Denotes the first quadrant as R + 2 , and its closure is R + 2 = R + 2 ¯ . For practical biological consideration, system (1) is defined on the domain R + 2 and all the solutions are positive. It is easy to prove lemma 1. Thus, any trajectory starting from R + 2 cannot cross the coordinate axes.

Lemma 1. All the solutions of system (1) are non-negative with the non-negative initial value ( x ( t 0 ) , y ( t 0 ) ) , i.e. R + 2 is an invariant set.

We will give some cases about invariant sets for system (1) in supplementary to lemma 1.

Remark. a) If A ≥ max { x 2 , 0 } , x 2 : = K 1 ( 1 − m 1 r 1 ) and B ≥ m a x { α e − m 2 d ,0 } , domain [ 0, A ] × [ 0, B ] is an invariant set; b) Domain [ A 1 , A 2 ] × [ 0, + ∞ ] and [ 0, + ∞ ] × [ B 1 , B 2 ] is not an invariant set, if A 1 , B 1 > 0 ; c) If r 1 ≤ m 1 or α e ≤ m 2 , domain [ A 1 , A 2 ] × [ B 1 , B 2 ] ( A 1 , B 1 > 0 ) is not an invariant set.

Theorem 1 All the solutions of system (1) are bounded with non-negative initial conditions ( x ( t 0 ) , y ( t 0 ) ) .

Proof. From the equation (1a) we have inequality x ˙ ≤ x ( r 1 − r 1 x / K 1 ) . With the help of [14], this implies that x ( t ) ≤ m a x { x ( t 0 ) , K 1 } . We denote the positive upper bound as M 1 . A similar argument, from the Equation (1b) we have inequality

y ˙ ≤ y ( α e x a + x − d y ) ≤ y ( α e M 1 a + M 1 − d y ) . (2)

Similarly, this implies that y ( t ) ≤ m a x { y ( t 0 ) , α e M 1 a + M 1 } : = M 2 . This completes the proof and the system under consideration is dissipative.

Lemma 2. All the solutions of system (1) satisfy

lim sup t → ∞ x ( t ) ≤ r 1 K 1 4 , lim sup t → ∞ y ( t ) ≤ r 1 K 1 4 e 1 . (3)

Proof. Taking an auxiliary function z = e x + y , it is obvious to see that

z ˙ ≤ e r 1 x ( 1 − x / K 1 ) − m z ≤ e r 1 K 1 / 4 − m z , (4)

where m = m i n { m 1 , m 2 } . By applying the theory of differential inequality [15], we obtain

z ( t ) ≤ e r 1 K 1 4 ( 1 − e − m ( t − t 0 ) ) + z ( t 0 ) e − m ( t − t 0 ) , (5)

which, upon letting t → ∞ , yields lim sup t → ∞ z ( t ) ≤ e r 1 K 1 / 4 . This completes the proof.

Remark. It is clear to see that all the solutions of system (0) are uniformly bounded.

Theorem 2. If r 1 ≤ m 1 , then the system (1) is Lagrange stable.

Proof. Taking the positive definite Lyapunov function V ( x , y ) = e x + y with infinitesimally small upper and large lower bounds, computing its Dini derivative along the trajectories of Equations (1), we have

D + V ≤ − e r 1 x 2 K 1 − m 2 y − d y 2 . (6)

Thus D + V is negative definite. This completes the proof.

Theorem 3. If parameters of system (1) satisfy

ω 1 : = K 1 r 1 ( r 1 − m 1 − α M 2 a ) > 0 , (7a)

ω 2 : = [ α e ( 1 − λ ) ω 1 a + e ( 1 − λ ) ω 1 − m 2 ] / d > 0 , (7b)

where λ ∈ ( 0,1 ) , then system (1) is permanent.

Proof. By theorem (2) we have a positive upper bound ξ 1 such that

m a x { lim sup t → ∞ x ( t ) , lim sup t → ∞ y ( t ) } ≤ ξ 1 . (8)

From Equation (1a) we have

x ˙ ≥ x ( r 1 − m 1 − α M 2 / a − r 1 x / K 1 ) ,

here M 2 is defined in theorem 2. With the help of [14], this implies that lim inf t → ∞ x ( t ) ≥ ω 1 . For ε 0 = λ ω 1 , there must exit T, such that for any t ≥ T we have x ( t ) ≥ ( 1 − λ ) ω 1 , thus

y ˙ ≥ [ α e ( 1 − λ ) ω 1 a + e ( 1 − λ ) ω 1 − m 2 ] y − d y 2 .

Similarly, this implies that lim inf t → ∞ y ( t ) ≥ ω 2 . This completes the proof and we get the permanence of the system.

Here we will use the Routh-Hurwitz criteria and the Perron’s theorems to analyze local stability and type of these equilibria by the nature of eigenvalues of Jacobian matrices around them, respectively. We observed that the Jacobian matrix of the system (1) is

J = [ r 1 − 2 r 1 x K 1 − α a y ( a + x ) 2 − m 1 − α x a + x α e a y ( a + x ) 2 α e x a + x − m 2 − 2 d y ] . (10)

For zero point E 0 , from its Jacobian matrix J ( E 0 ) we have: a) If r 1 < m 1 , E 0 is an asymptotically stable node; b) If r 1 > m 1 , E 0 is a saddle.

