Keywords
periodic orbits
models with competence
models with mutualism
models with capture
Funciones de Dulac
orbitas periódicas
modelos de competencia
modelos con mutualismo
modelos con captura
How to Cite
Abstract
To exclude the existence of sustained oscillations in mathematical ecology models is of vital interest to know the populational dynamics of interacting species. In this paper, Dulac functions are constructed for some generalizations of models commonly used in populational ecology. This result excludes the existence of periodic orbits for some models that describe intraspecific and / or interspecific competition, relationships of competition, predation and capture of individuals of any of the species.
References
L. Cherkas, A. Grin A., Algebraic aspects of finding a Dulac function for polynomial autonomous systems on the plane, Differ. Equa., 37(2001), no.3, 411–417, Doi: 10.1023/A:1019250717290
J.R. Cooley, D.C. Marshall, K.B.R. Hill, A specialized fungal parasite (Massospora cicadina) hijacks the sexual signals of periodical cicadas (Hemiptera: Cicadidae: Magicicada), Scientific Reports 8(2018), 1432, Doi: 10.1038/s41598-018-19813-0
M.P. Kahl Jr., Food ecology of the Wood Stork (Mycteria Americana) in Florida, Ecological Monographs 34(1964), no. 2, 97–117. Doi: 10.2307/1948449
J.D. Murray, Mathematical Biology - I. An Introduction, Springer-Verlag, New York, 2002. Doi: 10.1007/b98868
O. Osuna, C. Vargas, Construction of Dulac functions for mathematical models in population biology, International Journal of Biomathematics, 8(2015), no. 03. Doi: 10.1142/S1793524515500357
O. Osuna, G. Villaseñor, On the Dulac functions, Qualitative Theory of Dynamical Systems 10(2011), no. 1, 43 49. Doi: 10.1007/s12346-011-0036-y
O. Osuna, G. Villaseñor, Some properties of the Dulac functions set, E.J. of Qualitative Theory of Differential Equations 72(2011), 1–8. Doi: 10.14232/ejqtde.2011.1.72
O. Osuna, G. Villavicencio, Generating Dulac functions for differential equations and biological models, Avanza 6(2016) FM-IIT, UACJ, 1–8.
E. Saez, I. Szántó, On the construction of certain Dulac function, Trans. Automat. Control 33(1988), no. 9, 856. Doi: 10.1109/9.1315
R. Sarkar, J. Chattopadhayay, Occurrence of planktonic blooms under environmental fluctuations and its possible control mechanism–Mathematical models and experimental observations, Journal of Theoretical Biology, 224(2003), no. 4, 501–516. Doi: 10.1016/S0022-5193(03)00200-5
M. Scheffer, Fish and nutrients interplay determines algal biomass: A minimal model, Oikos 62(1991), no. 3, 271–282. Doi: 10.2307/3545491
T. Sota, S. Yamamoto, J.R. Cooley, K.B.R. Hill, C. Simon, J. Yoshimura, Independent divergence of 13- and 17-y life cycles among three periodical cicada lineages, PNAS 110(2013), no. 17, 6919–6924. Doi: 10.1073/ pnas.1220060110
E. Venturino, The effects of the diseases on competing species, Mathematical Biosciences 174(2001), no. 2, 111–131. Doi: 10.1016/S0025-5564(01)00081-5