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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.

When we study the evolution of population dynamics, two major types of mathematical modelings can be used: the continuous-time dynamical systems and the discrete-time dynamical systems. For the purpose of modeling small size population and non-overlapping generations, the discrete time systems are the appropriate model [

Definition 1.1 (Marotto-1978) Let f be differentiable in B r ′ ( z ) . The point z ∈ ℝ n is an expanding fixed point of f in B r ′ ( z ) , if f ( z ) = z and all eigenvalues of D f ( x ) exceed 1 in norm for all x ∈ B r ′ ( z ) .

Definition 1.2 (Marotto-1978) Assume that z is an expanding fixed point of f in B r ′ ( z ) for some r ′ > 0 . Then z is said to be an snap-back repeller of f if there exists a point z 0 ∈ B r ′ ( z ) with z 0 ≠ z and f M ( z 0 ) = z and | D f M ( z 0 ) | ≠ 0 for some positive integer M [

Under the assumptions for definitions (1.1) and (1.2), the following theorem by Marotto holds.

Theorem 1.3 (Marotto-1978) If f possesses a snap back repeller, then f is chaotic in the following sense: There exist 1) a positive integer N, such that f 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

a) f ( S ) ⊂ S ,

b) lim sup n → ∞ ‖ f n ( x ) − f n ( y ) ‖ > 0 , for all x , y ∈ S , with x ≠ y ,

c) lim sup n → ∞ ‖ f n ( x ) − f n ( y ) ‖ > 0 , for all x ∈ S and periodic point y of f,

3) an uncountable subset S 0 of S, such that lim inf n → ∞ ‖ f n ( x ) − f n ( y ) ‖ = 0 , for every x , y ∈ S 0 [

However, there was a minor technical flaw in his work [

Theorem 1.4 (Marotto-Li-Chen Theorem (2003)) Consider the following n-dimensional discrete dynamical system:

x n + 1 = f ( x n ) , x n ∈ ℝ n , n = 0 , 1 , 2 , ⋯

where f : ℝ n → ℝ n and z is a fixed point. Also assume that

1) f ( x ) is continuously differentiable in B r ′ ( z ) for some r ′ > 0 ,

2) All eigenvalues of ( D f ( z ) ) T D f ( z ) are greater than 1,

3) There exists a point z 0 = { x | ‖ x − z ‖ ≤ r ′ and all eigenvalues of ( D f ( x ) ) T D f ( x ) are larger than 1}, with z 0 ≠ z , such that f M ( z 0 ) = z where f i ( z 0 ) ∈ B r ′ ( z ) , i = 0 , 1 , 2 , ⋯ , M , and the determinant | D f M ( z 0 ) | ≠ 0 , for some positive integer M.

Then, the system is chaotic in the sense of Li-York [

Marotto refined his theorem in 2005 and he explained that a fixed point z is called a repelling fixed point under differentiable function f : ℝ n → ℝ n if all eigenvalues of D f ( z ) exceed 1 in magnitude, but z is expanding only if

‖ f ( x ) − f ( y ) ‖ > s ‖ x − y ‖

where s > 1 , for all x , y sufficiently close to z with x ≠ y (for x , y ∈ B r ′ ( z ) ). This implies that f is a 1-1 function in B r ′ ( z ) [

Definition 1.5 (Marotto-2005) Suppose z is a fixed point of f with all eigenvalues of D f ( z ) exceeding 1 in magnitude and suppose that there exists a point z 0 ≠ z in a repelling neighborhood of z and an integer M > 1 , such that x M = z and det ( D f ( x k ) ) ≠ 0 for 1 ≤ k ≤ M where x k = f k ( z 0 ) . Then z is called a snapback repeller of f [

He claimed that since det ( D f ( x k ) ) ≠ 0 for all 1 ≤ k ≤ M , then the homoclinic orbit is transversal in the sense that f for all k ≤ M is 1-1 map in a neighborhood of x k .

As Marotto explained in 1978, the condition det ( D f ( x k ) ) ≠ 0 guarantees the existence of the inverse of f M in B r ′ ( z ) . He mentioned that functions exhibit chaos and complex behavior when they possess snap-back repeller.

