In the past few decades, predator-prey systems involving the prey-taxis have attracted more and more scholars to investigate them. Chen et al. [1] reported stationary patterns of a predator-prey model with prey-taxis and investigated the stability of the nonconstant steady states by employing the Crandall-Rabinowitz bifurcation theory. Tu et al. [2] considered the asymptotic behaviors of a parabolic-elliptic chemotaxis system with competitive kinetics and loop of a predator-prey model. Bell and Haskell [3] established the global existence of positive classical solutions and the existence of nontrivial steady states via the bifurcation theory of a predator-prey system. The global existence and uniform boundedness of solutions to a predator-prey system with prey-taxis for general functional responses in any spatial dimensions have been investigated by Ahn and Yoon [4]. The existence of the unique global bounded classical solution is proven, and the steady-state bifurcation, the Hopf bifurcation, and Hopf/steady-state mode interaction are studied via the Lyapunov-Schmidt procedure by Qiu et al. [5]. We recommend more existing results about the predator-prey systems with directed prey-taxis, see Refs. [6] [7] [8] [9] [10], etc.

In this present paper, we focus on a predator-prey model with two predators and one prey as well as the prey-taxis as follows.

{ ∂ t u = ∂ x x u − χ ∂ x ⋅ ( u ( 1 + δ w ) 2 ∂ x w ) + β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u , x ∈ Ω ,     t > 0 , ∂ t v = ∂ x x v − ξ ∂ x ⋅ ( v ( 1 + δ w ) 2 ∂ x w ) + β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v , x ∈ Ω ,     t > 0 , ∂ t w = d ∂ x x w + r w ( 1 − w K ) − μ 1 u w α 1 + w + s u − μ 2 v w α 2 + w , x ∈ Ω ,     t > 0 , ∂ ν u = ∂ ν v = ∂ ν w = 0 , x ∈ ∂ Ω ,     t ≥ 0 , u ( x , 0 ) = u 0 ( x ) ≥ 0 ,     v ( x , 0 ) = v 0 ( x ) ≥ 0 ,     w ( x , 0 ) = w 0 ( x ) ≥ 0 , x ∈ Ω , (1)

where u = u ( x , t ) , v = v ( x , t ) and w = w ( x , t ) are predator and prey densities at position x and time t, respectively. Ω ⊂ ℝ N is a bounded domain with its smooth boundary ∂ Ω ; constant d describes the diffusive rate of prey; For j = 1 , 2 , β j are the ratios of biomass conversion of predators species; ρ j represent the rates of toxic substances produced by per unit biomass about predators due to prey species are toxic corresponding to the predators; δ j are the natural mortality of the predators u and v; α j describe the half-saturation constant of the predators; s describes the measure of mutual interference among the predator u; two constants r and K in the third equation are the intrinsic growth rate and the maximum environmental capacity of prey species, respectively.

Moreover, − χ ∂ x ⋅ ( u ( 1 + δ w ) 2 ∂ x w ) and − ξ ∂ x ⋅ ( v ( 1 + δ w ) 2 ∂ x w ) are prey-taxis terms. They imply the tendency of predators moving toward the positive direction of the increasing gradient of prey population as χ > 0 and ξ > 0 . If χ < 0 and ξ < 0 , we say that predators move toward the opposite direction of the increasing gradient of prey population to avoid group defense by a large number of prey species or volume-filling effect in predator species [11]. Consequently, χ , ξ > 0 and χ , ξ < 0 corresponding to attractive and repulsive prey-taxis, respectively. Moreover, u ( 1 + δ w ) 2 and v ( 1 + δ w ) 2 represent the distribution variations of the directed species dispersals [12]. Obviously, they depend on the density of the prey population. All parameters exhibited in the system (1) are set to be positive.

For system (1), define

f ( u , v , w ) = β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u ,

g ( u , v , w ) = β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v ,

and

h ( u , v , w ) = r w ( 1 − w K ) − μ 1 u w α 1 + w + s u − μ 2 v w α 2 + w ,

as well as some assumptions

(H1) β 1 − ρ 1 − δ 1 > 0 ,   w ^ > α 1 δ 1 β 1 − ρ 1 − δ 1     with     w ^ = − γ 2 + γ 2 2 − 4 γ 1 γ 3 2 γ 1 , γ 1 = r s ( β 1 − ρ 1 ) , γ 2 = μ 1 ( β 1 − ρ 1 − δ 1 ) K − r s ( β 1 − ρ 1 ) K ,   γ 3 = − μ 1 α 1 δ 1 K .

(H2) β 2 − ρ 2 − δ 2 > 0 ,   ( β 2 − ρ 2 ) K − ( α 2 + K ) δ 2 > 0.

