
-
Previous Article
Homogeneous solutions of stationary Navier-Stokes equations with isolated singularities on the unit sphere. Ⅲ. Two singularities
- DCDS Home
- This Issue
-
Next Article
The fractional Schrödinger equation with singular potential and measure data
Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions
1. | École des Hautes Études en Sciences Sociales, Centre d'analyse et de mathématique sociales (CAMS), CNRS, 54 bouvelard Raspail, 75006, Paris, France |
2. | Université Paris Diderot, Université de Paris, Laboratoire Jacques-Louis Lions (CNRS UMR 7598), 8 place Aurélie Nemours, 75205, Paris CEDEX 13, France |
For a stationary system representing prey and $ N $ groups of competing predators, we show classification results about the set of positive solutions. In particular, we show that if the number of components $ N $ is too large or if the competition between different groups is too small, then the system has only constant solutions, which we then completely characterize.
References:
[1] |
H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431–461, http://dx.doi.org/10.2307/1999245.
doi: 10.1090/S0002-9947-1984-0732100-6. |
[2] |
H. Berestycki and A. Zilio, Predators-prey models with competition, Part Ⅱ: uniform regularity estimates, In preparation. |
[3] |
H. Berestycki and A. Zilio, Predators-prey models with competition, part ⅰ: Existence, bifurcation and qualitative properties, Communications in Contemporary Mathematics, 20 (2018), 1850010, 53pp.
doi: 10.1142/S0219199718500104. |
[4] |
H. Berestycki and A. Zilio,
Predator-prey models with competition: The emergence of territoriality, The American Naturalist, 193 (2019), 436-446.
doi: 10.1086/701670. |
[5] |
L. Caffarelli, S. Patrizi and V. Quitalo,
On a long range segregation model, J. Eur. Math. Soc. (JEMS), 19 (2017), 3575-3628.
doi: 10.4171/JEMS/747. |
[6] |
L. A. Caffarelli, A. L. Karakhanyan and F.-H. Lin,
The geometry of solutions to a segregation problem for nondivergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351.
doi: 10.1007/s11784-009-0110-0. |
[7] |
L. A. Caffarelli and F.-H. Lin,
Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862.
doi: 10.1090/S0894-0347-08-00593-6. |
[8] |
L. A. Caffarelli and S. Salsa, A Geometric Approach to the Free Boundary Problems, Graduate Studies in Mathematics, 68. American Mathematical Society, Providence, RI, 2005.
doi: 10.1090/gsm/068. |
[9] |
M. Conti, S. Terracini and G. Verzini,
Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.
doi: 10.1016/j.aim.2004.08.006. |
[10] |
M. Conti, S. Terracini and G. Verzini,
A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815.
doi: 10.1512/iumj.2005.54.2506. |
[11] |
E. N. Dancer and Y. H. Du,
Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.
doi: 10.1006/jdeq.1994.1156. |
[12] |
E. N. Dancer, K. Wang and Z. Zhang,
Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005.
doi: 10.1090/S0002-9947-2011-05488-7. |
[13] |
E. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion–Ⅰ. general existence results, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 337–357, http://www.sciencedirect.com/science/article/pii/0362546X94E0063M.
doi: 10.1016/0362-546X(94)E0063-M. |
[14] |
E. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion–Ⅱ. the case of equal birth rates, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 359–373, http://www.sciencedirect.com/science/article/pii/0362546X94E0064N.
doi: 10.1016/0362-546X(94)E0064-N. |
[15] |
H. Jung,
Ueber die kleinste Kugel, die eine räumliche Figur einschliesst, J. Reine Angew. Math., 123 (1901), 241-257.
doi: 10.1515/crll.1901.123.241. |
[16] |
M. Mimura,
Asymptotic behaviors of a parabolic system related to a planktonic prey and predator model, SIAM J. Appl. Math., 37 (1979), 499-512.
