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December
2019, 39(12): 7141-7162.
doi: 10.3934/dcds.2019299
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.
Keywords:
Systems of elliptic equations,
asymptotic analysis,
stability of solutions,
classification of solutions.
Mathematics Subject Classification:
Primary: 35K57, 35J61; Secondary: 92D40, 92D25, 92D50.
Citation:
Henri Berestycki, Alessandro Zilio. Predator-prey models with competition, Part Ⅲ: Classification of stationary solutions. Discrete and Continuous Dynamical Systems,
2019, 39
(12)
: 7141-7162.
doi: 10.3934/dcds.2019299
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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. |
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