A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting
Abstract
:1. Introduction
2. Preliminaries
3. Main Results
3.1. Existence and Uniqueness
3.2. Boundedness and Non-Negativity
3.3. Local Stability
- The extinction point of both prey and predator population which is always feasible.
- The free predator point , which also always exists. Here, .
- The interior point where and . Notice that exists if .
- 1.
- is a saddle point.
- 2.
- If, thenis locally asymptotically stable and it is a saddle if.
- The Jacobian matrix in Equation (8) evaluated at isThe eigenvalues of are and , and consequently we have and for . Hence, is a saddle point.
- If is substituted into the Jacobian matrix in Equation (8), then we haveObviously, has eigenvalues and . We observe that . If , then and thus . On the other hand, if , then , and consequently . Therefore, is asymptotically stable (locally) if and is a saddle point if .
- 1.
- and
- 2.
- and.
- Since and , and . Therefore, is asymptotically stable.
- Suppose . If is an eigenvalue, then its complex conjugate is also an eigenvalue. We have that . Using the Matignon’s condition (see Theorem 2), it is obvious that is locally asymptotically stable if .
3.4. Hopf Bifurcation
- (i)
- The eigenvalues of the Jacobian matrix are a pair of complex-conjugate: ;
- (ii)
- ; and
- (iii)
3.5. Global Asymptotic Stability
4. Numerical Simulations
5. Concluding Remarks
Author Contributions
Funding
Conflicts of Interest
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Suryanto, A.; Darti, I.; S. Panigoro, H.; Kilicman, A. A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting. Mathematics 2019, 7, 1100. https://doi.org/10.3390/math7111100
Suryanto A, Darti I, S. Panigoro H, Kilicman A. A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting. Mathematics. 2019; 7(11):1100. https://doi.org/10.3390/math7111100
Chicago/Turabian StyleSuryanto, Agus, Isnani Darti, Hasan S. Panigoro, and Adem Kilicman. 2019. "A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting" Mathematics 7, no. 11: 1100. https://doi.org/10.3390/math7111100
APA StyleSuryanto, A., Darti, I., S. Panigoro, H., & Kilicman, A. (2019). A Fractional-Order Predator–Prey Model with Ratio-Dependent Functional Response and Linear Harvesting. Mathematics, 7(11), 1100. https://doi.org/10.3390/math7111100