doi: 10.3934/nhm.2021010

Multiple patterns formation for an aggregation/diffusion predator-prey system

1. 

DISIM - University of L'Aquila, Via Vetoio 1 (Coppito) 67100 L'Aquila (AQ), Italy

2. 

Department of Mathematics, An-Najah National University, Al-Junaid street P448 Nablus, Palestine

* Corresponding author: Simone Fagioli

Received  September 2020 Published  May 2021

We investigate existence of stationary solutions to an aggregation/diffusion system of PDEs, modelling a two species predator-prey interaction. In the model this interaction is described by non-local potentials that are mutually proportional by a negative constant $ -\alpha $, with $ \alpha>0 $. Each species is also subject to non-local self-attraction forces together with quadratic diffusion effects. The competition between the aforementioned mechanisms produce a rich asymptotic behavior, namely the formation of steady states that are composed of multiple bumps, i.e. sums of Barenblatt-type profiles. The existence of such stationary states, under some conditions on the positions of the bumps and the proportionality constant $ \alpha $, is showed for small diffusion, by using the functional version of the Implicit Function Theorem. We complement our results with some numerical simulations, that suggest a large variety in the possible strategies the two species use in order to interact each other.

Citation: Simone Fagioli, Yahya Jaafra. Multiple patterns formation for an aggregation/diffusion predator-prey system. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021010
References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Applied Mathematics Letters, 24 (2011), 1927-1932.  doi: 10.1016/j.aml.2011.05.022.  Google Scholar

[3]

N. Bellomo and S. -Y Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770. doi: 10.1142/S0218202517500154.  Google Scholar

[4]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006. doi: 10.1142/S0218202511400069.  Google Scholar

[5]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.  Google Scholar

[6]

M. BurgerM. Di FrancescoS. Fagioli and A. Stevens, Sorting phenomena in a mathematical model for two mutually attracting/repelling species, SIAM J. Math. Anal., 50 (2018), 3210-3250.  doi: 10.1137/17M1125716.  Google Scholar

[7]

M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2013), 709-738. doi: 10.4310/CMS. 2013. v11. n3. a3.  Google Scholar

[8]

M. Burger, R. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424. doi: 10.1137/130923786.  Google Scholar

[9]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens}, Nonlinear Anal. Real World Appl., 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar

[10]

G. Carlier and M. Laborde, Remarks on continuity equations with nonlinear diffusion and nonlocal drifts, J. Math. Anal. Appl., 444 (2016), 1690-1702.  doi: 10.1016/j.jmaa.2016.07.061.  Google Scholar

[11]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.  Google Scholar

[12]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.  Google Scholar

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J. A. CarrilloY. Huang and M. Schmidtchen, Zoology of a nonlocal cross-diffusion model for two species, SIAM J. Appl. Math., 78 (2018), 1078-1104.  doi: 10.1137/17M1128782.  Google Scholar

[14]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations., New Trends in Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2004, 234-244.  Google Scholar

[15]

Y. Chen and T. Kolokolnikov, A minimal model of predator-swarm interactions, J. R. Soc. Interface, 11 (2014). doi: 10.1098/rsif. 2013.1208.  Google Scholar

[16]

R. ChoksiR. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.  Google Scholar

[17]

M. CicaleseL. De LucaM. Novaga and M. Ponsiglione, Ground states of a two phase model with cross and self attractive interactions, SIAM J. Math. Anal., 48 (2016), 3412-3443.  doi: 10.1137/15M1033976.  Google Scholar

[18]

B. Düring, P. Markowich, J. -F. Pietschmann and M. -T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), 3687-3708. doi: 10.1098/rspa. 2009.0239.  Google Scholar

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K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[20]

M. Di FrancescoA. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Anal., 169 (2018), 94-117.  doi: 10.1016/j.na.2017.12.003.  Google Scholar

[21]

M. Di Francesco and S. Fagioli, A nonlocal swarm model for predators-prey interactions, Math. Models Methods Appl. Sci., 26 (2016), 319-355.  doi: 10.1142/S0218202516400042.  Google Scholar