But what will happen that existence of a transverse homoclinic map convince us that we have chaos? As it is mentioned by many authors, a point which is in intersection of stable manifold and unstable manifold of a hyperbolic fixed point is called homoclinic point [

A countable infinity of periodic orbits consists of orbits of all periods.

1) An uncountable infinity of non-periodic orbits.

2) A dense orbit.

Although, Wiggins in [

There are some researches which have more details about small neighborhood of a point on the homoclinic orbit [

As Gardini et al. discussed, in non-invertible maps homoclinic orbits may be associate with expanding fixed points and or expanding cycles. Also, they mentioned that in the neighborhood of such homoclinic orbits, there exists an invariant set on which the map is chaotic. They even for the case that They proved that even if det ( D f ( x k ) ) = 0 , there are some situations in which the map is chaotic although Marotto theorem does not work. Laura et al., provide a definition for non-critical expanding fixed points and then they defined when a homoclinic orbit is critical. They used those definitions to prove a generalization of Marotto theorem in the case that we do not need the homoclinic orbit to be non-degenerate [

Theorem 1.6 (L. Gardini. et al., (2011)) Let f be a piecewise smooth non-invertible map, f : X → X , X ∈ ℝ n . Let p be an expanding fixed point of f and O ( p ) a noncritical homoclinic orbit of p. Then in any neighborhood of O ( p ) , there exists an invariant cantor like set Λ on which the f is chaotic [

In [

The Ricker model is a well known population model which demonstrates stable, periodic and non-periodic and complex nonlinear dynamics [

f 1 = X 1 ( n + 1 ) = X 1 ( n ) e r 1 ( 1 − X 1 ( n ) k − X 2 ( n ) ) (1)

f 2 = X 2 ( n + 1 ) = X 2 ( n ) e r 2 ( 1 − X 2 ( n ) l − X 1 ( n ) ) (2)

here, X 1 demonstrates the population size of the first species, X 2 represents the population size of the second species, r 1 and r 2 are the intrinsic growth rate, k and l the carrying capacity of the environment.

The Jacobian matrix for (1)-(2) has the form

J : = [ ∂ f 1 ∂ X 1 ∂ f 1 ∂ X 2 ∂ f 2 ∂ X 1 ∂ f 2 ∂ X 2 ] (3)

where

∂ f 1 ∂ X 1 = ( 1 − r 1 X 1 k ) exp ( r 1 ( 1 − X 1 k − X 2 ) )

∂ f 1 ∂ X 2 = − r 1 X 1 exp ( r 1 ( 1 − X 1 k − X 2 ) )

∂ f 2 ∂ X 1 = − r 2 X 2 exp ( r 2 ( 1 − X 2 l − X 1 ) )

∂ f 2 ∂ X 2 = ( 1 − r 2 X 2 l ) exp ( r 2 ( 1 − X 2 l − X 1 ) )

Then, at the origin we have

J | ( 0,0 ) = ( e r 1 0 0 e r 2 )

and for the fixed point ( k ,0 ) we have

J | ( k , 0 ) = ( 1 − r 1 − k r 1 0 e r 2 ( 1 − l ) )

and for the fixed point ( 0 , l ) we have

J | ( 0 , l ) = ( e r 1 ( 1 − k ) 0 − l r 2 1 − r 2 )

and for the positive fixed point ( X 1 * , X 2 * ) = ( k ( 1 − l ) 1 − k l , l ( 1 − k ) 1 − k l ) , we have

J | ( X 1 * , X 2 * ) = ( − 1 + k l + r 1 − r 1 l − 1 + k l − k ( − 1 + l ) r 1 − 1 + k l − k ( − 1 + k ) r 2 − 1 + k l − 1 + k l + r 2 − r 2 k − 1 + k l ) (4)

Proposition 2.1 The local stability analysis results for the fixed points ( 0,0 ) , ( k ,0 ) , ( 0, l ) of (1)-(2) are summarized as below:

1) The equilibrium point ( 0,0 ) is always an unstable fixed point.

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

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

To study the global stability of the equilibrium points of system, at first we prove that all solutions in the first quadrant ℝ + 2 are eventually bounded.

