Global analysis in a predator-prey system with nonmonotonic functional response ∗

Q: What is the phase portrait of system (1.9)?

If 163 aD 2 < µ2 < 18+2√ 63 aD 2 and x3 < K < µ D , then system(1.9) has four equilibria: two hyperbolic saddles (0, 0) and (x2, y2), a hyperbolic stable node (K, 0), and an unstable focus (or node) (x1, y1).

Q: What is the phase portrait of the saddle-node?

The saddle-node (x0, y0) consists of two hyperbolic sectors andone parabolic sector; the parabolic sector is between the y-axis and the equilibrium.

Q: What is the corresponding phase portrait for system (3.15)?

When the parameter values lie on the Hopf bifurcation curve H, system (3.15) has two interior equilibria: one is a stable weak focus, the other is a hyperbolic saddle; and there is no periodic orbit.

Q: What is the simplest way to classify the dynamics of a homoclinic loop?

Dif a limit cycle exists, it is unique and stable if a homoclinic loop exists, it is stable a limit cycle and a homoclinic loop cannot coexist two equilibria: (x1, y1) unstable focus; (x2, y2) saddleµ

Q: What is the Dulac function for system (2.6)?

Taking the Dulac function D(x, y) = x−1y−1 for system (2.6) and noting thatK ≤ √3a if 4aD2 < µ2 ≤ 163 aD2 and x1 < K < x2, the authors can see that system (2.6) has no periodic orbits in the interior of the first quadrant.

Q: What is the proof of Lemma 3.1?

Rewriting system (3.19) as the form in system (3.9) and using the method in the proof of Lemma 3.1, the authors obtainẋ = y,ẏ = 4a0x0 µ0 λ2 + φ1(λ1, λ2) + φ2(λ1, λ2)x+ x0r2a0 λ1 − λ2√µ0r 8a0+ φ3(λ1, λ2) y+ x2 − r2x0√µ0r 8a0+ φ4(λ1, λ2) xy +R(x, y, λ1, λ2), (3.20)where φ1, φ2, and φ3 are smooth functions in variables (λ1, λ2) at least of the second order with respect to λ1 and λ2, φ4 is a smooth function of λ1 and λ2 at least of the first order, R is a C∞ function in variables (x, y) at least of the third order with respect to (x, y), and the coefficients depend smoothly on λ1 and λ2.

Q: What is the simplest explanation for the periodic orbit of system 1.9?

Theorem 2.5. If µ2 > 163 aD 2 and x2 > K > x3, then system (1.9) has at most one limit cycle in the interior of the first quadrant.

Question

1. What are the contributions mentioned in the paper "Global analysis in a predator-prey system with nonmonotonic functional response∗" ?

2. What is the formula for the third focal value of system (1.9)?

3. What is the phase portrait of system (1.9)?

4. What is the phase portrait of the saddle-node?

Accepted Answer

In this paper, a predator-prey system with non-monotonic functional response is considered and global qualitative and bifurcation analyses are combined to determine the global dynamics of the model.

Accepted Answer

when parameters pass from one side of the following surface to the other side, system (1.9) can undergo a supercritical Hopf bifurcation.

Accepted Answer

If 163 aD 2 < µ2 < 18+2√ 63 aD 2 and x3 < K < µ D , then system(1.9) has four equilibria: two hyperbolic saddles (0, 0) and (x2, y2), a hyperbolic stable node (K, 0), and an unstable focus (or node) (x1, y1).

Accepted Answer

The saddle-node (x0, y0) consists of two hyperbolic sectors andone parabolic sector; the parabolic sector is between the y-axis and the equilibrium.

Accepted Answer

When the parameter values lie on the Hopf bifurcation curve H, system (3.15) has two interior equilibria: one is a stable weak focus, the other is a hyperbolic saddle; and there is no periodic orbit.

Accepted Answer

Dif a limit cycle exists, it is unique and stable if a homoclinic loop exists, it is stable a limit cycle and a homoclinic loop cannot coexist two equilibria: (x1, y1) unstable focus; (x2, y2) saddleµ

Accepted Answer

Taking the Dulac function D(x, y) = x−1y−1 for system (2.6) and noting thatK ≤ √3a if 4aD2 < µ2 ≤ 163 aD2 and x1 < K < x2, the authors can see that system (2.6) has no periodic orbits in the interior of the first quadrant.

