an:07247462 Zbl 1453.39014 Baydemir, Pinar; Merdan, Huseyin; Karaoglu, Esra; Sucu, Gokce Complex dynamics of a discrete-time prey-predator system with Leslie type: stability, bifurcation analyses and chaos EN Int. J. Bifurcation Chaos Appl. Sci. Eng. 30, No. 10, Article ID 2050149, 21 p. (2020). 00452992 2020
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39A60 65L05 65L06 65P20 37N25 39A30 39A28 92D25 discrete prey-predator system; stability; flip bifurcation; Neimark-Sacker bifurcation; chaotic behavior Applying the forward Euler method to the continuous-time prey-predator model with Leslie type functional response, $\left\{ \begin{array}{rcl} \frac {dN(t)}{dt} &=& r_1N(t)-\epsilon P(t)N(t), \\ \frac {dP(t)}{dt}&=& P(t)\left(r_2-\theta\frac {P(t)}{N(t)}\right), \end {array} \right.$ the authors obtain the following discrete-time prey-predator system, $\left\{ \begin {array}{rcl} N_{t+1}&=& N_t+\delta N_t(r_1-\epsilon P_t), \\ P_{t+1} &=& P_t+\delta P_t\left(r_2-\theta\frac{P_{t+1}}{N_{t+1}}\right), \end {array} \right.$ where $$N$$ and $$P$$ represent prey and predator, respectively. Here $$\delta$$ is the integral step size. The discrete system has only a single positive equilibrium $$(\overline{N},\overline{P})=(\frac {\theta r_1}{\epsilon r_2},\frac {r_1}{\epsilon})$$. First, through linearization, conditions on the local stability of $$(\overline{N},\overline{P})$$ are obtained. Then, choosing $$\delta$$ as the bifurcation parameter, the flip bifurcation and the Neimark-Sacker bifurcation arising from $$(\overline{N},\overline{P})$$ are analyzed by employing the center manifold theorem and normal form theory. These theoretical results are not only supported but also extended by numerical simulations. For example, large values of $$\delta$$ can lead to chaotic behavior, which is impossible for the continuous-time counterpart. Yuming Chen (Waterloo)