Theorem 3.1 For r 1 , r 2 > 0 , k , l > 0 and initial conditions in the first quadrant ℝ + 2 , i.e. X 1 ( 0 ) > 0 and X 2 ( 0 ) > 0 , for the system of (1)-(2) we have: X 1 > 0 and X 2 > 0 for all n ∈ ℤ + . In addition, we can find some positive number M, such that m a x n ∈ ℤ + { X 1 ( n ) , X 2 ( n ) } ≤ M .

Proof. By induction.

Since X 1 ( 0 ) > 0 we have exp ( r 1 ( 1 − X 1 ( 0 ) k ) ) > 0 , hence

X 1 ( 1 ) = X 1 ( 0 ) e r 1 ( 1 − X 1 ( 0 ) k − X 2 ( 0 ) ) < X 1 ( 0 ) e r 1 ( 1 − X 1 ( 0 ) k ) > 0

Assume that for n ≤ j , we have X 1 ( j ) > 0 . Then for n = j + 1 we have

X 1 ( j + 1 ) = X 1 ( j ) e r 1 ( 1 − X 1 ( j ) k − X 2 ( j ) ) > 0

Therefore X 1 ( n ) > 0 for any n ∈ ℤ + . Similarly, since X 1 ( 0 ) > 0 and X 2 ( 0 ) > 0 , we automatically have exp ( r 2 ( 1 − X 2 ( 0 ) j ) ) > 0 is positive. Hence,

X 2 ( 1 ) = X 2 ( 0 ) e r 2 ( 1 − X 2 ( 0 ) j − X 1 ( 0 ) ) < X 2 ( 0 ) e r 2 ( 1 − X 2 ( 0 ) j ) > 0

Assume that for n ≤ j , we have X 2 ( j ) > 0 . Then for n = j + 1 we have

X 2 ( j + 1 ) = X 2 ( j ) e r 2 ( 1 − X 2 ( l ) j − X 1 ( j ) ) > 0

Therefore X 2 ( n ) > 0 for any n ∈ ℤ + .

To find an upper bound, we know,

X 1 ( n + 1 ) = X 1 ( n ) e r 1 ( 1 − X 1 ( n ) k ) ≤ m a x x ∈ ℝ + { f ( x ) }

If we define f 1 ( x ) = x e r 1 ( 1 − x k ) , then f ′ 1 ( x ) = ( 1 − r 1 x k ) e r 1 ( 1 − x k ) and f 1 ( x ) has critical points at x = k r 1 . Since f ′ 1 ( x ) > 0 if x < k r 1 and f ′ 1 ( x ) < 0 if x > k r 1 , then x = k r 1 is the maximal point of f 1 ( x ) , i.e. max x ∈ ℝ + { f 1 ( x ) } = f 1 ( k r 1 ) . Hence,

x 1 ( n + 1 ) = X 1 ( n ) e r 1 ( 1 − X 1 ( n ) k − X 2 ( n ) ) ≤ f 1 ( k r 1 ) = k e r 1 − 1 r 1 = M 1

Similarly, we define f 2 ( x ) = x e r 2 ( 1 − x l ) , then f ′ 2 ( x ) = ( 1 − r 2 x l ) e r 2 ( 1 − x l ) and f 2 ( x ) has critical points at x = l r 2 . Since f ′ 2 ( x ) > 0 if x < l r 2 and f ′ 2 ( x ) < 0 if x > l r 2 , then x = l r 2 is the maximal point of f 2 ( x ) , i.e. max x ∈ ℝ + { f 2 ( x ) } = f 2 ( l r 2 ) .

X 2 ( n + 1 ) = X 2 ( n ) e r 2 ( 1 − X 2 ( n ) l − X 1 ( n ) ) ≤ f 2 ( l r 2 ) = l e r 2 − 1 r 2 = M 2

Therefore, we can find some positive number M = max { M 1 , M 2 } , such that m a x n ∈ ℤ + { X 1 ( n ) , X 2 ( n ) } ≤ M .

To work on global stability, we need to study the persistence theory [

1) Persistence of system corresponding to ( k ,0 ) .

2) Persistence of system corresponding to ( 0, l ) .

For the first case, we have:

P = { ( X 1 , X 2 ) : X 1 ≥ 0 , X 2 ≥ 0 }

P k , 0 = { ( X 1 , X 2 ) ∈ P : X 1 > 0 }

∂ P k , 0 = P \ P k , 0

Proposition 3.2 The system is uniformly persistent with respect to ( P k ,0 , ∂ P k ,0 ) .

