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Global Dynamics of Modified Discrete Lotka-Volterra Model

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Advances in Discrete Dynamical Systems, Difference Equations and Applications (ICDEA 2021)

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Abstract

In this paper, we will prove general results regarding the global stability of planar monotone systems without minimal period-two solutions on a rectangular region \(\mathcal {R}\). We will illustrate the general results by an example of a well known system used in mathematical biology, which is a modified Leslie-Gower system of the form

$$\begin{aligned} x_{n+1} = \alpha x_n +(1-\alpha )\frac{cx_n}{a+cx_n+y_n}, \\ y_{n+1} = \beta y_n +(1-\beta )\frac{dy_n}{b+x_n+dy_n}, \end{aligned}$$

\(n=0,1,...,\) where the parameters a, b, c, d are positive numbers, \(\alpha , \beta \in (0,1)\), and the initial conditions \(x_{0},y_{0}\) are arbitrary nonnegative numbers. In most cases for different values of a, b, c, and d, there will either be one, two, three, or four equilibrium solutions present with at most one an interior equilibrium point. In the case when \(c=d=1\) and \(a=b\) there will exist an infinite number of interior equilibrium points, in which case we will find the basin of attraction for each of the equilibrium points.

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Authors and Affiliations

  1. University of Rhode Island, Kingston, RI, 02881, USA

    M. R. S. Kulenović & Sarah Van Beaver

Authors
  1. M. R. S. Kulenović
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  2. Sarah Van Beaver
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Correspondence to M. R. S. Kulenović .

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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. University of Rhode Island, Kingston, RI, USA

    Mustafa R. S. Kulenović

  3. University of Sarajevo, Sarajevo, Bosnia and Herzegovina

    Senada Kalabušić

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Kulenović, M.R.S., Van Beaver, S. (2023). Global Dynamics of Modified Discrete Lotka-Volterra Model. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_15

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