(H3) β 2 − ρ 2 − δ 2 > 0 ,   α 2 δ 2 ( β 1 − ρ 1 − δ 1 ) − α 1 δ 1 ( β 2 − ρ 2 − δ 2 ) > 0 , ( r − μ 1 ) ( β 2 − ρ 2 − δ 2 ) K − r α 2 δ 2 > 0.

As a result, we can conclude the classifications of the equilibria of system (1). 1) system (1) has a trivial equilibrium E 0 = ( 0,0,0 ) and a semi-trivial equilibrium E 1 = ( 0 , 0 , K ) ; 2) if (H1) holds, system (1) has a boundary equilibrium E 2 = ( u ^ , 0 , w ^ ) , where

w ^ = − γ 2 + γ 2 2 − 4 γ 1 γ 3 2 γ 1 ,     u ^ = ( β 1 − ρ 1 − δ 1 ) w ^ − α 1 δ 1 s δ 1 ;

3) if (H2) is valid, system (1) has a boundary equilibrium E 3 = ( 0 , v ˜ , w ˜ ) , where

w ˜ = α 2 δ 2 β 2 − ρ 2 − δ 2 ,     v ˜ = r α 2 ( β 2 − ρ 2 ) [ ( β 2 − ρ 2 ) K − ( α 2 + K ) δ 2 ] μ 2 ( β 2 − ρ 2 − δ 2 ) K ;

4) if (H3) is satisfied, system (1) has a unique positive equilibrium E ∗ = ( u * , v * , w * ) , where

u * = α 2 δ 2 ( β 1 − ρ 1 − δ 1 ) − α 1 δ 1 ( β 2 − ρ 2 − δ 2 ) s δ 1 ( β 2 − ρ 2 − δ 2 ) ,

v * = ( α 1 + w * ) γ 4 δ 2 μ 2 α 2 K ( β 1 − ρ 1 ) ( β 2 − ρ 2 − δ 2 ) ,

and

w * = α 2 δ 2 β 2 − ρ 2 − δ 2 ,

with γ 4 = α 2 δ 2 ( β 1 − ρ 1 ) [ ( r − μ 1 ) ( β 2 − ρ 2 − δ 2 ) K − r α 2 δ 2 ]     + μ 1 δ 1 K ( β 2 − ρ 2 − δ 2 ) [ α 2 δ 2 + α 1 ( β 2 − ρ 2 − δ 2 ) ] .

In this present paper, we will establish the global asymptotic stabilities of semi-trivial equilibrium E 1 = ( 0 , 0 , K ) , boundary equilibria E 2 = ( u ^ , 0 , w ^ ) , E 3 = ( 0 , v ˜ , w ˜ ) and the unique positive equilibrium E ∗ = ( u * , v * , w * ) by constructing some suitable time evolution Lyapunov functions, respectively.

This paper is structured as follows. In Section 2, we perform the main results of the present paper. In Section 3, the local-in-time existence of the classical solution of the model is given. In Section 4, the proofs of the main results are displayed. Finally, some conclusions are made in Section 5.

Theorem 2.1 Let Ω ⊂ ℝ N be a bounded domain with the smooth boundary ∂ Ω . Suppose ( u , v , w ) is a classical solution of system (1) with the initial conditions ( u 0 ( x ) , v 0 ( x ) , w 0 ( x ) ) ∈ [ W 1, p ( Ω ) ] 3 and u 0 ( x ) ≥ 0, v 0 ( x ) , w 0 ( x ) ≥ 0 for x ∈ Ω ¯ . We have the following global asymptotic stability results.

1) For any χ , ξ > 0 and

0 < β 1 ≤ ρ 1 + μ 1 ,     0 < β 2 ≤ ρ 2 + μ 2 ,     0 < K ≤ min { α 1 δ 1 μ 1 , α 2 δ 2 μ 2 } , (2)

then E 1 = ( 0,0, K ) is globally asymptotically stable.

2) If the condition (H1) holds and

β 2 ≤ μ 2 ≤ δ 2 α 2 w ^ ,     μ 1 = ( β 1 − ρ 1 ) ( α 1 + s u ^ ) α 1 + w ^ ,     0 < K ≤ r α 1 2 μ 1 u ^ ,     0 < χ 2 ≤ 4 d w ^ C 2 u ^ , (3)

then boundary equilibrium E 2 = ( u ^ , 0 , w ^ ) is globally asymptotically stable for any ξ > 0 .

