The dynamics of two interacting populations

Albrecht, Felix; Gatzke, Harry; Haddad, A.; Wax, Nelson

doi:10.1016/0022-247x(74)90267-4

Cited by 97 publications

(32 citation statements)

References 2 publications

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“…x' = xf(x, y), Under assumptions (pl), (p2), (p3), Albrecht et al [1] show that the solution (x(t), y(t)) of (3.1) is positive and bounded and there exists a unique equilibrium (x*,y*) in QO. Then (x*,y*) is a global attractor.…”

Section: Kolmogorov Model and Othersmentioning

confidence: 99%

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Some results on global stability of a predator-prey system

1982

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Abstract. In this paper we derive some results to ensure the global stability of a predator-prey system. The results cover most of the models which have been proposed in the ecological literature for predator-prey systems. The first result is very geometric and it is very easy to check from the graph of prey and predator isoclines. The second one is purely algebraic, however, it covers the defects of the first one especially in dealing with Holling's type-3 functional response in some sense. We also discuss the global stability of Kolmogorov's model. Some examples are presented in the discussion section.

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Section: Kolmogorov Model and Othersmentioning

confidence: 99%

“…The notations and assumptions of the Kolmogorov model basically follow from [1]. We first state the hypotheses of the model.…”

Section: Kolmogorov Model and Othersmentioning

confidence: 99%

Some results on global stability of a predator-prey system

1982

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“…The functions f and g describe the nature of the interactions between populations. Models in this form have a long history of investigation, for example, having been studied by Kolmogoroff, [20], Rescigno and Richardson, [26], [27], May, [23], and Albrecht, Gatzke, and Haddad and Wax, [1]. See Chapter 5 of Freedman, [6], for a thorough discussion of such models.…”

Section: Predator-prey Systemsmentioning

confidence: 99%

Did Something Change? Thresholds in Population Models

Hoppensteadt¹,

Waltman

2003

Trends in Nonlinear Analysis

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Dedication: This paper is dedicated to our good friend Professor Willi Jäger on the occasion of his 60th birthday. The three of us have collaborated since a momentous year (for us) at the Courant Institute of Mathematical Sciences in 1969-70.Abstract: The goal of this article is to illustrate several interesting bifurcations that can arise in population biology. These are of interest since it is often through bifurcation phenomena that changes significant enough to be measured occur. For example, a minor change in some environmental parameter can cause a system to change from being at rest to oscillating. We illustrate here the role of several canonical types of bifurcations in population modeling. IntroductionThe subject of mathematical population biology is a vast one, much too large to be surveyed in a single article. Since the thrust of this volume is to present material in a way to make the subject attractive to students, we have chosen to focus on specific examples and to illustrate the phenomenon of change. This forms one of the connections between theories and experiments since experiments often focus on when a moderate change in a parameter produces a major change in the outcome. Such changes in the model's solutions can reflect the occurrence of a bifurcation of solutions and with it an exchange of stabilities. For example, a stable static state can bifurcate into a new stable state or a stable oscillation as model parameters change. We present here several examples of where bifurcation phenomena have been observed in population biology and how they have been used to explain, predict and control outcomes. A major proponent of this approach is Rene Thom [30] who used canonical forms of singularities to identify bifurcation phenomena in developmental biology. In [14] a systematic development of certain bifurcations that arise in neuroscience is presented, and there are many excellent texts that present bifurcation phenomena for mathematicians and engineers, for example [2, 8,

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“…Ayala etal [21] studied the theoretical models through experimental tests. Later Albrecht etal [22], Gopalaswamy [23], Brown etal [24] studied the stability of two species systems.…”

Section: Introductionmentioning

confidence: 99%

“…Ayala etal [21] studied the theoretical models through experimental tests. Later Albrecht etal [22], Gopalaswamy [23], Brown etal [24] studied the stability of two species systems.The systems of non linear differential equations are perfect and an abstract representation of the real problem and attracted the attention of mathematicians [3], [4], [7] and [9] and ecologists. May [25] studied the nonlinear aspects of competition between three species.…”

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confidence: 99%

Two Species and Three Species Ecological Modeling - Homotopy Analysis

Putcha¹

2012

Diversity of Ecosystems

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11www.intechopen.com 2 Will-be-set-by-IN-TECH two-species systems and three-species systems. There are three main types of interaction, (i) If the growth rate of one population is decreased and the other increased the populations are in a predator-prey situation, (ii) If the growth rate of each population is decreased then it is competition, (iii) If each population's growth rate is enhanced then it is called mutualism or symbiosis.Eco system comprising organisms and a biotic environment is a functional unit of ecology and non linear differential equations of first order are used to represent complex ecological phenomena. In 1925 Volterra [20] first proposed a simple model for the predation of one species by another to explain the oscillatory levels of certain fish catches in the Adriatic with the following assumptions (i) The prey in the absence of any predation grows unboundedly in a Malthusian way (ii) The effect of the predation is to reduce the prey's per capita growth rate by a term proportional to the prey and predator populations (iii) In the absence of any prey for sustenance the predator's death rate results in exponential decay (iv) The prey's contribution to the predators' growth rate is proportional to the available prey as well as to the size of the predator population. Ayala etal [21] studied the theoretical models through experimental tests. Later Albrecht etal [22], Gopalaswamy [23], Brown etal [24] studied the stability of two species systems.The systems of non linear differential equations are perfect and an abstract representation of the real problem and attracted the attention of mathematicians [3], [4], [7] and [9] and ecologists. May [25] studied the nonlinear aspects of competition between three species. Schuster etal [26] and Zhang Jinyan [27] studied three species systems.So it is of great interest to formulate a mathematical models on Two Species and Three Species Ecological systems and find their approximate analytical solutions. In general construction of analytic approximations of nonlinear problems with strong nonlinearity is not easy and constructed Solutions are mainly determined by the type of nonlinearity and the corresponding technique. The region of convergence of obtained solutions depend on physical parameters rather than the analytical technique.Though numerical techniques can be used to solve the nonlinear system of differential equations the functional form of reasonably good solution, which is of vital use for detailed study of the system, can be constructed by Homotopy methods. In general it is difficult to obtain analytic approximations of nonlinear problems with strong nonlinearity. Traditionally, solution expressions of a nonlinear problem are mainly determined by the type of nonlinear equations and the employed analytic techniques, and the convergence regions of solution series are strongly dependent of physical parameters.The Homotopy analysis method HAM, initially proposed by Liao in his Ph.D. thesis [15], is a powerful analytic method for nonlinear problems. ...

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The dynamics of two interacting populations

Cited by 97 publications

References 2 publications

Some results on global stability of a predator-prey system

Some results on global stability of a predator-prey system

Did Something Change? Thresholds in Population Models

Two Species and Three Species Ecological Modeling - Homotopy Analysis

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