Rich dynamics and anticontrol of extinction in a prey–predator system

Abstract

This paper reveals some new and rich dynamics of a two-dimensional prey–predator system and to anticontrol the extinction of one of the species. For a particular value of the bifurcation parameter, one of the system variable dynamics is going to extinct, while another remains chaotic. To prevent the extinction, a simple anticontrol algorithm is applied so that the system or bits can escape from the vanishing trap. As the bifurcation parameter increases, the system presents quasiperiodic, stable, chaotic and also hyperchaotic orbits. Some of the chaotic attractors are Kaplan–Yorke type, in the sense that the sum of its Lyapunov exponents is positive. Also, atypically for undriven discrete systems, it is numerically found that, for some small parameter ranges, the system seemingly presents strange nonchaotic attractors. It is shown both analytically and by numerical simulations that the original system and the anticontrolled system undergo several Neimark–Sacker bifurcations. Beside the classical numerical tools for analyzing chaotic systems, such as phase portraits, time series and power spectral density, the ‘0–1’ test is used to differentiate regular attractors from chaotic attractors.

Appendices

Appendix

A. Stability of $X_{1,23}^*$

Denote by e the eigenvalues. As is well known, a fixed point $X^*$ of a discrete system is stable if and only if its eigenvalues, e, satisfy the condition of $|e|<1$.

(i) For $X_1^*$, eigenvalues are $e_1=0$ and $e_2(a)=a$. Therefore, $X_1^*$ is stable for $a<1$.

(ii) For $X_2^*$, eigenvalues are $e_1(a)=2-a$ and $e_2(a,d)=(a-1)d/a$. Therefore, for $a\in (1,3)$ and $d<a/(a-1)$, the fixed point $X_2^*$ is stable.

(iii) For $X_3^*$ with $a>1$, the graphs considered next are obtained by representing d as a function of a: $d=d(a)$ (Fig. 1b). Denote $\Delta (a,d)=(a/d+2)^2-4a$ (see the graph $\textcircled {4}$ in Fig. 1b). If $\Delta (a,b)\ge 0$ (regions $\textcircled {3}$, $\textcircled {5}$ and $\textcircled {8}$, including the blue region), the eigenvalues are real: $e_{1,2}^r(a,d)=\Big (1-\frac{a}{2d}\Big )\pm \frac{1}{2}\sqrt{\Delta (a,d)}$. If $\Delta (a,d)< 0$ (regions $\textcircled {2}$, $\textcircled {1}$ and $\textcircled {6}$), the eigenvalues $e_{1,2}^c$ are complex conjugated, $e_1^c=\overline{e_2^c}$: $e_{1,2}^c(a,d)=\Big (1-\frac{a}{2d}\Big )\pm \imath \frac{1}{2}\sqrt{-\Delta (a,d)}$. Modulus of the complex (and also of the real) eigenvalue is

$$\begin{aligned} A(a,d)=\sqrt{a-\frac{2a}{d}}. \end{aligned}$$

(A.1)

Finally, after some simple calculations, omitted here, the stability parametric domain is found (regions $\textcircled {2}$, $\textcircled {3}$ and $\textcircled {4}$, with boarders $\textcircled {1}$ and $\textcircled {5}$, respectively):

$$\begin{aligned} \max \bigg \{\frac{3a}{3+a},\frac{a}{a-1}\bigg \}<d<\frac{2a}{a-1}. \end{aligned}$$

In the remaining regions, $\textcircled {6}$ and $\textcircled {8}$, fixed points are unstable.

B. Proof of Theorem 1

(i) The bifurcation condition, $|e_{1,2}^c(a,d)|=1$, i.e., $A(a,d)=1$ [see (A.1)], leads to $d-2a/(a-1)=0$;

(ii) The derivative of $e_{1,2}^c$ with respect to d, determined along AB, is

$$\begin{aligned} \frac{\partial }{\partial d}\Big |&e_{1,2}^c(a,d)\Big |_{d=2a/(a-1)}=\frac{a}{d^2\sqrt{a-2a/d}}\Bigg |_{d=2a/(a-1)}\\&\quad =\frac{(a-1)^2}{4a}:=k>0~ \text {for~ all}~ a>1; \end{aligned}$$

(iii) On the curve AB, the equation $(e_{1,2}^c(a,d))^j=1$, for $d=2a/(a-1)$, has the following solutions: for $j=1$, $a=1$; for $j=2$, $a\in \{1,9\}$; for $j=3$, $a\in \{1,7\}$, and for $j=4$, $a\in \{1,5,9\}$. Therefore, on the considered domain $a\in (1,4]$ and $d=2a/(a-1)$ (see Fig. 1a), the condition (iii) is satisfied.