3) If the condition (H2)is valid and

β 1 ≤ μ 1 ≤ δ 1 α 1 w ˜ ,     μ 2 = α 2 ( β 2 − ρ 2 ) α 2 + w ˜ ,     0 < K ≤ r α 2 2 μ 2 v ˜ ,     0 < ξ 2 ≤ 4 d w ˜ C 2 v ˜ , (4)

then E 3 = ( 0 , v ˜ , w ˜ ) is globally asymptotically stable for any χ > 0 .

4) If the condition (H3) holds and

μ 1 = ( β 1 − ρ 1 ) ( α 1 + s u * ) α 1 + w * ,     μ 2 = α 2 ( β 2 − ρ 2 ) α 2 + w * ,     0 < K ≤ r α 1 2 μ 1 u * + r α 2 2 μ 2 v * , (5)

as well as

0 < χ 2 + ξ 2 ≤ 4 d w * max { u * , v * } C 2 , (6)

then E ∗ = ( u * , v * , w * ) is globally asymptotically stable, where C = max { ‖ w 0 ( x ) ‖ L ∞ ( Ω ) , K } .

Remark 2.1 From Theorem 2.1, we can find that the global asymptotic stability of the semi-trivial equilibrium E 1 = ( 0,0, K ) does not involve the prey-taxis coefficients χ and ξ . The global asymptotic stabilities of the boundary equilibria E 2 = ( u ^ , 0 , w ^ ) and E 3 = ( 0 , v ˜ , w ˜ ) only depend on prey-taxis coefficient χ and ξ , respectively. However, the global asymptotic stability of the unique positive equilibrium E ∗ = ( u * , v * , w * ) depends on prey-taxis coefficients χ and ξ .

Remark 2.2 The control conditions (3), (4) and (6) of the global asymptotic stabilities of the equilibria only involve the initial spatial density w 0 ( x ) of prey species but are independent of the initial spatial densities u 0 ( x ) and v 0 ( x ) of the predators.

The following conclusion is helpful to obtain the desired results.

Lemma 1 Suppose that Ω ⊂ ℝ N with the smooth boundary ∂ Ω . For any initial conditions ( u 0 ( x ) , v 0 ( x ) , w 0 ( x ) ) ∈ [ W 1, p ( Ω ) ] 3 satisfies u 0 ( x ) ≥ 0, v 0 ( x ) , w 0 ( x ) ≥ 0 for x ∈ Ω ¯ . Then there is a maximal existence time T max > 0 such that system (1) has a unique local non-negative classical solution ( u ( x , t ) , v ( x , t ) , w ( x , t ) ) ∈ [ C ( [ 0 , T max ) ; W 1 , p ( Ω ) ) ∩ C 2 , 1 ( Ω ¯ × ( 0 , T max ) ) ] 3 . Moreover, we have u ( x , t ) ,   v ( x , t ) > 0 ,   w ( x , t ) ≤ C for x ∈ Ω ¯ , t ∈ [ 0 , T max ) , where C = max { ‖ w 0 ( x ) ‖ L ∞ ( Ω ) , K } .

Proof. Denote by ψ ( x , t ) = ( u ( x , t ) , v ( x , t ) , w ( x , t ) ) . Then system (1) takes the form

{ ∂ ψ ∂ t = ∂ x ⋅ ( z ( ψ ) ∂ x ψ ) + Ψ ( ψ ) , x ∈ Ω ,     t > 0 , ∂ ψ ∂ ν = 0 , x ∈ ∂ Ω ,     t ≥ 0 , ψ ( ⋅ , 0 ) = ( u 0 ( x ) , v 0 ( x ) , w 0 ( x ) ) , x ∈ Ω .

where

z ( ψ ) = ( 1 0 − χ u ( 1 + δ w ) 2 0 1 − ξ u ( 1 + δ w ) 2 0 0 d ) ,     Ψ ( ψ ) = ( β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v r w ( 1 − w K ) − μ 1 u w α 1 + w + s u − μ 2 v w α 2 + w ) .

Obviously, z ( ψ ) is an upper-triangular matrix and is positive definite since d > 0 is valid. Therefore, the local existence can be checked by Amman’s fixed point argument [13]. Now rewrite the first equation of system (1) as follows.

