Qualitative Analysis for a Predator Prey System with Holling Type III Functional Response and Prey Refuge

Lemma 1.1. Any solution (x(t), y(t)) of system (1.2) with initial condition x(0) > m,   y(0) > 0 is positive and bounded for all t ≥ 0.

Lemma 2.1. Let (H₁) hold. Further assume that (H₃)  kα ≤ c and (H₄)  kα > c, m > max {0, (a − bx_*)/b}. Then E₀ is locally asymptotically stable, if any of (H₃) and (H₄) holds. When kα > c,   0 < m < (a − bx_*)/b,   E₀ is unstable, furthermore, E₀ is a saddle point.

Theorem 2.2. Assume kα > c. Then

• (I)

  E_* is locally asymptotically stable for 0 < m < (a − bx_*)/b if a(2c − kα) ≤ 2bcx_* holds.

• (II)

  E_* is locally asymptotically stable for m₁ < m < (a − bx_*)/b and E_* is locally unstable for 0 < m < m₁ if a(2c − kα) > 2bcx_* holds, where

  $$ (2.3)

• (III)

  system (2.1) undergoes Hopf bifurcation at m = m₁ if a(2c − kα) > 2bcx_* holds.

Proof. The Jacobian matrix of system (2.1) at E_* is

where $$. Then tr (J(E_*)) = −P/x_*   = R(m)/x_*, where $$, the discriminant of R(m) = 0 is $$. Hence, the equation R(m) = 0 has two roots m₁ and m₂, where $$.

Note that

and a(2c − kα)>(≤)2bcx_* implies m₁ > (≤)0. Consider Then

• (I)

  If a(2c − kα) ≤ 2bcx_* holds, then m₁ ≤ 0,   R(m) < 0 holds for m₁ < m < m₂. Considering (H₂) and m₂ > (a − bx_*)/b, for 0 < m < (a − bx_*)/b, tr (J(E_*)) < 0, which implies E_* is locally asymptotically stable.

• (II)

  If a(2c − kα) > 2bcx_* holds, then m₁ > 0, for m₁ < m < m₂,   R(m) < 0, since $$, by $$, we obtain m₁ < (a − bx_*)/b. Together with (H₂), for m₁ < m < (a − bx_*)/b,   tr (J(E_*)) < 0, which means E_* is locally asymptotically stable. On the other hand, for 0 < m < m₁,   tr (J(E_*)) > 0,   E_* is locally unstable.

• (III)

  We have

  $$ (2.7)
  these satisfy Liu’s Hopf bifurcation criterion (see [6], page 255); hence, the Hopf bifurcation occurs at m = m₁. This ends the proof.

Theorem 3.1. If E_*(x_*, y_*) is locally stable. Further assume that max {0, (a − 4bβ)/2b} < m < m₄, then the positive equilibrium E_*(x_*, y_*) of system (2.1) is globally asymptotically stable.

Proof. Take the Dulac function B(x, y) = x⁻²y⁻¹, for system (2.1) we have

where

If a = 2bm, ϕ(x) = 2b(x⁴ + m²β²) > 0 for x > 0.

On the other hand, there exist

The equation ϕ^′′(x) = 0 has two roots x₁ = 0,   x₂ = (a − 2bm)/4b.

Case  1. If a − 2bm > 0, then for 0 < x < x₂, ϕ^′′(x) < 0; for x > x₂, ϕ^′′(x) > 0. Hence, x = x₂ is the least value of the function ϕ^′(x). If β > x₂,   ϕ^′(x₂) = (a − 2bm)(β − x₂)(β + x₂) > 0, it has ϕ^′(x) > 0 for all x > 0, then ϕ(x) is increasing for x > 0, notice that ϕ(0) > 0. Therefore, ϕ(x) > 0 for x > 0. Since, T < 0 for x > 0, system (2.1) does not exist limit cycle.

Case  2. If bm < a < 2bm, then x₂ < 0, for x > 0,   ϕ^′′(x) > 0, hence, for x > 0,   ϕ^′(x) is increasing. Evidently, $$, then there exists $$ such that ϕ^′(x₀) = 0, where $$, hence, when 0 < x < x₀,     ϕ^′(x) < 0, when x > x₀,   ϕ^′(x) > 0. We know that ϕ(x) takes the least value at x = x₀, that is, ϕ(x) > ϕ(x₀). According to ϕ^′(x₀) = 0, for x > 0 we obtain $$, where $$.

To prove ϕ(x) > 0 for x > 0, it suffices to prove $$ for $$. Clearly, $$ takes the least value at x = β, and $$ is strictly decreasing at the interval (0, β). Hence, for $$ holds. Since $$. Therefore, for $$ holds if m₃ < m < m₄ holds, then for x > 0,   ϕ(x) > 0 holds.

