Chaos Induced by Snap-Back Repeller in a Two Species Competitive Model

In this paper, we investigate the complex dynamics of two-species Ricker-type discrete-time competitive model. We perform a local stability analysis for the fixed points and we will discuss about its persistence for boundary fixed points. This system inherits the dynamics of one-dimensional Ricker model such as cascade of period-doubling bifurcation, periodic windows and chaos. We explore the existence of chaos for the equilibrium points for a specific case of this system using Marotto theorem and proving the existence of snap-back repeller. We use several dynamical systems tools to demonstrate the qualitative behaviors of the system.

However, there was a minor technical flaw in his work [11] [12] [13]. Although he wanted to apply his theorem to any repelling fixed point, some of the conditions that he considered in the proof of his theorem were associated with only expanding fixed points. He incorrectly mentioned that if the absolute value for all eigenvalues of ( ) Df z is larger than 1, then the fixed point z is an expanding fixed point of f. As Then z is called a snapback repeller of f [11].
In [14], Gardini studied the homoclinic bifurcations in n dimensional endomorphisms (maps with a nonunique inverse) which are associated to expanding periodic orbits. The study of chaos for these kinds of map in one dimension was studied by Mira in 1987 [25]. Since, this topic is out of the discussion for this paper, so we avoid going through that. In this paper, we study the local dynamics of a two-species Ricker competitive model with four biological parameters. We will conduct a local stability analysis to study the local dynamics of the steady states of the system. We will use the persistence theory to study the global dynamics of the system. To study the chaotic dynamics of the system, we focus on a specific case with only three biological parameters. We provide the condition Then, at the origin we have and for the positive fixed point ( ) ( ) ( ) Proposition 2.1 The local stability analysis results for the fixed points ( ) (2) are summarized as below: 1) The equilibrium point ( ) 0, 0 is always an unstable fixed point.
2) The equilibrium point ( ) , 0 k for 1 l < and 1 0 2 r < < , has a stable manifold in 1 X direction and an unstable manifold in 2 X direction and is a saddle rection and a stable manifold in 2 X direction and is a stable node. Moreover,

( )
, 0 k for 1 l < and 1 2 r > , has an unstable manifold in 1 X direction and an unstable manifold in 2 X direction and is an unstable node. Finally, ( ) , 0 k for 1 l > and 1 2 r > , has an unstable manifold in 1 X direction and a stable manifold in 2 X direction and is a saddle point.
3) The equilibrium point ( ) 0,l for 1 k < and 1 0 2 r < < , has a stable manifold in 2 X direction and an unstable manifold in 1 X direction and is a saddle point. Also, ( ) 0,l for 1 k > and 2 0 2 r < < , has a stable manifold in 1 X direction and a stable manifold in 2 X direction and is a stable node. Moreover, ( ) 0,l for 1 k < and 2 2 r > , has an unstable manifold in 1 X direction and an unstable manifold in 2 X direction and is an unstable node. Finally, ( ) 0,l for 1 k > and 1 2 r > , has an unstable manifold in 2 X direction and a stable manifold in 1 X direction and is a saddle point.

Boundedness of the System Solutions
To study the global stability of the equilibrium points of system, at first we prove that all solutions in the first quadrant 2 +  are eventually bounded. Proof. By induction.
Since ( ) Assume that for n j ≤ , we have To find an upper bound, we know, Hence,

Persistence of the Species
To work on global stability, we need to study the persistence theory [28] [29]. American Journal of Computational Mathematics Here, we consider two cases: 1) Persistence of system corresponding to ( ) , 0 k .

Case 1: Persistence of System Corresponding to (k, 0)
For the first case, we have:

Case 2: Persistence of System Corresponding to (0, l)
For this case, we have:

Application of Snap-Back Repeller and Marroto Chaos in Study of Chaotic Dynamics of System
In this section, we explore analytically chaos in the sense of Marotto for a specific case of model (1) X n g X n X n X n r X n k F X n g X n X n X n r X n k The Jacobian matrix for (5) has the form : T. Azizi, B. Alali For this specific case, we have four fixed points ( ) 0, 0 , ( ) where ( ) and also, characteristic polynomial has the form ( ) 2) The equilibrium point ( ) , 0 k for 1 k < and 0 2 r < < , has a stable ma-American Journal of Computational Mathematics nifold in 1 X direction and an unstable manifold in 2 X direction and is a saddle point. Also, ( ) , 0 k for 1 k > and 0 2 r < < , has a stable manifold in 1 X direction and a stable manifold in 2 X direction and is a stable node. Moreover,

