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
j
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)