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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

* 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.

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
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. CaffarelliS. 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. CaffarelliA. 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. ContiS. 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. ContiS. 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. DancerK. 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. CaffarelliS. 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. CaffarelliA. 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. ContiS. 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. ContiS. 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. DancerK. 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.

Figure 1.  Pictorial description of Theorem 1.1
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