Proof. Here, ∂ P k ,0 is closed in P. For any positive solution of ( X 1 ( n ) , X 2 ( n ) ) of the system, as we proved in theorem (3.1), we have

X 1 ( n + 1 ) ≤ X 1 ( n ) e r 1 ( 1 − X 1 ( n ) k ) ≤ max X 1 ∈ ℝ + { f 1 ( X 1 ) } = k e r 1 − 1 r 1 = M 1

And for large enough n

X 2 ( n + 1 ) ≤ X 2 ( n ) e r 2 ( 1 − X 2 ( n ) l ) ≤ max X 2 ∈ ℝ + { f 2 ( X 2 ) } = l e r 2 − 1 r 2 = M 2

Therefore, system (1)-(2) is point dissipative. Assume for all n ≥ 0

Y ∂ = { ( X 1 ( 0 ) , X 2 ( 0 ) ) : ( X 1 ( n ) , X 2 ( n ) ) satisfies the system equations and ( X 1 ( n ) , X 2 ( n ) ) ∈ ∂ P k ,0 }

We see that

Y ∂ = { ( 0 , X 2 ) : X 2 ≥ 0 } = ∂ P k , 0

Moreover, ( 0,0 ) is the unique equilibrium in Y ∂ . Define W s ( 0,0 ) to be the stable manifold for ( 0,0 ) . We show that

W s ( 0,0 ) ∩ P k ,0 = ∅

Assume that in contradiction, there exist a solution ( X 1 ( n ) , X 2 ( n ) ) of system with X 1 ( n ) > 0 such that

( X 1 ( n ) , X 2 ( n ) ) → ( 0,0 ) as n → ∞

Then, for large n we have

X 1 ( n + 1 ) > X 1 ( n ) e r 1 / 2

Since r 1 > 0 , it follows that X 1 ( n ) → ∞ as n ∈ ∞ and contradiction. Also, every orbit in Y ∂ tends to ( 0,0 ) as n → ∞ . It means that ( 0,0 ) is an isolated invariant set in P and acyclic in Y ∂ . Note that Y ∂ repels uniformly the solution of systems with positive X 1 ( n ) [

Theorem 3.3. There exists s 1 > 0 such that for any X 1 ( 0 ) > 0 we have

s 1 < X 1 ( n ) < k e r 1 − 1 r 1

Proof. By proposition (3.2).

Theorem 3.4 All solutions { ( X 1 ( n ) , X 2 ( n ) ) } of system with X 1 ( 0 ) > 0 and X 2 ( 0 ) ≥ 0 , for l > 1 and 0 < r 1 < 2 , are decreasing to the fixed point ( k ,0 ) , i.e.

lim n → + ∞ X 1 ( n ) = k , lim n → + ∞ X 2 ( n ) = 0

Proof. By proposition (3.2) and theorem (3.3).

For this case, we have:

Q = { ( X 1 , X 2 ) : X 1 ≥ 0, X 2 ≥ 0 }

Q 0 , l = { ( X 1 , X 2 ) ∈ Q : X 2 > 0 }

∂ Q 0 , l = Q \ Q 0 , l

Proposition 3.5 The system is uniformly persistent with respect to ( Q 0, l , ∂ Q 0, l ) .

Proof. Here, ∂ Q 0, l is closed in Q. Similarly, for any positive solution of ( X 1 ( n ) , X 2 ( n ) ) of the system (1)-(2), similar to theorem (3.1), we can write

X 1 ( n + 1 ) ≤ X 1 ( n ) e r 1 ( 1 − X 1 ( n ) k ) ≤ max X 1 ∈ ℝ + { f ( X 1 ) } = k e r 1 − 1 r 1 = M 1

For large enough n

X 2 ( n + 1 ) ≤ X 2 ( n ) e r 2 ( 1 − X 2 ( n ) l ) ≤ max X 2 ∈ ℝ + { f ( X 2 ) } = l e r 2 − 1 r 2 = M 2