{ ∂ t u = ∂ x x u − χ ∂ x u ⋅ ∂ x w ( 1 + δ w ) 2 + 2 χ δ u ( 1 + δ w ) 3 ∂ x x w + u f 1 ( u , v , w ) , x ∈ Ω ,     t ∈ ( 0 , T max ) , ∂ ν u = 0 , x ∈ ∂ Ω ,     t ∈ ( 0 , T max ) , u ( x , 0 ) = u 0 ( x ) ≥ 0 , x ∈ Ω , (7)

where f 1 ( u , v , w ) = β 1 w α 1 + w + s u − ρ 1 w α 1 + w + s u − δ 1 . Obviously, 0 is a lower solution of (7). Therefore, the maximum principle shows u ( x , t ) ≥ 0 for all ( x , t ) ∈ Ω × ( 0 , T max ) . Combine u 0 ( x ) ≥ 0 ( ≡ 0 ) with the strong maximum principle, u ( x , t ) > 0 is valid. By the same way, we have v ( x , t ) , w ( x , t ) > 0 for all ( x , t ) ∈ Ω × ( 0 , T max ) . Finally, the maximum principle ensures that w ( x , t ) ≤ C for ( x , t ) ∈ Ω × ( 0 , T max ) . This ends the proof.

In the sequel, we shall give proof of Theorem 2.1 by constructing some suitable time evolution Lyapunov functions.

1) For E 1 = ( 0,0, K ) , define the following Lyapunov function

V 1 ( t ) = ∫ Ω     u ( ⋅ , t ) d x + ∫ Ω     v ( ⋅ , t ) d x + ∫ Ω ∫ K w w ( ⋅ , t ) − K w ( ⋅ , t ) d w d x . (8)

Then we deduce

V ˙ 1 ( t ) = ∫ Ω ( β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u ) d x     + ∫ Ω ( β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v ) d x − ∫ Ω d K | ∂ x w | 2 w 2 d x     + ∫ Ω ( w − K ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ]

≤ ∫ Ω ( ( β 1 − ρ 1 − μ 1 ) u w α 1 + w + s u − δ 1 u ) d x + ∫ Ω μ 1 K u α 1 + w + s u d x     + ∫ Ω μ 2 K v α 2 + w d x + ∫ Ω ( ( β 2 − ρ 2 − μ 2 ) v w α 2 + w − δ 2 v ) d x     − ∫ Ω d K | ∂ x w | 2 w 2 d x + ∫ Ω     r ( w − K ) ( 1 − w K ) d x ≤ ∫ Ω ( β 1 − ρ 1 − μ 1 ) u w α 1 + w + s u d x + ∫ Ω ( μ 1 K α 1 − δ 1 ) u d x + ∫ Ω ( β 2 − ρ 2 − μ 2 ) v w α 2 + w d x       + ∫ Ω ( μ 2 K α 2 − δ 2 ) v d x − ∫ Ω d K | ∂ x w | 2 w 2 d x − r K ∫ Ω ( w − K ) 2 d x .

Consequently, V ˙ 1 ( t ) ≤ 0 and E 1 = ( 0,0, K ) is globally asymptotically stable if (2) holds.

2) Define

V 2 ( t ) = ∫ Ω     ∫ u ^ u u ( ⋅ , t ) − u ^ u ( ⋅ , t ) d u d x + ∫ Ω     v ( ⋅ , t ) d x + ∫ Ω     ∫ w ^ w w ( ⋅ , t ) − w ^ w ( ⋅ , t ) d w d x . (9)

As a result, one deduces

V ˙ 2 ( t ) = ∫ Ω ( 1 − u ^ u ) ∂ t u d x + ∫ Ω     ∂ t v d x + ∫ Ω ( 1 − w ^ w ) ∂ t w d x = ∫ Ω ( u − u ^ ) ( β 1 w α 1 + w + s u − ρ 1 w α 1 + w + s u − δ 1 ) d x − ∫ Ω u ^ | ∂ x u | 2 u 2 d x     + ∫ Ω χ u ^ ∂ x w ⋅ ∂ x u u ( 1 + δ w ) 2 d x + ∫ Ω ( β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v ) d x     − ∫ Ω d w ^ | ∂ x w | 2 w 2 d x + ∫ Ω ( w − w ^ ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] = J ^ 1 ( t ) + J ^ 2 ( t ) ,

where

J ^ 1 ( t ) = ∫ Ω ( u − u ^ ) ( β 1 w α 1 + w + s u − ρ 1 w α 1 + w + s u − δ 1 ) d x     + ∫ Ω ( β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v ) d x     + ∫ Ω ( w − w ^ ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] ,

and

J ^ 2 ( t ) = − ∫ Ω u ^ | ∂ x u | 2 u 2 d x + ∫ Ω χ u ^ ∂ x w ⋅ ∂ x u u ( 1 + δ w ) 2 d x − ∫ Ω d w ^ | ∂ x w | 2 w 2 d x .