For the critical value r 1 = m 1 , E 0 is a higher order singular point. According to Frommer’s method, by using its Jacobian matrix and polar-coordinate-transformation: x = ρ cos θ , y = ρ sin θ , we derive R ( θ ) = − m 2 s i n 2 θ and characteristic equation Θ ( θ ) = − m 2 sin θ cos θ . For the first simple real root θ 1 = 0 of the characteristic equation, from the calculation R ( θ 1 ) = 0 , R ′ ( θ 1 ) = 0 , Θ ( θ 1 ) ≠ 0 and k = 1 , we know that θ = θ 1 is a characteristic direction. While for the second simple root θ 2 = π / 2 , it is clear to see an expansion Θ ( θ ) = m 2 ( θ − θ 2 ) + ⋯ , i.e. p = 1 (odd number) and C 2 ( 1 ) ( = m 2 ) has the opposite sign as R ( θ 2 ) . Thus θ = θ 2 is also a characteristic direction. By the sign of R ( θ 2 ) Θ ( θ ) = m 2 2 s i n θ c o s θ , we know that there is only one orbit tending to the critical point E 0 along the direction θ 2 in the normal regions of second type.

Taking the transformation t = − τ m 2 , we have following normal form (retain the symbol “ ⋅ ” for simplicity)

x ˙ = x + Φ ( x , y ) , y ˙ = Ψ ( x , y ) , (11)

where Φ ( x , y ) , Ψ ( x , y ) = o ( | x , y | ) in the right hand side are analytical functions, with terms starting from second order. From center manifold x = h ( y ) ,

the series of function ψ ( y ) = Ψ ( h ( y ) , y ) = r 1 m 2 K 1 y 2 + ⋯ with respect to y, and

m = 2 (even number), the critical point E 0 is a saddle node and the parabolic sector is on the right half (x,y)-plane.

For point E 2 , from its Jacobian matrix J ( E 2 ) and existence condition we have: a) If α e x 2 a + x 2 < m 2 , E 2 is an asymptotically stable node; b) If α e x 2 a + x 2 > m 2 , E 2 is a saddle.

For the critical value α e x 2 a + x 2 = m 2 , E 2 is a higher order singular point. By

using its Jacobian matrix and polar-coordinate-transformation, we derive R ( θ ) = c o s θ χ ( θ ) and characteristic equation Θ ( θ ) = − s i n θ χ ( θ ) , where

χ ( θ ) = ( m 1 − r 1 ) c o s θ − m 2 e s i n θ .

The real simple root θ 1 = 0 is in [ 0, π / 2 ] , it is clear to see an expansion Θ ( θ ) = − ( m 1 − r 1 ) θ + ⋯ and R ( θ 1 ) ≠ 0 , i.e. p = 1 (odd number) and C 1 ( 1 ) ( = r 1 − m 1 ) has contrary sign with R ( θ 1 ) . Thus θ = θ 1 is indeed a characteristic direction. While R ( θ 1 ) Θ ( θ ) = ( r 1 − m 1 ) χ ( θ ) sin θ changes its sign in the small neighbourhood of θ 1 , thus we know that there is only one orbit tending to the critical point E 2 along the direction θ 1 in the normal regions of second type. Another real simple root θ 2 is in ( π / 2 , π ) , it is clear to see that R ( θ 2 ) = 0 and

Θ ′ ( θ 2 ) = ( m 1 − r 1 ) [ 1 + ( m 1 − r 1 ) 2 e 2 m 2 2 ] c o s 2 θ 2 < 0,

i.e. k = 1 , thus θ = θ 1 is also a characteristic direction.

Taking transformations x → x + x 2 ; x = u + α x 2 a + x 2 v , y = ( m 1 − r 1 ) v and t = τ m 1 − r 1 , we have likewise real Jordan normal form

u ˙ = u + Φ ( u , v ) , v ˙ = Ψ ( u , v ) , (12)

in here Φ ( u , v ) , Ψ ( u , v ) = o ( | u , v | ) in the right hand side are analytical functions, with terms starting from second order. From the center manifold u = h ( v ) , the series of function ψ ( v ) = Ψ ( h ( v ) , v ) = a 2 v 2 + ⋯ with respect to v is derived, where

a 2 = − ( m 1 − r 1 ) 2 d + ( m 1 − r 1 ) α 2 a e x 2 ( a + x 2 ) 3 < 0 , (13)

Combining with m = 2 (even number), the critical point E * is a saddle node and the parabolic sector is on the left half (u, v)-plane.

For the interior equilibrium E * , from its Jacobian matrix J ( E * ) and existence conditions, denoting discriminant Δ = A 1 2 − 4 A 2 with notations

A 1 : = t r J ( E * ) = α x * y * ( a + x * ) 2 − r 1 x * K 1 − d y * , (14a)

A 2 : = det J ( E * ) = α 2 e a x * y * ( a + x * ) 3 − d y * ( A 1 + d y * ) , (14b)

we have:

(a) If A 1 < 0 and

(a1) A 2 > 0 , then E * is an asymptotically stable node; (a2) A 2 < 0 , then E * is a saddle;

(b) If A 1 = 0 and

(b1) A 2 > 0 , then E * is a center or a focus; (b2) A 2 < 0 , then E * is a saddle;

(c) If A 1 > 0 , then E * is unstable and

(c1) Δ = 0 , then E * is a node; (c2) Δ < 0 , then E * is a focus; (c3) Δ > 0 and A 2 > 0 , then E * is a node; (c4) Δ > 0 and A 2 < 0 , then E * is a saddle.

