with an intrinsic dynamics. Networks and Heterogeneous Media, 2015, 10 (4) : 787-807. doi: 10.3934/nhm.2015.10.787

[18]

Lining Ru, Xiaoping Xue. Flocking of Cucker-Smale model with intrinsic dynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6817-6835. doi: 10.3934/dcdsb.2019168

[19]

Steady Mushayabasa, Drew Posny, Jin Wang. Modeling the intrinsic dynamics of foot-and-mouth disease. Mathematical Biosciences & Engineering, 2016, 13 (2) : 425-442. doi: 10.3934/mbe.2015010

[20]

Qin Pan, Jicai Huang, Qihua Huang. Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. Discrete and Continuous Dynamical Systems - B, 2022, 27 (7) : 3533-3561. doi: 10.3934/dcdsb.2021195

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (550)
  • HTML views (172)
  • Cited by (1)

Other articles
by authors

[Back to Top]