[22]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808.  doi: 10.1088/0951-7715/26/10/2777.  Google Scholar

[23]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-Leader approximations of macroscopic models for vehicular and pedestrian flows, Active Particles, Vol. 1, Birkhäuser/Springer, Cham, 2017, 333-378.  Google Scholar

[24]

M. Di Francesco and Y. Jaafra, Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion, Kinet. Relat. Models, 12 (2019), 303-322.  doi: 10.3934/krm.2019013.  Google Scholar

[25]

M. Di Francesco and D. Matthes, Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations, Calc. Var. Partial Differential Equations, 50 (2014), 199-230.  doi: 10.1007/s00526-013-0633-5.  Google Scholar

[26]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774446.  Google Scholar

[27]

J. Evers, R. Fetecau and T. Kolokolnikov, Equilibria for an aggregation model with two species, SIAM J. Appl. Dyn. Syst., 16 (2017), 2287-2338. doi: 10.1137/16M1109527.  Google Scholar

[28]

S. Fagioli and E. Radici, Solutions to aggregation-diffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Math. Models Methods Appl. Sci., 28 (2018), 1801-1829.  doi: 10.1142/S0218202518400067.  Google Scholar

[29]

K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.  doi: 10.1142/S0218202510004921.  Google Scholar

[30]

K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), 1436-1450.  doi: 10.1016/j.mcm.2010.03.021.  Google Scholar

[31]

S. GottliebC. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[32]

S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477. doi: 10.1016/j. nonrwa. 2018.01.011.  Google Scholar

[33]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[34]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296.  doi: 10.1137/16M1072450.  Google Scholar

[35] H. Krause and G. D. Ruxton, Living in Groups, Oxford University Press, 2002.   Google Scholar
[36]

A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.  doi: 10.1021/j150111a004.  Google Scholar

[37]

D. MatthesR. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397.  doi: 10.1080/03605300903296256.  Google Scholar

[38]

M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys. 15, 19-65, 1982. doi: 10.1016/0065-227X(82)90004-1.  Google Scholar

[39]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[40]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.  Google Scholar

[41]

J. D. Murray, Mathematical Biology. I, 3rd edition, Interdisciplinary Applied Mathematics, Vol. 17, Springer-Verlag, New York, 2002.  Google Scholar

[42]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[43]

J. K. Parrish and L. Edelstein-Keshet, Complexity, patterns and evolutionary trade-offs in animal aggregation, Science, 254 (1999), 99-101.  doi: 10.1126/science.284.5411.99.  Google Scholar

[44]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2 (2003), 395-431.   Google Scholar

[45]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2.  Google Scholar

[46]

H. Tomkins and T. Kolokolnikov, Swarm shape and its dynamics in a predator-swarm model, preprint, 2014. Google Scholar

[47]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[48]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[49]

M. Torregrossa and G. Toscani, On a Fokker-Planck equation for wealth distribution, Kinet. Relat. Models, 11 (2018), 337-355. doi: 10.3934/krm. 2018016.  Google Scholar

[50]

M. Torregrossa and G. Toscani, Wealth distribution in presence of debts. A Fokker-Planck description, Commun. Math. Sci., 16 (2018), 537-560. doi: 10.4310/CMS. 2018. v16. n2. a11.  Google Scholar

[51]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS. 2006. v4. n3. a1.  Google Scholar

[52] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar
[53]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

[54]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Accad. Lincei Roma, 2 (1926), 31-113.   Google Scholar

show all references

References:
[1]

L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008.  Google Scholar

[2]

J. Bedrossian, Global minimizers for free energies of subcritical aggregation equations with degenerate diffusion, Applied Mathematics Letters, 24 (2011), 1927-1932.  doi: 10.1016/j.aml.2011.05.022.  Google Scholar

[3]

N. Bellomo and S. -Y Ha, A quest toward a mathematical theory of the dynamics of swarms, Math. Models Methods Appl. Sci., 27 (2017), 745-770. doi: 10.1142/S0218202517500154.  Google Scholar

[4]