By using

δ 1 = β 1 w ^ α 1 + w ^ + s u ^ + ρ 1 w ^ α 1 + w ^ + s u ^ ,     r = r w ^ K + μ 1 u ^ α 1 + w ^ + s u ^ ,

one yields

J ^ 1 ( t ) = ∫ Ω ( u − u ^ ) ( β 1 w α 1 + w + s u − ρ 1 w α 1 + w + s u − δ 1 ) d x     + ∫ Ω ( β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v ) d x     + ∫ Ω ( w − w ^ ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] = ∫ Ω ( β 1 − ρ 1 ) ( α 1 + s u ^ ) ( u − u ^ ) ( w − w ^ ) ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x     − ∫ Ω s w ^ ( β 1 − ρ 1 ) ( u − u ^ ) 2 ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x + ∫ Ω ( β 2 v w α 2 + w − ρ 2 v w α 2 + w − δ 2 v ) d x

    + ∫ Ω μ 1 u ^ ( w − w ^ ) 2 ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x − ∫ Ω μ 1 ( α 1 + w ^ ) ( u − u ^ ) ( w − w ^ ) ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x     − r K ∫ Ω ( w − w ^ ) 2 d x − ∫ Ω μ 2 v ( w − w ^ ) α 2 + w d x ≤ ∫ Ω [ ( β 1 − ρ 1 ) ( α 1 + s u ^ ) − μ 1 ( α 1 + w ^ ) ] ( u − u ^ ) ( w − w ^ ) ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x − ∫ Ω ρ 2 v w α 2 + w d x     + ∫ Ω μ 1 u ^ ( w − w ^ ) 2 ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x − r K ∫ Ω ( w − w ^ ) 2 d x + ∫ Ω ( β 2 − μ 2 ) v w α 2 + w d x     + ∫ Ω ( μ 2 w ^ α 2 + w − δ 2 ) v d x − ∫ Ω s w ^ ( β 1 − ρ 1 ) ( u − u ^ ) 2 ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x

≤ ∫ Ω [ ( β 1 − ρ 1 ) ( α 1 + s u ^ ) − μ 1 ( α 1 + w ^ ) ] ( u − u ^ ) ( w − w ^ ) ( α 1 + w + s u ) ( α 1 + w ^ + s u ^ ) d x     + ∫ Ω ( μ 1 s u ^ α 1 2 − r K ) ( w − w ^ ) 2 d x     + ∫ Ω ( β 2 − μ 2 ) v w α 2 + w d x + ∫ Ω ( μ 2 w ^ α 2 − δ 2 ) v d x ≤ 0 ,

due to (3) is valid. For J ^ 2 ( t ) , we have

J ^ 2 ( t ) = − ∫ Ω u ^ | ∂ x u | 2 u 2 d x + ∫ Ω χ u ^ ∂ x w ⋅ ∂ x u u ( 1 + δ w ) 2 d x − ∫ Ω d w ^ | ∂ x w | 2 w 2 d x ≤ − ∫ Ω u ^ | ∂ x u | 2 u 2 d x + ∫ Ω χ u ^ | ∂ x w | ⋅ | ∂ x u | u d x − ∫ Ω d w ^ | ∂ x w | 2 w 2 d x = − ∫ Ω     X 1 Q 1 X 1 T d x ,

where we define X 1 ( x , t ) = ( | ∂ x u ( x , t ) | , | ∂ x w ( x , t ) | ) in Ω × ( 0, ∞ ) , and the matrix Q 1 is

Q 1 = ( u ^ u 2 − χ u ^ 2 u − χ u ^ 2 u d w ^ w 2 ) .

Accordingly, we have u ^ u 2 > 0 and u ^ u 2 ( d w ^ w 2 − χ 2 u ^ 4 ) ≥ 0 as (3) holds. These

imply V ˙ 2 ( t ) = J ^ 1 ( t ) + J ^ 2 ( t ) ≤ 0 and E 2 = ( u ^ , 0 , w ^ ) is globally asymptotically stable.

3) Consider the following function

V 3 ( t ) = ∫ Ω     u ( ⋅ , t ) d x + ∫ Ω     ∫ v ˜ v v ( ⋅ , t ) − v ˜ v ( ⋅ , t ) d v d x + ∫ Ω     ∫ w ˜ w w ( ⋅ , t ) − w ˜ w ( ⋅ , t ) d w d x . (10)

