
-
Previous Article
Dynamics of phytoplankton species competition for light and nutrient with recycling in a water column
- DCDS-B Home
- This Issue
-
Next Article
On the semilinear fractional elliptic equations with singular weight functions
A spatially heterogeneous predator-prey model
Instituto de Matemática Interdisciplinar (IMI), Departamento de Análisis Matemático y Matemática Aplicada, Universidad Complutense de Madrid, Madrid, 28040, Spain |
This paper introduces a spatially heterogeneous diffusive predator–prey model unifying the classical Lotka–Volterra and Holling–Tanner ones through a prey saturation coefficient, $ m(x) $, which is spatially heterogenous and it is allowed to 'degenerate'. Thus, in some patches of the territory the species can interact according to a Lotka–Volterra kinetics, while in others the prey saturation effects play a significant role on the dynamics of the species. As we are working under general mixed boundary conditions of non-classical type, we must invoke to some very recent technical devices to get some of the main results of this paper.
References:
[1] |
D. Aleja, I. Antón and J. López-Gómez, Global structure of the periodic-positive solutions for a general class of periodic-parabolic logistic equations with indefinite weights, Submitted. Google Scholar |
[2] |
H. Amann and J. López-Gómez,
A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Maths., 47 (1994), 72-92.
doi: 10.1002/cpa.3160470105. |
[4] |
S. Cano-Casanova and J. López-Gómez,
Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Eqns., 178 (2002), 123-211.
doi: 10.1006/jdeq.2000.4003. |
[5] |
A. Casal, C. Eilbeck and J. López-Gómez,
Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439.
|
[6] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[7] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[8] |
E. N. Dancer, J. López-Gómez and R. Ortega,
On the spectrum of some linear noncooperative elliptic systems with radial symmetry, Diff. Int. Eqns., 8 (1995), 515-523.
|
[9] |
D. Daners and J. López-Gómez,
Global dynamics of generalized logistic equations, Adv. Nonl. Studies, 18 (2018), 217-236.
|
[10] |
Y. Du and S. B. Hsu,
A diffusive predator-prey model in heterogeneous environment, J. Diff. Eqns., 203 (2004), 331-364.
doi: 10.1016/j.jde.2004.05.010. |
[11] |
Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. math. Soc., 349 (1997), 2443-2475. Google Scholar |
[12] |
Y. Du and Y. Lou,
$S$-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Diff. Eqns., 144 (1998), 390-440.
doi: 10.1006/jdeq.1997.3394. |
[13] |
Y. Du and J. P. Shi,
A diffusive predator-prey model with a protection zone, J. Diff. Eqns., 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013. |
[14] |
Y. Du and J. P. Shi,
Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.
doi: 10.1090/S0002-9947-07-04262-6. |
[15] |
X. Feng, Y. Song and X. An,
Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod–Haldane functional response, Open Math., 16 (2018), 623-635.
|
[16] |
S. Fernández-Rincón and J. López-Gómez,
The singular perturbation problem for a class of generalized logistic equations under non-classical mixed boundary conditions, Adv. Nonl. Studies, 19 (2019), 1-27.
|
[17] |
J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino,
Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1995), 295-319.
doi: 10.1006/jdeq.1996.0071. |
[18] |
H. Freedman, Deterministic Mathematical Models in Population Biology, Marcel and Dekker, New York, 1980. |
[19] |
G. Guo and J. Wu,
Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonl. Anal., 72 (2010), 1632-1646.
doi: 10.1016/j.na.2009.09.003. |
[20] |
G. Guo and J. Wu,
The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194.
doi: 10.1016/j.jmaa.2011.11.044. |
[21] |
S. B. Hsu,
On global stability of a predator-prey system, Math. Biosc., 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
[22] |
S. B. Hsu and T. W. Hwang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
[23] |
Y. Jia, J. Wu and H. K. Xu,
Spatial pattern in an ecosystem of phytoplankton-nutrient from remote sensing, J. Math. Anal. Appl., 402 (2013), 23-34.
doi: 10.1016/j.jmaa.2012.12.071. |
[24] |
H. Jiang and L. Wang,
Analysis of steady-state for variable territory model with limited self-limitation, Acta Appl. Math., 148 (2017), 103-120.
doi: 10.1007/s10440-016-0080-3. |
[25] |
W. Ko and K. Ryu,
Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Diff. Eqns., 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[26] |
W. Ko and K. Ryu,
Coexistence states of a predator-prey system with non-monotonic functional response, Nonl. Anal. RWA, 8 (2007), 769-786.
doi: 10.1016/j.nonrwa.2006.03.003. |
[27] |
W. Ko and K. Ryu,
A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appns., 344 (2008), 217-230.