N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), 1140006. doi: 10.1142/S0218202511400069.  Google Scholar

[5]

M. Burger and M. Di Francesco, Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media, 3 (2008), 749-785.  doi: 10.3934/nhm.2008.3.749.  Google Scholar

[6]

M. BurgerM. Di FrancescoS. Fagioli and A. Stevens, Sorting phenomena in a mathematical model for two mutually attracting/repelling species, SIAM J. Math. Anal., 50 (2018), 3210-3250.  doi: 10.1137/17M1125716.  Google Scholar

[7]

M. Burger, M. Di Francesco and M. Franek, Stationary states of quadratic diffusion equations with long-range attraction, Commun. Math. Sci., 11 (2013), 709-738. doi: 10.4310/CMS. 2013. v11. n3. a3.  Google Scholar

[8]

M. Burger, R. Fetecau and Y. Huang, Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion, SIAM J. Appl. Dyn. Syst., 13 (2014), 397-424. doi: 10.1137/130923786.  Google Scholar

[9]

S. BoiV. Capasso and D. Morale, Modeling the aggregative behavior of ants of the species polyergus rufescens}, Nonlinear Anal. Real World Appl., 1 (2000), 163-176.  doi: 10.1016/S0362-546X(99)00399-5.  Google Scholar

[10]

G. Carlier and M. Laborde, Remarks on continuity equations with nonlinear diffusion and nonlocal drifts, J. Math. Anal. Appl., 444 (2016), 1690-1702.  doi: 10.1016/j.jmaa.2016.07.061.  Google Scholar

[11]

V. CalvezJ. A. Carrillo and F. Hoffmann, Equilibria of homogeneous functionals in the fair-competition regime, Nonlinear Anal., 159 (2017), 85-128.  doi: 10.1016/j.na.2017.03.008.  Google Scholar

[12]

J. A. CarrilloA. Chertock and Y. Huang, A finite-volume method for nonlinear nonlocal equations with a gradient flow structure, Commun. Comput. Phys., 17 (2015), 233-258.  doi: 10.4208/cicp.160214.010814a.  Google Scholar

[13]

J. A. CarrilloY. Huang and M. Schmidtchen, Zoology of a nonlocal cross-diffusion model for two species, SIAM J. Appl. Math., 78 (2018), 1078-1104.  doi: 10.1137/17M1128782.  Google Scholar

[14]

J. A. Carrillo and G. Toscani, Wasserstein metric and large-time asymptotics of nonlinear diffusion equations., New Trends in Mathematical Physics, World Sci. Publ., Hackensack, NJ, 2004, 234-244.  Google Scholar

[15]

Y. Chen and T. Kolokolnikov, A minimal model of predator-swarm interactions, J. R. Soc. Interface, 11 (2014). doi: 10.1098/rsif. 2013.1208.  Google Scholar

[16]

R. ChoksiR. C. Fetecau and I. Topaloglu, On minimizers of interaction functionals with competing attractive and repulsive potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1283-1305.  doi: 10.1016/j.anihpc.2014.09.004.  Google Scholar

[17]

M. CicaleseL. De LucaM. Novaga and M. Ponsiglione, Ground states of a two phase model with cross and self attractive interactions, SIAM J. Math. Anal., 48 (2016), 3412-3443.  doi: 10.1137/15M1033976.  Google Scholar

[18]

B. Düring, P. Markowich, J. -F. Pietschmann and M. -T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), 3687-3708. doi: 10.1098/rspa. 2009.0239.  Google Scholar

[19]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[20]

M. Di FrancescoA. Esposito and S. Fagioli, Nonlinear degenerate cross-diffusion systems with nonlocal interaction, Nonlinear Anal., 169 (2018), 94-117.  doi: 10.1016/j.na.2017.12.003.  Google Scholar

[21]

M. Di Francesco and S. Fagioli, A nonlocal swarm model for predators-prey interactions, Math. Models Methods Appl. Sci., 26 (2016), 319-355.  doi: 10.1142/S0218202516400042.  Google Scholar

[22]

