A nonlocal swarm model for predators-prey interactions. (English) Zbl 1334.35357
Summary: We consider a two-species system of nonlocal interaction PDEs modeling the swarming dynamics of predators and prey, in which all agents interact through attractive/repulsive forces of gradient type. In order to model the predator-prey interaction, we prescribed proportional potentials (with opposite signs) for the cross-interaction part. The model has a particle-based discrete (ODE) version and a continuum PDE version. We investigate the structure of particle stationary solution and their stability in the ODE system in a systematic form, and then consider simple examples. We then prove that the stable particle steady states are locally stable for the fully nonlinear continuum model, provided a slight reinforcement of the particle condition is required. The latter result holds in one space dimension. We complement all the particle examples with simple numerical simulations, and we provide some two-dimensional examples to highlight the complexity in the large time behavior of the system.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 45K05 | Integro-partial differential equations |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 92D25 | Population dynamics (general) |
| 35R09 | Integro-partial differential equations |
Keywords:
nonlocal interaction equations; systems with many species; Wasserstein distance; predators-prey model; stability of stationary statesReferences:
| [1] | Ambrosio, L., Gigli, N. and Savaré, G., Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edn., (Birkhäuser, 2008). · Zbl 1145.35001 |
| [2] | Ambrosio, L. and Savaré, G., Gradient flows of probability measures, in Handbook of Differential Equations: Evolutionary Equations, Vol. 3 (Elsevier/North-Holland, 2007), pp. 1-136. · Zbl 1203.35002 |
| [3] | Bellomo, N. and Soler, J., On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci.22 (2012) 1140006. · Zbl 1242.92065 |
| [4] | Bertozzi, A. L. and Brandman, J., Finite-time blow-up of \(L^\infty \)-weak solutions of an aggregation equation, Commun. Math. Sci.8 (2010) 45-65. · Zbl 1197.35061 |
| [5] | Bertozzi, A. L., Carrillo, J. A. and Laurent, T., Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity22 (2009) 683-710. · Zbl 1194.35053 |
| [6] | Bertozzi, A. L. and Laurent, T., The behavior of solutions of multidimensional aggregation equations with mildly singular interaction kernels, Chinese Ann. Math. Ser. B30 (2009) 463-482. · Zbl 1188.35141 |
| [7] | Bertozzi, A. L., Laurent, T. and Rosado, J., \( L^p\)-theory for the multidimensional aggregation equation, Commun. Pure Appl. Math.64 (2011) 45-83. · Zbl 1218.35075 |
| [8] | Bertozzi, A. L. and Slepcev, D., Existence and uniqueness of solutions to an aggregation equation with degenerate diffusion, Commun. Pure Appl. Anal.9 (2010) 1617-1637. · Zbl 1213.35266 |
| [9] | Bodnar, M. and Velazquez, J. J. L., An integro-differential equation arising as a limit of individual cell-based models, J. Differential Equations222 (2006) 341-380. · Zbl 1089.45002 |
| [10] | Boi, S., Capasso, V. and Morale, D., Modeling the aggregative behavior of ants of the speciesPolyergus Rufescens, Nonlinear Anal. Real World Appl.1 (2000) 163-176; Special Issue on Spatial Heterogeneity in Ecological Models (Alcalá de Henares, 1998). · Zbl 1011.92053 |
| [11] | Burger, M. and Di Francesco, M., Large time behavior of nonlocal aggregation models with nonlinear diffusion, Netw. Heterog. Media3 (2008) 749-785. · Zbl 1171.35328 |
| [12] | Carrillo, J. A., Di Francesco, M., Figalli, A., Laurent, T. and Slepčev, D., Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J.156 (2011) 229-271. · Zbl 1215.35045 |
| [13] | Carrillo, J. A., Di Francesco, M., Figalli, A., Laurent, T. and Slepčev, D., Confinement in nonlocal interaction equations, Nonlinear Anal.75 (2012) 550-558. · Zbl 1233.35032 |
| [14] | Carrillo, J. A., Lisini, S. and Mainini, E., Gradient flows for nonsmooth interaction potentials, Nonlinear Anal.100 (2014) 122-147. · Zbl 1285.49035 |
| [15] | Carrillo, J. A., McCann, R. J. and Villani, C., Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Rational Mech. Anal.179 (2006) 217-263. · Zbl 1082.76105 |
| [16] | Carrillo, J. A. and Toscani, G., Wasserstein metric and large-time asymptotics of nonlinear diffusion equations, in New Trends in Mathematical Physics (World Scientific, 2004), pp. 234-244. · Zbl 1089.76055 |
| [17] | Chen, Y. and Kolokolnikov, T., A minimal model of predator-swarm interaction, J. R. Soc. Interface11 (2014) 20131208. |
| [18] | Di Francesco, M. and Fagioli, S., Measure solutions for non-local interaction PDEs with two species, Nonlinearity26 (2013) 2777-2808. · Zbl 1278.35003 |
| [19] | Fellner, K. and Raoul, G., Stable stationary states of non-local interaction equations, Math. Models Methods Appl. Sci.20 (2010) 2267-2291. · Zbl 1213.35079 |
| [20] | Fellner, K. and Raoul, G., Stable stationary states of non-local interaction equations with singular interaction potentials, Math. Comput. Model.53 (2011) 1436-1450. · Zbl 1219.35342 |
| [21] | Jordan, R., Kinderlehrer, D. and Otto, F., The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal.29 (1998) 1-17. · Zbl 0915.35120 |
| [22] | Krause, H. and Ruxton, G. D., Living in Groups (Oxford Univ. Press, 2002). |
| [23] | Li, H. and Toscani, G., Long-time asymptotics of kinetic models of granular flows, Arch. Rational Mech. Anal.172 (2004) 407-428. · Zbl 1116.82025 |
| [24] | Lotka, A. J., Contribution to the theory of periodic reaction, J. Phys. Chem.14 (1910) 271-274. |
| [25] | Mogilner, A. and Edelstein-Keshet, L., A non-local model for a swarm, J. Math. Biol.38 (1999) 534-570. · Zbl 0940.92032 |
| [26] | Murray, J. D., Mathematical Biology. I, 3rd edn., , Vol. 17 (Springer, 2002), An introduction. · Zbl 1006.92001 |
| [27] | Okubo, A. and Levin, S. A., Diffusion and Ecological Problems: Modern Perspectives, , Vol. 14 (Springer, 2001). |
| [28] | Parrish, J. K. and Edelstein-Keshet, L., Complexity, patterns and evolutionary trade-offs in animal aggregation, Science254 (1999) 99-101. |
| [29] | H. Tomkins and T. Kolokolnikov, Swarm shape and its dynamics in a predator-swarm model, preprint (2014). |
| [30] | Topaz, C. M. and Bertozzi, A. L., Swarming patterns in a two-dimensional kinematic model for biological groups, SIAM J. Appl. Math.65 (2004) 152-174. · Zbl 1071.92048 |
| [31] | Topaz, M., Bertozzi, A. L. and Lewis, M. A., A nonlocal continuum model for biological aggregation, Bull. Math. Biol.68 (2006) 1601-1623. · Zbl 1334.92468 |
| [32] | Villani, C., Topics in Optimal Transportation, , Vol. 58 (Amer. Math. Soc., 2003). · Zbl 1106.90001 |
| [33] | Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. Accad. Lincei Roma2 (1926) 31-113. · JFM 52.0450.06 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