Thus, system (1)-(2) is point dissipative. Now, for all n ≥ 0 , we set

L ∂ = { ( X 1 ( 0 ) , X 2 ( 0 ) ) : ( X 1 ( n ) , X 2 ( n ) ) satisfies the systeme quations and ( X 1 ( n ) , X 2 ( n ) ) ∈ ∂ Q 0, l }

for which

L ∂ = { ( X 1 , 0 ) : X 2 ≥ 0 } = ∂ Q 0 , l

Moreover, ( 0,0 ) is the unique equilibrium in L ∂ . Set W s ( 0,0 ) to be the stable manifold for ( 0,0 ) . We prove that

W s ( 0,0 ) ∩ Q 0, l = ∅

By contradiction, there exist a solution ( X 1 ( n ) , X 2 ( n ) ) of system with X 2 ( n ) > 0 such that

( X 1 ( n ) , X 2 ( n ) ) → ( 0,0 ) as n → ∞

For large n we have

X 2 ( n + 1 ) > X 2 ( n ) e r 2 / 2

Since r 2 > 0 , it leads to X 2 ( n ) → ∞ as n ∈ ∞ which is a contradiction. Also, every orbit in L ∂ tends to ( 0,0 ) as n → ∞ . It implies that ( 0,0 ) is an isolated invariant set in Q and acyclic in L ∂ . Here, l ∂ repels uniformly the solutions of system with positive X 2 ( n ) [

Theorem 3.6 There exists s 2 > 0 such that for any X 2 ( 0 ) > 0 we have

s 2 < X 2 ( n ) < l e r 2 − 1 r 2

Proof. By proposition (3.5).

Theorem 3.7 All solutions { ( X 1 ( n ) , X 2 ( n ) ) } of system with X 1 ( 0 ) ≥ 0 and X 2 ( 0 ) > 0 , for k > 1 and 0 < r 2 < 2 , are decreasing to the fixed point ( 0, l ) , i.e.

lim n → + ∞ X 1 ( n ) = 0 , lim n → + ∞ X 2 ( n ) = l

Proof. By proposition (3.5) and theorem (3.6).

Finally, we have the following result

Theorem 3.8 If there are positive constants s 1 , s 2 > 0 and M 1 , M 2 > 0 such that the solution ( X 1 ( n ) , X 2 ( n ) ) of system satisfies

0 < s 1 ≤ lim n → + ∞ inf X 1 ( n ) ≤ lim n → + ∞ sup X 1 ( n ) ≤ M 1 = k e r 1 − 1 r 1

0 < s 2 ≤ lim n → + ∞ inf X 2 ( n ) ≤ lim n → + ∞ sup X 2 ( n ) ≤ M 2 = l e r 2 − 1 r 2

Then, system (1)-(2) is persistent. If system is not persistent, it is called non-persistent smith2011dynamical.

In this section, we explore analytically chaos in the sense of Marotto for a specific case of model (1)-(2). Without loss of generality, we consider k = l , then we have

F : = { g 1 ( X 1 ( n ) , X 2 ( n ) ) = X 1 ( n ) exp ( r 1 ( 1 − X 1 ( n ) k − X 2 ( n ) ) ) g 2 ( X 1 ( n ) , X 2 ( n ) ) = X 2 ( n ) exp ( r 2 ( 1 − X 2 ( n ) k − X 1 ( n ) ) ) (5)

The Jacobian matrix for (5) has the form

J : = [ ∂ g 1 ∂ X 1 ∂ g 1 ∂ X 2 ∂ g 2 ∂ X 1 ∂ g 2 ∂ X 2 ] (6)

where

∂ g 1 ∂ X 1 = ( 1 − r 1 X 1 k ) exp ( r 1 ( 1 − X 1 k − X 2 ) ) (7)

∂ g 1 ∂ X 2 = − r 1 X 1 exp ( r 1 ( 1 − X 1 k − X 2 ) ) (8)

∂ g 2 ∂ X 1 = − r 2 X 2 exp ( r 2 ( 1 − X 2 k − X 1 ) ) (9)

∂ g 2 ∂ X 2 = ( 1 − r 2 X 2 k ) exp ( r 2 ( 1 − X 2 k − X 1 ) ) (10)

For this specific case, we have four fixed points ( 0,0 ) , ( k ,0 ) , ( 0, k ) and ( X 1 * , X 2 * ) = ( k k + 1 , k k + 1 ) . At ( 0,0 ) we have

