# Von Bertalanffy's growth dynamics with strong Allee effect

J. Leonel Rocha; Sandra M. Aleixo

Discussiones Mathematicae Probability and Statistics (2012)

- Volume: 32, Issue: 1-2, page 35-45
- ISSN: 1509-9423

## Access Full Article

top## Abstract

top## How to cite

topJ. Leonel Rocha, and Sandra M. Aleixo. "Von Bertalanffy's growth dynamics with strong Allee effect." Discussiones Mathematicae Probability and Statistics 32.1-2 (2012): 35-45. <http://eudml.org/doc/270990>.

@article{J2012,

abstract = {Von Bertalanffy’s model is one of the most popular differential equation used in order to study the increase in average length or weight of fish. However, this model does not include demographic Allee effect. This phenomenon is known in the fisheries literature as “depensation”, which arises when populations decline rapidly at low densities. In this paper we develop and investigate new corrected von Bertalanffy’s models with Allee effects. The generalization that we propose results from considering correction factors, one of rational type and the others of polynomial type, where two parameters are considered. The use of a parameter C > 0 leads the presented generalization, which yields some more flexible models with variable extinction rates. An Allee limit or unstable equilibrium $E^u$ is incorporated so that the models under study have strong Allee effect. We analyze and show the transition from the strong Allee effect to the inexistence of this effect, through by a “weakening” of the Allee effect, depending on the parameters C and $E^u$. Finally, we discuss the flexibility of corrected von Bertalanffy’s sigmoid growth curves. So, the correspondents inflexion points are variable, i.e., the fish mass when the growth rate is maximum may be varied.},

author = {J. Leonel Rocha, Sandra M. Aleixo},

journal = {Discussiones Mathematicae Probability and Statistics},

keywords = {Von Bertalanffy's model; population dynamics; Allee effect; von Bertalanffy's model population dynamics},

language = {eng},

number = {1-2},

pages = {35-45},

title = {Von Bertalanffy's growth dynamics with strong Allee effect},

url = {http://eudml.org/doc/270990},

volume = {32},

year = {2012},

}

TY - JOUR

AU - J. Leonel Rocha

AU - Sandra M. Aleixo

TI - Von Bertalanffy's growth dynamics with strong Allee effect

JO - Discussiones Mathematicae Probability and Statistics

PY - 2012

VL - 32

IS - 1-2

SP - 35

EP - 45

AB - Von Bertalanffy’s model is one of the most popular differential equation used in order to study the increase in average length or weight of fish. However, this model does not include demographic Allee effect. This phenomenon is known in the fisheries literature as “depensation”, which arises when populations decline rapidly at low densities. In this paper we develop and investigate new corrected von Bertalanffy’s models with Allee effects. The generalization that we propose results from considering correction factors, one of rational type and the others of polynomial type, where two parameters are considered. The use of a parameter C > 0 leads the presented generalization, which yields some more flexible models with variable extinction rates. An Allee limit or unstable equilibrium $E^u$ is incorporated so that the models under study have strong Allee effect. We analyze and show the transition from the strong Allee effect to the inexistence of this effect, through by a “weakening” of the Allee effect, depending on the parameters C and $E^u$. Finally, we discuss the flexibility of corrected von Bertalanffy’s sigmoid growth curves. So, the correspondents inflexion points are variable, i.e., the fish mass when the growth rate is maximum may be varied.

LA - eng

KW - Von Bertalanffy's model; population dynamics; Allee effect; von Bertalanffy's model population dynamics

UR - http://eudml.org/doc/270990

ER -

## References

top- [1] S.M. Aleixo, J.L. Rocha and D.D. Pestana , Populational growth models proportional to beta densities with Allee effect , Am. Inst. Phys. (1124) 2009 3-12
- [2] S.M. Aleixo and J.L. Rocha, Generalized models from Beta(p,2) densities with strong Allee effect: dynamical approach , J. Comput. Inf. Technol 3 (2012) 201-207
- [3] W.C. Allee, Animal aggregations (University of Chicago Press, Chicago, 1931)
- [4] D.S. Boukal and L. Berec, Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters , J. Theor. Biol. 218 (2002) 375-394. doi: 10.1006/jtbi.2002.3084
- [5] C.E. Brassil, Mean time to extinction of a metapopulation with an Allee effect, Ecol. Model. 143 (2001) 9-16. doi: 10.1016/S0304-3800(01)00351-9
- [6] G.M. Cailliet, W.D. Smith, H.F. Mollet and K.J. Goldman, Age and growth studies of chondrichthyan fishes: the need for consistency in terminology, verification, validation and growth function fitting, Environ. Biol. Fish. 77 (2006) 211-228. doi: 10.1007/s10641-006-9105-5
- [7] A.B. Cooper, A guide to fisheries stock assessment. From data to recommendations (New Hampshire Sea Grant, 2006)
- [8] T.E. Essington, J.F. Kitchell and C.J. Walters , The von Bertalanffy growth function, bioenergetics and the consumption rates of fish, Can. J. Fish. Aquat. Sci. 58 (2001) 2129-2138. doi: 10.1139/f01-151
- [9] J.A. Hutchings and J.D. Reynolds, Marine fish population collapses: consequences for recovery and extinction risk, BioSci. 54 (2004) 297-309. doi: 10.1641/0006-3568(2004)054[0297:MFPCCF]2.0.CO;2
- [10] A.M. Kramer, B. Dennis, A.M. Liebhold and J.M. Drake, The evidence for Allee effects, Popul. Ecol. 51 (2009) 341-354. doi: 10.1007/s10144-009-0152-6
- [11] J.A. Musick, Criteria to define extinction risk in marine fishes, Fisheries 24 (1999) 6-14. doi: 10.1577/1548-8446(1999)024<0006:CTDERI>2.0.CO;2
- [12] H.T. Odum and W.C. Allee, A note on the stable point of populations showing both intraspecific cooperation and disoperation , Ecology 35 (1954) 95-97. doi: 10.2307/1931412
- [13] J.L. Rocha and S.M. Aleixo, Modeling Allee effect from Beta(p,2) densities, Proc. ITI 2012, 34th Int. Conf. Information Technology Interfaces (2012) 461-466
- [14] S.J. Schreiber, Allee effects, extinctions and chaotic transients in simple population models, Theor. Popul. Biol. 64 (2003) 201-209. doi: 10.1016/S0040-5809(03)00072-8 Zbl1104.92053
- [15] A. Tsoularis, Analysis of logistic growth models , Res. Lett. Inf. Math. Sci. 2 (2001) 23-46
- [16] M. Wang and M. Kot, Speeds of invasion in a model with strong or weak Allee effects, Math. Biosci. 171 (2001) 83-97. doi: 10.1016/S0025-5564(01)00048-7
- [17] L. Von Bertalanffy, A quantitative theory of organic growth, Human Biology 10 (1938) 181-213
- [18] L. Von Bertalanffy, Quantitative laws in metabolism and growth, The Quarterly Review of Biology 32 (1957) 217-231. doi: 10.1086/401873

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.