M. Di Francesco and S. Fagioli, Measure solutions for non-local interaction PDEs with two species, Nonlinearity, 26 (2013), 2777-2808.  doi: 10.1088/0951-7715/26/10/2777.  Google Scholar

[23]

M. Di Francesco, S. Fagioli, M. D. Rosini and G. Russo, Follow-the-Leader approximations of macroscopic models for vehicular and pedestrian flows, Active Particles, Vol. 1, Birkhäuser/Springer, Cham, 2017, 333-378.  Google Scholar

[24]

M. Di Francesco and Y. Jaafra, Multiple large-time behavior of nonlocal interaction equations with quadratic diffusion, Kinet. Relat. Models, 12 (2019), 303-322.  doi: 10.3934/krm.2019013.  Google Scholar

[25]

M. Di Francesco and D. Matthes, Curves of steepest descent are entropy solutions for a class of degenerate convection-diffusion equations, Calc. Var. Partial Differential Equations, 50 (2014), 199-230.  doi: 10.1007/s00526-013-0633-5.  Google Scholar

[26]

Y. Du, Order Structure and Topological Methods in Nonlinear Partial Differential Equations, Vol. 1, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812774446.  Google Scholar

[27]

J. Evers, R. Fetecau and T. Kolokolnikov, Equilibria for an aggregation model with two species, SIAM J. Appl. Dyn. Syst., 16 (2017), 2287-2338. doi: 10.1137/16M1109527.  Google Scholar

[28]

S. Fagioli and E. Radici, Solutions to aggregation-diffusion equations with nonlinear mobility constructed via a deterministic particle approximation, Math. Models Methods Appl. Sci., 28 (2018), 1801-1829.  doi: 10.1142/S0218202518400067.  Google Scholar

[29]

K. Fellner and G. Raoul, Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci., 20 (2010), 2267-2291.  doi: 10.1142/S0218202510004921.  Google Scholar

[30]

K. Fellner and G. Raoul, Stability of stationary states of non-local equations with singular interaction potentials, Math. Comput. Modelling, 53 (2011), 1436-1450.  doi: 10.1016/j.mcm.2010.03.021.  Google Scholar

[31]

S. GottliebC. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112.  doi: 10.1137/S003614450036757X.  Google Scholar

[32]

S. Guo, Bifurcation and spatio-temporal patterns in a diffusive predator-prey system, Nonlinear Anal. Real World Appl., 42 (2018), 448-477. doi: 10.1016/j. nonrwa. 2018.01.011.  Google Scholar

[33]

R. JordanD. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.  doi: 10.1137/S0036141096303359.  Google Scholar

[34]

G. Kaib, Stationary states of an aggregation equation with degenerate diffusion and bounded attractive potential, SIAM J. Math. Anal., 49 (2017), 272-296.  doi: 10.1137/16M1072450.  Google Scholar

[35] H. Krause and G. D. Ruxton, Living in Groups, Oxford University Press, 2002.   Google Scholar
[36]

A. J. Lotka, Contribution to the theory of periodic reaction, J. Phys. Chem., 14 (1910), 271-274.  doi: 10.1021/j150111a004.  Google Scholar

[37]

D. MatthesR. J. McCann and G. Savaré, A family of nonlinear fourth order equations of gradient flow type, Comm. Partial Differential Equations, 34 (2009), 1352-1397.  doi: 10.1080/03605300903296256.  Google Scholar

[38]

M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys. 15, 19-65, 1982. doi: 10.1016/0065-227X(82)90004-1.  Google Scholar

[39]

A. Mogilner and L. Edelstein-Keshet, A non-local model for a swarm, J. Math. Biol., 38 (1999), 534-570.  doi: 10.1007/s002850050158.  Google Scholar

[40]

D. Morale, V. Capasso and K. Oelschläger, An interacting particle system modelling aggregation behavior: From individuals to populations, J. Math. Biol., 50 (2005), 49-66. doi: 10.1007/s00285-004-0279-1.  Google Scholar

[41]