J | ( 0 , 0 ) = ( e r 1 0 0 e r 2 )

and at ( k ,0 ) we have

J | ( k , 0 ) = ( 1 − r 1 − k r 1 0 e r 2 ( 1 − k ) )

and also for the fixed point ( 0, k ) we have

J | ( 0, k ) = ( e r 1 ( 1 − k ) 0 − k r 2 1 − r 2 )

and finally for the positive fixed point ( X 1 * , X 2 * ) = ( k k + 1 , k k + 1 ) , we have

J | ( X 1 * , X 2 * ) = ( k + 1 − r 1 k + 1 − k r 1 k + 1 − k r 2 k + 1 k + 1 − r 2 k + 1 ) (11)

where

d e t ( J | ( X 1 * , X 2 * ) ) = − k r 1 r 2 − r 1 r 2 − k + r 1 + r 2 − 1 k + 1 (12)

tr ( J | ( X 1 * , X 2 * ) ) = 2 k + 2 − r 2 − r 1 k + 1 (13)

and also, characteristic polynomial has the form

P ( X ) : = X 2 − 2 k + 2 − r 2 − r 1 k + 1 X − k r 1 r 2 − r 1 r 2 − k + r 1 + r 2 − 1 k + 1 (14)

Proposition 4.1 The local stability analysis results for the fixed points ( 0,0 ) , ( k ,0 ) , ( 0, k ) of (5) are summarized as below:

1) The equilibrium point ( 0,0 ) is always an unstable fixed point.

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

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

Proposition 4.2 The local stability analysis results for the fixed points ( X 1 * , X 2 * ) = ( k k + 1 , k k + 1 ) of (5) are summarized as below:

1) The equilibrium point ( X 1 * , X 2 * ) is an unstable fixed point if and only if

r 1 r 2 ( 1 − k ) + 2 ( k + 1 ) < ( r 1 + r 2 ) , 4 ( k + 1 ) − 2 ( r 1 + r 2 ) + r 1 r 2 ( 1 − k ) > 0 , k < 1

or

k < r 1 r 2 − r 1 − r 2 r 1 r 2 , 4 ( k + 1 ) − 2 ( r 1 + r 2 ) + r 1 r 2 ( 1 − k ) > 0 , k < 1

2) The equilibrium point ( X 1 * , X 2 * ) is a stable fixed point if and only if

k > r 1 r 2 − r 1 − r 2 r 1 r 2 , 4 ( k + 1 ) − 2 ( r 1 + r 2 ) + r 1 r 2 ( 1 − k ) > 0 , k < 1

3) The equilibrium point ( X 1 * , X 2 * ) is a saddle point if and only if

4 ( k + 1 ) − 2 ( r 1 + r 2 ) + r 1 r 2 ( 1 − k ) < 0 , k > 1

Proof. Using Theorem 1.1.1 (Linearized Stability) in [

The equilibrium point ( X 1 * , X 2 * ) is an unstable fixed point if and only if | d e t ( J ) | > 1 and | tr ( J ) | < | 1 + d e t ( J ) | . tr ( J | ( X 1 * , X 2 * ) ) − d e t ( J | ( X 1 * , X 2 * ) ) − 1 < 0 gives us:

r 1 r 2 ( k − 1 ) k + 1 < 0 → k < 1 (15)

Also, tr ( J | ( X 1 * , X 2 * ) ) + d e t ( J | ( X 1 * , X 2 * ) ) + 1 < 0 gives us:

4 ( k + 1 ) − 2 ( r 1 + r 2 ) + r 1 r 2 ( 1 − k ) k + 1 > 0 (16)

and d e t ( J | ( X 1 * , X 2 * ) ) > 1 gives us

r 1 r 2 ( 1 − k ) − ( r 1 + r 2 ) k + 1 < 0

that is to say

k > r 1 r 2 − r 1 − r 2 r 1 r 2

Moreover, d e t ( J | ( X 1 * , X 2 * ) ) < − 1 gives us

r 1 r 2 ( 1 − k ) + 2 ( k + 1 ) − ( r 1 + r 2 ) k + 1 < 0

The positive fixed point of system (5) is asymptotically stable if and only if

| tr ( J ) | < 1 + d e t ( J ) < 2 (17)