Remark. If A 1 + d y * < 0 , E * is asymptotically stable.

If the interior equilibrium E * exists and A 1 = 0, A 2 > 0 , i.e. the Jacobian matrix J ( E * ) has a pair of purely imaginary eigenvalues λ 1,2 = ± i β , in here β = A 2 , or E * is a center of local linear approximation of system (1). We consider this critical case or the center and focus problem by the H. Poincare’s formal series method. With the application of linear transforms

x → x + x * , y → y + y * and x = d y * u + β v , y = α e a y * ( a + x * ) 2 u (or x = − β u + d y * v , y = α e a y * ( a + x * ) 2 v ), the above system (1) transforms into following normal form

u ˙ = β v + P ( u , v ) , v ˙ = − β u + Q ( u , v ) , (15)

where higher order terms P ( u , v ) and Q ( u , v ) in the right hand side of the above system are all real analytical functions of u and v, with terms starting from second order.

Suppose that the above system has a first integral in the form of F ( u , v ) = u 2 + v 2 + F 3 ( u , v ) + F 4 ( u , v ) + ⋯ , where F k ( u , v ) is a homogeneous polynomial with order k ( k = 3,4, ⋯ ). Computing its derivative along the above system, we have its total derivative

d F d t = ( 2 u + F 3 u + ⋯ ) ( β v + P 2 + ⋯ ) + ( 2 v + F 3 v + ⋯ ) ( − β u + Q 2 + ⋯ ) , (16)

where P k ( u , v ) and Q k ( u , v ) are homogeneous polynomials in u, v of degree k ( k = 2,3, ⋯ ). In order to get Fourier series for further analysis, we firstly take the polar-coordinate-transformation u = r cos θ , v = r sin θ and assume F k = r k Φ k ( θ ) ( k = 3,4, ⋯ ), then following recurrence relations are derived by comparing coefficients:

d Φ 3 d θ = ϕ 3 ( θ ) , ϕ 3 ( θ ) : = 2 β r 3 ( u P 2 + v Q 2 ) , d Φ 4 d θ = ϕ 4 ( θ ) , ϕ 4 ( θ ) : = 1 β r 4 ( 2 u P 3 + P 2 F 3 u + 2 v Q 3 + Q 2 F 3 v ) ,                                     ⋮

From a complicated expression of function ϕ 3 ( θ ) or the integral ∫ 0 2 π   ϕ 3 ( θ ) d θ = 0 , we know that ϕ 3 ( θ ) is precisely a periodic function of period 2 π . Thus the focus value c 0 ( 1 ) of E * reads

c 0 ( 4 ) = − e α 2 s 1 2 a s 2 2 K 1 3 a ¯ 11 β 3 [ r 1 3 s 1 8 + 6 a r 1 3 s 1 7 + ( a α e K 1 r 1 2 + 15 a 2 r 1 3 − α K 1 r 1 2 s 2 ) s 1 6       + 20 a ( 1 4 a e α K 1 + a 2 r 1 − 9 40 K 1 s 2 α ) r 1 2 s 1 5     + ( − 1 2 a e α 2 K 1 2 r 1 s 2 + 10 a 3 e α K 1 r 1 2 − 8 a 2 α K 1 r 1 2 s 2 + 15 a 4 r 1 3 ) s 1 4     + 6 a r 1 ( 5 3 a 3 e α K 1 r 1 − 1 12 α 2 e a s 2 K 1 2 + a 4 r 1 2 − 7 6 a 2 α K 1 r 1 s 2 + 1 6 α 2 K 1 2 s 2 2 ) s 1 3

    + a ( 3 2 a 2 e α 2 K 1 2 r 1 s 2 + 5 a 4 e α K 1 r 1 2 + 1 2 α 3 e K 1 3 s 2 2 − 3 a 3 α K 1 r 1 2 s 2       + 2 a α 2 K 1 2 r 1 s 2 2 + a 5 r 1 3 ) s 1 2 + a α K 1 ( 5 2 a 3 e α K 1 r 1 s 2 + a 5 e r 1 2 + 1 2 a e α 2 K 1 2 s 2 2     − 1 2 a 4 r 1 2 s 2 + a 2 α K 1 r 1 s 2 2 − 1 2 α 2 K 1 2 s 2 3 ) s 1 + a 5 α 2 e K 1 2 r 1 s 2 ] , (17)

where a ¯ : = a + x * , s 1 : = x * and s 2 : = y * for later convenience. When c 0 ( 4 ) ≠ 0 , the point E * is a fine focus of order 1. If c 0 ( 4 ) > 0 , E * is an unstable fine focus; If c 0 ( 4 ) < 0 , E * is a stable fine focus. When c 0 ( 4 ) = 0 , continuing this procedure to obtain c 0 ( 6 ) .