J. D. Murray, Mathematical Biology. I, 3rd edition, Interdisciplinary Applied Mathematics, Vol. 17, Springer-Verlag, New York, 2002.  Google Scholar

[42]

A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, 2nd edition, Interdisciplinary Applied Mathematics, Vol. 14, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-4978-6.  Google Scholar

[43]

J. K. Parrish and L. Edelstein-Keshet, Complexity, patterns and evolutionary trade-offs in animal aggregation, Science, 254 (1999), 99-101.  doi: 10.1126/science.284.5411.99.  Google Scholar

[44]

R. Rossi and G. Savaré, Tightness, integral equicontinuity and compactness for evolution problems in Banach spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2 (2003), 395-431.   Google Scholar

[45]

F. Santambrogio, Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential Equations and Their Applications, Vol. 87, Birkhäuser/Springer, Cham, 2015. doi: 10.1007/978-3-319-20828-2.  Google Scholar

[46]

H. Tomkins and T. Kolokolnikov, Swarm shape and its dynamics in a predator-swarm model, preprint, 2014. Google Scholar

[47]

C. M. Topaz and A. L. Bertozzi, Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math., 65 (2004), 152-174.  doi: 10.1137/S0036139903437424.  Google Scholar

[48]

C. M. TopazA. L. Bertozzi and M. A. Lewis, A nonlocal continuum model for biological aggregation, Bull. Math. Biol., 68 (2006), 1601-1623.  doi: 10.1007/s11538-006-9088-6.  Google Scholar

[49]

M. Torregrossa and G. Toscani, On a Fokker-Planck equation for wealth distribution, Kinet. Relat. Models, 11 (2018), 337-355. doi: 10.3934/krm. 2018016.  Google Scholar

[50]

M. Torregrossa and G. Toscani, Wealth distribution in presence of debts. A Fokker-Planck description, Commun. Math. Sci., 16 (2018), 537-560. doi: 10.4310/CMS. 2018. v16. n2. a11.  Google Scholar

[51]

G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496. doi: 10.4310/CMS. 2006. v4. n3. a1.  Google Scholar

[52] J. L. Vazquez, The Porous Medium Equation. Mathematical Theory, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar
[53]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics, Vol. 58, American Mathematical Society, Providence, RI, 2003. doi: 10.1090/gsm/058.  Google Scholar

[54]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Accad. Lincei Roma, 2 (1926), 31-113.   Google Scholar

Figure 1.  A possible example of a stationary solution to (3) with $ N_\rho = N_\eta = 3 $ is plotted as described in Definition 1.1
Figure 2.  Example of mixed stationary state. Note that by symmetry $ L_\rho=-R_\rho $ and $ L_\eta=-R_\eta $
Figure 3.  An example of a separated stationary state
Figure 4.  In this figure, a mixed steady state is plotted by using initial data given by (70), $ \alpha = 0.1 $, $ \theta = 0.4 $. Number of particles are chosen equal to number of cells in the finite volume method, which is $ N = 71 $
Figure 5.  A separated steady state is presented in this figure. Initial data are given by (71). The parameters are $ \alpha = 0.2 $ and $ \theta = 0.4 $ with $ N = 91 $
Figure 6.  This figure shows how from the initial densities $ \rho_0,\eta_0 $ and $ \theta $, as in Figure 4, a transition between mixed and a sort of separated steady state appears by choosing the value of $ \alpha = 6 $. This large value of $ \alpha $ suggests an unstable behavior in the profile, see Remark 3.1
Figure 7.  A steady state of four bumps is showed in this figure starting from initial data as in (72) with $ \alpha = 0.05 $ and $ \theta = 0.3 $. The number of particles $ N = 181 $, which is the same as number of cells
Figure 8.  Starting from initial data as in (73) with $ \alpha = 1 $ and $ \theta = 0.3 $, we get a five bumps steady state
Figure 9.  This last figure shows a possible existence of traveling waves by choosing initial data as in (74), $ \alpha = 1 $ and $ \theta = 0.2 $. The first two plots are performed by particles method, while the third one is done by finite volume method. Here we fix $ N = 101 $
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