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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

Henri Berestycki 1,, and  Alessandro Zilio 2,

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

* Corresponding author: Henri Berestycki

To Luis Caffarelli, with admiration and affection

Received  December 2018 Revised  May 2019 Published  September 2019

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 & Continuous Dynamical Systems - A, 2019, 39 (12) : 7141-7162. doi: 10.3934/dcds.2019299
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.  Google Scholar

[2]

H. Berestycki and A. Zilio, Predators-prey models with competition, Part Ⅱ: uniform regularity estimates, In preparation. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

H. Berestycki and A. Zilio, Predators-prey models with competition, Part Ⅱ: uniform regularity estimates, In preparation. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Pictorial description of Theorem 1.1
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