In this section, we consider a critical case: A 2 = 0, A 1 ≠ 0 , i.e. E * is a critical point of higher order and its Jacobian matrix with zero eigenvalue can be rewritten as

J ( E * ) = [ α λ x * a + x * − α x * a + x * λ d y * − d y * ] , (18)

where positive parameter λ : = α e a d ( a + x * ) . By using its Jacobian matrix and the polar-coordinate-transformation, we derive function

R ( θ ) = ( α x * a + x * c o s θ + d y * s i n θ ) ( λ c o s θ − s i n θ ) (19)

and its characteristic equation

Θ ( θ ) = ( d y * cos θ − α x * a + x * sin θ ) ( λ cos θ − sin θ ) = 0. (20)

It is clear to see that the real roots θ 1,2 in ( 0, π / 2 ) ∪ ( π , 3 π / 2 ) respectively satisfy

tan θ 1 = d y * ( a + x * ) α x * , tan θ 2 = λ .

Case I: θ = θ 1

In this case, for θ = θ 1 , R ( θ 1 ) ≠ 0 if and only if t a n θ 1 ≠ λ . With some calculation, we derive R ( θ 1 ) = − c 1 ( 1 ) and series of function Θ ( θ ) at θ 1 : Θ ( θ ) = c 1 ( 1 ) ( θ − θ 1 ) + ⋯ , where

c 1 ( 1 ) = [ d 2 y * 2 ( a + x * ) 2 + α 2 x * 2 ] ( a d y * − α λ x * + d x * y * ) s i n ( θ 1 ) 2 ( a + x * ) 3 d 2 y * 2 , (21)

i.e. p = 1 (odd number), R ( θ 1 ) and c 1 ( 1 ) have opposite sign. Thus θ = θ 1 is a characteristic direction. While R ( θ 1 ) Θ ( θ ) changes its sign in the small neighbourhood of θ 1 , thus we know that there is only one orbit tending to the critical point E 2 along the direction θ 1 in the normal regions of second type.

If t a n θ 1 = λ , then R ( θ 1 ) = c 1 ( 1 ) = 0 , while

R ′ ( θ 1 ) = − [ ( d 2 y * 2 + α 2 ) x * 2 + 2 a d 2 x * y * 2 + a 2 d 2 y * 2 ] [ ( d λ y * + α ) x * + λ d y * a ] sin ( θ 1 ) 2 ( a + x * ) 3 d 2 y * 2 < 0.

Thus θ = θ 1 is also a characteristic direction.

Case II: θ = θ 2

In this case, for θ = θ 2 , we have R ( θ 2 ) = 0 ,

R ′ ( θ 1 ) = − ( λ 2 + 1 ) [ a x * + λ d y * ( a + x * ) ] sin ( θ 2 ) 2 λ 2 ( a + x * ) < 0

and expression of function Θ ( θ ) at θ 2 : Θ ( θ ) = c 2 ( 1 ) ( θ − θ 2 ) + ⋯ , where

c 2 ( 1 ) = − ( λ 2 + 1 ) ( a d y * − α λ x * + d x * y * ) s i n ( θ 2 ) 2 λ 2 ( a + x * ) . (22)

If t a n θ 1 ≠ λ , then c 2 ( 1 ) ≠ 0 and θ = θ 1 is not a special direction; If t a n θ 1 = λ , then θ = θ 1 is a characteristic direction.

From now on, making a change of variables x → x + x * , y → y + y * , and taking transformations x = α x * u a + x * + v , y = d y * u + λ v and t = τ A 1 . Substituting them into the system (1), we have following normal form

u ˙ = u + Φ ( u , v ) , v ˙ = Ψ ( u , v ) , (23)

where Φ ( u , v ) , Ψ ( u , v ) = o ( | u , v | ) in the right hand side are analytical functions, with terms starting from second order. From center manifold u = h ( v ) and series of function ψ ( v ) = Ψ ( h ( v ) , v ) = a 2 v 2 + a 3 v 3 + ⋯ with respect to v, where

a 2 = − d y * ( r 1 d a ¯ 4 − a α K 1 ( α e + d y * ) a ¯ + 2 α 2 e a 2 K 1 ) a ¯ K 1 ( d 2 a ¯ 3 y * − a α 2 e x * ) A 1 , (24)

one can judge the type of critical point E * in this special case. When a 2 ≠ 0 , i.e. r 1 d a ¯ 4 + 2 α 2 e a 2 K 1 ≠ a α K 1 ( α e + d y * ) a ¯ , combining with m = 2 (even number), the critical point E * is a saddle node. Furthermore, if the coefficient a 2 > 0 (or < 0), then the parabolic sector is on the right (or left) half (u,v)-plane. If a 2 = 0 , continuing the above series to obtain

a 3 = a α d K 1 2 A 1 2 a ¯ 6 ( − s 1 a e α 2 + d 2 s 2 a ¯ 3 ) 2 [ a 4 e 4 K 1 2 s 1 2 α 6 + d K 1 2 a ¯ s 1 s 2 a 3 ( 4 a − 3 a ¯ ) e 3 α 5     − d K 1 a ¯ 2 s 1 a 2 ( − r 1 e a ¯ 3 + ( ( d s 2 + A 1 ) K 1 + r 1 a ) e a ¯ 2 − K 1 ( a e A 1 + 2 d s 2 2 ) a ¯     + 5 a s 2 2 d K 1 ) e 2 α 4 + d 2 s 2 K 1 a ¯ 3 ( − 2 r 1 e a ¯ 4 + ( ( 2 d s 2 + A 1 ) K 1 + 7 r 1 a ) e a ¯ 3     − ( ( 4 d s 2 + 3 A 1 ) K 1 + 5 r 1 a ) a e a ¯ 2 + ( a ( 3 d s 2 + 2 A 1 ) e + 2 d s 2 2 ) K 1 a a ¯