C. Neimark–Sacker ($\hbox {N}-\hbox {S}$) bifurcation of the anticontrolled system

(i) $A(\gamma )=1$ if and only if

$$\begin{aligned} \gamma =\frac{a d-2 a-d}{(a-1) d}. \end{aligned}$$

The graph of $\gamma$ in the space $(a,d,\gamma )$, ${\mathcal {S}}$, is the N–S surface (Fig. 1d).

Note that the intersection with the plane $\gamma =0$ (Fig. 1d) represents the N–S curve for the uncontrolled system (1), at the fixed point $X_3^*$.

(ii) Next, one has

$$\begin{aligned} \frac{\partial }{\partial \gamma }\Big |e_{1,2}^c(\gamma )\Big |_{\gamma }=-\frac{(a-1)^2 d}{2 a}:=k<0. \end{aligned}$$

(iii) To ensure $\gamma \in (0,1)$, suppose $d>2a/(a-1)$. For $a\in (1,9)$, the eigenvalues $e_{1,2}^c(\gamma )$ are: $e_{1,2}^c(\gamma )=\frac{5-a}{4}\pm \imath \frac{1}{4}\sqrt{(9-a)(a-1)}$. Note that the image of the fixed point $e_{1}^c(\gamma )=\frac{5-a}{4}+\imath \frac{1}{4} \sqrt{(9-a)(a-1)}$ is in the half upper complex-plane, hence [compare with (3)]

$$\begin{aligned} \arg (e_{1,2}^c(\gamma ))=\arccos \frac{5-a}{4}. \end{aligned}$$

Since $\arccos \frac{5-a}{4}$ is increasing from 0 to $\pi$ for $a\in (1,9)$, the equation $(e_{1,2}(\gamma ))^{j}=1$ has only the following solutions: $j=3$, $a=7$, $j=4$, $a=5$.

D. The ‘0–1’ test

The ‘0–1’ test, proposed in [29], is designed to distinguish chaotic behavior from regular behavior in deterministic systems. Consider a discrete or continuous-time dynamical system and a one-dimensional observable data set $\phi (j)$, $j=1,2,\ldots ,N$, of the underlying system, constructed from time series. The 0–1 test has a theorem [30], which states that a nonchaotic motion is bounded, while a chaotic dynamic behaves like a Brownian motion.

(1) First, compute the translation variables (for some $c\in (0,\pi )$, [29])

$$\begin{aligned} p(n)=\sum _{j=1}^n\phi (j)\cos (jc),~~~ q(n)=\sum _{j=1}^n\phi (j)\sin (jc), \end{aligned}$$

for $n=1,2,\ldots ,N$.

(2) To determine the growths of p and q, the mean-square displacement is determined:

$$\begin{aligned} M(n)= & {} \lim _{N\rightarrow \infty }\frac{1}{N}\sum _{j=1}^N[p(j+n)-p(j)]^2\\&+[q(j+n)-q(j)]^2. \end{aligned}$$

where $n\ll N$ (in practice, $n=N/10$ gives good results).

(3) The asymptotic growth rate is defined as

$$\begin{aligned} K=\lim _{n\rightarrow \infty }\log M(n)/\log n. \end{aligned}$$

Because of occurrence of resonances for isolated values of c (where K is larger), the median of the computed values of K is used, since the median is robust against outliers associated with resonances [29]. If the underlying dynamics is regular (i.e., periodic or quasiperiodic), then $K = 0$; if the underlying dynamics is chaotic, then $K = 1$. Improved variants can be found in [29, 31].

In Fig. 18, the GOPY model are presented. Figure 18a represents the plot of q versus p and in Fig. 18b the mean-square displacement M as a function of n. Figure 18(i) represents the regular orbit of the logistic map $x_{n+1}=rx_n(1-x_n)$ for $r=3.55$; Fig. 18(ii) presents the chaotic orbit of the logistic map for $r=4$, while Fig. 18(iii) the SNA of the GOPY map $x_{n+1}=2a\tanh (x_n)\cos (2\pi \theta _n)$, $\theta _{n+1}=\theta _n+\omega$, with $a=1.5$, and $\omega =(\sqrt{5}-1)/2$ [12].