doi: 10.1016/j.jmaa.2008.03.006. |
[28] |
S. Li, J. Wu and Y. Dong,
Effects of degeneracy and response in a diffusion predator-prey model, Nonlinearity, 31 (2018), 1461-1483.
doi: 10.1088/1361-6544/aaa2de. |
[29] |
S. Li, J. Wu and Y. Dong,
Effects of degeneracy in a diffusion predator-prey model with Holling type-Ⅱ functional response, Nonl. Anal. RWA, 43 (2018), 78-95.
doi: 10.1016/j.nonrwa.2018.02.003. |
[30] |
J. López-Gómez,
Positive periodic solutions of Lotka–Volterra reaction-diffusion systems, Diff. Int. Eqns., 5 (1992), 55-72.
|
[31] |
J. López-Gómez,
Nonlinear eigenvalues and global bifurcation. Application to the search of positive solutions for general Lotka-Volterra Reaction-Diffusion systems with two species, Diff. Int. Eqns., 7 (1994), 1427-1452.
|
[32] |
J. López-Gómez,
A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.
|
[33] |
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC Press, Boca Raton, FL, 2001.
doi: 10.1201/9781420035506.![]() ![]() |
[34] |
J. López-Gómez,
Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns., 8 (2003), 1025-1042.
|
[35] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific, Singapore, 2013.
doi: 10.1142/8664. |
[36] |
J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2016.
![]() |
[37] |
J. López-Gómez and M. Molina-Meyer,
The maximum principle for cooperative weakly elliptic systems and some applications, Diff. Int. Eqns, 7 (1994), 383-398.
|
[38] |
J. López-Gómez and E. Muñoz-Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity, 33 (2020), 34-71. Google Scholar |
[39] |
J. López-Gómez and R. M. Pardo,
Coexistence regions in Lotka-Volterra systems with diffusion, Nonl. Anal., 19 (1992), 11-28.
doi: 10.1016/0362-546X(92)90027-C. |
[40] |
J. López-Gómez and R. M. Pardo,
The existence and the uniqueness for the predator-prey model with diffusion, Diff. Int. Eqns., 6 (1993), 1025-1031.
|
[41] |
J. López-Gómez and R. M. Pardo,
Invertibility of linear noncooperative elliptic systems, Nonl. Anal., 31 (1998), 687-699.
doi: 10.1016/S0362-546X(97)00640-8. |
[42] |
J. López-Gómez,
A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.
|
[43] | R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1974. Google Scholar |
[44] |
H. Nie and J. Wu,
Multiplicity and stability of a predator-prey model with non-monotonic conversion rate, Nonl. Anal. RWA, 10 (2009), 154-171.
doi: 10.1016/j.nonrwa.2007.08.020. |
[45] |
P. Y. H. Pang and M. Wang,
Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Royal Soc. Edinburgh, 133 (2003), 919-942.
doi: 10.1017/S0308210500002742. |
[46] |
P. Y. H. Pang and M. Wang,
Non-constant positive steady-states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.
doi: 10.1112/S0024611503014321. |
[47] |
R. Peng, M. X. Wang and W. Y. Chen,
Positive steady-states of a predator-prey model with diffusion and non-monotonic conversion rate, Acta Math. Sinica, 23 (2007), 749-760.
doi: 10.1007/s10114-005-0789-9. |
[48] |
K. Ryu and I. Ahn,
Positive solutions for ratio-dependent predator-prey interaction systems, J. Diff. Eqns., 218 (2005), 117-135.
doi: 10.1016/j.jde.2005.06.020. |
[49] |
M. X. Wang and Q. Wu,
Positive solutions fo a predator-prey model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.
doi: 10.1016/j.jmaa.2008.04.054. |
[50] |
H. Yuan, J. Wu, Y. Jia and H. Nie,
Coexistence states of a predator-prey model with cross-diffusion, Nonl. Anal. RWA, 41 (2018), 179-203.
doi: 10.1016/j.nonrwa.2017.10.009. |
[51] |
X. Zeng, W. Zeng and L. Liu,
Effects of the protection zone on coexistence of the species for a ratio-dependent predator-prey model, J. Math. Anal. Appl., 462 (2018), 1605-1626.
doi: 10.1016/j.jmaa.2018.02.060. |
[52] |
J. Zhou and C. Mu,
Coexistence states of a Holling type-Ⅱ predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.
doi: 10.1016/j.jmaa.2010.04.001. |
[53] |
J. Zhou and C. Mu,
Coexistence of a diffusive predator-prey model with Holling-type Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.
