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**Three-dimensional bifurcation analysis of a predator-prey model with uncertain formulation.**
*(English)*
Zbl 1414.34031

This paper focuses on structural sensitivity in using alternative choices in model formulation. An example of a predator-prey system with an uncertain functional response that exhibits complex bifurcations is introduced to illustrate how qualitative changes in bifurcations occur during a small continuous change between two acceptable model formulations. A bifurcation analysis reveals that most changes that occur with the change in formulation are relevant to a codimension-three degenerated Bogdanov-Takens bifurcation. Its canonical unfolding is derived, and its analysis highlights the differences in the phase portraits predicted with the two model formulations. The concept and method in this paper are novel and general and can be applied to other fields involving the modeling of complex systems.

Reviewer: Xinyu Song (Xinyang)

### MSC:

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

34C23 | Bifurcation theory for ordinary differential equations |

34D30 | Structural stability and analogous concepts of solutions to ordinary differential equations |

37G05 | Normal forms for dynamical systems |

92D25 | Population dynamics (general) |

### Keywords:

bifurcation analysis; codimension-three Bogdanov-Takens bifurcation; structural sensitivity; population models; functional response
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\textit{C. Aldebert} et al., SIAM J. Appl. Math. 79, No. 1, 377--395 (2019; Zbl 1414.34031)

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### References:

[1] | M. Adamson and A. Morozov, When can we trust our model predictions? Unearthing structural sensitivity in biological systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 469 (2013), 20120500. · Zbl 1371.92057 |

[2] | M. Adamson and A. Morozov, Bifurcation analysis of models with uncertain function specification: How should we proceed?, Bull. Math. Biol., 76 (2014), pp. 1218–1240. · Zbl 1297.92059 |

[3] | M. Adamson and A. Morozov, Defining and detecting structural sensitivity in biological models: Developing a new framework, J. Math. Biol., 69 (2014), pp. 1815–1848. · Zbl 1308.34073 |

[4] | C. Aldebert, Uncertainty in Predictive Ecology: Consequence of Choices in Model Construction, Ph.D. thesis, Aix-Marseille University, Marseille, France, 2016. |

[5] | C. Aldebert, B. Kooi, D. Nerini, and J. Poggiale, Is structural sensitivity a problem of oversimplified biological models? Insights from nested dynamic energy budget models, J. Theoret. Biol., 448 (2018), pp. 1–8. · Zbl 1397.92706 |

[6] | C. Aldebert, D. Nerini, M. Gauduchon, and J. Poggiale, Does structural sensitivity alter complexity-stability relationships?, Ecol. Complex., 28 (2016), pp. 104–112. |

[7] | C. Aldebert, D. Nerini, M. Gauduchon, and J. Poggiale, Structural sensitivity and resilience in a predator-prey model with density-dependent mortality, Ecol. Complex., 28 (2016), pp. 163–173. |

[8] | C. Aldebert and D. Stouffer, Community dynamics and sensitivity to model structure: Toward a probabilistic view of process-based model predictions, J. Royal Soc. Interface, 15 (2018), 20180741. |

[9] | T. Anderson, W. Gentleman, and B. Sinha, Influence of grazing formulations on the emergent properties of a complex ecosystem model in a global ocean general circulation model, Prog. Oceanogr., 87 (2010), pp. 201–213. |

[10] | A. Andronov and L. Pontryagin, Coarse systems, Dokl. Akad. Nauk SSSR, 14 (1937), pp. 247–251. |

[11] | S. M. Baer, B. W. Kooi, Y. A. Kuznetsov, and H. R. Thieme, Multiparametric bifurcation analysis of a basic two-stage population model, SIAM J. Appl. Math., 66 (2006), pp. 1339–1365, . · Zbl 1106.34030 |

[12] | M. Baklouti, F. Diaz, C. Pinazo, V. Faure, and B. Quéguiner, Investigation of mechanistic formulations depicting phytoplankton dynamics for models of marine pelagic ecosystems and description of a new model, Prog. Oceanogr., 71 (2006), pp. 1–33. |

[13] | M. Baklouti, V. Faure, L. Pawlowski, and A. Sciandra, Investigation and sensitivity analysis of a mechanistic phytoplankton model implemented in a new modular numerical tool (eco\textup3m) dedicated to biogeochemical modelling, Prog. Oceanogr., 71 (2006), pp. 34–58. |

[14] | A. Bazykin, Nonlinear Dynamics of Interacting Populations, World Scientific, Singapore, 1998. |

[15] | L. Breiman, Statistical modeling: The two cultures, Statist. Sci., 16 (2001), pp. 199–231. · Zbl 1059.62505 |

[16] | U. Brose, R. Williams, and N. Martinez, Allometric scaling enhances stability in complex food webs, Ecol. Lett., 9 (2006), pp. 1228–1236. |

[17] | J. Brown, J. Gillooly, A. Allen, V. Savage, and G. West, Toward a metabolic theory of ecology, Ecology, 85 (2004), pp. 1771–1789. |

[18] | F. Cordoleani, D. Nerini, M. Gauduchon, A. Morozov, and J. Poggiale, Structural sensitivity of biological models revisited, J. Theoret. Biol., 283 (2011), pp. 82–91. · Zbl 1397.92565 |

[19] | A. Dhooge, W. Govaerts, Y. Kuznetsov, W. Mestrom, A. Riet, and B. Sautois, MATCONT and CL_MATCONT: Continuation Toolboxes in MATLAB, , 2011. |

[20] | R. Dickinson and J. Gelinas, Sensitivity analysis of ordinary differential equation systems—A direct method, J. Comput. Phys., 21 (1976), pp. 123–143. · Zbl 0333.65038 |