We check (17) using (12) and (13). tr ( J | ( X 1 * , X 2 * ) ) − d e t ( J | ( X 1 * , X 2 * ) ) − 1 < 0 and tr ( J | ( X 1 * , X 2 * ) ) + d e t ( J | ( X 1 * , X 2 * ) ) + 1 < 0 give us (15) and (16). and d e t ( J | ( X 1 * , X 2 * ) ) < 1 gives us

r 1 r 2 ( 1 − k ) − ( r 1 + r 2 ) k + 1 < 0

that is to say

k > r 1 r 2 − r 1 − r 2 r 1 r 2

Finally, The equilibrium point ( X 1 * , X 2 * ) is a saddle point if and only if tr 2 ( J ) − 4 d e t ( J ) > 0 and | tr ( J ) | > | 1 + d e t ( J ) | . The first condition gives us

( r 1 − r 2 ) 2 + 4 k 2 ( k + 1 ) 2 > 0

which is always true. Another conditions to check are: tr ( J | ( X 1 * , X 2 * ) ) − d e t ( J | ( X 1 * , X 2 * ) ) − 1 > 0 gives us:

r 1 r 2 ( k − 1 ) k + 1 > 0 (18)

and, tr ( J | ( X 1 * , X 2 * ) ) + d e t ( J | ( X 1 * , X 2 * ) ) + 1 < 0 which gives us:

4 ( k + 1 ) − 2 ( r 1 + r 2 ) + r 1 r 2 ( 1 − k ) k + 1 < 0 (19)

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

Also, if we look at

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 Z * = ( X 1 * , X 2 * ) of the system is a snap-back repeller. According to definition (1.5) and

Lemma 4.3 Assume that the conditions of the first part of the proposition (4.2) are satisfied. The fixed point Z * = ( X 1 * , X 2 * ) of map F is called snap-back repeller if there exists a point Z 0 = ( X 1 , X 2 ) in the neighborhood of Z * such that Z 0 ≠ Z * , F ( Z 0 ) = Z * , | d e t ( J | ( X 1 , X 2 ) ) | ≠ 0 , that is to say, at first, the following system of equations has a unique solution

{ X 1 * = X 1 exp ( r 1 ( 1 − X 1 k − X 2 ) ) X 2 * = X 2 exp ( r 2 ( 1 − X 2 k − X 1 ) ) (20)

and

k 2 − k ( r 2 X 2 + r 1 X 1 + r 1 r 2 X 1 X 2 ) + r 1 r 2 X 1 X 2 ≠ 0 (21)

Then Z * for some parameter values ( r 1 , r 2 ) and k, is a snap-back repeller for map (5).

Proof. From | d e t ( J | ( X 1 , X 2 ) ) | ≠ 0 , we have:

∂ g 1 ∂ X 1 ∂ g 2 ∂ X 2 − ∂ g 1 ∂ X 2 ∂ g 2 ∂ X 1 ≠ 0

which

( ( 1 − r 1 X 1 k ) ( 1 − r 2 X 2 k ) − r 1 r 2 X 1 X 2 ) e ( r 1 ( 1 − X 1 k − X 2 ) + r 2 ( 1 − X 2 k − X 1 ) ) ≠ 0

and it gives us

k 2 − k ( r 2 X 2 + r 1 X 1 + r 1 r 2 X 1 X 2 ) + r 1 r 2 X 1 X 2 ≠ 0

Therefore, any solution Z * = ( X 1 * , X 2 * ) ≠ ( X 1 , X 2 ) = Z 0 of system (20) which satisfies the first part of the proposition (4.2) and (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

a) F ( S ) ⊂ S ,

b) lim sup n → ∞ ‖ F n ( x ) − F n ( y ) ‖ > 0 , for all x , y ∈ S , with x ≠ y ,

c) lim sup n → ∞ ‖ F n ( x ) − F n ( y ) ‖ > 0 , for all x ∈ S and periodic point y of f,

3) an uncountable subset S 0 of S, such that lim inf n → ∞ ‖ F n ( x ) − F n ( y ) ‖ = 0 , for every x , y ∈ S 0 .

Proof. By theorem (1.3).

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Azizi, T. and Alali, B. (2020) Chaos Induced by Snap-Back Repeller in a Two Species Competitive Model. American Journal of Computational Mathematics, 10, 311-328. https://doi.org/10.4236/ajcm.2020.102017