doi: 10.1137/0137039. |
[17] |
N. Soave and A. Zilio,
Uniform bounds for strongly competing systems: The optimal Lipschitz case, Arch. Ration. Mech. Anal., 218 (2015), 647-697.
doi: 10.1007/s00205-015-0867-9. |
[18] |
S. Terracini, G. Verzini and A. Zilio, Spiraling asymptotic profiles of competition-diffusion systems, Communications on Pure and Applied Mathematics, 2019.
doi: 10.1002/cpa.21823. |
[19] |
G. Verzini and A. Zilio,
Strong competition versus fractional diffusion: The case of Lotka-Volterra interaction, Comm. Partial Differential Equations, 39 (2014), 2284-2313.
doi: 10.1080/03605302.2014.890627. |
[20] |
V. Volterra,
Variations and fluctuations of the number of individuals in animal species living together, Journal du Cons. Int. Explor. Mer, 3 (1928), 3-51.
doi: 10.1093/icesjms/3.1.3. |
show all references
References:
[1] |
H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc., 282 (1984), 431–461, http://dx.doi.org/10.2307/1999245.
doi: 10.1090/S0002-9947-1984-0732100-6. |
[2] |
H. Berestycki and A. Zilio, Predators-prey models with competition, Part Ⅱ: uniform regularity estimates, In preparation. |
[3] |
H. Berestycki and A. Zilio, Predators-prey models with competition, part ⅰ: Existence, bifurcation and qualitative properties, Communications in Contemporary Mathematics, 20 (2018), 1850010, 53pp.
doi: 10.1142/S0219199718500104. |
[4] |
H. Berestycki and A. Zilio,
Predator-prey models with competition: The emergence of territoriality, The American Naturalist, 193 (2019), 436-446.
doi: 10.1086/701670. |
[5] |
L. Caffarelli, S. Patrizi and V. Quitalo,
On a long range segregation model, J. Eur. Math. Soc. (JEMS), 19 (2017), 3575-3628.
doi: 10.4171/JEMS/747. |
[6] |
L. A. Caffarelli, A. L. Karakhanyan and F.-H. Lin,
The geometry of solutions to a segregation problem for nondivergence systems, J. Fixed Point Theory Appl., 5 (2009), 319-351.
doi: 10.1007/s11784-009-0110-0. |
[7] |
L. A. Caffarelli and F.-H. Lin,
Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries, J. Amer. Math. Soc., 21 (2008), 847-862.
doi: 10.1090/S0894-0347-08-00593-6. |
[8] |
L. A. Caffarelli and S. Salsa, A Geometric Approach to the Free Boundary Problems, Graduate Studies in Mathematics, 68. American Mathematical Society, Providence, RI, 2005.
doi: 10.1090/gsm/068. |
[9] |
M. Conti, S. Terracini and G. Verzini,
Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.
doi: 10.1016/j.aim.2004.08.006. |
[10] |
M. Conti, S. Terracini and G. Verzini,
A variational problem for the spatial segregation of reaction-diffusion systems, Indiana Univ. Math. J., 54 (2005), 779-815.
doi: 10.1512/iumj.2005.54.2506. |
[11] |
E. N. Dancer and Y. H. Du,
Competing species equations with diffusion, large interactions, and jumping nonlinearities, J. Differential Equations, 114 (1994), 434-475.
doi: 10.1006/jdeq.1994.1156. |
[12] |
E. N. Dancer, K. Wang and Z. Zhang,
Dynamics of strongly competing systems with many species, Trans. Amer. Math. Soc., 364 (2012), 961-1005.
doi: 10.1090/S0002-9947-2011-05488-7. |
[13] |
E. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion–Ⅰ. general existence results, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 337–357, http://www.sciencedirect.com/science/article/pii/0362546X94E0063M.
doi: 10.1016/0362-546X(94)E0063-M. |
[14] |
E. Dancer and Y. Du, Positive solutions for a three-species competition system with diffusion–Ⅱ. the case of equal birth rates, Nonlinear Analysis: Theory, Methods & Applications, 24 (1995), 359–373, http://www.sciencedirect.com/science/article/pii/0362546X94E0064N.