  − 2 a 2 s 2 2 d K 1 ) a e α 3 − d 3 s 2 K 1 a ¯ 5 a ( r 1 e a ¯ 3 + ( ( ( − d s 2 − A 1 ) K 1 − r 1 a ) e + 4 r 1 s 2 ) a ¯ 2   + ( a K 1 ( d s 2 + A 1 ) e + ( − 2 d s 2 2 − A 1 s 2 ) K 1 − 4 r 1 s 2 a ) a ¯ + K 1 s 2 a ( 3 d s 2 + A 1 ) ) e α 2   − d 3 s 2 a ¯ 7 ( − 2 r 1 2 a ¯ 3 + ( 2 d K 1 r 1 s 2 + 2 a r 1 2 ) a ¯ 2 − d K 1 s 2 ( 3 r 1 a − A 1 K 1 ) a ¯   + a s 2 d K 1 2 ( d s 2 − 2 A 1 ) ) e α − s 2 3 d 5 a ¯ 8 A 1 K 1 2 ] . (25)

If a 3 > 0 , E * is an unstable node; If a 3 < 0 , E * is a saddle. When a 3 = 0 , continuing the above procedure to obtain a 4 and using the same criteria of a 2 .

In this section, we consider a critical case: A 1 = A 2 = 0 , i.e. E * is a critical point of higher order and its Jacobian matrix with two zero eigenvalues can be rewritten as

J ( E * ) = d y * [ 1 − 1 λ λ − 1 ] . (26)

From the above section we know that its characteristic equation is Θ ( θ ) = λ cos θ − sin θ = 0 , and function

R ( θ ) = d y * ( 1 λ cos θ + sin θ ) ( λ cos θ − sin θ ) .

Similarly, θ = θ 1 is a special direction, where t a n θ 1 = λ .

Making a change of variables and taking transformations x = u + v , y = λ u ; t = τ d y * , we have following normal form

u ˙ = v + Φ ( u , v ) , v ˙ = Ψ ( u , v ) . (27)

From center manifold v = h ( u ) , we have series of function ψ ( v ) = Ψ ( u , h ( u ) ) : ψ ( v ) = a 2 u 2 + a 3 u 3 + ⋯ and series of following function:

[ Φ u ( u , v ) + Ψ v ( u , v ) ] | v = h ( u ) = b 1 u + b 2 u 2 + ⋯ .

where coefficients

a 2 = 2 a α ( e + λ ) [ ( a + s 1 ) λ − 3 2 s 2 ] ( a + s 1 ) 4 λ d s 2 , b 1 = − α a ( − e a ¯ − λ a ¯ + 2 s 2 ) K 1 − 2 a ¯ 3 r 1 K 1 a ¯ 3 d s 2 ,                                                     ⋮ (28)

If a 2 , b 1 ≠ 0 , i.e. k = 2 (even number), n = m = 1 , or a 2 ≠ 0 , b 1 = 0 , then E * is a degenerate singular point. If a 2 = 0 , b 1 ≠ 0 , but

a 3 = − ( e + λ ) α a K 1 − 6 a ¯ 2 r 1 2 e α a a ¯ 2 K 1 < 0, (29)

i.e. k = 3 (odd number), n = m = 1 , and combining a discriminant μ = b 1 2 + 8 a 3 ≥ 0 , where

μ = 1 4 a ¯ 6 K 1 2 d 2 λ 2 [ ( 25 ( e 2 + ( 22 / 25 ) e λ + ( 1 / 25 ) λ 2 ) a 2 α 2 − 68 a ¯ 2 λ d ( e + ( 11 / 17 ) λ ) a α + 36 d 2 a ¯ 4 λ 2 ) K 1 2 + 72 a ¯ 2 ( − ( 11 / 6 ( e + ( 1 / 11 ) λ ) ) a α + a ¯ 2 λ d ) r 1 K 1 + 36 a ¯ 4 r 1 2 ] (30)

then E * is a complicated singular point and its neighbourhood U ( E * , δ ) consists of one hyperbolic sector and one elliptic sector, if μ < 0 and b 1 ≠ 0 , or just b 1 = 0 , then E * is a center or a focus.

In this section, we will give following theorems to illustrate the global stability by constructing proper Lyapunov functions.

Theorem 4. If r 1 ≤ m 1 , then the zero point E 0 is globally asymptotical stable.

Proof. Taking an unbounded positive definite Lyapunov function in lemma 2, we know that D + V is negative definite, similarly. This completes the proof.