doi: 10.1016/j.jmaa.2011.07.027. |
[54] |
J. Zhou and J. P. Shi,
The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie–Gower predator-prey model with Holling type-Ⅱ functional responses, J. Math. Anal. Appl., 405 (2013), 618-630.
doi: 10.1016/j.jmaa.2013.03.064. |
show all references
References:
[1] |
D. Aleja, I. Antón and J. López-Gómez, Global structure of the periodic-positive solutions for a general class of periodic-parabolic logistic equations with indefinite weights, Submitted. Google Scholar |
[2] |
H. Amann and J. López-Gómez,
A priori bounds and multiple solutions for superlinear indefinite elliptic problems, J. Diff. Eqns., 146 (1998), 336-374.
doi: 10.1006/jdeq.1998.3440. |
[3] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan,
The principal eigenvalue and maximum principle for second order elliptic operators in general domains, Comm. Pure Appl. Maths., 47 (1994), 72-92.
doi: 10.1002/cpa.3160470105. |
[4] |
S. Cano-Casanova and J. López-Gómez,
Properties of the principal eigenvalues of a general class of non-classical mixed boundary value problems, J. Diff. Eqns., 178 (2002), 123-211.
doi: 10.1006/jdeq.2000.4003. |
[5] |
A. Casal, C. Eilbeck and J. López-Gómez,
Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns., 7 (1994), 411-439.
|
[6] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation from simple eigenvalues, J. Funct. Anal., 8 (1971), 321-340.
doi: 10.1016/0022-1236(71)90015-2. |
[7] |
M. G. Crandall and P. H. Rabinowitz,
Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rat. Mech. Anal., 52 (1973), 161-180.
doi: 10.1007/BF00282325. |
[8] |
E. N. Dancer, J. López-Gómez and R. Ortega,
On the spectrum of some linear noncooperative elliptic systems with radial symmetry, Diff. Int. Eqns., 8 (1995), 515-523.
|
[9] |
D. Daners and J. López-Gómez,
Global dynamics of generalized logistic equations, Adv. Nonl. Studies, 18 (2018), 217-236.
|
[10] |
Y. Du and S. B. Hsu,
A diffusive predator-prey model in heterogeneous environment, J. Diff. Eqns., 203 (2004), 331-364.
doi: 10.1016/j.jde.2004.05.010. |
[11] |
Y. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. math. Soc., 349 (1997), 2443-2475. Google Scholar |
[12] |
Y. Du and Y. Lou,
$S$-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Diff. Eqns., 144 (1998), 390-440.
doi: 10.1006/jdeq.1997.3394. |
[13] |
Y. Du and J. P. Shi,
A diffusive predator-prey model with a protection zone, J. Diff. Eqns., 229 (2006), 63-91.
doi: 10.1016/j.jde.2006.01.013. |
[14] |
Y. Du and J. P. Shi,
Allee effect and bistability in a spatially heterogeneous predator-prey model, Trans. Amer. Math. Soc., 359 (2007), 4557-4593.
doi: 10.1090/S0002-9947-07-04262-6. |
[15] |
X. Feng, Y. Song and X. An,
Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod–Haldane functional response, Open Math., 16 (2018), 623-635.
|
[16] |
S. Fernández-Rincón and J. López-Gómez,
The singular perturbation problem for a class of generalized logistic equations under non-classical mixed boundary conditions, Adv. Nonl. Studies, 19 (2019), 1-27.
|
[17] |
J. M. Fraile, P. Koch Medina, J. López-Gómez and S. Merino,
Elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation, J. Diff. Eqns., 127 (1995), 295-319.
doi: 10.1006/jdeq.1996.0071. |
[18] |
H. Freedman, Deterministic Mathematical Models in Population Biology, Marcel and Dekker, New York, 1980. |
[19] |
G. Guo and J. Wu,
Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonl. Anal., 72 (2010), 1632-1646.
doi: 10.1016/j.na.2009.09.003. |
[20] |
G. Guo and J. Wu,
The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl., 389 (2012), 179-194.
doi: 10.1016/j.jmaa.2011.11.044. |
[21] |
S. B. Hsu,
On global stability of a predator-prey system, Math. Biosc., 39 (1978), 1-10.
doi: 10.1016/0025-5564(78)90025-1. |
[22] |
S. B. Hsu and T. W. Hwang,
Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.
doi: 10.1137/S0036139993253201. |
[23] |
Y. Jia, J. Wu and H. K. Xu,
Spatial pattern in an ecosystem of phytoplankton-nutrient from remote sensing, J. Math. Anal. Appl., 402 (2013), 23-34.