Straightforward computation showing

V ˙ 3 ( t ) = ∫ Ω     ∂ t u d x + ∫ Ω ( 1 − v ˜ v ) ∂ t v d x + ∫ Ω ( 1 − w ˜ w ) ∂ t w d x = ∫ Ω ( β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u ) d x − ∫ Ω v ˜ | ∂ x v | 2 v 2 d x     + ∫ Ω ξ v ˜ ∂ x w ⋅ ∂ x v v ( 1 + δ w ) 2 d x + ∫ Ω ( v − v ˜ ) ( β 2 w α 2 + w − ρ 2 w α 2 + w − δ 2 ) d x     − ∫ Ω d w ˜ | ∂ x w | 2 w 2 d x + ∫ Ω ( w − w ˜ ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] = J ˜ 1 ( t ) + J ˜ 2 ( t ) ,

where

J ˜ 1 ( t ) = ∫ Ω ( β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u ) d x     + ∫ Ω ( v − v ˜ ) ( β 2 w α 2 + w − ρ 2 w α 2 + w − δ 2 ) d x     + ∫ Ω ( w − w ˜ ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] ,

and

J ˜ 2 ( t ) = − ∫ Ω v ˜ | ∂ x v | 2 v 2 d x + ∫ Ω ξ v ˜ ∂ x w ⋅ ∂ x v v ( 1 + δ w ) 2 d x − ∫ Ω d w ˜ | ∂ x w | 2 w 2 d x .

Note that

δ 2 = β 2 w ˜ α 2 + w ˜ + ρ 2 w ˜ α 2 + w ˜ ,     r = r w ˜ K + μ 2 v α 2 + w ˜ ,

we get

J ˜ 1 ( t ) = ∫ Ω ( β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u ) d x     + ∫ Ω ( v − v ˜ ) ( β 2 w α 2 + w − ρ 2 w α 2 + w − δ 2 ) d x     + ∫ Ω ( w − w ˜ ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] = ∫ Ω ( β 1 u w α 1 + w + s u − ρ 1 u w α 1 + w + s u − δ 1 u ) d x     + ∫ Ω α 2 ( β 2 − ρ 2 ) ( v − v ˜ ) ( w − w ˜ ) ( α 2 + w ) ( α 2 + w ˜ ) d x

    + ∫ Ω μ 2 v ˜ ( w − w ˜ ) 2 ( α 2 + w ) ( α 2 + w ˜ ) d x − ∫ Ω μ 2 ( α 2 + w ˜ ) ( v − v ˜ ) ( w − w ˜ ) ( α 2 + w ) ( α 2 + w ˜ ) d x     − r K ∫ Ω ( w − w ˜ ) 2 d x − ∫ Ω μ 1 u ( w − w ˜ ) α 1 + w + s u d x = − ∫ Ω ρ 1 u w α 1 + w + s u d x + ∫ Ω [ α 2 ( β 2 − ρ 2 ) − μ 2 ( α 2 + w ˜ ) ] ( v − v ˜ ) ( w − w ˜ ) ( α 2 + w ) ( α 2 + w ˜ ) d x     + ∫ Ω ( μ 2 v ˜ ( α 2 + w ) ( α 2 + w ˜ ) − r K ) ( w − w ˜ ) 2 d x     + ∫ Ω ( β 1 − μ 1 ) u w α 1 + w + s u d x + ∫ Ω ( μ 1 w ˜ α 1 + w + s u − δ 1 ) u d x

≤ ∫ Ω [ α 2 ( β 2 − ρ 2 ) − μ 2 ( α 2 + w ˜ ) ] ( v − v ˜ ) ( w − w ˜ ) ( α 2 + w ) ( α 2 + w ˜ ) d x     + ∫ Ω ( μ 2 v ˜ α 2 2 − r K ) ( w − w ˜ ) 2 d x + ∫ Ω ( β 1 − μ 1 ) u w α 1 + w + s u d x     + ∫ Ω ( μ 1 w ˜ α 1 − δ 1 ) u d x ≤ 0 ,

if (4) is satisfied. For J ˜ 2 ( t ) , we have

J ˜ 2 ( t ) = − ∫ Ω v ˜ | ∂ x v | 2 v 2 d x + ∫ Ω ξ v ˜ ∂ x w ⋅ ∂ x v v ( 1 + δ w ) 2 d x − ∫ Ω d w ˜ | ∂ x w | 2 w 2 d x ≤ − ∫ Ω v ˜ | ∂ x v | 2 v 2 d x + ∫ Ω ξ v ˜ | ∂ x w | ⋅ | ∂ x v | v d x − ∫ Ω d w ˜ | ∂ x w | 2 w 2 d x = − ∫ Ω     X 2 Q 2 X 2 T d x ,

where denote by X 2 ( x , t ) = ( | ∂ x v ( x , t ) | , | ∂ x w ( x , t ) | ) in Ω × ( 0, ∞ ) and

Q 2 = ( v ˜ v 2 − ξ v ˜ 2 v − ξ v ˜ 2 v d w ˜ w 2 ) .

