Global dynamics of some system of second-order difference equations

<p style='text-indent:20px;'>In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n},\\ x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the parameters <inline-formula><tex-math id="M1">\begin{document}$ \alpha_i,\ \beta_i,\ \gamma_i $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M2">\begin{document}$ i \in \{1,2\} $\end{document}</tex-math></inline-formula> and the initial conditions <inline-formula><tex-math id="M3">\begin{document}$ x_{-1}, x_0, y_{-1}, y_0 $\end{document}</tex-math></inline-formula> are positive real numbers. Some numerical example are given to illustrate our theoretical results.</p>

Hopf bifurcation and chaos in a Leslie–Gower prey–predator model with discrete delays

In this work, a Leslie–Gower prey-predator model with two discrete delays has been investigated. The positivity, boundedness and persistence of the delayed system have been discussed. The system exhibits the phenomenon of Hopf bifurcation with respect to both delays. The conditions for occurrence of Hopf bifurcation are obtained for different combinations of delays. It is shown that delay induces the complexity in the system and brings the periodic oscillations, quasi-periodic oscillations and chaos. The properties of periodic solution have been determined using central manifold and normal form theory. Further, the global stability of the system has been established for different cases of discrete delays. The numerical computation has also been performed to verify analytical results.

Periodic Solution for a Max-Type Fuzzy Difference Equation

The paper is concerned with the dynamics behavior of positive solutions for the following max-type fuzzy difference equation system: xn+1=maxA/xn, A/xn−1, xn−2, n=0,1,2,…, where xn is a sequence of positive fuzzy numbers, and the parameter A and the initial conditions x−2, x−1, x0 are also positive fuzzy numbers. Firstly, the fuzzy set theory is used to transform the fuzzy difference equation into the corresponding ordinary difference equations with parameters. Then, the expression for the periodic solution of the max-type ordinary difference equations is obtained by the iteration, the inequality technique, and the mathematical induction. Moreover, we can obtain the expression for the periodic solution of the max-type fuzzy difference equation. In addition, the boundedness and persistence of solutions for the fuzzy difference equation is proved. Finally, the results of this paper are simulated and verified by using MATLAB 2016 software package.

Global Dynamics of Some Exponential Type Systems

We explore the boundedness and persistence, existence of an invariant rectangle, local dynamical properties about the unique positive fixed point, global dynamics by the discrete-time Lyapunov function, and the rate of convergence of some 2,3-type exponential systems of difference equations. Finally, theoretical results are numerically verified.

Age-dependent intra-specific competition in pre-adult life stages and its effects on adult population dynamics

Intra-specific competition in insect and amphibian species is often experienced in completely different ways in their distinct life stages. Competition among larvae is important because it can impact on adult traits that affect disease transmission, yet mathematical models often ignore larval competition. We present two models of larval competition in the form of delay differential equations for the adult population derived from age-structured models that include larval competition. We present a simple prototype equation that models larval competition in a simplistic way. Recognising that individual larvae experience competition from other larvae at various stages of development, we then derive a more complex equation containing an integral with a kernel that quantifies the competitive effect of larvae of ageāon larvae of agea. In some parameter regimes, this model and the famous spruce budworm model have similar dynamics, with the possibility of multiple co-existing equilibria. Results on boundedness and persistence are also proved.

Asymptotic Behavior of a Second-Order Fuzzy Rational Difference Equation

We study the qualitative behavior of the positive solutions of a second-order rational fuzzy difference equation with initial conditions being positive fuzzy numbers, and parameters are positive fuzzy numbers. More precisely, we investigate existence of positive solutions, boundedness and persistence, and stability analysis of a second-order fuzzy rational difference equation. Some numerical examples are given to verify our theoretical results.

Behavior of a Competitive System of Second-Order Difference Equations

We study the boundedness and persistence, existence, and uniqueness of positive equilibrium, local and global behavior of positive equilibrium point, and rate of convergence of positive solutions of the following system of rational difference equations:xn+1=(α1+β1xn-1)/(a1+b1yn),yn+1=(α2+β2yn-1)/(a2+b2xn), where the parametersαi,βi,ai, andbifori∈{1,2}and initial conditionsx0,x-1,y0, andy-1are positive real numbers. Some numerical examples are given to verify our theoretical results.