( )
, 0 k for 1 k < and 2 r > , has an unstable manifold in 1 X direction and an unstable manifold in 2 X direction and is an unstable node. Finally, ( ) , 0 k for 1 k > and 2 r > , has an unstable manifold in 1 X direction and a stable manifold in 2 X direction and is a saddle point.
3) The equilibrium point ( ) 0, k for 1 k < and 0 2 r < < , has a stable manifold in 2 X direction and an unstable manifold in 1 X direction and is a saddle point. Also, ( ) 0, k for 1 k > and 0 2 r < < , has a stable manifold in 1 X direction and a stable manifold in 2 X direction and is a stable node. Moreover, ( ) 0, k for 1 k < and 2 r > , has an unstable manifold in 1 X direction and an unstable manifold in 2 X direction and is an unstable node. Finally, ( ) 0, k for 1 k > and 2 r > , has an unstable manifold in 2 X direction and a stable manifold in 1 X direction and is a saddle point. Proposition 4.2 The local stability analysis results for the fixed points ( ) (5) are summarized as below: 1) The equilibrium point ( ) We check (17) using (12) and (13).
give us (15) and (16). and and, Numerical simulations, including bifurcation diagrams and time series display that this model demonstrates chaotic oscillations after a cascade of period-doubling bifurcations. As we can see in Figure 2, there are chaotic regions which are embedded in periodic windows regions. The periodic behaviors which appear alternately in the chaotic area, contain a copy of bifurcation diagram and it is repeating when we are changing the bifurcation parameter r. The bifurcation diagram for system (5) with respect to r displays the same qualitative dynamics for different values of k. Moreover, we have run bifurcation analysis with respect to k with different r values in Figure 3.
Also, if we look at Figure 4, at first, the equilibrium point is stable, when we increase r, it loses stability, from one cycle to two cycles, and produces a flip bifurcation. As r continues to increase, periodic oscillations are observed with periods 4, …, which eventually leads to chaos.
To prove the existence of chaos for the map (5) in the sense of Marotto, we need to find the conditions under which the fixed point ( ) of the system is a snap-back repeller. According to definition (1.5) and Figure 1     , , det 0 X X J ≠ , that is to say, at first, the following system of equations has a unique solution Proof. From and it gives us ( )  (21), is snap-back repeller for system (5). Theorem 4.4 Under the assumptions of the first part of proposition (4.2) and lemma (4.3), the map (4.1) is chaotic in the sense of Li-York, which means that: There exist 1) a positive integer N, such that map (4.1) has a point of period p, for each integer p N ≥ , 2) a scrambled set of F, i.e., an uncountable set S containing no periodic points of F, such that Proof. By theorem (1.3).

Conclusion
Studying the evolution of population models and complex dynamics of competitive models has attracted many researchers during several past decades. In this paper, we studied the complex dynamics of a two-species Ricker model which consists of four different biological parameters. We explored the stability of the origin and two other boundary fixed points using local stability theorem. Also, we provided the condition under which the solutions are bounded. We have seen that this model undergoes period doubling bifurcation but it does not show Neimark-Sacker bifurcation. We used the persistence theory to reveal the global behavior of system and we discovered the persistence of the system for two boundary fixed points. Afterward, we changed the model to a specific case with only three biological parameters and we discussed about the local stability of extinction and boundary fixed points of the system. Moreover, we discovered the chaotic dynamics of the new model using Marotto theorem. As we discussed, Marotto theorem is a rigorous theorem to study chaotic dynamics for systems with higher dimensions and can be used to study the chaotic dynamics of competitive models. We presented the conditions under which the new system undergoes snap-back repeller and as a result, it is chaotic in the sense of Li-York. Finally, we used bifurcation diagram to demonstrate the interesting dynamics of new system and the role of biological parameters r and k in appearance of different types of complicated dynamics. The new system has the same number of fixed points as the first system and the bifurcation analysis displayed the same qualitative dynamics for both species as we expected.