It is clear that v ˜ v 2 > 0 and | Q 2 | = v ˜ v 2 ( d w ˜ w 2 − ξ 2 v ˜ 4 ) ≥ 0 since 0 < ξ 2 ≤ 4 d w ˜ C 2 v ˜ . Consequently, V ˙ 3 ( t ) = J ˜ 1 ( t ) + J ˜ 2 ( t ) ≤ 0 and thus E 3 = ( 0 , v ˜ , w ˜ ) is globally asymptotically stable.

4) Introducing the following time evolution Lyapunov function

V 4 ( t ) = ∫ Ω     ∫ u * u u ( ⋅ , t ) − u * u ( ⋅ , t ) d u d x + ∫ Ω     ∫ v * v v ( ⋅ , t ) − v * v ( ⋅ , t ) d v d x     + ∫ Ω     ∫ w * w w ( ⋅ , t ) − w * w ( ⋅ , t ) d w d x . (11)

Direct computations illustrate that

V ˙ 4 ( t ) = ∫ Ω ( 1 − u * u ) ∂ t u d x + ∫ Ω ( 1 − v * v ) ∂ t v d x + ∫ Ω ( 1 − w * w ) ∂ t w d x = ∫ Ω ( u − u * ) ( β 1 w α 1 + w + s u − ρ 1 w α 1 + w + s u − δ 1 ) d x     + ∫ Ω ( v − v * ) ( β 2 w α 2 + w − ρ 2 w α 2 + w − δ 2 ) d x     + ∫ Ω ( w − w * ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ]

    − ∫ Ω u * | ∂ x u | 2 u 2 d x − ∫ Ω v * | ∂ x v | 2 v 2 d x − ∫ Ω d w * | ∂ x w | 2 w 2 d x     + ∫ Ω χ u * ∂ x w ⋅ ∂ x u u ( 1 + δ w ) 2 d x + ∫ Ω ξ v * ∂ x w ⋅ ∂ x v v ( 1 + δ w ) 2 d x = J 1 * ( t ) + J 2 * ( t ) ,

where

J 1 * ( t ) = ∫ Ω ( u − u * ) ( β 1 w α 1 + w + s u − ρ 1 w α 1 + w + s u − δ 1 ) d x     + ∫ Ω ( v − v * ) ( β 2 w α 2 + w − ρ 2 w α 2 + w − δ 2 ) d x     + ∫ Ω ( w − w * ) [ r ( 1 − w K ) − μ 1 u α 1 + w + s u − μ 2 v α 2 + w ] ,

and

J 2 * ( t ) = − ∫ Ω u * | ∂ x u | 2 u 2 d x − ∫ Ω v * | ∂ x v | 2 v 2 d x − ∫ Ω d w * | ∂ x w | 2 w 2 d x     + ∫ Ω χ u * ∂ x w ⋅ ∂ x u u ( 1 + δ w ) 2 d x + ∫ Ω ξ v * ∂ x w ⋅ ∂ x v v ( 1 + δ w ) 2 d x .

By employing these facts

δ 1 = β 1 w * α 1 + w * + s u * + ρ 1 w * α 1 + w * + s u * ,     δ 2 = β 2 w * α 2 + w * + ρ 2 w * α 2 + w * ,

r = w * K + μ 1 u * α 1 + w * + s u * + μ 2 v * α 2 + w * ,

we can obtain

J 1 * ( t ) = ∫ Ω ( β 1 − ρ 1 ) ( α 1 + s u * ) ( u − u * ) ( w − w * ) ( α 1 + w + s u ) ( α 1 + w * + s u * ) d x     − ∫ Ω s w * ( β 1 − ρ 1 ) ( u − u * ) 2 ( α 1 + w + s u ) ( α 1 + w * + s u * ) d x     + ∫ Ω α 2 ( β 2 − ρ 2 ) ( v − v * ) ( w − w * ) ( α 2 + w ) ( α 2 + w * ) d x     − ∫ Ω μ 2 ( α 2 + w * ) ( v − v * ) ( w − w * ) ( α 2 + w ) ( α 2 + w * ) d x

  + ∫ Ω μ 2 v * ( w − w * ) 2 ( α 2 + w ) ( α 2 + w * ) d x − r K ∫ Ω ( w − w * ) 2 d x   + ∫ Ω μ 1 u * ( w − w * ) 2 ( α 1 + w + s u ) ( α 1 + w * + s u * ) d x   − ∫ Ω μ 1 ( α 1 + w * ) ( u − u * ) ( w − w * ) ( α 1 + w + s u ) ( α 1 + w * + s u * ) d x