Remark. Taking an unbounded positive definite Lyapunov function V ( x , y ) = 1 2 ( u x 2 + y 2 ) , where positive constant u > ( m 2 − α e ) 2 4 a α d , and computing its Dini derivative along orbits of Equations (1), we have

D + V = − K 1 g ( x , y ) + u r 1 x 3 ( a + x ) + u K 1 ( m 1 − r 1 ) x 3 K 1 ( a + x ) , (31)

where auxiliary function

g ( x , y ) = [ u a ( m 1 − r 1 ) x 2 + d x y 3 + a m 2 y 2 ] + [ u α x 2 + ( m 2 − α e ) x y + a d y 2 ] y .

On account of the discriminant Δ = ( m 2 − α e ) 2 − 4 u α a d < 0 of a quadratic function, it is obvious that g ( x , y ) ≥ 0 or D + V is negative definite. This also completes the proof.

Theorem 5. Assume that the equilibrium E 2 exists and α e x 2 a ≤ m 2 , then E 2 is globally asymptotical stable.

Proof. Taking an unbounded positive definite Lyapunov function

V ( x , y ) = x − x 2 − x 2 l n ( x / x 2 ) + y / e , (32)

and computing its Dini derivative along orbits of Equations (1), we have

D + V ≤ − r 1 K 1 ( x − x 2 ) 2 + y e ( α e x 2 a − m 2 ) − d e y 2 (33)

This completes the proof.

Theorem 6. Assume that the equilibrium E 2 exists and α e ≤ m 2 , x 2 > 2 a , then E 2 is globally asymptotical stable.

Proof. Taking an unbounded positive definite Lyapunov function V ( x , y ) = 1 2 [ u ( x − x 2 ) 2 + y 2 ] with positive constant

u ≤ 4 a m 2 r 1 ( x 2 − 2 a ) α 2 x 2 K 1 , (34)

and computing its Dini derivative along orbits of Equations (1), we have D + V = g ( x , y ) K 1 ( a + x ) , g ( x , y ) = g 1 ( x ) + g 2 ( x , y ) + g 3 ( x , y ) , where auxiliary functions

g 1 ( x ) = − u r 1 x 3 [ x − ( 2 x 2 − a ) ] , g 2 ( x , y ) = u x 2 r 1 ( 2 a − x 2 ) x 2 + u α x 2 K 1 x y − a K 1 m 2 y 2 , g 3 ( x , y ) = − d K 1 x y 3 + K 1 ( α e − m 2 ) x y 2 − u a x 2 2 r 1 x − a d K 1 y 3 − u α K 1 x 2 y .

Conditions in this theorem imply that discriminant of a quadratic function in g 2 ( x , y ) is

Δ 2 = ( u α x 2 K 1 ) 2 + 4 u x 2 ( 2 a − x 2 ) a K 1 m 2 ≤ 0,

thus g 2 ( x , y ) ≤ 0 , i.e. g ( x , y ) ≤ 0 . By using the Krasovskii’s theorem, the proof is completed.

Theorem 7. Assume that the equilibrium E * exists and η : = α y * a ( a + x * ) − r 1 K 1 ≤ 0 , then E * is globally asymptotical stable.

Proof. Taking an unbounded positive definite Lyapunov function

V ( x , y ) = x − x * − x * ln ( x / x * ) + u ( y − y 2 − y * ln ( y / y * ) ) , u = a + x * e a ≤ 0 , (35)

and computing its Dini derivative along orbits of Equations (1), we have

D + V = [ α y * ( a + x ) ( a + x * ) − r 1 K 1 ] ( x − x * ) 2 − u d ( y − y * ) 2 ≤ η ( x − x * ) 2 − u d ( y − y * ) 2 . (36)

From the condition we know that D + V is negative definite. By using the Krasovskii’s theorem, the proof is completed.

As we see, if r 1 ≤ m 1 , i.e. the point E * does not exist, there are no closed orbits in R + 2 . The trace of Jacobian matrix J ( E * ) is non-positive if r 1 − m 1 ≤ m 2 − α e whether the existence condition r 1 − m 1 ≤ 0 holds or not. By using the Bendixson criteria, there are no closed orbits in R + 2 . Such closed orbits and limit cycles surrounding the existed point E * cannot cross the x and y coordinate axes and are confined in an invariant domain [ 0, A ] × [ 0, B ] , where the upper bounds A, B are all sufficiently large. The point E * is not a saddle point for index j ( E * ) = 1 on this occasion. In this section, we will construct proper Dulac functions to further illustrate the existence problem of closed orbits and limit cycles.

Theorem 8. If x 2 < a or r 1 ≤ m 1 , then for system (1), there are no closed orbits in R + 2 .

Proof. Taking the diffeomorphism φ : u = x , v = y , d τ = d t / ( a + x ) which preserving the orientation of time, the system (1) is topologically equivalent to following system [16] [17]

{ x ˙ = P ( x , y ) : = x [ ( a + x ) ( r 1 − m 1 − r 1 x K 1 ) − α y ] , y ˙ = Q ( x , y ) : = y [ α e a x − ( a + x ) ( m 2 + d y ) ] , (37)

For notation simplicity, we still retain the symbols x, y, t. Define a Dulac function B ( x , y ) = x − 1 y − 1 , along the above system, we have

∂ ( B P ) ∂ x + ∂ ( B Q ) ∂ y ≤ g ( x ) y ,

where the numerator

g ( x ) = ( r 1 − m 1 − a r 1 K 1 ) − 2 r 1 x K 1 ≤ r 1 K 1 ( x 2 − a ) .