In sum, if one of the following three conditions holds (1) m = a/2b; (2) 0 < m < a/2b,   x₂ < β⇒max {0, (a − 4bβ)/2b} < m < a/2b; (3) a/2b < m < a/b, m₃ < m < m₄⇒a/2b < m < m₄, the function T does not change the sign for x > 0, then system (2.1) does not exist limit cycle. It is easy to see that the conditions (1), (2), and (3) are equal to max {0, (a − 4bβ)/2b} < m < m₄. The proof is completed.

Theorem 4.1. If a(2c − kα) > 2bcx_* holds, for 0 < m < m₁ system (2.1) admits at least one limit cycle in Ω.

Proof. We construct a Bendixson loop $$ which includes E_* of system (2.1). Let $$ be a length of the line L₁:$$ be a length of line L₂:b(x + m) − a = 0. Define

where $$. The orbit of system (4.1) with initial value ((a/b) − m, a₀) intersects with the line x = x_* and the intersection point C(x_*, y₁), we obtain the orbit arc $$. Let $$ be a length of line L₃:$$ be a length of line L₄:x = 0. Because $$ is a length of orbit line of system (2.1) and (dL₂/dt)|_(2.1) = −b((a/b) − m) ²y < 0(y > 0), (dL₃/dt)|_(2.1) = y₁(−cβ² + (kα − c)x²) < 0  (0 < x < x_*),   (dL₄/dt)|_(2.1) = mβ²(a − bm) > 0, the orbits of system (2.1) tend to the interior of the Bendixson loop from the outer of $$, and $$, by comparing system (2.1) to system (4.1): dx/dt|_(2.1) < dx/dt|_(4.1) < 0 and dy/dt|_(2.1) = dy/dt|_(4.1) > 0. Then the orbits of system (2.1) tend to the interior of the Bendixson loop from the outer of $$. On the other hand, under the condition of Theorem 4.1, E_*(x_*, y_*) is unstable, by Poincaré-Bendixson Theorem, system (2.1) admits at least one limit cycle in the region $$. This ends the proof.

Lemma 4.2 (see [7].)Let f(x), g(x) be continuously differentiable functions on the open interval (r1, r2), and φ(y) be continuously differentiable functions on R in

such that

• (1)

  dφ(y)/dy > 0,

• (2)

  having a unique x₀ ∈ (r₁, r₂), such that (x − x₀)g(x − x₀) > 0 for x ≠ x₀ and g(x₀) = 0,

• (3)

  f(x₀)d/dx(f(x)/g(x)) < 0 for x ≠ x₀,

then system (4.1) has at most one limit cycle.

Theorem 4.3. If a(2c − kα) > 2bcx_* holds, for $$ system (2.1) exists exactly one limit cycle which is globally asymptotically stable in Ω.

Proof. Let u = x,   v = ln y,   τ = −x²t, still denote u, v, τ, as x,   y,    t, then system (2.1) becomes

the positive equilibrium E_*(x_*, y_*) changes $$.

Let $$, then $$ transform to the origin O(0,0), still denote $$, as x,   y yield

where F(x) = ((x + x_* + m)(a − b(x + x_* + m))(β² + (x + x_*) ²))/(x + x_*) ² − y_*.

Clearly, F(0) = 0. It is easy to see that the conditions (1) and (2) of Lemma 4.2 for x₀ = 0 are satisfied. Consider

Note that by the assumption of Theorem 4.3, E_* is unstable equilibrium and then $$. Consider where where

Then, we have

where

By a simple computation, we obtain

It is easy to verify that $$ and $$ has two roots $$ and $$ defined by, respectively, Obviously, $$. Therefore, $$ for $$ and $$ for $$ which indicates that $$ is the minimum point of the function $$ when x⩾−x_*. Substituting $$ into $$, we obtain It is easy to see that if $$, then $$, which implies $$ for all x⩾−x_*. That is, the function $$ is a strictly increasing function for x⩾−x_*.

Note that $$ for 0 < m < a/b and $$. It follows from (4.6) that

Hence, there exists a point $$, such that $$, that is,

This, together with the monotonicity of $$ when x⩾−x_*, we may conclude that $$ for $$ and $$ for $$. Therefore, $$ is the minimum point of the function $$ for −x_* < x < ∞.

Together with (4.16), we obtain

It follows from (4.6), we have $$. This indicates $$ for all x > −x_*.

Then all the conditions of Lemma 4.2 are satisfied, considering Theorem 4.1, we obtain the conclusion of this theorem. The proof is completed.