≤ ∫ Ω [ ( β 1 − ρ 1 ) ( α 1 + s u * ) − μ 1 ( α 1 + w * ) ] ( u − u * ) ( w − w * ) ( α 1 + w + s u ) ( α 1 + w * + s u * ) d x     + ∫ Ω [ α 2 ( β 2 − ρ 2 ) − μ 2 ( α 2 + w * ) ] ( v − v * ) ( w − w * ) ( α 2 + w ) ( α 2 + w * ) d x     + ∫ Ω ( μ 1 u * α 1 2 + μ 2 v * α 2 2 − r K ) ( w − w * ) 2 d x ≤ 0 ,

here we use the Condition (5). Moreover

J 2 * ( t ) = − ∫ Ω u * | ∂ x u | 2 u 2 d x − ∫ Ω v * | ∂ x v | 2 v 2 d x − ∫ Ω d w * | ∂ x w | 2 w 2 d x     + ∫ Ω χ u * ∂ x w ⋅ ∂ x u u ( 1 + δ w ) 2 d x + ∫ Ω ξ v * ∂ x w ⋅ ∂ x v v ( 1 + δ w ) 2 d x . ≤ − ∫ Ω u * | ∂ x u | 2 u 2 d x − ∫ Ω v * | ∂ x v | 2 v 2 d x − ∫ Ω d w * | ∂ x w | 2 w 2 d x     + ∫ Ω χ u * | ∂ x w | ⋅ | ∂ x u | u d x + ∫ Ω ξ v * | ∂ x w | ⋅ | ∂ x v | v d x = − ∫ Ω     X 3 Q 3 X 3 T d x ,

where the vector function X 3 ( x , t ) is given by X 3 ( x , t ) = ( | ∂ x u ( x , t ) | , | ∂ x v ( x , t ) | , | ∂ x w ( x , t ) | ) in Ω × ( 0, ∞ ) and

Q 3 = ( u * u 2 0 − χ u * 2 u 0 v * v 2 − ξ v * 2 v − χ u * 2 u − ξ v * 2 v d w * w 2 ) .

We can obtain

| u * u 2 | > 0 ,     | u * u 2 0 0 v * v 2 | = u * v * u 2 v 2 > 0.

as well as

| Q 3 | = u * u 2 | v * v 2 − ξ v * 2 v − ξ v * 2 v d w * w 2 | − χ u * 2 u | 0 v * v 2 − χ u * 2 u − ξ v * 2 v | = u * v * u 2 v 2 ( d w * w 2 − ξ 2 v * 4 − χ 2 u * 4 ) ≥ 0

if

0 < χ 2 + ξ 2 ≤ 4 d w * max { u * , v * } C 2 .

Thence A 3 is a nonnegative definite matrix, which gives J 2 * ( t ) = − ∫ Ω     X 3 Q 3 X 3 T d x ≤ 0 . We conclude that E ∗ = ( u * , v * , w * ) is globally asymptotically stable. These end the proof.

This present paper deals with the global asymptotic stability of a three-species predator-prey model with prey-taxis. This system possesses a semi-trivial equilibrium E 1 = ( 0 , 0 , K ) , two boundary equilibria E 2 = ( u ^ , 0 , w ^ ) and E 3 = ( 0 , v ˜ , w ˜ ) , and a unique positive equilibrium E ∗ = ( u * , v * , w * ) . By constructing some suitable Lyapunov functions, we establish their global asymptotic stability, respectively. It is concluded that the prey-taxis coefficients χ , ξ can not influence the global asymptotic stability of the semi-trivial equilibrium E 1 = ( 0 , 0 , K ) . Also, the global asymptotic stability of two boundary equilibria E 2 = ( u ^ , 0 , w ^ ) and E 3 = ( 0 , v ˜ , w ˜ ) rely on the single prey-taxis coefficient χ and ξ , respectively. However, the global asymptotic stability of the unique positive equilibrium E ∗ = ( u * , v * , w * ) is determined by prey-taxis coefficients χ and ξ . These phenomena suggest that the prey-taxis has an influence on the global asymptotic stability of the equilibria of the System (1). Consequently, we will continuously explore the complicated dynamics of the System (1) with prey-taxis effect in the future.

This work was supported by the National Natural Science Foundation of China (No. 12002297) and the China Postdoctoral Science Foundation (No. 2021M701118).

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

Chen, M.X. and Zheng, Q.Q. (2022) Global Stability of a Three-Species System with Attractive Prey-Taxis. Applied Mathematics, 13, 658-671. https://doi.org/10.4236/am.2022.138041