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Mustafa Kulenović

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This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto’s theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations.

This paper investigates the dynamics of non-autonomous competitive systems of difference equations with asymptotically constant coefficients. We are mainly interested in global attractivity results for such systems and the application of such results to the evolutionary population of competition models of two species.

M. Kulenović, Connor O’Loughlin, E. Pilav

<jats:p>We present the bifurcation results for the difference equation <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> </mrow> <mo>/</mo> <mfenced open="(" close=")" separators="|"> <mrow> <mi>a</mi> <msubsup> <mi>x</mi> <mi>n</mi> <mn>2</mn> </msubsup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mn>2</mn> </msubsup> <mo>+</mo> <mi>f</mi> </mrow> </mfenced> </math> </jats:inline-formula> where <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>a</mi> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi>f</mi> </math> </jats:inline-formula> are positive numbers and the initial conditions <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> </math> </jats:inline-formula> and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> </jats:inline-formula> are nonnegative numbers. This difference equation is one of the perturbations of the sigmoid Beverton–Holt difference equation, which is a major mathematical model in population dynamics. We will show that this difference equation exhibits transcritical and Neimark–Sacker bifurcations but not flip (period-doubling) bifurcation since this difference equation cannot have period-two solutions. Furthermore, we give the asymptotic approximation of the invariant manifolds, stable, unstable, and center manifolds of the equilibrium solutions. We give the necessary and sufficient conditions for global asymptotic stability of the zero equilibrium as well as sufficient conditions for global asymptotic stability of the positive equilibrium.</jats:p>

M. Kulenović, J. Marcotte, O. Merino

It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction \begin{document}$ \mathcal{B} $\end{document} with relatively simple geometry: the boundary of each component of \begin{document}$ \mathcal{B} $\end{document} consists of the union of two unordered curves, and the components of \begin{document}$ \mathcal{B} $\end{document} are not comparable as sets. The boundary curves are Lipschitz if the map is of class \begin{document}$ C^1 $\end{document} . Further, if a periodic point is in \begin{document}$ \partial \mathcal{B} $\end{document} , then \begin{document}$ \partial\mathcal{B} $\end{document} is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

Toufik Khyat, M. Kulenović

<jats:p>In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0,1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:math>, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mi>f</mml:mi></mml:math> is decreasing in the variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math> and increasing in the variable <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math>. As a case study, we use the difference equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:msubsup><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>/</mml:mo><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>c</mml:mi><mml:msubsup><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>+</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>f</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0,1</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo></mml:math>, where the initial conditions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>≥</mml:mo><mml:mn>0</mml:mn></mml:math> and the parameters satisfy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mml:mi>c</mml:mi><mml:mo>,</mml:mo><mml:mi>d</mml:mi><mml:mo>,</mml:mo><mml:mi>f</mml:mi><mml:mo>></mml:mo><mml:mn>0</mml:mn></mml:math>. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.</jats:p>

We investigate the period-doubling and Naimark–Sacker bifurcations of the equilibrium of the difference equation xn+1 = γxn−1 + δxn Cxn−1 + xn where the parameters γ, δ, C are positive numbers and the initial conditions x−1 and x0 are arbitrary nonnegative numbers such that x−1 + x0 > 0. AMS Subject Classifications: 39A10, 39A20, 39A60, 37B25.

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