[21] | E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X. Wang, Auto 97: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, Canada, 1997. |

[22] | F. Dumortier, R. Roussarie, J. Sotomayor, and H. Żoladek, Bifurcations of Planar Vector Fields. Nilpotent Singularities and Abelian Integrals, Springer-Verlag, Berlin, 1991. · Zbl 0755.58002 |

[23] | E. Fulton, A. Smith, and C. Johnson, Effect of complexity on marine ecosystem models, Mar. Ecol. Prog. Ser., 253 (2003), pp. 1–16. |

[24] | G. Fussmann and B. Blasius, Community response to enrichment is highly sensitive to model structure, Biol. Lett., 1 (2005), pp. 9–12. |

[25] | G. Fussmann, S. Ellner, K. Shertzer, and N. Hairston, Jr., Crossing the Hopf bifurcation in a live predator-prey system, Science, 290 (2000), pp. 1358–1360. |

[26] | T. Gross, W. Ebenhöh, and U. Feudel, Enrichment and foodchain stability: The impact of different forms of predator-prey interaction, J. Theoret. Biol., 227 (2004), pp. 349–358. |

[27] | T. Gross and U. Feudel, Generalized models as a universal approach to the analysis of nonlinear dynamical systems, Phys. Rev. E (3), 73 (2006), 016205. |

[28] | J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. · Zbl 0515.34001 |

[29] | R. Hall and A. Hastings, Minimizing invader impacts: Striking the right balance between removal and restoration, J. Theoret. Biol., 249 (2007), pp. 437–444. |

[30] | L. Heckmann, B. Drossel, U. Brose, and C. Guill, Interactive effects of body-size structure and adaptative foraging on food-web stability, Ecol. Lett., 15 (2012), pp. 243–250. |

[31] | C. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), pp. 385–398. |

[32] | C. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 45 (1965), pp. 3–60. |

[33] | V. Ivlev, Experimental Ecology of the Feeding of Fishes, Pischepromizdat, Moscow, 1955. |

[34] | J. Jeschke, M. Kopp, and R. Tollrian, Predator functional response: Discriminating between handling and digesting prey, Ecol. Monogr., 72 (2002), pp. 95–112. |

[35] | S. Jø rgensen, B. Fath, S. Bastianoni, J. Marques, F. Müller, S. Nielsen, B. Patten, E. Tiezzi, and R. Ulanowicz, A New Ecology Systems Perspective, Elsevier, Oxford, UK, 2007. |

[36] | B. Kartascheff, C. Guill, and B. Drossel, Positive complexity-stability relations in food web models without foraging adaptation, J. Theoret. Biol., 259 (2009), pp. 12–23. · Zbl 1402.92424 |

[37] | B. Kartascheff, L. Heckmann, B. Drossel, and C. Guill, Why allometric scaling enhances stability in food web models, Theor. Ecol., 3 (2010), pp. 195–208. |

[38] | S. Kooijman, Dynamic Energy Budget Theory for Metabolic Organisation, 3rd ed., Cambridge University Press, Cambridge, UK, 2010. |

[39] | Y. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed., Springer, New York, 2004. · Zbl 1082.37002 |

[40] | M. Loreau, From Populations to Ecosystems: Theoretical Foundations for a New Ecological Synthesis, Princeton University Press, Princeton, NJ, 2010. |

[41] | D. Ludwig, D. Jones, and C. Holling, Qualitative analysis of insect outbreak systems: The spruce budworm and forest, J. Anim. Ecol., 47 (1978), pp. 315–332. |

[42] | M. Myerscough, M. Darwen, and W. Hogarth, Stability, persistence and structural stability in a classical predator-prey model, Ecol. Model., 89 (1996), pp. 31–42. |

[43] | V. Nozdracheva, Bifurcation of structurally unstable separatrix loop, Differ. Uravn., 18 (1982), pp. 1551–1558. · Zbl 0514.58030 |

[44] | J. Poggiale, M. Baklouti, B. Queguiner, and S. Kooijman, How far details are important in ecosystem modelling: The case of multi-limiting nutrients in phytoplankton-zooplankton interactions, Philos. Trans. Roy. Soc. London Ser. B, 365 (2010), pp. 3495–3507. |

[45] | M. Rosenzweig, Paradox of enrichment: Destablilization of exploitation ecosystems in ecological time, Science, 171 (1971), pp. 385–387. |

[46] | M. Rosenzweig and R. MacArthur, Graphical representation and stability conditions of predator-prey interaction, Am. Nat., 97 (1963), pp. 209–223. |

[47] | F. Scharf, F. Juanes, and R. Rountree, Predator size-prey size relationships of marine fish predators: Interspecific variation and effects of ontogeny and body size on trophic-niche breadth, Mar. Ecol. Prog. Ser., 208 (2000), pp. 229–248. |

[48] | M. Scheffer, J. Bascompte, W. Brock, V. Brokvin, S. Carpenter, V. Dakos, H. Held, E. van Nes, M. Rietkerk, and G. Sugihara, Early-warning signals for critical transitions, Nature, 461 (2009), pp. 53–59. |

[49] | G. van Voorn and B. Kooi, Combining bifurcation and sensitivity analysis for ecological models: Model analysis, and the allegory of the cave, Eur. Phys. J. Spec. Top., 226 (2017), pp. 2101–2118. |

[50] | A. K. Wilkins, B. Tidor, J. White, and P. I. Barton, Sensitivity analysis for oscillating dynamical systems, SIAM J. Sci. Comput., 31 (2009), pp. 2706–2732, . · Zbl 1201.65122 |

[51] | S. Wood and M. Thomas, Super–sensitivity to structure in biological models, Philos. Trans. Roy. Soc. London Ser. B, 266 (1999), pp. 565–570. |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.