By using the Bendixson-Dulac criteria, the proof is completed.

Remark. (a) If x 2 < a , taking Dulac function B ( x , y ) = 1 ( x + c ) y , c ≥ 0 , then there are no closed orbits; (b) If 2 x 2 ≤ a , taking Dulac function B ( x , y ) = a + x x y , then there are also no closed orbits.

Theorem 9. If K 1 ≤ 4 a m 1 , r 1 − m 1 ≤ m 2 , and 2 a r 1 > K 1 [ 2 ( r 1 − m 1 ) + α e − m 2 ] , then for system (1), there are no closed orbits in R + 2 .

Proof. We can take another type of Dulac function e u x , u > 0 for our discussion. Obviously, along system (37), we have

∂ ( B P ) ∂ x + ∂ ( B Q ) ∂ y = 1 K 1 ∑ i + j = 0 3     a i j x i y j , (38)

where these coefficients a 30 , a 20 , a 10 , ⋯ in the summation are

a 30 = − u r 1 , a 20 : = ( − a r 1 − K 1 m 1 + K 1 r 1 ) u − 3 r 1 , a 10 : = a K 1 ( r 1 − m 1 ) u + ( α e − m 2 ) K 1 − 2 a r 1 + 2 ( r 1 − m 1 ) K 1 ,             ⋮ a 00 = − K 1 a ( m 1 + m 2 − r 1 ) ,

For a quadratic function a 30 x 3 + K 1 r 1 u x 2 − a K 1 u m 1 x of variable x, its discriminant implies that it is non-positive. Since other coefficients preserved are all non-positive, this completes the proof.

Remark. For the system (0) and the above form Dulac function e u x , u > 0 , if following conditions hold: K 1 ≤ 2 a , r 1 − m 1 ≤ m 2 , and 4 a r 1 > K 1 [ ( r 1 − m 1 ) + α e − m 2 ] , then there are no closed orbits. This procedure can also derive our theorem while it is uncomplicated but tedious.

Corollary 1. If E * is asymptotically stable, then combining with conditions in theorem 8 or 9, E * is globally asymptotical stable.

Remark. See reference [18] for the above corollary.

Theorem 10. If x 2 > a and the equilibrium point E * is unstable, then system (1) has limit cycle(s).

Proof. We see that d x d t | x = x 2 = − α x 2 y a + x 2 < 0 with y > 0 . Thus the straight line

x = x 2 is an untangent line of the system (37). Taking the Dulac function w ( x , y ) = e x + y − l , where l > 0 is sufficiently large, and computing w = 0 along the orbits of Equations (37), we derive

d w d t = − m 2 ( l − e x ) − d ( l − e x ) 2 + e r 1 x ( 1 − x K 1 ) − e m 1 x < 0 ,

where 0 < x < x 2 . For this system, we can construct a Bendixson ring including unstable singular point E * . By the Poincare-Bendixson theorem, system (37) has at least one limit cycle in the first quadrant. This completes the proof.

Without loss of generality, denoting the system (1) as

( x ˙ y ˙ ) = f ( x , y ) . (39)

Firstly, by fixing values of all the parameters and varying the bifurcation parameter m 1 , we observed that the two equilibrium points E 0 and E 2 collide with each other as m 1 cross the critical value m 1 = r 1 . Thus, there is a chance of bifurcation around this point E 0 .

Theorem 11. The system (39) undergoes a transcritical bifurcation around E 0 when the parameters satisfy m 1 = m 1 [ T C ] : = r 1 .

Proof. For the Jacobian matrix at E 0 , we have eigenvectors v and w corresponding to matrix J ( E 0 ) and its transpose, respectively: v = w = ( 1 0 ) . Then the following transversality conditions are hold:

w T f m 1 ( E 0 , m 1 [ T C ] ) = ( 1 0 ) ( 0 0 ) = 0 , w T [ D f m 1 ( E 0 , m 1 [ T C ] ) v ] = ( 1 0 ) ( − 1 0 ) = − 1 ≠ 0 , w T [ D 2 f ( E 0 , m 1 [ T C ] ) ( v , v ) ] = ( 1 0 ) ( − α a 0 ) = − α a ≠ 0. (40)

Clearly, one of the eigenvalues of the Jacobian matrix J ( E 0 ) is zero and the other is negative. From the Sotomayor’s theorem( see [19] [20]), the system (39) undergoes a transcritical bifurcation around E 0 at m 1 = m 1 [ T C ] . This completes the proof.

Here we take a as bifurcation parameter for the consideration of J ( E 2 ) .

Theorem 12. Assume that the parameters satisfy conditions for the existence of E 2 and m 1 < r 1 , then the system (39) undergoes a transcritical bifurcation around E 2 when the parameters satisfy:

a = a [ T C ] : = ( α e − m 2 ) x 2 m 2 . (41)

Proof. For the Jacobian matrix at E 2 , we have eigenvectors v and w corresponding to matrix J ( E 2 ) and its transpose, respectively: v = ( 1 m 1 − r 1 m 2 ) , w = ( 0 1 ) . Then the following transversality conditions are hold:

w T f a ( E 0 , a [ T C ] ) = ( 0 1 ) ( 0 0 ) = 0 , w T [ D f a ( E 0 , a [ T C ] ) v ] = ( 0 1 ) m 1 − r 1 e ( a [ T C ] + x 2 ) ( 1 − e ) = m 1 − r 1 e ( a [ T C ] + x 2 ) ≠ 0 , w T [ D 2 f ( E 0 , a [ T C ] ) ( v , v ) ] = 2 ( m 1 − r 1 ) m 2 [ α e a ( a [ T C ] + x 2 ) 2 + d ( r 1 − m 1 ) m 2 ] < 0. (42)

It is clear to see that one of the eigenvalues of the Jacobian matrix J ( E 2 ) is zero and the other is negative. From the Sotomayor’s theorem, the system (39) undergoes a transcritical bifurcation around E 2 at a = a [ T C ] . This completes the proof.

