Population Dynamics Model for Coexistence of Three Interacting Species
Applied and Computational Mathematics
Volume 4, Issue 4, August 2015, Pages: 258-263
Received: May 31, 2015;
Accepted: Jun. 11, 2015;
Published: Jun. 29, 2015
Views 3604 Downloads 120
M. Rafique, Department of Mathematics, Faculty of Science, HITEC University, Taxila, Pakistan
M. Abdul Qader, Department of Mathematics, Faculty of Science, HITEC University, Taxila, Pakistan
Follow on us
Over the years applications of mathematics in the form of mathematical modeling in a whole range of different fields including physical, social, management, biological, and medical sciences have broken all bounds. In particular, the mathematical models to study population dynamics of various interacting species in an isolated environment have attracted the attention of mathematical biologists. In nature, there may be two, three, or more species interacting within themselves giving rise to the corresponding predator-prey models. In each case, both predator and prey evolve their own strategies to deal with the situation. The parameters which influence both the predator and the pry to evoke strategies for their survival include environmental conditions, predator’s appetite, aggressiveness, liking for some particular prey, its physical fitness versus that of the prey, prey’s agility, active prudence to run away or hide, etc. In the literature interactions between, two, three or more species, sharing the same habitat have been discussed in detail. In this paper we present a model pertaining to the interaction between three species. It is a realistic model in which three species, x, y and z, interact within themselves in such a way that species y (predator) preys on species x (prey), while the species z preys on both the species x and y. Accordingly, the resulting situation has been analyzed. The objective of this paper is to analyze the possibility for three interacting species to live in an isolated environment harmoniously. The model presented here has three equilibrium points, however, only one of them has been ascertained to be locally stable. The existence of this equilibrium point signifies amicable coexistence of the three species, if no outside intervention accrues any destabilization to the existing environment.
Malthusian Growth Model, Carrying Capacity of Environment, Logistic Equation, Malthus-Verhulst Equation, Lotka-Volterra Equations, Equilibrium Point, Jacobian Matrix
To cite this article
M. Abdul Qader,
Population Dynamics Model for Coexistence of Three Interacting Species, Applied and Computational Mathematics.
Vol. 4, No. 4,
2015, pp. 258-263.
T. R. Malthus: An Essay on the Principles of Population, Oxford World’s Classics reprint (1798)
P. F. Verhulst: Recherches mathématiques sur la loi d'accroissement de la population, Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1-41, (1845).
P.F.Verhulst: Deuxième mémoire sur la loi d'accroissement de la population, Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20, 1-32, (1847)
Eric W. Weisstein: Logistic Equation, From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogisticEquation.html
A. J. Lotka: Elements of Physical Biology, Williams & Wilkins, Baltimore (1925)
V. Volterra: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. R. Accad. Naz. dei Lincei, Ser. VI, vol. 2. (1926)
M. L. Rosenzweig: Exploitation in three trophic levels, Americal Naturalist, 107, 275-294 (1973)
D. J. Wolkind: Exploitation in three trophic levels: an extension allowing intraspecies carnivore interaction, American Naturalist, 110, 431-447 (1976)
H. C. Hilborn: Chaos and Nonlinear Dynamics, Oxford University Press (1994)
R. M. May: Stability and Complexity in Model Ecosystems, Princeton University press (2001)
E. Chauvet et. al. : A Lotka-Volterra Three-Species Food Chain, mathematics Magazine, vol. 75, No. 4 (2002)
M. Mamat et. al. : Numerical Simulation Dynamical Model of Three-Species Food Chain with Lotka-Volterra Linear Functional Response, Journal of Sustainability Science and Management, Vol. 6. No. 1, 44-50 (2011)
Islam Sallam et. al.: Finding the Balance: Population, Natural Resources and Sustainability, International Journal of Innovative research in Science, Engineering and Technology, 7701-7715 (2013)
Alan Hastings: Chaos in Three-Species Food Chain, Ecology, Vol. 72, No. 3,896-903 (1991)
A. Korobeinikov , and G. C. Wake: Global Properties of the Three-Dimensional Predator-Prey Lotka-Volterrs Systems, Journal of applied Mathematics & Decision Sciences, 3(2), 155-162 (1999)