doi: 10.1016/j.jmaa.2012.12.071. |
[24] |
H. Jiang and L. Wang,
Analysis of steady-state for variable territory model with limited self-limitation, Acta Appl. Math., 148 (2017), 103-120.
doi: 10.1007/s10440-016-0080-3. |
[25] |
W. Ko and K. Ryu,
Qualitative analysis of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge, J. Diff. Eqns., 231 (2006), 534-550.
doi: 10.1016/j.jde.2006.08.001. |
[26] |
W. Ko and K. Ryu,
Coexistence states of a predator-prey system with non-monotonic functional response, Nonl. Anal. RWA, 8 (2007), 769-786.
doi: 10.1016/j.nonrwa.2006.03.003. |
[27] |
W. Ko and K. Ryu,
A qualitative study on general Gause-type predator-prey models with constant diffusion rates, J. Math. Anal. Appns., 344 (2008), 217-230.
doi: 10.1016/j.jmaa.2008.03.006. |
[28] |
S. Li, J. Wu and Y. Dong,
Effects of degeneracy and response in a diffusion predator-prey model, Nonlinearity, 31 (2018), 1461-1483.
doi: 10.1088/1361-6544/aaa2de. |
[29] |
S. Li, J. Wu and Y. Dong,
Effects of degeneracy in a diffusion predator-prey model with Holling type-Ⅱ functional response, Nonl. Anal. RWA, 43 (2018), 78-95.
doi: 10.1016/j.nonrwa.2018.02.003. |
[30] |
J. López-Gómez,
Positive periodic solutions of Lotka–Volterra reaction-diffusion systems, Diff. Int. Eqns., 5 (1992), 55-72.
|
[31] |
J. López-Gómez,
Nonlinear eigenvalues and global bifurcation. Application to the search of positive solutions for general Lotka-Volterra Reaction-Diffusion systems with two species, Diff. Int. Eqns., 7 (1994), 1427-1452.
|
[32] |
J. López-Gómez,
A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.
|
[33] |
J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Research Notes in Mathematics 426, Chapman & Hall/CRC Press, Boca Raton, FL, 2001.
doi: 10.1201/9781420035506.![]() ![]() |
[34] |
J. López-Gómez,
Classifying smooth supersolutions for a general class of elliptic boundary value problems, Adv. Diff. Eqns., 8 (2003), 1025-1042.
|
[35] |
J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific, Singapore, 2013.
doi: 10.1142/8664. |
[36] |
J. López-Gómez, Metasolutions of Parabolic Equations in Population Dynamics, CRC Press, Boca Raton, 2016.
![]() |
[37] |
J. López-Gómez and M. Molina-Meyer,
The maximum principle for cooperative weakly elliptic systems and some applications, Diff. Int. Eqns, 7 (1994), 383-398.
|
[38] |
J. López-Gómez and E. Muñoz-Hernández, Global structure of subharmonics in a class of periodic predator-prey models, Nonlinearity, 33 (2020), 34-71. Google Scholar |
[39] |
J. López-Gómez and R. M. Pardo,
Coexistence regions in Lotka-Volterra systems with diffusion, Nonl. Anal., 19 (1992), 11-28.
doi: 10.1016/0362-546X(92)90027-C. |
[40] |
J. López-Gómez and R. M. Pardo,
The existence and the uniqueness for the predator-prey model with diffusion, Diff. Int. Eqns., 6 (1993), 1025-1031.
|
[41] |
J. López-Gómez and R. M. Pardo,
Invertibility of linear noncooperative elliptic systems, Nonl. Anal., 31 (1998), 687-699.
doi: 10.1016/S0362-546X(97)00640-8. |
[42] |
J. López-Gómez,
A bridge between operator theory and mathematical biology, Fields Inst. Comm., 25 (2000), 383-397.
|
[43] | R. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 1974. Google Scholar |
[44] |
H. Nie and J. Wu,
Multiplicity and stability of a predator-prey model with non-monotonic conversion rate, Nonl. Anal. RWA, 10 (2009), 154-171.
doi: 10.1016/j.nonrwa.2007.08.020. |
[45] |
P. Y. H. Pang and M. Wang,
Qualitative analysis of a ratio-dependent predator-prey system with diffusion, Proc. Royal Soc. Edinburgh, 133 (2003), 919-942.
doi: 10.1017/S0308210500002742. |
[46] |
P. Y. H. Pang and M. Wang,
Non-constant positive steady-states of a predator-prey system with non-monotonic functional response and diffusion, Proc. London Math. Soc., 88 (2004), 135-157.