We choose parameter d as a bifurcation parameter. Assume that the parameters satisfy conditions for the existence of E * and

d = d [ H ] : = α x * ( a + x * ) 2 − r 1 x * K 1 y * < α 2 e a x * y * ( a + x * ) 3 . (43)

Define λ ( d ) = u ( d ) ± i v ( d ) as a pair of purely imaginary eigenvalues of matrix J ( E * ) , where u ( d ) = A 1 / 2 . It is clear to see that, if following conditions hold at d = d [ H ] : A 1 = 0 , A 2 > 0 , and transversality condition

u ′ ( d ) = y * [ a 3 r 1 + ( K 1 α e + 4 r 1 x * ) a 2 + ( 5 r 1 x * 2 + K 1 α ( e x * − y * ) ) a + 2 r 1 x * 3 ] α 2 ( a + x * ) [ − ( a + x * ) ( − K 1 α y * + a 2 r 1 + 2 a r 1 x * + r 1 x * 2 ) d − K 1 a α 2 e ] ≠ 0 , (44)

for instance, r 1 x * 2 ≥ K 1 α y * . Then E * changes its stability through Hopf bifurcation threshold the critical value d = d [ H ] .

Theorem 13. Assume that the parameters satisfy conditions for the existence of equilibrium E * , the condition (43) and the transversality condition (44), then the system (1) undergoes a Hopf bifurcation around equilibrium E * as parameter d passes through the bifurcation value d [ H ] .

We will calculate the first Lyapunov number σ at the point E * of the system which is used to determine the stability of limit cycles around Hopf bifurcation. Therefore we first translate the point E * to the origin by a non-singular linear transformation x ^ = x − x * , y ^ = y − y * , then the system (39) in power series around the origin(drop the hats for the sake of convenience as usual) is:

{ x ˙ = a 10 x + a 01 y + a 20 x 2 + a 11 x y + a 02 y 2 + a 30 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 + F 1 ( x , y ) , y ˙ = b 10 x + b 01 y + b 20 x 2 + b 11 x y + b 02 y 2 + b 30 x 3 + b 21 x 2 y + b 12 x y 2 + b 03 y 3 + F 2 ( x , y ) , (45)

with coefficients

a 10 = − r 1 x * K 1 + α x * y * ( a + x * ) 2 , a 01 = − α x * a + x * , a 20 = − r 1 K 1 + α y * ( a + x * ) 2 − α x * y * ( a + x * ) 3 , a 02 = 0 , a 11 = − α a ( a + x * ) 2 , a 30 = − α y * a ( a + x * ) 4 , a 03 = 0 , a 12 = 0 , a 21 = α a ( a + x * ) 3 ; b 10 = α e y * a ( a + x * ) 2 , b 01 = − d y * , b 20 = − α e y * a ( a + x * ) 3 , b 02 = − d , b 11 = α e a ( a + x * ) 2 , b 30 = α e y * a ( a + x * ) 4 , b 03 = 0 , b 12 = 0 , b 21 = − α e a ( a + x * ) 3 . (46)

and smooth functions F 1 ( x , y ) = ∑ i + j = 4 ∞     a i j x i y j and F 2 ( x , y ) = ∑ i + j = 4 ∞     b i j x i y j . From [19] [21], we know that the first Lyapunov number for a planar system is given by

σ = − 3 π 2 a 01 A 2 3 / 2 { [ a 10 b 10 ( a 11 2 + a 11 b 02 + a 02 b 11 ) + a 10 a 01 ( b 11 2 + a 20 b 11 + a 11 b 02 )     + b 10 2 ( a 11 a 02 + 2 a 02 b 02 ) − 2 a 10 b 10 ( b 02 2 − a 20 a 02 ) − 2 a 10 a 01 ( a 20 2 − b 20 b 02 )     − a 01 2 ( 2 a 20 b 20 + b 11 b 20 ) + ( a 01 b 01 − 2 a 10 2 ) ( b 11 b 02 − a 11 a 20 ) ]     − ( a 10 2 + a 01 b 10 ) [ 3 ( b 10 b 03 − a 10 a 30 ) + 2 a 10 ( a 21 + b 12 ) + ( b 10 a 12 − a 01 b 21 ) ] } , (47)

By computing the first Lyapunov number at the point E * , we will know that the Hopf bifurcation is supercritical and limit cycle(s) is(are) stable if σ < 0 ; the Hopf bifurcation is subcritical and limit cycle(s) is(are) unstable if σ > 0 . Hence we have to give the following numerical simulations in the next section, since the above expression is much too complicated.