Modeling herd behavior in population systems. (English) Zbl 1225.49037
Summary: We show that under suitable simple assumptions the classical two populations system may exhibit unexpected behavior. Considering a more elaborated social model, in which the individuals of one population gather together in herds, while the other one shows a more individualistic behavior, we model the fact that interactions among the two occur mainly through the perimeter of the herd. We account for all types of populations’ interactions, symbiosis, competition and the predator-prey interactions. There is a situation in which competitive exclusion does not hold: the socialized herd behavior prevents the competing individualistic population from becoming extinct. For the predator-prey case, sustained limit cycles are possible, the existence of Hopf bifurcations representing a distinctive feature of this model compared with other classical predator-prey models. The system’s behavior is fully captured by just one suitably introduced new threshold parameter, defined in terms of the original model parameters.
MSC:
| 49N75 | Pursuit and evasion games |
| 92D25 | Population dynamics (general) |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 34H05 | Control problems involving ordinary differential equations |
References:
| [1] | V. Volterra, U. D’Ancona, La concorrenza vitale tra le specie nell’ambiente marino, VIIe Congr. int. acqui. et de pêche, Paris, 1931 1-14.; V. Volterra, U. D’Ancona, La concorrenza vitale tra le specie nell’ambiente marino, VIIe Congr. int. acqui. et de pêche, Paris, 1931 1-14. |
| [2] | Lotka, A. J., Elements of Mathematical Biology (1956), Dover: Dover New York · Zbl 0074.14404 |
| [3] | Malchow, H.; Petrovskii, S.; Venturino, E., Spatiotemporal Patterns in Ecology and Epidemiology (2008), CRC: CRC Boca Raton · Zbl 1298.92004 |
| [4] | Murray, J. D., Mathematical Biology (1989), Springer Verlag: Springer Verlag New York · Zbl 0682.92001 |
| [5] | Leslie, P. H.; Gower, J. C., The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47, 219-234 (1960) · Zbl 0103.12502 |
| [6] | Holling, C. S., The functional response of invertebrate predators to prey density, Mem. Entomol. Soc. Can., 45, 3-60 (1965) |
| [7] | Tanner, J. T., The stability and intrinsic growth rates of prey and predator populations, Ecology, 56, 855-867 (1975) |
| [8] | Braza, P. A., The bifurcations structure for the Holling Tanner model for predator-prey interections using two-timing, SIAM. J. Appl. Math., 63, 889-904 (2003) · Zbl 1035.34043 |
| [9] | Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, J. Math. Biol., 42, 490-506 (2001) · Zbl 0984.92035 |
| [10] | Hsu, S. B.; Hwang, T. W.; Kuang, Y., A ratio-dependent food chain model and its application to biological control, Math. Biosci., 181, 55-83 (2003) · Zbl 1036.92033 |
| [11] | Abrams, P. A., The fallacies of ratio-dependent predation, Ecology, 75, 6, 1842-1850 (2003) |
| [12] | Abrams, P. A.; Ginzburg, L. R., The nature of predation: prey dependent, ratio dependent or neither?, Trends. Ecol. Evol., 15, 337-341 (2000) |
| [13] | Akcakaya, H. R., Population cycles of mammals: evidence for ratio-dependent predator-prey hypothesis, Ecol. Monogr., 62, 119-142 (1992) |
| [14] | Matsuoka, T.; Seno, H., Possibly longest food chain: analysis of a mathematical model, Math. Model. Nat. Phenom., 3, 4, 131-160 (2008) · Zbl 1337.92186 |
| [15] | Freedman, H. I.; Wolkowitz, G., Predator-prey systems with group defence: the paradox of enrichment revisited, Bull. Math. Biol., 48, 493-508 (1986) · Zbl 0612.92017 |
| [16] | Chattopadhyay, J.; Chatterjee, S.; Venturino, E., Patchy agglomeration as a transition from monospecies to recurrent plankton blooms, Journal of Theoretical Biology, 253, 289-295 (2008) · Zbl 1398.92270 |
| [17] | V. Ajraldi, E. Venturino, Mimicking spatial effects in predator-prey models with group defense, in: J. Vigo Aguiar, P. Alonso, S. Oharu, E. Venturino, B. Wade (Eds.), Proceedings of the International Conference CMMSE 2009, 1, 2009, pp. 57-66.; V. Ajraldi, E. Venturino, Mimicking spatial effects in predator-prey models with group defense, in: J. Vigo Aguiar, P. Alonso, S. Oharu, E. Venturino, B. Wade (Eds.), Proceedings of the International Conference CMMSE 2009, 1, 2009, pp. 57-66. |
| [18] | Apreutesei, N.; Ducrot, A.; Volpert, V., Competition of species with intra-specific competition, Math. Model. Nat. Phenom., 3, 4, 1-27 (2008) · Zbl 1337.35068 |
| [19] | Waltman, P., Competition Models in Population Biology (1983), SIAM: SIAM Philadelphia · Zbl 0572.92019 |
| [20] | Hirsch, M.; Smale, S., Differential Equations, Dynamical Systems, and Linear Algebra (1974), Academic Press: Academic Press New York · Zbl 0309.34001 |
| [21] | Ackleh, A. S.; Zhang, P., Competitive exclusion in a discrete stage-structured two species model, Math. Model. Nat. Phenom., 4, 6, 156-175 (2009) · Zbl 1183.92078 |
| [22] | Goh, B. S., Stability in models of mutualism, Am. Nat., 113, 261-275 (1979) |
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.