doi: 10.1112/S0024611503014321. |
[47] |
R. Peng, M. X. Wang and W. Y. Chen,
Positive steady-states of a predator-prey model with diffusion and non-monotonic conversion rate, Acta Math. Sinica, 23 (2007), 749-760.
doi: 10.1007/s10114-005-0789-9. |
[48] |
K. Ryu and I. Ahn,
Positive solutions for ratio-dependent predator-prey interaction systems, J. Diff. Eqns., 218 (2005), 117-135.
doi: 10.1016/j.jde.2005.06.020. |
[49] |
M. X. Wang and Q. Wu,
Positive solutions fo a predator-prey model with predator saturation and competition, J. Math. Anal. Appl., 345 (2008), 708-718.
doi: 10.1016/j.jmaa.2008.04.054. |
[50] |
H. Yuan, J. Wu, Y. Jia and H. Nie,
Coexistence states of a predator-prey model with cross-diffusion, Nonl. Anal. RWA, 41 (2018), 179-203.
doi: 10.1016/j.nonrwa.2017.10.009. |
[51] |
X. Zeng, W. Zeng and L. Liu,
Effects of the protection zone on coexistence of the species for a ratio-dependent predator-prey model, J. Math. Anal. Appl., 462 (2018), 1605-1626.
doi: 10.1016/j.jmaa.2018.02.060. |
[52] |
J. Zhou and C. Mu,
Coexistence states of a Holling type-Ⅱ predator-prey system, J. Math. Anal. Appl., 369 (2010), 555-563.
doi: 10.1016/j.jmaa.2010.04.001. |
[53] |
J. Zhou and C. Mu,
Coexistence of a diffusive predator-prey model with Holling-type Ⅱ functional response and density dependent mortality, J. Math. Anal. Appl., 385 (2012), 913-927.
doi: 10.1016/j.jmaa.2011.07.027. |
[54] |
J. Zhou and J. P. Shi,
The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie–Gower predator-prey model with Holling type-Ⅱ functional responses, J. Math. Anal. Appl., 405 (2013), 618-630.
doi: 10.1016/j.jmaa.2013.03.064. |




[1] |
Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035 |
[2] |
Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197 |
[3] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[4] |
Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301 |
[5] |
Zhouxin Li, Yimin Zhang. Ground states for a class of quasilinear Schrödinger equations with vanishing potentials. Communications on Pure & Applied Analysis, 2021, 20 (2) : 933-954. doi: 10.3934/cpaa.2020298 |
[6] |
Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1693-1716. doi: 10.3934/dcdss.2020450 |
[7] |
Carmen Cortázar, M. García-Huidobro, Pilar Herreros, Satoshi Tanaka. On the uniqueness of solutions of a semilinear equation in an annulus. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021029 |
[8] |
Junichi Minagawa. On the uniqueness of Nash equilibrium in strategic-form games. Journal of Dynamics & Games, 2020, 7 (2) : 97-104. doi: 10.3934/jdg.2020006 |
[9] |
Tuan Hiep Pham, Jérôme Laverne, Jean-Jacques Marigo. Stress gradient effects on the nucleation and propagation of cohesive cracks. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 557-584. doi: 10.3934/dcdss.2016012 |
[10] |
Min Li, Jiahua Zhang, Yifan Xu, Wei Wang. Effects of disruption risk on a supply chain with a risk-averse retailer. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021024 |
[11] |
Benrong Zheng, Xianpei Hong. Effects of take-back legislation on pricing and coordination in a closed-loop supply chain. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021035 |
[12] |
Vo Anh Khoa, Thi Kim Thoa Thieu, Ekeoma Rowland Ijioma. On a pore-scale stationary diffusion equation: Scaling effects and correctors for the homogenization limit. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2451-2477. doi: 10.3934/dcdsb.2020190 |
[13] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[14] |
Wenmin Gong, Guangcun Lu. On coupled Dirac systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4329-4346. doi: 10.3934/dcds.2017185 |
[15] |
Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810 |
[16] |
Tuvi Etzion, Alexander Vardy. On $q$-analogs of Steiner systems and covering designs. Advances in Mathematics of Communications, 2011, 5 (2) : 161-176. doi: 10.3934/amc.2011.5.161 |
[17] |
Alberto Bressan, Ke Han, Franco Rampazzo. On the control of non holonomic systems by active constraints. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3329-3353. doi: 10.3934/dcds.2013.33.3329 |
[18] |
Lekbir Afraites, Abdelghafour Atlas, Fahd Karami, Driss Meskine. Some class of parabolic systems applied to image processing. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1671-1687. doi: 10.3934/dcdsb.2016017 |
[19] |
Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513 |
[20] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1717-1746. doi: 10.3934/dcdss.2020451 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]