Accepted Answer

Rewriting system (3.19) as the form in system (3.9) and using the method in the proof of Lemma 3.1, the authors obtainẋ = y,ẏ = 4a0x0 µ0 λ2 + φ1(λ1, λ2) + φ2(λ1, λ2)x+ x0r2a0 λ1 − λ2√µ0r 8a0+ φ3(λ1, λ2) y+ x2 − r2x0√µ0r 8a0+ φ4(λ1, λ2) xy +R(x, y, λ1, λ2), (3.20)where φ1, φ2, and φ3 are smooth functions in variables (λ1, λ2) at least of the second order with respect to λ1 and λ2, φ4 is a smooth function of λ1 and λ2 at least of the first order, R is a C∞ function in variables (x, y) at least of the third order with respect to (x, y), and the coefficients depend smoothly on λ1 and λ2.

Accepted Answer

Theorem 2.5. If µ2 > 163 aD 2 and x2 > K > x3, then system (1.9) has at most one limit cycle in the interior of the first quadrant.

Accepted Answer

−KD = 0,(3.1) µ2 − 4aD2 = 0.(3.2)By bifurcation analysis the authors show that system (1.9) has a homoclinic loop for some parameters values.

Accepted Answer

On the other hand, the equilibrium (x1, y1) is an unstable focus when µ2 > 18+2 √ 63 aD 2 and K ≥ x3; however, it is a stable focus as µ2 > 18+2√ 63 aD 2 andx1 < K < x3.

Accepted Answer

From the corresponding phase portrait the authors can see that there exist only two initial population densities such that the predator and prey can coexist.

Accepted Answer

It is easy to prove that the orbit (x∗(t), y∗(t)) must cross the line l2 at point C(x1, yc) and yc ≥ yb, where l2 = {(x, y) : x = x1, y > 0}.

Accepted Answer

Then when K = x3, system (1.9) can be transformed intodx̄ dτ =− x1ȳ + r ( − a K + 3x1 − 6x 2 1 K ) x̄2 − x̄ȳ − r ( 1− 4x1 K ) x̄3 − r K x̄4,dȳ dτ = (µ− 2Dx1)y1x̄−Dy1x̄2 + (µ− 2Dx1)x̄ȳ −Dx̄2ȳ.(2.5)According to the formula for the third focal value (i.e., the Liapunov number) of a multiple focus on p. 253 of Andronov [2], the authors obtain, after elementary but lengthy computations, the following expression for the third focal value α3 of the multiple focus (0, 0) in system (2.5):α3 = 3πx1y1r 2KD(x1y1(µ− 2Dx1)) 32 ( 2µx1 − 10aD + µ 2 D )= 3πr2KD(µ− 2Dx1)(x1y1(µ− 2Dx1)) 12 (2µ2 − µ√ µ2 − 4aD2 − 10aD2).

Accepted Answer

It is clear that the periodic orbit of system (1.9) must be in the domain E1 if it exists, whereE1 = {(x, y) : 0 < x < K, 0 < y < +∞}.Taking a ray line l1 beginning at the point A(K, 0)l1 = {(x, y) : x = K, y >

Global analysis in a predator-prey system with nonmonotonic functional response ∗

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Most frequently asked questions

1. What are the contributions mentioned in the paper "Global analysis in a predator-prey system with nonmonotonic functional response∗" ?

2. What is the formula for the third focal value of system (1.9)?

3. What is the phase portrait of system (1.9)?

4. What is the phase portrait of the saddle-node?

5. What is the corresponding phase portrait for system (3.15)?

6. What is the simplest way to classify the dynamics of a homoclinic loop?

7. What is the Dulac function for system (2.6)?

8. What is the proof of Lemma 3.1?

9. What is the simplest explanation for the periodic orbit of system 1.9?

10. What is the simplest way to show that system (1.9) has a homoclinic?

11. What is the equilibrium of the multiple focus?

12. What is the phase portrait of the predator-prey system?

13. What is the simplest way to prove that the periodic orbit of system (1.9) must be?

14. what is the formula for the third focal value of a multiple focus?

15. What is the axis of system (1.9)?

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Citations

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Dynamic study of a ratio-dependent predator–prey model with strong allee effect and nonlinear harvesting

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Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting

References

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

A Reflection on Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Elements of applied bifurcation theory

Elements of Applied Bifurcation Theory

The functional response of predators to prey density and its role in mimicry and population regulation.

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