**3. Complexity in prey-predator system with Allee effect**

In recent years many type of predator-prey problems, originated in Biological sciences, investigated which depend on various environmental and social conditions, [8–11]. Some problems solved by the application of Allee effect, which is an interesting phenomenon, to some predator-prey systems appear to be very interesting, [12–16]. The Allee effect on prey-predator system is a phenomenon in biology which characterizes certain correlation between population size or density and the mean individual fitness of a population or species. In the following study we investigate the complexity in a predator – prey problem with the Allee effect.

### **3.1 Discrete prey-predator model**

**Figure 5.**

**Figure 6.**

**Figure 7.**

**38**

*Regular two periodic motion of the swing when F* ¼ 0*:*9, *β* ¼ 0*:*5, *k* ¼ 0*:*3, *λ* ¼ 1, *ω* ¼ 1, *ϑ* ¼ 2*=*3*.*

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

*Chaotic oscillation of the swing when F* ¼ 0*:*2, *β* ¼ 0, *k* ¼ 0*:*1, *λ* ¼ 0*:*05, *ω* ¼ 1, *ϑ* ¼ 2*=*3*.*

*Chaotic oscillation of the swing when F* ¼ 0*:*2, *β* ¼ 0*:*01, *k* ¼ 0*:*1, *λ* ¼ 0*:*05, *ω* ¼ 1, *ϑ* ¼ 2*=*3*.*

A model for the prey-predator problem with Allee effect can represented as

$$\begin{aligned} X\_{n+1} &= X\_n + rX\_n(\mathbf{1} - X\_n)(\mathbf{1} - \exp\left(-\epsilon X\_n\right)) - aX\_n Y\_n\\ Y\_{n+1} &= Y\_n + aY\_n(X\_n - Y\_n)\left(\frac{Y\_n}{\mu + Y\_n}\right) \end{aligned} \tag{9}$$

where *Xn* and *Yn* refers to the density of prey and predators. Further, *r* correspond to the growth rate parameter of the prey population and *a* the predation prameter. Here,


For assumed values of parameters *a* ¼ 2*:*0, *r* ¼ 2*:*4, fixed points of system (9) are obtained, approximately, as *P*<sup>∗</sup> <sup>1</sup> ð Þ 0, 0 , *<sup>P</sup>*<sup>∗</sup> <sup>2</sup> ð Þ 1, 0 , *<sup>P</sup>*<sup>∗</sup> <sup>3</sup> ð Þ 0*:*545455, 0*:*545455 and by using stability analysis, we find all are unstable.

### **3.2 Bifurcation diagrams**

The phenomena of bifurcation provide a qualitative change in the behavior of a system during evolution. Such a change occurs when a particular parameter is varied while keeping other parameters constant. Bifurcation diagram shows the splitting of stable solutions within a certain range of values of the parameter. During the processes of bifurcation, one observes different cycles of evolution which leading to the chaotic situation. Phenomena like bistability, periodic windows within chaos etc. may also be observed for some systems. A bifurcation can be taken as a

tool to analyze the regular, chaotic as well as complexity within the system. For *a* ¼ 2*:*0 and 1*:*8≤*r*≤2*:*4 , **Figure 8** shows bifurcation of system (9), where some interesting phenomena observed that the system is not producing a period doubling bifurcation scenario which is very common for many nonlinear systems.