doi: 10.1016/0362-546X(94)E0064-N. |
[15] |
H. Jung,
Ueber die kleinste Kugel, die eine räumliche Figur einschliesst, J. Reine Angew. Math., 123 (1901), 241-257.
doi: 10.1515/crll.1901.123.241. |
[16] |
M. Mimura,
Asymptotic behaviors of a parabolic system related to a planktonic prey and predator model, SIAM J. Appl. Math., 37 (1979), 499-512.
doi: 10.1137/0137039. |
[17] |
N. Soave and A. Zilio,
Uniform bounds for strongly competing systems: The optimal Lipschitz case, Arch. Ration. Mech. Anal., 218 (2015), 647-697.
doi: 10.1007/s00205-015-0867-9. |
[18] |
S. Terracini, G. Verzini and A. Zilio, Spiraling asymptotic profiles of competition-diffusion systems, Communications on Pure and Applied Mathematics, 2019.
doi: 10.1002/cpa.21823. |
[19] |
G. Verzini and A. Zilio,
Strong competition versus fractional diffusion: The case of Lotka-Volterra interaction, Comm. Partial Differential Equations, 39 (2014), 2284-2313.
doi: 10.1080/03605302.2014.890627. |
[20] |
V. Volterra,
Variations and fluctuations of the number of individuals in animal species living together, Journal du Cons. Int. Explor. Mer, 3 (1928), 3-51.
doi: 10.1093/icesjms/3.1.3. |

[1] |
Haiyun Deng, Hairong Liu, Long Tian. Classification of singular sets of solutions to elliptic equations. Communications on Pure and Applied Analysis, 2020, 19 (6) : 2949-2964. doi: 10.3934/cpaa.2020129 |
[2] |
Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31 |
[3] |
Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019 |
[4] |
Jingyu Li. Asymptotic behavior of solutions to elliptic equations in a coated body. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1251-1267. doi: 10.3934/cpaa.2009.8.1251 |
[5] |
Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146 |
[6] |
Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886 |
[7] |
Yuxia Guo, Jianjun Nie. Classification for positive solutions of degenerate elliptic system. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1457-1475. doi: 10.3934/dcds.2018130 |
[8] |
Limei Dai. Entire solutions with asymptotic behavior of fully nonlinear uniformly elliptic equations. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1707-1714. doi: 10.3934/cpaa.2011.10.1707 |
[9] |
Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure and Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97 |
[10] |
La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 |
[11] |
Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032 |
[12] |
Chang-Shou Lin, Lei Zhang. Classification of radial solutions to Liouville systems with singularities. Discrete and Continuous Dynamical Systems, 2014, 34 (6) : 2617-2637. doi: 10.3934/dcds.2014.34.2617 |
[13] |
Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166 |
[14] |
Mustafa Hasanbulli, Yuri V. Rogovchenko. Classification of nonoscillatory solutions of nonlinear neutral differential equations. Conference Publications, 2009, 2009 (Special) : 340-348. doi: 10.3934/proc.2009.2009.340 |
[15] |
Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1757-1774. doi: 10.3934/dcdsb.2020001 |
[16] |
Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521 |
[17] |
Dumitru Motreanu. Three solutions with precise sign properties for systems of quasilinear elliptic equations. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 831-843. doi: 10.3934/dcdss.2012.5.831 |
[18] |
Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations. Networks and Heterogeneous Media, 2019, 14 (3) : 567-587. doi: 10.3934/nhm.2019022 |
[19] |
Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure and Applied Analysis, 2016, 15 (2) : 519-533. doi: 10.3934/cpaa.2016.15.519 |
[20] |
Fabrício Cristófani, Ademir Pastor. Nonlinear stability of periodic-wave solutions for systems of dispersive equations. Communications on Pure and Applied Analysis, 2020, 19 (10) : 5015-5032. doi: 10.3934/cpaa.2020225 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]